Free FRM Part I Practice Exam: Questions and Explanations

This full-length practice set contains 100 four-choice, single-answer questions. It covers Foundations of Risk Management (20), Quantitative Analysis (20), Financial Markets and Products (30), and Valuation and Risk Models (30). Start at Question 1 .

These are original Finance Prep practice questions. They are not official GARP questions, copied live-exam content or exam dumps. Mastery Exam Prep and Finance Prep are independent from GARP.

Pace your attempt

For a timed exercise, allow four hours, the working-time allowance stated in GARP’s exam information . Record one choice per question before opening its explanation. This static set does not reproduce the official testing interface or establish equivalent difficulty. Use the 2026 official materials alongside this practice.

Practice questions

Questions 1-25

Question 1

Topic: Financial Markets and Products

A life insurer’s risk committee is assessing whether its term-life business offsets longevity exposure assumed from a defined-benefit pension plan. It is also reviewing underwriting for new individual annuities.

Existing contracts:

• Pension benefits: 10,000 members each receive $20,000 at the end of the coming year if alive then.
• Term-life benefits: 1,000 insured people each have a $100,000 death benefit payable at year-end if death occurs during the year.
• For both populations, actuaries revise the probability of surviving the year from 95% to 97%. Contract counts and benefit amounts remain unchanged.

Application evidence: Applicants who privately expect unusually long lives based on family medical histories disproportionately request larger lifetime annuities. Current underwriting does not capture this information. The pattern concerns selection before coverage begins, not changes in behavior afterward.

Considering only benefit payments on existing contracts, which assessment correctly states the change in expected year-end payments and identifies the underwriting problem?

• A. Expected benefit payments increase by $6 million; the application pattern indicates moral hazard.
• B. Expected benefit payments increase by $2 million; the application pattern indicates adverse selection.
• C. Expected benefit payments increase by $2 million; the application pattern indicates moral hazard.
• D. Expected benefit payments increase by $6 million; the application pattern indicates adverse selection.

Best answer: B

Explanation: Guaranteed pension payments expose the insurer to longevity risk: more survivors mean more benefits payable. Term-life insurance creates mortality exposure: fewer deaths mean fewer death benefits. For the stated one-year horizon, the changes are:

\[ \begin{aligned} \Delta E[\text{pension payments}] &= 10{,}000 \times 20{,}000 \times 0.02 = 4{,}000{,}000, \\ \Delta E[\text{death benefits}] &= 1{,}000 \times 100{,}000 \times (-0.02) = -2{,}000{,}000. \end{aligned} \]

Expected total benefit payments therefore increase by $2 million. The term-life book partially offsets the pension exposure to this survival revision; it does not eliminate the increase or establish a complete lifetime hedge.

The application pattern is adverse selection because customers use private information about expected longevity when choosing coverage before purchase. Moral hazard instead concerns behavior changed by insurance coverage. Collecting relevant medical and family-history information can improve annuity underwriting and pricing.

• A. Higher survival reduces rather than increases death benefits, and pre-contract selection based on private longevity information is adverse selection.
• B. Expected pension payments rise by $4 million while death benefits fall by $2 million; privately informed annuity purchases create adverse selection.
• C. The payment increase is $2 million, but selection based on private information before coverage begins is adverse selection rather than moral hazard.
• D. Adverse selection describes the application pattern, but higher survival reduces expected death benefits, leaving a net payment increase of $2 million.

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Question 2

Topic: Valuation and Risk Models

A risk manager is choosing between unweighted historical simulation with full revaluation and Gaussian delta-normal VaR to estimate 99% one-day loss VaR for two trading books.

Both methods use current positions. All material risk factors have synchronized observations and can be mapped to current exposures. Treat each history as representative of current conditions, with independent daily factor changes, and assume the distribution assessments are reliable.

Validation exhibit: Book A holds short equity-index options; Book B holds futures. Within each book, both repricing calculations use the same large joint factor shock drawn from its history. Positive amounts denote losses, measured in millions of US dollars.

Which allocation of VaR methods is best supported by the exhibit?

• A. Book A: historical simulation; Book B: delta-normal VaR.
• B. Book A: delta-normal VaR; Book B: delta-normal VaR.
• C. Book A: delta-normal VaR; Book B: historical simulation.
• D. Book A: historical simulation; Book B: historical simulation.

Best answer: A

Explanation: Full-revaluation historical simulation applies past joint factor shocks to current positions. It can preserve nonlinear price responses and empirical skewness or fat tails when relevant factors are available and the history is representative. Book A’s full-revaluation loss is $3.8 million versus $1.2 million under first-order repricing, demonstrating material curvature. Its 2,000 observations provide about 20 observations in the worst 1% of the sample.

Delta-normal VaR assumes approximately linear portfolio losses and Gaussian factor changes. Both assumptions are supported for Book B. Its 80-observation history implies only 0.8 expected observations beyond the true 99th percentile, making empirical tail estimation unstable. Under the supported Gaussian assumption, estimating covariance from the entire sample provides a more defensible basis for its 99% VaR.

These preferences remain conditional: historical simulation cannot create unobserved scenarios, while delta-normal remains vulnerable to parameter-estimation error and changes in distributional assumptions.

• A. Book A’s curvature and non-Gaussian factors favor full-revaluation historical simulation; Book B’s linear response, adequate Gaussian fit, and short history favor delta-normal VaR.
• B. For Book A, first-order repricing misses substantial curvature, and Gaussian factor assumptions conflict with the observed skewness and fat tails.
• C. Delta-normal misses Book A’s nonlinear loss response, while historical simulation estimates Book B’s extreme tail from a very short sample despite an adequate Gaussian fit.
• D. Book B’s 80 observations provide sparse coverage of the 1% tail, while its Gaussian factors and linear loss response support delta-normal estimation.

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Question 3

Topic: Valuation and Risk Models

A risk manager is calculating a trading book’s 10-trading-day, 99% VaR for a position-limit review. The model measures dollar profit and loss, with gains positive.

Model assumptions:

• Daily profit and loss is normally distributed with mean $50,000 and standard deviation $500,000.
• Daily profit and loss amounts are independent and identically distributed.
• The 10-day profit and loss is the sum of the daily amounts.

Loss is defined as the negative of profit and loss, and VaR is the 99th percentile of the loss distribution. The 99th percentile of the standard normal distribution is 2.326.

Which 10-day VaR should the manager report, rounded to the nearest $0.01 million?

• A. $3.18 million.
• B. $3.52 million.
• C. $4.18 million.
• D. $3.68 million.

Best answer: A

Explanation: Independent normal daily profit and loss amounts sum to a normal distribution. Over 10 days, the expected profit becomes $500,000, while the standard deviation becomes \(500{,}000\sqrt{10}\), approximately $1,581,139. Because loss is negative profit and loss, the expected loss is -$500,000.

The 99th-percentile loss is therefore:

\[ \operatorname{VaR}_{0.99,10} = -500{,}000 + 2.326 \times 500{,}000 \times \sqrt{10} \approx 3{,}177{,}729 \]

The reported VaR is $3.18 million. Only the standard deviation scales with the square root of time; the mean scales linearly. Directly multiplying one-day VaR by square-root time is not appropriate here because expected daily profit is nonzero.

• A. The 10-day expected loss is -$500,000, giving VaR of \( -500{,}000 + 2.326 \times 500{,}000 \times \sqrt{10} \), approximately $3.18 million.
• B. This multiplies the mean-adjusted one-day VaR by \(\sqrt{10}\), incorrectly scaling the expected profit by square-root time rather than linearly.
• C. This treats the $500,000 expected 10-day profit as an expected loss, adding it rather than subtracting it from the normal quantile component.
• D. This uses the 10-day standard deviation but omits the $500,000 expected profit, effectively assuming a zero loss mean.

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Question 4

Topic: Foundations of Risk Management

A bank holds a $100 million pool of five-year subordinated corporate loans. It needs $80 million immediately, must remain servicer, and must retain some credit exposure to support monitoring incentives. Its priority is minimizing retained loan principal loss in the specified stress.

Contractual and funding conditions:

• Only approved banks, insurers, and bankruptcy-remote securitization vehicles may participate; all quoted structures qualify.
• Borrower consents permit the quoted transactions. Asset transfers are true sales with no recourse beyond retained interests.
• A committed $80 million five-year borrowing facility can fund guarantee or CDS structures.
• Ignore fees and funding costs. Protection sellers perform fully, and protection covers the remaining five-year loan life.

Stress and quoted terms: Every borrower defaults just before maturity, triggering guarantee and CDS protection. The CDS reference the same borrowers. Recoveries apply uniformly across the pool.

Which structure best meets the bank’s stated priority while satisfying its funding and contractual requirements?

• A. Purchase $100 million of cash-settled CDS protection and draw $80 million from the committed borrowing facility.
• B. Sell 80% of the loan pool at par and retain the remaining 20% pro rata.
• C. Sell the $80 million senior tranche at par and retain the $20 million first-loss tranche.
• D. Obtain the 70% loan-loss guarantee and draw $80 million from the committed borrowing facility.

Best answer: B

Explanation: Credit risk transfer depends on loss allocation, not simply on cash proceeds or protection notional. With 40% loan recovery, the pool loses $60 million. Selling 80% pro rata transfers $48 million of that loss and leaves $12 million with the bank. The sale also supplies the required $80 million immediately.

Pro rata retention exposes the bank to the same fraction of each loan loss, preserving an economic incentive to monitor borrowers while it continues servicing.

Protection must be evaluated against the actual exposure. CDS settled using senior bond recoveries can underhedge subordinated loans even when notionals match. Likewise, selling senior securitization securities can raise substantial liquidity while leaving the seller exposed to the entire retained first-loss layer. Funding raised and credit risk transferred are therefore distinct measures.

• A. Senior bond recovery of 70% produces a $30 million CDS payment, leaving $30 million of loan loss because subordinated loan recovery is only 40%.
• B. The sale raises $80 million and leaves the bank bearing 20% of the $60 million pool loss, or $12 million, the smallest retained loss.
• C. The senior sale raises $80 million, but the $60 million pool loss exhausts the bank’s entire $20 million retained first-loss tranche.
• D. The guarantee pays $42 million against a $60 million loan loss, leaving $18 million of retained loss despite satisfying the funding requirement.

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Question 5

Topic: Financial Markets and Products

A derivatives desk has sold a one-year note whose maturity payment equals a reference long variance swap’s settlement, floored at zero and capped at $50,000. The desk wants to replicate that payment with options.

Annualized realized volatility \(\sigma\) is expressed as a numerical percentage, so 20% is entered as 20. Realized variance is \(v=\sigma^2\), measured in squared percentage points. All contracts reference the same realization period and cash-settle at maturity. European calls on these realized measures are available.

Contract exhibit:

Ignore initial premiums and compare terminal cash flows only. Which portfolio, established now and held unchanged, exactly replicates the note’s payment for every nonnegative realized volatility?

• A. Buy five volatility calls at strike 20 and sell five volatility calls at strike 30.
• B. Buy four volatility calls at strike 20 and sell four volatility calls at strike 32.5.
• C. Buy one variance call at strike 400 and sell one variance call at strike 900.
• D. Buy five variance calls at strike 400 and sell five variance calls at strike 500.

Best answer: C

Explanation: A volatility swap is linear in realized volatility, while a variance swap is linear in realized variance and therefore quadratic in volatility. Here, the reference settlement is \(100(v-400)\) dollars.

A variance call spread produces the required floor and cap:

\[ 100\left[\max(v-400,0)-\max(v-900,0)\right]. \]

The upper strike is \(400+50{,}000/100=900\). Below variance 400, both calls expire worthless. Between 400 and 900, the portfolio pays the positive reference settlement. Above 900, its payment remains $50,000.

These variance thresholds correspond to realized volatilities of 20% and 30%. Between them, the note’s payment is quadratic in volatility, whereas a volatility call spread is linear between its strikes. Matching endpoint payments or local sensitivity therefore does not establish exact replication. Holding the variance call spread unchanged makes this a static replication.

• A. At 25% realized volatility, this spread pays $25,000, whereas the note pays $22,500; matching the floor and cap does not match the intervening curvature.
• B. This spread matches sensitivity just above 20%, but at 25% it pays $20,000 rather than $22,500; matching local sensitivity is not exact replication.
• C. The spread preserves the $100 variance multiplier, begins paying above variance 400, and caps its payment at $100 times 500, or $50,000.
• D. Five contracts increase the effective variance multiplier to $500, making the spread pay $50,000 at variance 500 when the note pays only $10,000.

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Question 6

Topic: Financial Markets and Products

A stock will pay a certain $6 cash dividend in three months and no other dividends before month 6. European calls and puts on the stock have a strike price of $80 and expire in six months.

Executable market prices:

Both claims are risk-free zero-coupon securities. All securities can be bought or sold short at these prices, with immediate access to short-sale proceeds. Borrowing and lending are implemented through the claims. Ignore bid-ask spreads, transaction costs, taxes, and margin requirements.

Which trade generates a positive initial cash receipt and exactly offsets all subsequent cash flows for every possible stock price at expiration?

• A. Buy the stock and call; sell the put and both claims, for an initial receipt of $7.34.
• B. Buy the stock and put; sell the call and both claims, for an initial receipt of $1.34.
• C. Sell the stock and put; buy the call and expiration claim, for an initial receipt of $4.60.
• D. Sell the stock and put; buy the call and expiration claim and sell the dividend claim, for an initial receipt of $10.54.

Best answer: B

Explanation: European put-call parity incorporates the present value of certain dividends. For stock price \(S_0\), put price \(P\), call price \(C\), strike \(K\), and dividend \(D\), it requires:

\[ S_0+P=C+PV(K)+PV(D). \]

The stock-plus-put portfolio costs $88.00. The call plus the two risk-free claims costs $89.34. These portfolios have identical future cash flows, so buying the cheaper portfolio and shorting the more expensive portfolio produces an initial receipt of $1.34.

At month 3, the stock’s $6 dividend repays the short dividend claim. At expiration, the stock plus the long put minus the short call delivers exactly $80, regardless of the stock price. That amount repays the expiration claim. The initial receipt is therefore retained without any subsequent net payment. Individually plausible option prices can still violate the relative pricing required by parity.

• A. The terminal net cash flow is \(2(S_T-80)\), where \(S_T\) is the expiration stock price, so losses remain possible.
• B. The stock dividend offsets the $6 claim repayment, and the stock-plus-put-minus-call payoff of $80 offsets the expiration claim.
• C. The expiration cash flows cancel, but the short stock creates an unhedged $6 payment at the dividend date.
• D. At the dividend date, the short stock and short dividend claim each require $6, leaving an unhedged $12 outflow.

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Question 7

Topic: Quantitative Analysis

A risk analyst compares two forecasts of annual return volatility for a long-only portfolio. Portfolio weights and individual asset volatilities remain fixed.

Portfolio inputs:

Correlation estimates:

Define the diversification benefit as the difference between portfolio volatility with all pairwise correlations set to +1 and portfolio volatility under the forecast correlations.

Which conclusion is supported by the exhibit? Round annual portfolio volatilities to two decimal places.

• A. Annual volatility rises from 17.76% to 19.22%, and the diversification benefit decreases.
• B. Annual volatility falls from 18.65% to 17.84%, and the diversification benefit increases.
• C. Annual volatility remains at 22.50% in both forecasts, and the diversification benefit is unchanged.
• D. Annual volatility rises from 17.76% to 18.29%, and the diversification benefit decreases.

Best answer: D

Explanation: Portfolio variance aggregates individual variances and pairwise covariances:

\[ \sigma_p^2 = \sum_i w_i^2\sigma_i^2 + 2\sum_{i\lt j}w_iw_j\sigma_i\sigma_j\rho_{ij}. \]

The weighted asset volatilities are 0.120, 0.075, and 0.030. Their squared values sum to 0.020925. The covariance contributions, including the factor of two, total 0.010620 before the update and 0.012510 afterward. Therefore:

• Before: \(\sigma_p = \sqrt{0.031545} = 17.76\%\).
• After: \(\sigma_p = \sqrt{0.033435} = 18.29\%\).

Although the average pairwise correlation falls, the increase in A-B correlation receives the largest covariance weight and more than offsets the reductions elsewhere. An unweighted average correlation therefore gives a misleading risk assessment.

With all correlations at +1, portfolio volatility equals the weighted sum of asset volatilities, 22.50%. The diversification benefit narrows from approximately 4.74 to 4.21 percentage points.

• A. The 19.22% result includes the rise in A-B correlation while leaving A-C and B-C correlations at their earlier values.
• B. These figures replace distinct pairwise correlations with their arithmetic averages, ignoring the unequal covariance weights determined by portfolio weights and asset volatilities.
• C. The weighted sum of volatilities, 22.50%, applies when all pairwise correlations equal +1, not under either forecast in the exhibit.
• D. Covariance-based aggregation gives variances of 0.031545 and 0.033435, raising volatility and narrowing the gap to the 22.50% perfectly correlated benchmark.

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Question 8

Topic: Valuation and Risk Models

A risk committee is preparing a report that identifies mean-variance-efficient strategies and assesses whether this screening establishes their 99% one-year VaR ranking. VaR is expressed as a percentage of initial portfolio value.

Only four fixed-weight strategies are feasible: Defensive, Core, Growth, and Blend. No additional combinations are permitted. Their one-year return moments are given below.

Blend’s holdings:

• 50% in a sleeve with an expected return of 8% and standard deviation of 12%.
• 50% in a sleeve with an expected return of 12% and standard deviation of 20%.
• Correlation between the two sleeve returns: -0.25.

Returns have finite variances but may be skewed or fat-tailed; no distributional family is specified.

Which conclusion is supported for the report?

• A. The efficient set is Blend and Growth; their 99% VaR ranking cannot be inferred from the supplied moments.
• B. The efficient set is Blend and Growth; Blend’s 99% VaR must be lower than Growth’s.
• C. The efficient set is Defensive, Blend, and Growth; their 99% VaR ranking cannot be inferred from the supplied moments.
• D. The efficient set is Defensive, Core, and Growth; their 99% VaR ranking cannot be inferred from the supplied moments.

Best answer: A

Explanation: Within a fixed feasible set, a strategy is mean-variance efficient if no alternative has at least as high an expected return and no greater variance, with at least one strict improvement.

Blend’s expected return is 10%. Using decimal returns, its variance is:

\[ \sigma_{\text{Blend}}^2 = 0.5^2(0.12^2) + 0.5^2(0.20^2) + 2(0.5)(0.5)(-0.25)(0.12)(0.20) = 0.0106. \]

Its standard deviation is therefore approximately 10.30%. Blend dominates Defensive by offering a higher expected return with lower variance. It dominates Core by offering the same expected return with lower variance. Growth remains efficient because its higher expected return requires greater variance. Thus, Blend and Growth form the efficient set.

This screening compares only expected return and variance. A 99% VaR comparison requires information about the loss distributions’ upper-tail quantiles. Skewness and fat tails can change those quantiles without changing the first two moments, so the supplied data do not establish the VaR ranking.

• A. Blend’s 10% expected return and approximately 10.30% standard deviation dominate Defensive and Core; the supplied moments do not determine tail quantiles.
• B. The efficient set is correctly identified, but lower standard deviation does not establish lower 99% VaR when loss distributions can have different shapes.
• C. Including Defensive overlooks Blend’s covariance benefit: Blend has a higher expected return and lower standard deviation, so Defensive is dominated.
• D. Averaging sleeve standard deviations incorrectly gives Blend 16%; its covariance-based standard deviation is approximately 10.30%, making Blend efficient and Core dominated.

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Question 9

Topic: Quantitative Analysis

A lender estimates one-year default risk for a flagged borrower. The following screening rates apply to the borrower’s population.

Analyst’s calculation:

• Prior odds: \(0.05/0.95\).
• Likelihood ratio for the observed flag: \(0.80/0.135\).
• Reported default probability after multiplying these quantities and converting the resulting odds to a probability: 23.77%.

Which single revision to the analyst’s calculation yields the correct one-year default probability, rounded to two decimal places?

• A. Use 10% rather than 13.5% as the likelihood-ratio denominator; the revised default probability is 29.63%.
• B. Use the updated odds directly rather than converting them to a probability; the revised default probability is 31.19%.
• C. Use prior probability instead of prior odds, retaining the likelihood ratio and final conversion; the revised default probability is 22.86%.
• D. Use 20% rather than 13.5% as the likelihood-ratio denominator; the revised default probability is 17.39%.

Best answer: A

Explanation: An odds-based Bayesian update compares the likelihood of the same evidence under default and nondefault. Its denominator is the flag rate conditional on nondefault, not the overall flag rate.

Let \(D\) denote default within one year and \(F\) denote a flag. The correct likelihood ratio is \(P(F\mid D)/P(F\mid D^c)=0.80/0.10=8\). Multiplying prior odds by this ratio gives posterior odds of \((0.05/0.95)\times8=8/19\). Converting odds to probability gives \((8/19)/(1+8/19)=8/27\), or 29.63%.

The overall flag rate belongs in the probability form of Bayes’ rule: \(P(D\mid F)=(0.05\times0.80)/0.135\). Mixing that denominator with an odds-based prior understates the borrower’s default probability, producing 23.77% instead of 29.63%.

• A. The correct likelihood ratio is 0.80/0.10 = 8, giving posterior odds of 8/19 and a posterior probability of 8/27.
• B. The analyst’s computed 0.3119 represents odds, not probability, and removing the conversion leaves the incorrect likelihood-ratio denominator unchanged.
• C. An odds update must begin with prior odds; this substitution also leaves the incorrect marginal flag-rate denominator unchanged.
• D. The 20% false-negative rate conditions on default; the denominator must instead be the 10% flag rate conditional on nondefault.

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Question 10

Topic: Valuation and Risk Models

A bank reviews its internal stress-testing program against the Basel Committee’s 2018 Stress testing principles.

Purpose: The board wants to assess one-year group losses and business contributions under a severe but plausible recession to inform capital allocations.

Program review:

• The board approves the objectives, receives results, and uses them in capital planning.
• Staffing, data access, documentation, and independent review of each business model are adequate.
• Each business selects the scenario producing its largest standalone loss. Lending selects a deep recession; trading selects a market dislocation that assumes continued economic growth.
• The bank sums these business-level maximum losses and reports the total as its group recession loss. All models use a one-year horizon.

Which change to the aggregation methodology would most directly support the board’s purpose?

• A. Assign occurrence probabilities to the business scenarios and use their probability-weighted total as the group estimate.
• B. Align business scenarios to the same marginal loss percentile and sum their losses for the group estimate.
• C. Evaluate all businesses under a common recession scenario and aggregate their losses, incorporating material cross-risk interactions.
• D. Apply normal-period loss correlations to the business maxima and use the diversification-adjusted total as the group estimate.

Best answer: C

Explanation: Firm-wide stress testing should capture material risks under coherent adverse conditions and support the decisions for which it is designed. A group recession estimate therefore requires consistent macroeconomic assumptions across businesses, together with relevant dependencies and cross-risk effects.

The bank currently combines losses from two different economic states: a deep recession and a market dislocation with continued growth. Their sum may serve as a conservative envelope of standalone outcomes, but it does not measure losses under the board’s specified recession. Adequate governance, resources, documentation, and standalone model review do not resolve this inconsistency. The bank should assess all businesses under a common severe but plausible recession and use the resulting group losses and business contributions as inputs to capital allocation.

• A. Probability weighting produces an average across different conditions rather than estimating group losses conditional on the specified severe recession.
• B. Matching marginal loss percentiles standardizes tail severity but does not ensure that the business losses arise under consistent recession assumptions.
• C. A coherent recession scenario connects business losses to the same adverse conditions, while incorporating interactions supports a meaningful assessment of group losses.
• D. Normal-period correlations neither reconcile incompatible scenario assumptions nor reliably represent dependencies during the recession the board wants to assess.

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Question 11

Topic: Financial Markets and Products

A refinery plans to buy 1,000,000 barrels of local crude oil in 18 months. It hedges by buying 1,000 exchange-traded crude oil futures contracts, each covering 1,000 barrels, with three months to expiry. Before each expiry, it will replace the position with new three-month contracts until the purchase occurs.

One-day observations: All prices are in dollars per barrel. No futures contracts were rolled during the day.

Valuation and cash conditions:

• The refinery uses the local forward quotation to estimate its future purchase cost; it holds no local forward contract.
• Futures losses are settled daily in cash, with today’s variation margin due at the close.
• The refinery has $5 million available for variation margin and cannot raise additional cash before payment.
• Changes in estimated physical purchase cost generate no cash before procurement. Ignore discounting, transaction costs, and changes in initial margin.

Which conclusion about the change in estimated hedged procurement cost and today’s margin-funding shortfall is supported?

• A. Estimated hedged procurement cost increases by $7 million, with a $2 million margin-funding shortfall.
• B. Estimated hedged procurement cost increases by $1 million, with no margin-funding shortfall.
• C. Estimated hedged procurement cost increases by $1 million, with a $2 million margin-funding shortfall.
• D. Estimated hedged procurement cost decreases by $1 million, with a $2 million margin-funding shortfall.

Best answer: C

Explanation: A long futures hedge offsets a future buyer’s exposure to rising commodity prices, but imperfect matching and daily settlement introduce other risks. Here, estimated physical purchase cost falls by $6 million. The futures position covers 1,000,000 barrels and loses $7 million. Estimated hedged procurement cost therefore becomes $74 million plus $7 million, or $81 million, compared with $80 million initially.

Daily settlement requires $7 million in cash now, not merely the $1 million net economic change. The refinery has only $5 million available, leaving a $2 million liquidity shortfall. The favorable change in future purchase cost cannot fund today’s margin payment.

The local crude benchmark and exchange futures have different underlying specifications and maturities, creating residual basis risk. Repeatedly replacing shorter-dated futures is stack-and-roll hedging, which also exposes the refinery to changing calendar spreads and rollover conditions.

• A. The $7 million futures loss must be combined with the $6 million decline in estimated physical purchase cost, leaving a $1 million increase.
• B. The lower future purchase-cost estimate provides no cash today; the $7 million margin payment exceeds available cash by $2 million.
• C. The $7 million futures loss exceeds the $6 million purchase-cost reduction, while $5 million of available cash leaves $2 million of margin unfunded.
• D. The $6 million decline in estimated purchase cost is outweighed by the $7 million futures loss, so hedged cost increases rather than decreases.

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Question 12

Topic: Foundations of Risk Management

An internal auditor examines a bank’s ERM implementation over the latest quarter. Board-approved policy assigns desk heads responsibility for daily trading-limit compliance. It also requires the CRO to report any unapproved breach lasting more than one business day directly to the board risk committee, even if subsequently resolved.

Review exhibit:

• Data validation: Desk and central-risk records agreed on exposures, thresholds, and breach timing.
• Exceptions: Six unapproved breaches lasted three to five business days. Each desk returned within limits before month-end.
• Challenge: The CRO documented timely requests to reduce exposures. Desk heads deferred action while positions remained profitable.
• Compensation: Desk-head bonuses depended on trading revenue; unapproved breaches did not reduce awards.
• Committee reporting: At a business executive’s request, final reports omitted the breaches and displayed only month-end compliance.

Which conclusion and corrective response most directly address the principal ERM implementation barrier shown by this evidence?

• A. Risk accountability is undermined by incentives and filtered escalation; link desk-head pay to breach history and enforce the CRO’s reporting duty.
• B. Overly restrictive limits are encouraging exceptions; recalibrate desk thresholds and link bonuses to compliance with the revised limits.
• C. Risk responsibilities lack formal clarity; document desk-head ownership of remediation and give the CRO independent authority to escalate unapproved breaches.
• D. Risk data are insufficiently integrated; centralize exposure reconciliation and require a consolidated breach dashboard for the board risk committee.

Best answer: A

Explanation: A strong enterprise risk culture is demonstrated through decisions, incentives, and responses to challenge, not policy documentation alone. Here, accurate records and explicit responsibilities did not prevent profitable positions from remaining outside limits. Revenue-based compensation imposed no consequences for unapproved risk taking, while management filtered breach information before it reached the board.

The principal barrier is failure to implement and enforce accountability. Desk heads should retain responsibility for compliance and remediation, with compensation reflecting breach behavior. The CRO’s independent reporting obligation must be enforced so the committee receives all breaches meeting the policy’s reporting criterion, including those resolved before month-end. A compliant month-end position does not erase earlier violations or demonstrate that challenge was effective.

• A. Revenue-based bonuses and omitted breach reports weakened existing accountability and challenge, so compensation consequences and enforced escalation target the observed behavior.
• B. Repeated breaches do not establish that limits are too restrictive; recalibration would not address deferred remediation or suppression of required breach reporting.
• C. Formal ownership and independent CRO escalation are already specified; the gap lies in following and enforcing those responsibilities, not documenting additional authority.
• D. Matching desk and central-risk records show that breaches were captured consistently; consolidation misses their deliberate removal from committee reports.

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Question 13

Topic: Valuation and Risk Models

A bank defines economic credit capital \(K\) as one-year 99% expected shortfall of credit losses minus expected loss. Assume \(K\) is differentiable at the current portfolio and homogeneous of degree one in loan exposures. Current portfolio capital is $12.0 million.

Model output: Exposure and standalone capital are in millions of US dollars. Marginal sensitivity is measured in dollars of capital per dollar of exposure and evaluated at the current portfolio. Standalone capital uses the same measure with each book held alone.

Any infinitesimal exposure-neutral transfer between books is feasible, with credit quality and dependence parameters held fixed.

Which conclusion correctly identifies the largest Euler component contribution and the transfer yielding the greatest first-order capital reduction per dollar moved?

• A. The property book has the largest component contribution; transferring exposure from the property book to the corporate book maximizes the first-order capital reduction.
• B. The corporate book has the largest component contribution; transferring exposure from the consumer book to the corporate book maximizes the first-order capital reduction.
• C. The consumer book has the largest component contribution; transferring exposure from the consumer book to the corporate book maximizes the first-order capital reduction.
• D. The property book has the largest component contribution; transferring exposure from the consumer book to the corporate book maximizes the first-order capital reduction.

Best answer: D

Explanation: Euler allocation multiplies each exposure by its marginal capital sensitivity: \(K_i=x_i\,\partial K/\partial x_i\). Differentiability and degree-one homogeneity imply that these components sum to total portfolio capital.

The component contributions are:

• Corporate: \(100\times0.040=4.0\) million dollars.
• Property: \(50\times0.100=5.0\) million dollars.
• Consumer: \(25\times0.120=3.0\) million dollars.

They reconcile to $12.0 million. Property accounts for approximately 41.7% of total capital, the largest current contribution.

For an exposure-neutral transfer, the first-order capital change per dollar moved equals the recipient’s marginal sensitivity minus the donor’s. Moving exposure from consumer to corporate produces \(0.040-0.120=-0.080\), the greatest available reduction.

Component contribution measures current allocated risk; marginal sensitivity measures incremental risk. Standalone capital measures a book in isolation and does not allocate portfolio diversification benefits. The transfer ranking is local and need not remain optimal after a large reallocation.

• A. Property-to-corporate transfers reduce capital by $0.060 per dollar moved, less than the $0.080 reduction from consumer-to-corporate transfers.
• B. Corporate has the largest standalone capital, but its Euler contribution is only $4.0 million, below the property book’s $5.0 million.
• C. Consumer has the highest marginal sensitivity, but its smaller exposure produces a component contribution of only $3.0 million.
• D. Property contributes $5.0 million, the largest component, while consumer-to-corporate transfers exploit the largest marginal sensitivity difference, 0.080.

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Question 14

Topic: Foundations of Risk Management

A risk manager is considering replacing a concentrated equity portfolio with a diversified equity portfolio to reduce risk while preserving its CAPM expected return. Assume CAPM holds and both portfolios follow a single-index model, with each portfolio’s residual return uncorrelated with the market return.

The annual risk-free rate is 3.0%, the expected annual market return is 8.0%, and annual market return volatility is 20.0%.

Which comparison of annual expected returns and total return volatilities should the manager report, rounded to one decimal place?

• A. Expected returns are 9.0% for both portfolios; total volatilities are 30.0% for the concentrated portfolio and 24.7% for the diversified portfolio.
• B. Expected returns are 9.0% for both portfolios; total volatilities are 42.0% for the concentrated portfolio and 30.0% for the diversified portfolio.
• C. Expected returns are 10.5% for the concentrated portfolio and 9.2% for the diversified portfolio; total volatilities are 30.0% and 24.7%, respectively.
• D. Expected returns are 9.0% for both portfolios; total volatilities are 30.0% for the concentrated portfolio and 10.0% for the diversified portfolio.

Best answer: A

Explanation: CAPM prices systematic risk, measured by beta, rather than total risk. Both portfolios therefore have the same expected annual return:

\[ E[R_p] = 3.0\% + 1.20(8.0\% - 3.0\%) = 9.0\%. \]

In the single-index model, market-driven and residual returns are uncorrelated, so total variance is \(\sigma_p^2 = \beta_p^2\sigma_m^2 + \sigma_{\epsilon,p}^2\). Using decimal returns:

• Concentrated portfolio: \(\sqrt{(1.20 \times 0.20)^2 + 0.18^2} = 30.0\%\).
• Diversified portfolio: \(\sqrt{(1.20 \times 0.20)^2 + 0.06^2} \approx 24.7\%\).

Diversification reduces firm-specific residual risk, but it does not eliminate exposure to common market movements. At the unchanged beta, even eliminating residual risk entirely would leave annual volatility of \(1.20 \times 20.0\% = 24.0\%\). Thus, the replacement reduces total risk without changing the CAPM expected return.

• A. Equal betas imply equal CAPM expected returns, while combining the common market-driven variance with each residual variance gives the stated total volatilities.
• B. These volatilities add market-driven and residual standard deviations directly; uncorrelated return components require adding their variances instead.
• C. These expected returns substitute total volatility relative to market volatility for beta, incorrectly treating residual risk as priced market exposure.
• D. Reducing residual volatility by a factor of three does not reduce total volatility proportionally because the common market exposure remains unchanged.

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Question 15

Topic: Quantitative Analysis

A risk analyst estimates dependence between the daily P&Ls of a $2,000,000 long position in asset A and a $1,000,000 short position in asset B.

Daily asset returns are measured in percentage points, so a recorded return of 1 means 1%. The following variance estimates use the same observations and are expressed in squared percentage points:

Each position’s P&L equals its signed initial dollar exposure multiplied by the asset’s decimal return. Ignore financing costs. The risk report measures P&L in thousands of dollars and covariance in (thousand dollars) squared.

Which covariance and correlation estimates for the two position P&Ls are supported by these data?

• A. Covariance: -6; correlation: -0.50.
• B. Covariance: -600; correlation: -0.50.
• C. Covariance: -1,200; correlation: -1.00.
• D. Covariance: 600; correlation: 0.50.

Best answer: B

Explanation: Portfolio variance includes both individual variances and the covariance cross-term. For the equally weighted portfolio, \(4.75 = 0.25(4) + 0.25(9) + 2(0.5)(0.5)\operatorname{Cov}(R_A,R_B)\). Therefore, the asset-return covariance is \((19-4-9)/2 = 3\), and their correlation is \(3/(\sqrt{4}\sqrt{9})=0.50\).

A one-percentage-point return produces $20,000 on the long position and -$10,000 on the short position. Thus, in thousands of dollars, the P&Ls are \(X=20R_A\) and \(Y=-10R_B\). Their covariance is \(20(-10)(3)=-600\). Their standard deviations are \(20\sqrt{4}=40\) and \(10\sqrt{9}=30\), giving correlation \(-600/(40\times30)=-0.50\).

Covariance changes with the product of the scaling factors. Correlation is unchanged by positive rescaling, but reverses sign when exactly one variable is multiplied by a negative factor.

• A. Using multipliers of 2 and -1 mixes million-dollar exposures with percentage-point returns; the report requires multipliers of 20 and -10.
• B. The return covariance is 3; applying P&L multipliers of 20 and -10 gives covariance -600 and correlation -0.50.
• C. Recovering return covariance requires dividing the cross-term by two; omitting that division doubles both the P&L covariance and the correlation.
• D. The short position reverses the sign of asset B’s contribution to P&L, making both covariance and correlation negative.

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Question 16

Topic: Valuation and Risk Models

A risk manager evaluates a proposal to expand a long-only equity portfolio. Within each portfolio, holdings have equal weights and come from distinct issuers.

For every holding in both portfolios, the model for its mean-centered annual return is \(r_i = F + \epsilon_i\), where:

• \(F\) is the same funding-conditions factor for every holding, with an annual standard deviation of 8%.
• Each residual \(\epsilon_i\) has an annual standard deviation of 30%.
• Residuals are mutually independent and independent of \(F\).

Assume these relationships remain unchanged after expansion. Which conclusion about the proposed portfolio is supported? Round annual volatility to the nearest 0.1% and the factor’s share of variance to the nearest whole percent.

• A. Annual volatility is 3.1%, and the funding factor accounts for 7% of portfolio variance.
• B. Annual volatility is 8.5%, and the funding factor accounts for 88% of portfolio variance.
• C. Annual volatility is 8.0%, and the funding factor accounts for 100% of portfolio variance.
• D. Annual volatility is 5.0%, and the funding factor accounts for 64% of portfolio variance.

Best answer: B

Explanation: Independent issuer-specific shocks diversify as the number of equally weighted holdings increases. A shared factor does not: every holding has a loading of one, so the portfolio also has a funding-factor loading of one.

The proposed portfolio’s annual variance is:

\[ \sigma_p^2 = 0.08^2 + \frac{0.30^2}{100} = 0.0073. \]

Therefore, annual volatility is \(\sqrt{0.0073} \approx 8.5\%\), and the funding factor’s variance share is \(0.0064/0.0073 \approx 88\%\).

The current portfolio has annual variance \(0.08^2 + 0.30^2/25 = 0.0100\), giving 10.0% volatility and a 64% funding-factor share. Expansion reduces total risk, but the shared funding factor becomes more dominant. More issuers and sectors diversify residual risk without eliminating concentration in the common risk driver.

• A. Dividing both factor and residual variances by 100 treats the shared funding shock as independent across holdings, understating its contribution.
• B. The common factor contributes \(0.08^2\) of variance, while independent residuals contribute \(0.30^2/100\), yielding 8.5% volatility and 88% factor attribution.
• C. With 100 holdings, residual volatility remains 3.0%, so total volatility exceeds the 8.0% common-factor floor and residual risk still contributes.
• D. Scaling the current portfolio’s entire variance by 25/100 incorrectly diversifies the common factor; only independent residual variance shrinks with holding count.

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Question 17

Topic: Foundations of Risk Management

A risk manager uses the security market line to assess an equity analyst’s return forecast. Use the market index as the CAPM market portfolio. All returns are one-year arithmetic returns, and the volatility and correlation estimates refer to the same return horizon.

Which conclusion correctly compares the analyst’s forecast with the stock’s CAPM expected return?

• A. The CAPM expected return is 18.0%; the analyst’s forecast is 5.0 percentage points below that return.
• B. The CAPM expected return is 8.2%; the analyst’s forecast is 4.8 percentage points above that return.
• C. The CAPM expected return is 12.4%; the analyst’s forecast is 0.6 percentage points above that return.
• D. The CAPM expected return is 16.0%; the analyst’s forecast is 3.0 percentage points below that return.

Best answer: C

Explanation: CAPM compensates investors for systematic risk, measured by beta, rather than total volatility alone. Beta can be obtained from correlation and relative volatility:

\[ \beta = \rho_{S,M}\frac{\sigma_S}{\sigma_M} = 0.70\frac{0.30}{0.15} = 1.40. \]

The market risk premium is 10.0% minus 4.0%, or 6.0%. Therefore:

\[ E[R_S] = R_f + \beta(E[R_M]-R_f) = 4.0\% + 1.40(6.0\%) = 12.4\%. \]

At beta 1.40, the analyst’s 13.0% forecast lies 0.6 percentage points above the security market line. This is a positive forecast alpha relative to CAPM, not evidence that the stock will actually outperform. Neither the CAPM expected return nor the analyst’s forecast guarantees a realized return.

• A. This calculation treats the 10.0% expected market return as the market risk premium instead of subtracting the 4.0% risk-free return.
• B. Using correlation of 0.70 as beta ignores relative volatility; beta also incorporates the stock-to-market standard deviation ratio.
• C. Beta is 1.40, so applying it to the 6.0% market risk premium gives a 12.4% expected return, below the 13.0% forecast.
• D. Using the volatility ratio of 2.00 as beta ignores the correlation of 0.70, overstating the stock’s systematic risk.

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Question 18

Topic: Quantitative Analysis

A risk analyst must report a frequentist one-sided 95% upper confidence bound for a trading desk’s population mean daily loss. Daily loss is positive for a loss and negative for a profit. Assume losses are independent observations from an unchanged normal distribution with unknown mean and variance.

Sample data:

• Number of trading days: 25
• Sample mean daily loss: $42,000
• Sample standard deviation of daily loss: $30,000

For a Student’s t-distribution with 24 degrees of freedom, the upper-tail 5% and 2.5% critical values are 1.711 and 2.064, respectively.

Using these critical values and rounding to the nearest dollar, which report correctly states and interprets the required bound?

• A. The upper bound is $52,266; the fixed population mean has a 95% probability of being below this observed bound.
• B. The upper bound is $54,384; the method produces bounds covering the fixed population mean in 95% of repeated samples.
• C. The upper bound is $54,384; the fixed population mean has a 95% probability of being below this observed bound.
• D. The upper bound is $52,266; the method produces bounds covering the fixed population mean in 95% of repeated samples.

Best answer: D

Explanation: A one-sided upper confidence bound accounts for sampling uncertainty in the estimated population mean. Because the population variance is unknown and observations are normal, use Student’s t-distribution with \(25-1=24\) degrees of freedom.

The estimated standard error is \(30{,}000/\sqrt{25}=6{,}000\) dollars. The 95% upper bound is therefore \(42{,}000+1.711\times6{,}000=52{,}266\) dollars. A one-sided 95% bound places the entire 5% error probability in one tail; the 2.5% upper-tail critical value would instead produce a one-sided 97.5% bound.

The population mean is fixed, while the sample and calculated bound vary. Across repeated samples under the stated assumptions, 95% of bounds constructed by this procedure would be at least as large as the true mean. Once observed, a particular bound either covers that mean or does not.

• A. The numerical bound is correct, but frequentist confidence does not assign a probability to the fixed mean after observing the bound.
• B. Using the upper-tail 2.5% critical value produces a one-sided 97.5% upper confidence bound, not the required 95% bound.
• C. The calculation uses the critical value for 97.5% one-sided coverage, and the probability statement incorrectly treats the fixed population mean as random.
• D. The upper-tail 5% critical value gives $52,266, and 95% coverage describes the procedure across repeated samples.

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Question 19

Topic: Foundations of Risk Management

A financial group delegates trade approval to its three business units.

Governance facts:

• Each unit has a $65 million one-year stress-loss limit.
• The board caps net firmwide losses at $110 million in any approved one-year stress scenario. The group has $180 million of loss-absorbing capital.
• Monitoring certifies business-unit limit compliance separately. Bonuses depend only on business-unit profit.

A risk analyst compiles the following estimates for the group’s two approved scenarios. Each scenario affects all units simultaneously. Amounts are in USD millions, after eliminating intragroup exposures; positive values indicate losses and negative values indicate gains.

Which assessment of the group’s stress exposure and risk governance is best supported?

• A. The maximum firmwide stress loss is $105 million, indicating appetite compliance alongside compliance with business-unit limits.
• B. The maximum firmwide stress loss is $160 million, indicating an appetite breach despite compliance with business-unit limits.
• C. The maximum firmwide stress loss is $120 million, indicating appetite compliance because losses remain below available capital.
• D. The maximum firmwide stress loss is $120 million, indicating an appetite breach despite compliance with business-unit limits.

Best answer: D

Explanation: Enterprise risk management evaluates combined exposures to common shocks against firmwide risk appetite. Passing each business-unit limit is not sufficient.

The property downturn produces a $90 million net loss after including the macro trading gain. The rate spike produces a $120 million net loss, exceeding the $110 million appetite cap by $10 million, although every unit remains within its $65 million limit. Available capital does not replace the board’s lower risk-appetite threshold.

An enterprise response would coordinate shared-factor exposures and business-unit limits and align incentives with contributions to aggregate risk. Unit heads can retain trade-approval authority and responsibility for their positions. ERM requires coordinated oversight, not centralized approval of every trade.

• A. The $105 million figure averages the $90 million and $120 million scenario totals; appetite applies to each scenario, not their average.
• B. Adding each unit’s largest loss gives $160 million, but those maxima occur in different scenarios and do not represent the maximum joint scenario loss.
• C. The $180 million capital amount exceeds the modeled loss, but the separate $110 million risk-appetite cap is still breached.
• D. The rate-spike scenario produces a $120 million net loss, exceeding the $110 million appetite cap while no unit exceeds its $65 million limit.

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Question 20

Topic: Valuation and Risk Models

A USD-based risk analyst evaluates two FX forwards with the same remaining maturity. Each contract exchanges the indicated foreign-currency leg for a fixed USD amount at maturity. Neither contract has a cash flow during the next trading day.

The one-day arithmetic returns on the current market forward quotes are jointly normally distributed. Quotes are expressed in USD per unit of foreign currency.

Position and return data:

Model assumptions:

• Hold the USD discount factor to maturity fixed at 0.98.
• The correlation between the two quote returns is 0.60.
• Use 2.326 as the rounded standard normal 99th-percentile value.

Define loss as negative USD portfolio P&L and VaR as the 99th percentile of that loss. What is the portfolio’s one-day 99% parametric VaR, rounded to the nearest $1,000?

• A. $157,000
• B. $169,000
• C. $320,000
• D. $117,000

Best answer: A

Explanation: With a fixed USD exchange leg and discount factor, an FX forward’s value change equals its signed foreign-currency amount times the market forward quote change times the discount factor. Its monetary exposure to a quote return is therefore not simply its foreign-currency notional.

The EUR return exposure is $9,800,000, calculated as 0.98 times EUR8,000,000 times 1.25 USD/EUR. The GBP return exposure is -$7,840,000 because that contract delivers GBP.

Expected USD P&L is \(9{,}800{,}000(0.001)-7{,}840{,}000(0.0005)=5{,}880\), so expected loss is -$5,880. Each position’s standalone P&L standard deviation is $78,400. Opposite exposure signs make the covariance contribution negative:

\[ \sigma_L=\sqrt{78{,}400^2+78{,}400^2-2(0.60)(78{,}400)^2}\approx70{,}123.89. \]

For normally distributed losses, \(\operatorname{VaR}_{0.99}=\mu_L+2.326\sigma_L\). Using unrounded intermediate values gives approximately $157,228, which rounds to $157,000. The expected gain reduces VaR, while the positive return correlation provides diversification because the positions have opposite signs.

• A. The signed, discounted USD return exposures give expected loss of -$5,880 and loss volatility of approximately $70,124, producing a $157,000 VaR.
• B. Adding the expected portfolio gain to the volatility term produces approximately $169,000; a positive expected gain instead reduces the loss quantile.
• C. Using a positive covariance contribution despite the opposite position signs overstates loss volatility and produces approximately $320,000.
• D. Using the foreign-currency amounts directly as USD return exposures gives approximately $117,000; the exposures require conversion using the forward quotes and discounting.

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Question 21

Topic: Quantitative Analysis

A risk analyst fits an OLS regression of daily portfolio returns on market-index returns, including an intercept, to model the portfolio’s conditional expected return. The 300 observations are divided into five equally sized groups by market return.

The residual summary below reports means and within-group standard deviations. One basis point (bp) equals 0.01 percentage points of return.

Which response best addresses the model weakness most clearly indicated by this summary?

• A. Retain the linear specification and use heteroskedasticity-consistent standard errors to evaluate the estimated market exposure.
• B. Retain the linear specification and use robust regression to downweight observations with large absolute residuals.
• C. Retain the linear specification and refit using weights inversely proportional to each group’s residual variance.
• D. Add squared market return to the specification and estimate the expanded model using ordinary least squares.

Best answer: D

Explanation: Residual diagnostics distinguish misspecification of the conditional mean from problems with error variance or unusual observations. A suitably specified conditional-mean model should not leave a systematic residual pattern as its predictors change.

Here, average residuals are positive at both market-return extremes and negative near the center. The linear model therefore underpredicts returns at the extremes and overpredicts them near the center. Adding squared market return allows curvature while retaining a regression that is linear in its coefficients. The expanded model should then undergo further residual checks.

Within-group residual standard deviations are nearly constant. This evidence favors revising the functional form rather than changing variance-based weights, standard errors, or the treatment of individual large residuals.

• A. Heteroskedasticity-consistent standard errors adjust inference, but they do not correct the systematic curvature in the model’s conditional mean.
• B. Downweighting large residuals targets outlier sensitivity rather than the systematic U-shaped pattern across market-return groups.
• C. Inverse-variance weighting addresses unequal error variances, whereas the nearly constant within-group dispersion and curved residual means indicate a specification problem.
• D. Positive residual means at both extremes and negative means near the center support adding a quadratic term to capture conditional-mean curvature.

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Question 22

Topic: Financial Markets and Products

A central counterparty (CCP) has novated OTC swaps between its clearing members. Member Orion defaults after completing its latest variation-margin settlement, which included a $12 million payment to the CCP. Subsequent price movements produce a $60 million closeout loss, measured before applying default-waterfall resources.

Default waterfall: The CCP must use these resources in the stated order:

• Orion’s initial margin: $30 million.
• Orion’s default-fund contribution: $10 million.
• The CCP’s dedicated capital tranche: $5 million.
• Surviving members’ mutualized default fund: $40 million available.

Some collateral cannot be liquidated until after scheduled payments to nondefaulting members fall due.

Which assessment of the loss allocation and remaining clearing risk is supported?

• A. The mutualized default fund absorbs $15 million, and nondefaulting clearing members resume bilateral counterparty exposure and responsibility for closing out their original trades.
• B. The mutualized default fund absorbs $20 million, and the CCP retains payment-liquidity and default-management responsibilities toward nondefaulting clearing members.
• C. The mutualized default fund absorbs $3 million, and the CCP retains payment-liquidity and default-management responsibilities toward nondefaulting clearing members.
• D. The mutualized default fund absorbs $15 million, and the CCP retains payment-liquidity and default-management responsibilities toward nondefaulting clearing members.

Best answer: D

Explanation: Novation replaces the original bilateral contracts with contracts facing the CCP. Orion’s default therefore leaves the CCP responsible for managing the defaulted portfolio and fulfilling its obligations to nondefaulting clearing members.

Variation margin settles exposure from price changes up to the settlement time. The $60 million loss arises afterward, so the earlier $12 million payment is not an additional loss-absorbing resource. Applying the specified waterfall, the surviving members’ fund absorbs \(60 - 30 - 10 - 5 = 15\) million dollars.

Loss coverage and payment liquidity are different. Collateral may ultimately cover losses yet be unavailable when cash payments are due. Clearing reduces bilateral counterparty exposure but concentrates liquidity demands, operational continuity, and default-management responsibilities at the CCP. Mutualization also transmits residual losses to surviving members rather than eliminating system risk.

• A. Novation substitutes the CCP as counterparty; a clearing-member default does not automatically restore the original bilateral contracts.
• B. The dedicated $5 million CCP capital tranche must absorb losses before mutualized resources, reducing their required contribution from $20 million to $15 million.
• C. The settled $12 million variation-margin payment cannot be deducted again from a closeout loss arising after that settlement.
• D. The waterfall leaves $15 million for surviving members, while delayed collateral liquidation leaves the CCP responsible for meeting payments before proceeds arrive.

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Question 23

Topic: Quantitative Analysis

A bank is validating a finite one-year stress-scenario probability model for a credit portfolio. Trigger A is a funding shortfall, and trigger B is a collateral-value breach.

The model documentation states:

The two triggers are independent. The probability of an obligor’s default conditional on neither trigger occurring is 0%.

Which validation conclusion is supported by the model’s probabilities?

• A. The triggers are dependent, and the conditional default probability given neither trigger is zero.
• B. The triggers are dependent, and the conditional default probability given neither trigger is undefined.
• C. The triggers are independent, and the conditional default probability given neither trigger is undefined.
• D. The triggers are independent, and the conditional default probability given neither trigger is zero.

Best answer: B

Explanation: Independence requires the joint probability to equal the product of the marginal probabilities. Here, \(P(A)P(B)=0.60\times0.55=0.33\), whereas \(P(A\cap B)=0.15\), so the triggers are dependent.

Inclusion-exclusion gives \(P(A\cup B)=0.60+0.55-0.15=1\). Consequently, the probability that neither trigger occurs is zero.

For any default event D and conditioning event N, elementary conditional probability is \(P(D\mid N)=P(D\cap N)/P(N)\), requiring \(P(N)>0\). With N representing neither trigger, both numerator and denominator are zero. The resulting undefined ratio does not establish a zero conditional default probability; the model assigns no probability mass to the conditioning state.

• A. The dependence conclusion is supported, but zero probability for the conditioning event makes the conditional default probability undefined rather than zero.
• B. The joint probability does not equal the product of the marginals, and inclusion-exclusion gives zero probability for the no-trigger conditioning event.
• C. The conditional probability is undefined, but independence fails because the observed 15% joint probability differs from the required 33%.
• D. Independence would require a 33% joint probability, while the no-trigger event has zero probability and therefore cannot support the stated conditional probability.

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Question 24

Topic: Quantitative Analysis

A risk analyst fits two OLS models using the same 250 independent daily observations. Portfolio and market returns are recorded in percentage points (1% is entered as 1.00). Credit spread changes are in basis points, with widening positive.

Model conditions:

• The multiple-regression error has zero conditional mean given both regressors.
• The regressors are negatively correlated, but the design matrix has full column rank.
• Conditional error variance increases with the absolute spread change.

Consider a market return of -2.00% and credit spread widening of 30 basis points. Which conclusion about the multiple-regression fitted return and the properties of its OLS estimators is supported?

• A. The fitted return is -2.38%; the multiple-regression OLS estimators are biased and are not guaranteed to be BLUE under heteroskedasticity.
• B. The fitted return is -2.38%; the multiple-regression OLS estimators are unbiased and are not guaranteed to be BLUE under heteroskedasticity.
• C. The fitted return is -1.58%; the multiple-regression OLS estimators are biased and are not guaranteed to be BLUE under heteroskedasticity.
• D. The fitted return is -1.58%; the multiple-regression OLS estimators are unbiased and are not guaranteed to be BLUE under heteroskedasticity.

Best answer: D

Explanation: Multiple regression estimates a partial association for each included regressor. Here, a one-percentage-point increase in market return is associated with a 0.50-percentage-point increase in fitted portfolio return, holding the spread change fixed.

The scenario’s fitted return is \( \widehat{R} = 0.02 + 0.50(-2.00) - 0.02(30) = -1.58 \) percentage points, or -1.58%.

The simple-regression slope of 0.90 also captures associated spread movements. Because spreads have a negative coefficient and are negatively correlated with market returns, their omission raises the market-only slope relative to the partial slope. Combining that univariate slope with the separate spread coefficient would double-count part of the association.

Zero conditional error mean and full column rank support unbiased OLS estimation in the multiple model. Heteroskedasticity does not introduce coefficient bias, but OLS is no longer guaranteed to be the best linear unbiased estimator (BLUE). Conventional homoskedastic standard errors are also generally inappropriate; heteroskedasticity-robust standard errors address this inference problem.

• A. The forecast substitutes the market-only slope for the partial slope, and heteroskedasticity does not cause bias under the stated exogeneity and rank conditions.
• B. The -2.38% forecast combines the market-only slope of 0.90 with the spread slope; the joint model requires its partial market slope of 0.50.
• C. The fitted return is valid, but unequal error variances do not bias OLS estimators when conditional error means are zero and regressors have full rank.
• D. Using the partial market slope gives -1.58%; zero conditional error mean preserves unbiasedness, while heteroskedasticity removes the Gauss-Markov efficiency guarantee.

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Question 25

Topic: Quantitative Analysis

A bank compares separate marginal models for its daily number of failed payments and the amount of an individual failed payment. Failure indicators are independent and have the same constant probability.

Treat the following exhibit as exact population statistics for this exercise. Payment amounts are in US dollars.

Which distribution pairing is consistent with the payment-failure mechanism and all the reported statistics?

• A. A binomial distribution for the daily failure count and a lognormal distribution for the amount per failed payment.
• B. A Poisson distribution for the daily failure count and a lognormal distribution for the amount per failed payment.
• C. A binomial distribution for the daily failure count and a normal distribution for the amount per failed payment.
• D. A Poisson distribution for the daily failure count and a normal distribution for the amount per failed payment.

Best answer: A

Explanation: A binomial distribution models the number of failures in a fixed number of independent trials with a constant failure probability. Here, \(p = 8/40 = 0.20\), so the count variance is \(40 \times 0.20 \times 0.80 = 6.4\), matching the exhibit. A Poisson count with mean 8 would instead have variance 8.

For a lognormal amount, the logarithm is normally distributed. Complementary percentiles therefore have equal logarithmic distances from the median, or equal multiplicative spacing. The amount increases by a factor of 1.25 from $80,000 to $100,000 and again from $100,000 to $125,000. This fits a lognormal model. A normal model would require equal dollar distances from the median, which these quantiles do not exhibit.

• A. The binomial count matches the fixed independent trials and reported variance, while the amount quantiles have equal multiplicative spacing of 1.25.
• B. A Poisson count has equal mean and variance, whereas the reported mean is 8 and the variance is 6.4.
• C. Normal amounts require complementary percentiles to be equally spaced around the median, but the reported dollar deviations are $20,000 and $25,000.
• D. Poisson counts require equal mean and variance, and normal amounts require symmetric dollar deviations around the median; neither property matches the exhibit.

Questions 26-50

Question 26

Topic: Valuation and Risk Models

A risk manager evaluates a long American call with strike $100 and a long American put with strike $140 on the same stock. Each option covers one share, and their exercise decisions are evaluated independently.

Model conditions:

• The remaining exercise opportunities represented in the tree are immediately before the dividend, immediately after it, and expiration one period later.
• A $5 cash dividend is paid to shareholders between the first two nodes, with an equal stock-price drop and no elapsed time. There are no further dividends.
• The risk-free accumulation factor for the remaining period is 1.02. Branch probabilities are risk-neutral, and transaction costs are zero.

The exhibit gives exercise values and terminal payoffs in dollars per share.

Which policy maximizes the value of both options in this model?

• A. Hold the call until expiration; hold the put until expiration.
• B. Exercise the call before the dividend; exercise the put immediately after the dividend.
• C. Hold the call until expiration; exercise the put immediately after the dividend.
• D. Exercise the call before the dividend; hold the put until expiration.

Best answer: B

Explanation: At each American-option node, compare immediate exercise with continuation using optimal decisions at later nodes. Discount expected terminal payoffs using risk-neutral probabilities.

After the dividend, call continuation is \( [(17/35)\times38]/1.02 \approx 18.10 \) dollars, exceeding its $15 exercise value. Before the dividend, exercise produces $20, exceeding that unchanged continuation value. The call should therefore be exercised before the stock goes ex-dividend.

Put continuation after the dividend is \( [(17/35)\times2+(18/35)\times42.25]/1.02 \approx 22.25 \) dollars. Exercising there gives $25, so the post-dividend American put value is $25. Before the dividend, waiting preserves that $25 value with no discounting, whereas exercising gives $20.

A dividend can favor pre-dividend call exercise because exercising secures shareholder dividend eligibility. For puts, waiting through the dividend raises intrinsic value; positive interest rates can then favor exercising rather than delaying receipt of the strike.

• A. Expiration-only exercise gives continuation values of $18.10 for the call and $22.25 for the put, below their optimal exercise values of $20 and $25.
• B. The call’s $20 pre-dividend exercise value exceeds $18.10 continuation, while waiting through the dividend lets the put exercise for $25 rather than $20.
• C. Post-dividend put exercise is optimal, but holding the call replaces its $20 pre-dividend exercise value with continuation worth $18.10.
• D. Pre-dividend call exercise is optimal, but the put’s post-dividend continuation value of $22.25 is below its $25 exercise value.

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Question 27

Topic: Valuation and Risk Models

A risk committee is comparing one-year 95% VaR and 95% expected shortfall (ES) as internal capital measures for two credit books. It wants combining the books not to increase capital above the sum of their standalone charges.

The joint loss distribution is shown below. Losses are in millions of dollars, and combined loss equals the sum of the two book losses in each scenario.

VaR is the smallest loss with cumulative probability of at least 95%. ES averages exactly the worst 5% probability mass, including any required fraction of a boundary scenario.

Which calculation and interpretation correctly compare combined capital with the sum of standalone capital?

• A. Combined VaR is $10 million versus a standalone sum of $0; combined ES is $10 million versus a standalone sum of $20 million. Only VaR penalizes aggregation.
• B. Combined VaR is $10 million versus a standalone sum of $0; combined ES is $10 million versus a standalone sum of $12 million. Only VaR penalizes aggregation.
• C. Combined VaR is $10 million versus a standalone sum of $20 million; combined ES is $10 million versus a standalone sum of $12 million. Both measures reward aggregation.
• D. Combined VaR is $0 versus a standalone sum of $0; combined ES is $12 million versus a standalone sum of $12 million. Neither measure changes capital on aggregation.

Best answer: B

Explanation: Subadditivity requires \(\rho(L_A+L_B) \le \rho(L_A)+\rho(L_B)\): combining portfolios must not produce measured risk above their summed standalone risk.

Each book has zero loss with 97% probability, so its 95% VaR is zero. Combined loss is zero with only 94% probability, making combined VaR $10 million. VaR therefore violates subadditivity here and creates an incentive to keep the books separate.

Each book’s ES averages 3% probability mass at a loss of $10 million and 2% at zero:

\[ \mathrm{ES}_{95\%}=\frac{0.03\times10+0.02\times0}{0.05}=6 \]

Amounts are in millions of dollars. The combined worst 5% consists entirely of $10 million losses, so combined ES is $10 million versus $12 million separately, a $2 million diversification benefit.

Coherence also requires monotonicity, positive homogeneity, and translation invariance. These mean that higher scenario losses cannot reduce risk, scaling losses scales risk proportionately, and adding a certain loss increases risk by that amount. ES satisfies all four properties. VaR can be subadditive for some distributions, but it is not generally coherent.

• A. Each book’s worst 5% includes 3% at $10 million and 2% at zero, so standalone ES is $6 million, not $10 million.
• B. Combined VaR exceeds the zero standalone sum, while combined ES of $10 million is below the $12 million standalone sum.
• C. Each book has a 97% probability of zero loss, making its 95% VaR zero rather than $10 million.
• D. The aggregate distribution has a 6% probability of a $10 million loss, giving both combined VaR and combined ES of $10 million.

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Question 28

Topic: Financial Markets and Products

An investor holds two European call option contracts. Each contract covers 100 shares, has a strike price of $60, and was purchased for a premium of $7.50 per share.

One month before expiration, the stock price is $64 and the call premium is $6.75 per share. The investor evaluates a scenario in which the stock price at expiration is $66. Ignore fees, taxes, and financing costs.

Which report correctly states the calls’ current moneyness, the position’s current total intrinsic and time values, and its total profit or loss at expiration under this scenario?

• A. Out of the money; $0 intrinsic value, $1,350 time value, and a $300 expiration loss.
• B. In the money; $800 intrinsic value, $550 time value, and a $1,200 expiration profit.
• C. In the money; $800 intrinsic value, $700 time value, and a $300 expiration loss.
• D. In the money; $800 intrinsic value, $550 time value, and a $300 expiration loss.

Best answer: D

Explanation: A call is in the money when the stock price exceeds its strike price. Current intrinsic value is therefore $4 per share, or $800 across the 200 shares covered by the contracts. Time value uses the current option quote: $6.75 minus $4 equals $2.75 per share, or $550 for the position. Together, these components equal its current market value of $1,350, not its original purchase cost of $1,500.

At expiration, a $66 stock price produces a payoff of $6 per share, totaling $1,200. Subtracting the original $1,500 premium gives a $300 loss. An in-the-money option can still produce a loss because moneyness ignores the premium paid; this investment’s expiration break-even stock price is $67.50.

• A. Moneyness compares the stock price with the strike price, not the break-even price; a $64 stock price gives these calls positive intrinsic value.
• B. The $1,200 amount is the expiration payoff, not profit; profit also deducts the $1,500 premium paid for the position.
• C. The $700 time value uses the original purchase premium; current time value must use the current $6.75 quote less $4 intrinsic value per share.
• D. Across 200 shares, intrinsic value is $800 and time value is $550; the $1,200 expiration payoff less the $1,500 purchase cost produces a $300 loss.

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Question 29

Topic: Valuation and Risk Models

A risk manager compares four strategies using one-share units. A European call and put each have a strike of $100 and expire at the final observation. Their initial premiums are $6 and $10, respectively. Both options are cash-settled.

Strategies:

• Naked: Write the call and hold no stock.
• Covered: Write the call, buy one share initially, and hold it until expiration.
• Stop-loss: Write the call and initially hold no stock. At each observation, rebalance to one share if the observed price exceeds $100 and zero shares if it is below $100.
• Put-insured: Buy one share and the put initially, holding both until expiration.

Observed stock prices:

Stock trades execute only at the displayed prices. Each stock purchase or sale incurs a $1 commission per share. There are no additional option trading or settlement costs. Ignore interest and dividends. At expiration, mark any stock held to market rather than selling it.

Which set of terminal dollar profits, including premiums and commissions, is consistent with this record?

• A. Naked: $6; covered: -$3; stop-loss: -$10; put-insured: -$7.
• B. Naked: $0; covered: -$9; stop-loss: -$16; put-insured: $3.
• C. Naked: $6; covered: -$3; stop-loss: $4; put-insured: -$7.
• D. Naked: $6; covered: -$2; stop-loss: -$8; put-insured: -$6.

Best answer: A

Explanation: A naked written call has no stock hedge; a covered call has one share backing the obligation. The stop-loss position starts naked, becomes covered at $108, and becomes naked again at $94. Crossing the $100 trigger does not imply execution at $100.

At expiration, the call payoff is zero and the put payoff is $12. Profits per one-share unit are:

• Naked: $6.
• Covered: \(6 + 88 - 96 - 1 = -3\) dollars.
• Stop-loss: \(6 + 94 - 108 - 2 = -10\) dollars.
• Put-insured: \(88 + 12 - 96 - 10 - 1 = -7\) dollars.

The purchased put establishes a $100 terminal value floor for the stock-plus-put position, not recovery of its $107 initial outlay. Dynamic portfolio insurance instead attempts to reproduce protection through changing stock and cash holdings. Discrete rebalancing, price gaps, and transaction costs can undermine that protection. The binary stop-loss rule also differs from delta hedging, which adjusts the share position according to the option’s delta rather than switching solely between zero and one share.

• A. The call expires worthless, the stop-loss stock round trip loses $16 after commissions, and the insured share’s $100 terminal value follows a $107 outlay.
• B. These figures omit the initial $6 call premium receipts and $10 put premium payment, which must be included in profit.
• C. The $4 stop-loss profit assumes both stock trades execute at $100; the observed purchase and sale prices are $108 and $94.
• D. These figures omit stock commissions: $1 for each initially purchased-and-held share and $2 for the stop-loss strategy’s round trip.

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Question 30

Topic: Financial Markets and Products

A pension fund plans to meet a $9 million liability due in two years entirely with principal receipts from fixed-rate, agency-guaranteed mortgage-backed securities. Principal receipts are held in a non-interest-bearing reserve; coupons fund separate obligations.

Immediately after market interest rates rise, the fund updates its assessment:

Borrowers remain current, and the guarantee of contractual principal and interest is unchanged. The fund does not plan to sell the securities or borrow to meet the liability.

Which risk most directly explains the projected cash-flow funding gap, and how large is that gap?

• A. Credit risk, with a $3 million funding gap.
• B. Interest-rate price risk, with a $2 million funding gap.
• C. Extension risk, with a $3 million funding gap.
• D. Contraction risk from faster prepayments, with a $3 million funding gap.

Best answer: C

Explanation: Extension risk arises when slower mortgage prepayments delay principal receipts beyond an investor’s funding horizon. Rising mortgage rates generally reduce borrowers’ incentive to refinance, potentially extending the effective life of an MBS. Here, expected principal receipts before the two-year liability date fall from $9 million to $6 million, leaving a $3 million funding gap.

The separate $2 million decline in market value reflects interest-rate price risk. Because the funding plan does not involve selling the securities, that valuation loss does not measure the cash-flow gap. Credit guarantees protect contractual payments against covered credit losses; they do not preserve an assumed prepayment schedule. Conversely, faster prepayments return principal earlier and can create reinvestment risk when receipts must be invested until a later liability date.

• A. Borrowers remain current and credit protection is unchanged; the projected shortfall reflects repayment timing rather than borrower default.
• B. The $2 million market-value decline measures a valuation loss, not the principal-receipt shortfall funding the liability.
• C. Slower principal repayment leaves only $6 million available against the $9 million liability, producing a $3 million shortfall.
• D. Contraction risk brings principal forward through faster prepayments. The reduced receipts before the liability date instead indicate extension risk from slower prepayments.

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Question 31

Topic: Quantitative Analysis

A risk analyst must select a conditional-mean forecasting model for 120 quarterly observations of a covariance-stationary credit-risk indicator. All models include an intercept and use the same observations.

Observed dependence: The series ACF has local peaks at lags 4 and 8. Its PACF has prominent spikes at lags 1 and 4, with negligible values beyond lag 4.

Selection rule: Choose the lowest-BIC model among those for which an eight-lag Box–Pierce test does not reject zero residual autocorrelation at 5%. Use degrees of freedom equal to eight minus the number of estimated AR and MA coefficients, excluding the intercept. The chi-square critical values are 14.07, 12.59, and 9.49 for 7, 6, and 4 degrees of freedom, respectively.

For the ARMA(1,1) residuals, autocorrelations at lags 4 and 8 are 0.30 and 0.20. The sum of squared residual autocorrelations at the other six tested lags is 0.0050.

Which specification should the analyst retain?

• A. Retain the fitted ARMA(1,1) specification.
• B. Retain the fitted AR(4) specification with all four lags.
• C. Retain the fitted AR(1) specification.
• D. Retain the fitted AR specification with only lags 1 and 4.

Best answer: D

Explanation: BIC balances model fit against parameter count, but a low BIC does not guarantee that residual serial dependence has been removed. The Box–Pierce statistic jointly assesses residual autocorrelations:

\[ Q=n\sum_{k=1}^{m}\hat{\rho}_k^2=120(0.0050+0.30^2+0.20^2)=16.2. \]

For the ARMA(1,1) model, eight tested lags minus two fitted dynamic coefficients gives six degrees of freedom. Since 16.2 exceeds 12.59, this model fails the specified diagnostic despite having the lowest BIC.

Both AR specifications containing lag 4 pass the diagnostic. The model using only lags 1 and 4 has the lower BIC, so it satisfies the selection rule. Lag 4 captures dependence one year earlier in quarterly data, consistent with the seasonal ACF evidence. An intercept captures a constant mean, not seasonal dependence. Failure to reject indicates no detected autocorrelation at the tested lags, rather than proof that residuals are white noise.

• A. Its Box–Pierce statistic is \(120(0.0050+0.30^2+0.20^2)=16.2\), exceeding the six-degree-of-freedom cutoff of 12.59 despite its lowest BIC.
• B. Its statistic of 2.8 passes the four-degree-of-freedom test, but its BIC exceeds that of the model using only lags 1 and 4.
• C. Its residual statistic of 24.0 exceeds the seven-degree-of-freedom cutoff of 14.07, so it fails the required residual diagnostic.
• D. Its statistic of 3.6 is below the six-degree-of-freedom cutoff of 12.59, and its BIC is lowest among models passing the diagnostic.

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Question 32

Topic: Valuation and Risk Models

A credit fund holds corporate bonds and has bought CDS protection on the same issuers. A risk manager wants to identify an economically coherent joint spread shock that exhausts the fund’s $40 million mark-to-market loss buffer.

The exhibit shows current portfolio sensitivities and spread changes from a selected historical episode. All shocks are measured relative to current spreads. Treat sensitivities as constant over the shock ranges considered, and ignore defaults, interest-rate changes, carry, and other risk factors.

A market review finds that forced cash-bond sales can cause cash spreads to widen more than matched CDS spreads.

Which scenario design best meets the risk manager’s objective?

• A. Use a reverse-stress scenario with cash spreads widening 100 bp and CDS spreads widening 100 bp.
• B. Use a historical replay with cash spreads widening 300 bp and CDS spreads widening 280 bp.
• C. Use a hypothetical amplification with cash spreads widening 600 bp and CDS spreads widening 560 bp.
• D. Use a reverse-stress scenario with cash spreads widening 300 bp and CDS spreads widening 200 bp.

Best answer: D

Explanation: Reverse stress testing starts with a specified adverse outcome and identifies conditions that could produce it. Here, the endpoint is a $40 million portfolio loss. With equal and opposite spread sensitivities, common spread widening is largely hedged; the critical exposure is the cash-CDS basis.

For spread changes measured in basis points, the loss in millions of dollars is \(L = 0.40(\Delta s_{\text{cash}} - \Delta s_{\text{CDS}})\). Exhausting the buffer therefore requires cash spreads to widen 100 bp more than CDS spreads. A 300 bp cash-spread widening paired with a 200 bp CDS-spread widening meets that endpoint. Forced cash-bond sales provide a coherent mechanism for this divergence. Larger historical or hypothetical market-wide shocks need not challenge a hedged portfolio more severely.

• A. The $40 million cash-bond loss is fully offset by a $40 million CDS gain, so equal widening does not exhaust the portfolio’s buffer.
• B. The historical replay produces a net loss of $8 million, since CDS gains offset most of the cash-bond losses.
• C. Doubling both historical spread changes produces only a $16 million net loss, leaving the buffer unexhausted despite larger individual shocks.
• D. The 100 bp excess widening of cash spreads produces a $40 million net loss and is consistent with the forced-selling mechanism.

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Question 33

Topic: Valuation and Risk Models

A risk manager reviews a desk’s one-day trading loss. The approved strategy combines an equity-index portfolio with a partial futures hedge and deliberately retains some market exposure.

P&L reconstruction (USD): The approved-execution column assumes timely execution of all authorized trades using the same observed market prices. Positive amounts are gains; negative amounts are losses.

Audit findings:

• An internal interface defect discarded part of the hedge order before exchange submission.
• All executed trades settled in full and on time, with no counterparty default.

For internal loss-event reporting, which allocation of the total trading loss by primary risk source is best supported?

• A. Operational risk: $0; market risk: $200,000; credit risk: $400,000.
• B. Operational risk: $600,000; market risk: $0; credit risk: $0.
• C. Operational risk: $0; market risk: $600,000; credit risk: $0.
• D. Operational risk: $400,000; market risk: $200,000; credit risk: $0.

Best answer: D

Explanation: Operational risk arises from failed people, processes, systems, or external events. A market movement can turn an operational failure into a financial loss, so attribution requires distinguishing the failure’s incremental effect from the intended market exposure.

The approved strategy would have produced a $200,000 net loss: an $800,000 portfolio loss less a $600,000 hedge gain. The actual net loss was $600,000 because the executed hedge gained only $200,000. The intended residual exposure therefore accounts for $200,000 of market-risk loss, while the interface defect accounts for the additional $400,000 of operational-risk loss. There is no credit-risk loss because counterparties met every obligation on executed trades.

• A. The missing $400,000 hedge gain was never owed on executed trades; it reflects an internal execution failure rather than counterparty nonperformance.
• B. The approved strategy would still lose $200,000 under the observed market movement, so the interface defect did not cause the entire loss.
• C. The adverse index move explains the price change, but $400,000 of the loss arose from unintended exposure created by the interface defect.
• D. The approved strategy would lose $200,000; failed hedge transmission caused the additional $400,000 loss, while counterparties fulfilled all obligations.

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Question 34

Topic: Foundations of Risk Management

A risk manager compares a market-only model with a three-factor model for two well-diversified portfolios, P and Q.

Assumptions: All inputs refer to one-year arithmetic returns. The three-factor APT pricing relation holds. Portfolio residuals have zero means, are uncorrelated with the factors, and have negligible variance.

Estimated loadings (dimensionless):

Factor data:

The correlation between size and value shocks is 0.50. Both are uncorrelated with market shocks.

The manager buys P and shorts Q in equal dollar amounts. Define the spread return as \(R_P-R_Q\), measured per dollar of long-side notional. Ignore transaction and borrowing costs.

Which expected annual spread return and annual systematic volatility are supported by the exhibit, rounded to one decimal place?

• A. Expected return: 2.0%; systematic volatility: 10.8%.
• B. Expected return: 4.0%; systematic volatility: 12.5%.
• C. Expected return: 2.0%; systematic volatility: 8.7%.
• D. Expected return: 0.0%; systematic volatility: 0.0%.

Best answer: C

Explanation: In a multifactor model, market neutrality does not eliminate exposure to other systematic factors. Subtracting Q’s loadings from P’s gives market, size, and value exposures of 0, 1.0, and -0.5. The risk-free components cancel, so APT implies:

\[ E[R_P-R_Q]=1.0(3.0\%)-0.5(2.0\%)=2.0\%. \]

The variance includes the size-value covariance term:

\[ \begin{aligned} \sigma_{\text{spread}}^2 &=0.10^2+(-0.5\times0.08)^2\\ &\quad+2(1.0)(-0.5)(0.50)(0.10)(0.08)\\ &=0.0076. \end{aligned} \]

Thus systematic volatility is approximately 8.7%. Positive correlation reduces variance because the size and value exposures have opposite signs.

The positive expected return is not a riskless arbitrage: two systematic exposures remain. A market-only model would place this nonmarket risk in its residual component rather than eliminate it. APT uses no-arbitrage restrictions for diversified portfolios, whereas CAPM relies on market-equilibrium assumptions. Adding factors alone does not establish unbiased estimates.

• A. This volatility omits the size-value covariance term; the negative value loading makes that term reduce the spread’s variance.
• B. These figures replace the negative value loading with a positive loading, reversing its contribution to both the expected premium and covariance.
• C. Net size and value loadings of 1.0 and -0.5 produce a 2.0% expected return and variance of 0.0076, giving 8.7% volatility.
• D. Using market beta alone ignores the spread’s net size and value exposures, which remain systematic despite zero market exposure.

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Question 35

Topic: Valuation and Risk Models

A fixed-income analyst must determine the clean value of a 6% annual-coupon bond by bootstrapping spot discount factors from two benchmark securities. All securities are option-free and have the same credit risk. Prices and cash flows are per $100 face value, with settlement at time 0.

Coupons are paid semiannually. Both coupon bonds settle exactly halfway through their current six-month coupon periods, so accrued interest equals one-half of a semiannual coupon. The zero-coupon benchmark has no accrued interest. Ignore transaction costs and taxes.

Observed clean prices:

• Zero-coupon benchmark: $98.50.
• 4% annual-coupon benchmark: $99.30.

Remaining cash flows:

What clean value should the analyst assign to the 6% coupon bond, rounded to the nearest $0.01?

• A. $102.25 per $100 face value.
• B. $102.74 per $100 face value.
• C. $99.74 per $100 face value.
• D. $100.75 per $100 face value.

Best answer: D

Explanation: Bootstrapping uses dirty prices because the present value of remaining cash flows includes accrued interest. Clean prices exclude it.

The zero-coupon benchmark gives the three-month discount factor:

\[ D_{0.25}=98.50/100=0.985. \]

The 4% coupon benchmark has accrued interest of $1.00, making its dirty price $100.30. Deduct the present value of its first coupon before isolating the nine-month discount factor:

\[ D_{0.75}=\frac{100.30-2(0.985)}{102}=0.9640196. \]

Apply these factors to the target bond’s remaining cash flows:

\[ P_{\text{dirty}}=3(0.985)+103(0.9640196)=102.2490196. \]

Its accrued interest is $1.50, so its clean value is $102.2490196 minus $1.50, or $100.75. A coupon rate is not a spot rate: coupon-bond valuation combines discount factors for multiple payment dates.

• A. This is the target bond’s dirty value; its $1.50 accrued interest must be deducted to obtain the clean value.
• B. This result omits the benchmark’s $2 payment in three months when isolating the discount factor for its nine-month payment.
• C. This result treats the benchmark’s $99.30 clean price as its full present value, omitting its $1.00 accrued interest.
• D. Using the benchmark’s $100.30 dirty price gives a target dirty value of $102.2490; subtracting $1.50 accrued interest gives $100.75.

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Question 36

Topic: Quantitative Analysis

A risk analyst reviews the following daily returns for a trading strategy. The 12-observation sample has a mean return of 0.50%.

Use denominator \(n-1\) for sample variance. For kurtosis, use empirical central moments with denominator \(n\), without a finite-sample bias correction.

Which statement correctly characterizes the sample’s standard deviation and excess kurtosis? Round both statistics to two decimal places.

• A. The sample standard deviation is 2.17 percentage points, and excess kurtosis is 5.56.
• B. The sample standard deviation is 2.08 percentage points, and excess kurtosis is 5.56.
• C. The sample standard deviation is 2.08 percentage points, and excess kurtosis is 2.56.
• D. The sample standard deviation is 2.17 percentage points, and excess kurtosis is 2.56.

Best answer: D

Explanation: Sample standard deviation uses the specified variance denominator of \(n-1\), while empirical kurtosis uses central moments calculated with denominator \(n\).

Subtracting the mean of 0.50% gives deviations of -5, -1, 0, 1, and 5 percentage points, with zero occurring eight times. The squared deviations sum to 52, and the fourth powers sum to 1,252. Therefore,

\[ s=\sqrt{\frac{52}{11}}\approx 2.17\text{ percentage points}. \]

Excess kurtosis is the fourth central moment divided by the squared second central moment, minus 3:

\[ \frac{1{,}252/12}{(52/12)^2}-3\approx 2.56. \]

A normal distribution has ordinary kurtosis of 3 and excess kurtosis of zero. The positive sample excess kurtosis indicates greater standardized fourth-moment tail weight than the normal benchmark, despite the sample’s symmetry. It does not establish the strategy’s population tail probabilities.

• A. The standard deviation follows the specified convention, but 5.56 is ordinary kurtosis; subtracting 3 gives excess kurtosis of 2.56.
• B. The standard deviation uses denominator 12 instead of 11, and 5.56 reports ordinary kurtosis rather than excess kurtosis.
• C. The standard deviation of 2.08 uses denominator 12 for variance; the specified sample standard deviation requires denominator 11.
• D. Squared deviations sum to 52, giving \(\sqrt{52/11}=2.17\); the empirical fourth-moment ratio minus 3 gives excess kurtosis of 2.56.

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Question 37

Topic: Valuation and Risk Models

A risk analyst generates 1,000 independent one-day portfolio loss scenarios under a fixed model using real-world probabilities. Positive amounts represent losses.

The 40 losses above $4.0 million sum to $260.0 million. Use nearest-rank empirical 95% VaR. For 95% expected shortfall (ES), average exactly the worst 5% of scenarios, including only the necessary observations tied at the VaR boundary.

The analyst reports a nominal 95% bootstrap percentile confidence interval for the model’s 95% VaR of $3.7 million to $4.6 million. This uses 2,000 resamples, each consisting of 1,000 draws with replacement from the simulated losses, with model inputs unchanged.

Which conclusion correctly combines the ES estimate with the interpretation of the interval?

• A. The 95% ES estimate is $6.0 million; the interval describes a range containing 95% of the model’s one-day losses.
• B. The 95% ES estimate is $6.5 million; the interval describes sampling uncertainty in the model’s 95% VaR.
• C. The 95% ES estimate is $6.0 million; the interval describes sampling uncertainty in the model’s 95% VaR.
• D. The 95% ES estimate is $6.5 million; the interval describes a range containing 95% of the model’s one-day losses.

Best answer: C

Explanation: Empirical VaR is a quantile of the simulated loss distribution. The 950th ordered loss is $4.0 million: 940 losses are lower, and the next 20 equal $4.0 million.

The worst 5% of 1,000 scenarios contains 50 losses. Include all 40 losses above VaR and 10 of the losses tied at VaR. Working in millions of US dollars:

\[ \operatorname{ES}_{0.95} = \frac{260 + 10 \times 4}{50} = 6.0. \]

Averaging losses strictly above VaR would exclude part of the required tail probability mass.

Bootstrap resampling holds the model fixed and assesses how the estimated VaR varies with simulation-sampling noise. The interval’s nominal 95% level concerns intended coverage of the model’s true loss quantile across repeated samples, not the proportion of future losses between its endpoints. It does not capture model misspecification.

• A. The ES calculation includes the required 50 scenarios, but the bootstrap interval concerns uncertainty in VaR estimation, not coverage of future losses.
• B. The $6.5 million figure averages only the 40 losses strictly above VaR; 95% ES must include 10 additional losses of $4.0 million.
• C. Adding 10 boundary losses to the 40 exceedances gives ES of $6.0 million; bootstrap resampling assesses sampling uncertainty in the estimated loss quantile.
• D. Averaging only strict VaR exceedances omits required tail observations, and the bootstrap confidence interval describes estimation uncertainty rather than future-loss coverage.

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Question 38

Topic: Financial Markets and Products

A risk manager is classifying a mark-to-market loss on a fixed-rate corporate bond immediately after a weaker outlook is announced for the issuer’s industry. The bond’s price falls, while the issuer remains current on all contractual payments.

Annual-compounded yields immediately before and after the announcement are:

The bond’s contractual cash flows, quoted bid-ask spread, and executable trading sizes are unchanged.

Which risk most directly explains the observed mark-to-market loss?

• A. Interest rate risk from movements in the benchmark yield.
• B. Market liquidity risk from deteriorating secondary-market trading conditions.
• C. Credit spread risk from repricing the issuer’s credit exposure.
• D. Default risk realized through missed contractual debt payments.

Best answer: C

Explanation: Credit spread risk is the risk that a bond’s value falls as its required yield relative to a benchmark increases. Here, the spread rises from 5.50% minus 3.20%, or 2.30%, to 6.10% minus 2.80%, or 3.30%. The 100-basis-point widening outweighs the 40-basis-point decline in the Treasury yield, leaving the corporate yield 60 basis points higher and its price lower.

A sector downturn can increase compensation demanded for expected credit losses and uncertainty before any payment is missed. Spread widening therefore does not itself establish that default has occurred. The unchanged trading conditions also make a liquidity-driven interpretation less supported by the evidence.

• A. The Treasury yield fell by 40 basis points, which would support the fixed-rate bond’s price rather than explain its decline.
• B. Unchanged bid-ask spreads and executable trading sizes do not support deteriorating trading liquidity as the principal source of the loss.
• C. The corporate spread widened from 230 to 330 basis points, more than offsetting the decline in the Treasury yield.
• D. The issuer remains current on payments, so the observed loss is a valuation change rather than a realized payment default.

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Question 39

Topic: Foundations of Risk Management

An investment committee is selecting an active equity manager to complement an existing index holding. It wants to compare risk-adjusted active performance against the assigned index, rather than standalone portfolio efficiency or performance adjusted only for market exposure.

All estimates below are annualized and come from the same sample. The risk-free rate is 3.0%, and the assigned index has a mean return of 8.0%. Treat that index as the market portfolio for CAPM calculations.

Which recommendation follows the committee’s evaluation criterion?

• A. Select Pine using its Treynor ratio of 0.20.
• B. Select Pine using its Sharpe ratio of 1.00.
• C. Select Cedar using its information ratio of 0.75.
• D. Select Pine using its Jensen alpha of 7.5%.

Best answer: C

Explanation: The information ratio measures average return above an assigned benchmark divided by tracking error, the standard deviation of portfolio returns minus benchmark returns. It therefore matches an evaluation of risk-adjusted active performance.

Cedar’s information ratio is \( (0.11 - 0.08)/0.04 = 0.75 \), while Pine’s is \( (0.13 - 0.08)/0.10 = 0.50 \). Cedar produces less average benchmark outperformance, but more outperformance per unit of active risk.

The benchmark and denominator distinguish the measures. Sharpe uses excess return over the risk-free rate divided by total portfolio volatility. Treynor uses the same excess-return numerator divided by beta. Jensen alpha subtracts the CAPM-required return from portfolio return and has no risk denominator. A manager can rank highly on these measures yet rank lower on the information ratio.

• A. Pine’s Treynor ratio is 0.10/0.50 = 0.20, but beta measures systematic exposure rather than variability of benchmark-relative returns.
• B. Pine’s Sharpe ratio is (13% - 3%)/10% = 1.00, but it measures excess return relative to total volatility, not active risk.
• C. Cedar’s information ratio is (11% - 8%)/4% = 0.75, exceeding Pine’s 0.50 and matching the benchmark-relative criterion.
• D. Pine’s CAPM-required return is 5.5%, giving alpha of 7.5%, but alpha does not scale performance by active-return volatility.

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Question 40

Topic: Financial Markets and Products

A corporate treasurer starts with $1,000,000 and evaluates a simultaneous spot round trip intended to generate a risk-free dollar profit:

USD -> EUR -> CHF -> USD

Executable dealer quotes:

The bids and asks are from the dealers’ perspectives. All three trades can be locked simultaneously for the same settlement date. A total transaction fee of $2,500 is deducted from the final USD proceeds. Assume full execution at the quoted rates, no interim rounding, no other costs, and no counterparty or settlement risk.

Which interpretation of the proposed round trip is supported? Round the net result to the nearest dollar.

• A. The round trip generates a net gain of approximately $934, so it is an arbitrage opportunity.
• B. The round trip generates a net loss of approximately $168, so it is not an arbitrage opportunity.
• C. The round trip generates a net loss of approximately $5,923, so it is not an arbitrage opportunity.
• D. The round trip generates a net gain of approximately $2,332, so it is an arbitrage opportunity.

Best answer: B

Explanation: Triangular arbitrage requires a positive final profit after applying executable bid-ask quotes and transaction costs. A customer buys the currency being priced at the dealer’s ask and sells it at the dealer’s bid.

For this route, the treasurer buys EUR at 1.1004 USD per EUR, sells EUR at 0.9930 CHF per EUR, and buys USD at 0.9003 CHF per USD. The net dollar profit is:

\[ \frac{1,000,000}{1.1004}\times\frac{0.9930}{0.9003}-1,000,000-2,500\approx -168.41 \]

The conversions produce a gross gain of approximately $2,331.59, but the fee exceeds that gain. Final proceeds are approximately $999,831.59, so the proposed round trip is not an arbitrage opportunity.

• A. This gain incorrectly uses the bid when buying EUR and USD and the ask when selling EUR, reversing the executable sides.
• B. Buying EUR at 1.1004, selling EUR at 0.9930, and buying USD at 0.9003 leaves approximately $999,832 after the fee.
• C. This loss corresponds to the reverse sequence, USD to CHF to EUR to USD, rather than the proposed trading direction.
• D. The approximately $2,332 gain is before transaction costs; deducting the $2,500 fee turns it into a net loss.

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Question 41

Topic: Quantitative Analysis

A market risk analyst is deciding how to transform two daily log-price indexes before modeling their dynamics. The exhibit gives exact population moments, not sample estimates, for every trading day \(t \ge 1\). The integer lag \(k\) satisfies \(1 \le k < t\). Levels are measured in log units; variances and autocovariances are measured in squared log units.

Model-implied moments:

• Index \(X_t\):
  • Mean: \(4 + 0.0002t\); variance: \(0.0025\).
  • Autocovariance: \(\operatorname{Cov}(X_t,X_{t-k}) = 0.0025(0.8)^k\).
• Index \(Y_t\):
  • Mean: \(4 + 0.0002t\); variance: \(0.0004t\).
  • Autocovariance: \(\operatorname{Cov}(Y_t,Y_{t-k}) = 0.0004(t-k)\).

Which conclusion correctly identifies when linear detrending suffices and when first differencing is required to obtain covariance stationarity?

• A. First differencing is required for X; linear detrending suffices for Y.
• B. Linear detrending suffices for X; first differencing is required for Y.
• C. First differencing is required for X; first differencing is required for Y.
• D. Linear detrending suffices for X; linear detrending suffices for Y.

Best answer: B

Explanation: Covariance stationarity requires a time-invariant mean and variance, with autocovariance depending only on lag rather than calendar time. Neither level series is stationary because both means increase with time.

For X, subtracting \(4 + 0.0002t\) gives a zero-mean series with variance \(0.0025\) and autocovariance \(0.0025(0.8)^k\). These satisfy covariance stationarity. Autocovariance declining with lag is compatible with stationarity; it need not be identical across different lags.

For Y, subtracting the same deterministic trend removes the changing mean but leaves variance \(0.0004t\) and autocovariance \(0.0004(t-k)\). Its moments are consistent with a random walk with drift. First differences, \(\Delta Y_t = Y_t-Y_{t-1}\), have constant mean \(0.0002\), constant variance \(0.0004\), and zero autocovariance at positive lags. Thus detrending addresses X’s deterministic trend, while differencing addresses Y’s stochastic trend.

• A. X becomes stationary after linear detrending, whereas detrending Y leaves its variance and autocovariance time-dependent.
• B. Detrending X leaves constant variance and lag-dependent autocovariance, while differencing Y produces constant variance and zero autocovariance at positive lags.
• C. Differencing produces stationary transformations of both indexes, but X does not require it because removing its deterministic mean trend already establishes stationarity.
• D. Removing Y’s linear mean trend leaves variance and autocovariance dependent on calendar time, so the detrended series remains nonstationary.

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Question 42

Topic: Valuation and Risk Models

A bank uses the following exhaustive joint distribution to evaluate diversification between two portfolios over a one-month horizon. All losses and risk measures are in USD millions; positive values represent losses. The combined loss is \(L_A + L_B\).

Define 95% VaR as the smallest loss \(v\) such that \(\Pr(L \le v) \ge 0.95\). Define 95% expected shortfall as the probability-weighted average of the worst 5% of losses, taking only the required fraction of any boundary probability mass.

For this distribution, which statement correctly compares the combined risk measures with the sums of the corresponding standalone measures?

• A. VaR violates subadditivity (50 > 0); expected shortfall satisfies subadditivity (70 < 80).
• B. VaR violates subadditivity (50 > 0); expected shortfall violates subadditivity (100 > 80).
• C. VaR violates subadditivity (50 > 0); expected shortfall satisfies subadditivity (80 = 80).
• D. VaR satisfies subadditivity (0 = 0); expected shortfall satisfies subadditivity (70 < 80).

Best answer: A

Explanation: Subadditivity requires \(\rho(L_A+L_B) \le \rho(L_A)+\rho(L_B)\). Each portfolio has at least 95% probability of zero loss, so both standalone VaRs are zero. The combined loss is zero with probability 94%, 50 with probability 4%, and 100 with probability 2%. Its 95% VaR is therefore 50, violating subadditivity.

Expected shortfall averages a fixed 5% tail probability. The standalone values are \(0.04 \times 50 / 0.05 = 40\) and \(0.02 \times 100 / 0.05 = 40\). The combined worst 5% includes all 2% probability at 100 and only 3% of the 4% probability at 50:

\[ \operatorname{ES}_{95\%}(L_A+L_B)=\frac{0.02\times100+0.03\times50}{0.05}=70. \]

Thus, combined expected shortfall is below the standalone sum of 80. This counterexample establishes that VaR is not coherent in general, not that it always violates subadditivity. Expected shortfall is coherent for integrable losses.

• A. Combined VaR exceeds the zero standalone sum, while combined expected shortfall includes 2% probability at 100 and 3% at 50, giving 70.
• B. Averaging losses strictly above VaR gives 100 but includes only 2% probability; expected shortfall must also include 3% probability at 50.
• C. Standalone expected shortfalls sum to 80, but the combined worst 5% produces 70; subadditivity does not require exact additivity.
• D. Although each standalone VaR is zero, the combined portfolio has only 94% probability of zero loss, making its 95% VaR 50.

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Question 43

Topic: Financial Markets and Products

A credit analyst tracks a fixed cohort of corporate bond issues over one calendar year. No issues are added, and no scheduled principal repayments occur. The exhibit refers to the same opening cohort; monetary amounts are in USD millions.

All defaults are fully resolved by year-end. Recoveries are final cash proceeds attributable to principal; ignore interest, costs, and discounting.

Which summary correctly reports the combined cohort’s dollar-weighted default rate, principal-weighted recovery rate on defaulted bonds, and realized principal loss rate relative to opening portfolio principal?

• A. Default rate: 10.0%; recovery rate: 52.0%; loss rate: 4.8%.
• B. Default rate: 12.5%; recovery rate: 52.0%; loss rate: 6.0%.
• C. Default rate: 12.5%; recovery rate: 32.0%; loss rate: 8.5%.
• D. Default rate: 10.0%; recovery rate: 32.0%; loss rate: 6.8%.

Best answer: B

Explanation: Issue-based and dollar-weighted default rates describe different characteristics of the same cohort. Ten of 100 issues default, giving an issue-based rate of 10.0%. Principal-based measures instead reflect the amounts exposed and recovered:

• Dollar-weighted default rate: \( (100 + 25)/(500 + 500) = 12.5\% \).
• Principal-weighted recovery rate: \( (60 + 5)/(100 + 25) = 52.0\% \).
• Realized principal loss rate: \( (125 - 65)/1{,}000 = 6.0\% \).

Recovery is conditional on default, so its denominator is defaulted principal. Portfolio loss uses total opening principal as its denominator. The recovery rate implies LGD of 48.0%, and multiplying the dollar-weighted default rate by this LGD gives the same 6.0% realized loss rate.

• A. The 10.0% default rate is issue-based; multiplying it by the correctly calculated 48.0% LGD understates the principal-based portfolio loss rate.
• B. Defaulted principal totals $125 million, recoveries total $65 million, and losses total $60 million, producing these rates with the appropriate denominators.
• C. The 32.0% recovery rate weights category recovery rates by defaulted issue counts rather than defaulted principal, overstating LGD and the loss rate.
• D. These figures combine an issue-based default rate with recovery rates weighted by defaulted issue counts, rather than measuring defaults and recoveries using principal weights.

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Question 44

Topic: Valuation and Risk Models

A risk manager is preparing a daily variance-attribution report for a rates portfolio. Its signed key-rate sensitivities to the 2-year, 5-year, and 10-year zero rates are +$100, $0, and -$60 per basis point, respectively. Each sensitivity measures the first-order portfolio P&L from a 1 bp increase at that tenor. Ignore convexity.

Daily yield changes, measured in basis points, equal the sum of the following loading vectors multiplied by their respective principal-component factor scores. The scores are mutually uncorrelated, and vector entries follow the maturity order 2-year, 5-year, 10-year:

• Level: Loading vector \( (1,1,1)/\sqrt{3} \); daily factor variance of 25 bp².
• Slope: Loading vector \( (-1,0,1)/\sqrt{2} \); daily factor variance of 9 bp².
• Curvature: Loading vector \( (1,-2,1)/\sqrt{6} \); daily factor variance of 4 bp².

Which factor contributes the most to daily portfolio P&L variance, and what percentage of total variance does it contribute, rounded to one decimal place?

• A. The slope factor, contributing approximately 88.9%.
• B. The level factor, contributing approximately 94.6%.
• C. The level factor, contributing approximately 65.8%.
• D. The slope factor, contributing approximately 94.1%.

Best answer: A

Explanation: Principal-component variance attribution combines portfolio exposure to each factor with that factor’s variance. Projecting the signed key-rate sensitivities onto the loading vectors gives level, slope, and curvature exposures of \( 40/\sqrt{3} \), \( -160/\sqrt{2} \), and \( 40/\sqrt{6} \) dollars per basis point, respectively.

Because the scores are uncorrelated, their P&L variance contributions add:

\[ \operatorname{Var}(\Delta P)=25\left(\frac{40}{\sqrt{3}}\right)^2+9\left(\frac{-160}{\sqrt{2}}\right)^2+4\left(\frac{40}{\sqrt{6}}\right)^2=129{,}600. \]

The slope contribution is 115,200 dollars squared, so its share is \( 115{,}200/129{,}600=88.9\% \). Daily portfolio P&L volatility is therefore $360.

Although level has the greatest factor variance, the portfolio’s opposing short- and long-maturity sensitivities create much greater slope exposure. Key-rate sensitivities describe local tenor exposure; they are not themselves exposures to independent principal components.

• A. The slope exposure is \( -160/\sqrt{2} \) dollars per bp, giving variance of 115,200 dollars squared out of total variance of 129,600 dollars squared.
• B. This percentage treats the tenor sensitivities as factor exposures, producing 250,000/(250,000 + 14,400); factor exposures instead require projection onto the loading vectors.
• C. The ratio 25/38 measures the level factor’s share of total factor variance, not portfolio P&L variance, which also depends on portfolio exposures.
• D. This percentage uses squared projected sensitivities alone; each squared sensitivity must be weighted by its corresponding factor variance.

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Question 45

Topic: Foundations of Risk Management

A portfolio manager can invest in two risky funds and a risk-free asset yielding 2% annually. The mandate is to maximize expected annual return subject to annual volatility no greater than 7.5%. Borrowing and short selling are prohibited.

The correlation between the funds’ annual returns is 0.125. Assume these estimates apply over the one-year investment horizon.

Which allocation best meets the mandate?

• A. 55% in Fund A, 20% in Fund B, and 25% in the risk-free asset.
• B. 50% in Fund A, 25% in Fund B, and 25% in the risk-free asset.
• C. 25% in Fund A, 30% in Fund B, and 45% in the risk-free asset.
• D. 60% in Fund A, 20% in Fund B, and 20% in the risk-free asset.

Best answer: B

Explanation: Portfolio volatility incorporates covariance rather than simply averaging individual volatilities. Because the risk-free asset has zero variance, total portfolio variance is

\[ \sigma_p^2=w_A^2\sigma_A^2+w_B^2\sigma_B^2+2w_Aw_B\rho\sigma_A\sigma_B. \]

With 50% in Fund A and 25% in Fund B, variance is \(0.0025+0.0025+0.000625=0.005625\), giving volatility of 7.5%. Expected return is \(0.50(8\%)+0.25(14\%)+0.25(2\%)=8\%\).

The risky portion holds two-thirds in Fund A and one-third in Fund B. Under these inputs, this is the tangency portfolio, with expected return and volatility both 10%. Investing 75% in it and 25% risk-free produces the selected allocation on the capital allocation line through this tangency portfolio, not the risky-asset-only frontier. Calling this the capital market line would additionally require the relevant market-portfolio equilibrium assumptions. Feasibility alone is insufficient: the mandate requires the highest expected return within the risk limit.

• A. The allocation is feasible, with volatility approximately 7.19%, but its expected return of 7.7% is below the achievable 8.0%.
• B. The allocation earns an expected 8.0% with volatility exactly 7.5%, maximizing expected return within the permitted risk budget.
• C. Its volatility of approximately 6.78% satisfies the limit, but the expected return of 7.1% does not maximize return within that limit.
• D. Its expected return is 8.0%, but volatility is approximately 7.62%, exceeding the mandate’s 7.5% limit.

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Question 46

Topic: Valuation and Risk Models

A risk analyst evaluates a six-month holding period for a long position in a corporate bond. All prices and cash amounts are per $100 face value. The exit valuations cover only the bond’s remaining cash flows.

Valuation exhibit:

Cash flows and conventions:

• Accrued interest is $1.50 at both purchase and exit.
• The only coupon during the holding period is $3.00, received after three months and reinvested for the remaining three months at 4% annually, using simple interest.
• Total transaction costs, accumulated to exit, are $0.20. There are no other cash flows or costs.

For carry/roll-down, hold the initial curves unchanged as functions of remaining maturity; do not evolve them using implied forwards. Attribute market effects by updating the benchmark curve first, then the credit-spread curve. Express all components relative to the initial dirty purchase price, with transaction costs as a separate deduction.

Which report correctly states the net holding-period return and its attribution?

• A. Net return 2.93%; carry/roll-down +4.23%; benchmark-rate effect -2.00%; credit-spread effect +0.90%.
• B. Net return 4.43%; carry/roll-down +5.73%; benchmark-rate effect -2.00%; credit-spread effect +0.90%.
• C. Net return 2.93%; carry/roll-down +4.23%; benchmark-rate effect -1.10%; credit-spread effect 0.00%.
• D. Net return 0.93%; carry/roll-down +4.23%; benchmark-rate effect -2.00%; credit-spread effect -1.10%.

Best answer: A

Explanation: Realized bond return uses dirty prices and counts each coupon once, including its reinvestment income. The purchase dirty price is $100.00, and the actual exit dirty price is $100.10. The coupon’s exit value is \(3.00[1+0.04(3/12)]=3.03\).

Thus, the net holding-period return is:

\[ R_{\text{net}}=\frac{100.10+3.03-100.00-0.20}{100.00}=2.93\%. \]

The unchanged-curve exit dirty price is $101.20. Its $1.20 price gain represents roll-down under the stated reference assumption. Adding $3.03 of coupon carry gives a combined carry/roll-down contribution of 4.23%.

Updating only the benchmark curve lowers the dirty value from $101.20 to $99.20, producing a -2.00% contribution. Updating credit spreads then raises it to $100.10, producing +0.90%. Deducting transaction costs of 0.20% reconciles the total: \(4.23-2.00+0.90-0.20=2.93\%\). These sequential price differences allocate market effects without counting the benchmark-rate movement again in the spread component.

• A. The unchanged-curve gain including the reinvested coupon is 4.23%; sequential revaluations give -2.00% for benchmark rates and +0.90% for credit spreads.
• B. The 5.73% carry/roll-down figure adds exit accrued interest a second time; the dirty exit valuation already includes that $1.50.
• C. The -1.10% difference between actual and unchanged-curve exit values combines benchmark-rate and credit-spread effects rather than isolating the benchmark-rate effect.
• D. Using the actual-versus-unchanged-curve difference as the spread component counts the benchmark-rate loss twice; spreads must be updated from the benchmark-adjusted valuation.

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Question 47

Topic: Foundations of Risk Management

A bank’s board-approved governance framework specifies the following responsibilities:

• Risk appetite: Aggregate stressed credit losses must not exceed $40 million.
• Senior management: Has delegated authority to set business-line exposure limits within that appetite.
• Board risk committee: Reviews the risk profile and recommends appetite changes to the full board.
• Audit committee: Oversees financial reporting and internal-control assurance.

Senior management proposes increasing a corporate lending exposure limit from $600 million to $750 million. Independent risk has challenged the assumptions and estimates aggregate stressed credit losses of $36 million after the expansion, using the unchanged board-approved stress scenario. The audit committee has received a satisfactory internal audit report on exposure data and limit-monitoring controls.

Which interpretation of decision authority is supported?

• A. The proposal changes an operating limit, with approval resting with the board risk committee; independent risk provides measurement and challenge.
• B. The proposal changes an operating limit, with approval resting with senior management; independent risk provides measurement and challenge.
• C. The proposal changes an operating limit, with approval resting with independent risk; senior management implements the limit and manages the business.
• D. The proposal requires an appetite revision, with approval resting with the full board; independent risk provides measurement and challenge.

Best answer: B

Explanation: Risk appetite establishes the board’s boundaries for risk-taking; operating limits translate those boundaries into business constraints. An increase in an operating limit does not necessarily change risk appetite. Here, projected aggregate stressed losses remain at $36 million against an unchanged $40 million ceiling, so approval falls within senior management’s delegated authority.

Independent risk measures exposures, challenges assumptions, monitors compliance, and escalates concerns. Its independence does not make it the owner of commercial decisions. The board risk committee reviews the risk profile and recommends appetite changes under its stated mandate. The audit committee oversees financial reporting and internal-control assurance. Satisfactory audit findings support confidence in the controls but neither authorize additional risk-taking nor transfer approval rights. The full board retains responsibility for setting risk appetite, rather than approving every operating-limit adjustment within it.

• A. The risk committee’s review-and-recommendation mandate does not give it approval authority over operating limits delegated to senior management.
• B. Senior management has delegated limit-setting authority, and the proposed expansion remains within the board-approved stressed-loss ceiling.
• C. Independent measurement and challenge do not transfer management’s delegated limit-setting authority to the risk-control function.
• D. The projected $36 million stressed loss remains below the unchanged $40 million appetite ceiling, so increasing the exposure limit does not require an appetite revision.

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Question 48

Topic: Quantitative Analysis

A risk analyst is validating a normal model for an equity’s daily continuously compounded returns. For one observation, the adjusted closing price fell from $125 to $100, with no intervening cash distribution.

Sample statistics for daily continuously compounded returns:

• Number of independent observations: 240
• Sample skewness: -0.30
• Sample kurtosis: 3.80, measured so that a normal distribution has kurtosis 3

The analyst uses the standard asymptotic Jarque-Bera test at the 5% significance level. The chi-squared critical value with two degrees of freedom is 5.99.

Which report of the price observation and distribution diagnostic is supported?

• A. Report a log return of -22.31% and a Jarque-Bera statistic of 5.00; normality is not rejected, and no specific alternative distribution is established.
• B. Report a log return of -22.31% and a Jarque-Bera statistic of 10.00; normality is rejected, and no specific alternative distribution is established.
• C. Report a log return of -22.31% and a Jarque-Bera statistic of 10.00; normality is rejected, and a Student’s t distribution is established.
• D. Report a log return of -18.13% and a Jarque-Bera statistic of 10.00; normality is rejected, and no specific alternative distribution is established.

Best answer: B

Explanation: Simple and continuously compounded returns describe the same price change using different conventions. The simple return is \(100/125-1=-0.20\). The continuously compounded return is therefore

\[ r=\ln(1-0.20)=\ln(0.80)\approx-0.22314. \]

For sample size \(n\), skewness \(S\), and ordinary kurtosis \(K\), the Jarque-Bera statistic is

\[ JB=\frac{n}{6}\left[S^2+\frac{(K-3)^2}{4}\right] =40\left[0.09+\frac{0.64}{4}\right]=10.00. \]

Because 10.00 exceeds 5.99, the normal-return assumption is rejected at 5%. The test combines deviations in skewness and kurtosis from their normal values. It does not determine which alternative distribution fits the returns; that requires additional modeling and diagnostic evidence.

• A. The standard Jarque-Bera statistic uses a sample-size multiplier of 240/6, not 240/12, producing 10.00 and rejection at the stated significance level.
• B. The log price ratio is -22.31%, and the Jarque-Bera statistic of 10.00 exceeds 5.99; rejection does not identify a replacement distribution.
• C. Jarque-Bera rejection indicates incompatibility with normality but cannot establish that the returns follow a Student’s t distribution.
• D. The -18.13% figure applies the exponential conversion to the simple return of -20%; converting a simple return to a log return requires the logarithm.

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Question 49

Topic: Financial Markets and Products

A trader holds a short position in five exchange-traded equity-index futures contracts. Each contract has a multiplier of $50 per index point.

Margin terms:

• The account opens with $60,000, equal to the required initial margin for the entire position.
• Maintenance margin for the entire position is $45,000.
• Daily settlement gains and losses are credited to or debited from the account.
• If the post-settlement balance falls below maintenance margin, the trader must restore it to initial margin before the next trading session.
• Required deposits are made on time. Gains remain in the account, the position is unchanged, and there are no fees or interest.

Settlement record:

Which schedule of additional cash deposits is required for margin calls arising through Day 4?

• A. Deposit $7,000 before the Day 3 session and $28,000 before the Day 5 session.
• B. Deposit $20,000 before the Day 3 session and $17,000 before the Day 5 session.
• C. Deposit $20,000 before the Day 3 session and $37,000 before the Day 5 session.
• D. Deposit $5,000 before the Day 3 session and $17,000 before the Day 5 session.

Best answer: B

Explanation: Margin calls depend on the post-settlement account balance, not simply the day’s loss. For this short position, each one-point increase in the index causes a $250 loss across five contracts; a decrease produces an equal gain.

• Day 1: The 52-point increase produces a $13,000 loss. The balance falls to $47,000, so no deposit is required.
• Day 2: The 28-point increase produces a $7,000 loss. The balance falls to $40,000, requiring a $20,000 deposit before Day 3 to restore $60,000.
• Day 3: The 44-point decrease produces an $11,000 gain. The balance rises to $71,000.
• Day 4: The 112-point increase produces a $28,000 loss. The balance falls to $43,000, requiring a $17,000 deposit before Day 5.

The first deposit remains in the account and must be included when calculating the second call.

• A. These amounts are the daily settlement losses on Days 2 and 4, not the deposits needed to restore initial margin.
• B. Balances of $40,000 after Day 2 and $43,000 after Day 4 require deposits of $20,000 and $17,000 to restore initial margin.
• C. The $37,000 second deposit results from omitting the first $20,000 deposit when carrying the margin balance forward.
• D. The $5,000 first deposit restores the Day 2 balance only to maintenance margin of $45,000, rather than required initial margin of $60,000.

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Question 50

Topic: Financial Markets and Products

A trader proposes a one-year reverse cash-and-carry trade in a stock. The desk can short 1,000 shares today at $100 per share and simultaneously enter a long forward to purchase 1,000 shares in one year at $104 per share.

Contract and funding terms:

• The stock loan is secured through forward maturity with no recall, and all short-sale proceeds are available for investment.
• A certain dividend of $2 per share is paid in six months. The short seller must reimburse the stock lender on that date.
• A total stock-borrowing fee of $1,200 is payable at the end of the year.
• The desk invests all sale proceeds for one year and borrows separately to fund the dividend reimbursement. Lending and borrowing both use 8% annual simple interest.
• Shares delivered under the forward are returned to the stock lender. Ignore other costs and default risk.

What net profit does the strategy lock in at the one-year maturity?

• A. A net profit of $800.
• B. A net profit of $2,800.
• C. A net profit of $1,920.
• D. A net profit of $720.

Best answer: D

Explanation: Reverse cash-and-carry combines a short sale of the underlying asset, investment of the sale proceeds, and a long forward that supplies the asset for return to its lender. The short seller remains responsible for dividends and stock-borrowing charges.

The $100,000 sale proceeds grow to $108,000. The $2,000 dividend reimbursement occurs after six months, so its financing liability at maturity is \(2{,}000(1 + 0.08 \times 0.5) = 2{,}080\). The forward purchase costs $104,000, and the borrowing fee is $1,200.

Thus, maturity profit is \(108{,}000 - 104{,}000 - 2{,}080 - 1{,}200 = 720\) dollars. Future spot prices do not affect this result because the forward fixes the repurchase price. Secured stock borrowing and access to the sale proceeds are crucial: borrowing restrictions, recall risk, or restricted proceeds can prevent an apparent pricing discrepancy from becoming an executable arbitrage.

• A. This amount deducts the dividend reimbursement but omits the $80 interest incurred by financing it for six months.
• B. This amount deducts the forward purchase and stock-borrowing fee but omits the $2,080 maturity liability associated with the dividend reimbursement.
• C. This amount accounts for the forward purchase and financed dividend reimbursement but omits the $1,200 stock-borrowing fee.
• D. The invested proceeds reach $108,000, while the forward purchase, financed dividend reimbursement, and stock-borrowing fee total $107,280.

Questions 51-75

Question 51

Topic: Financial Markets and Products

A copper processor wants to hedge the benchmark copper-price exposure of its budgeted processing margin for a batch completed in three months.

Batch assumptions:

• Expected refined copper sales are 1,000 metric tonnes, priced at the benchmark copper price on the completion date.
• The concentrate supplier’s invoice equals 85% of that same benchmark price multiplied by 1,000 metric tonnes.
• Other processing costs are fixed.

Each copper futures contract covers 25 metric tonnes and expires on the completion date. Assume futures price changes equal changes in the cash benchmark, and ignore financing costs. The hedge targets the planned production volume, not unexpected recovery losses or production interruptions.

Which futures position offsets the benchmark-price sensitivity of the budgeted processing margin?

• A. Sell 40 copper futures contracts.
• B. Buy 6 copper futures contracts.
• C. Sell 6 copper futures contracts.
• D. Buy 34 copper futures contracts.

Best answer: C

Explanation: A commodity processor’s price exposure depends on both its input costs and output revenues. Here, both are tied to the same copper benchmark, so their sensitivities can be netted.

Let \(P\) denote the benchmark price in dollars per metric tonne and \(C\) the fixed processing costs. The budgeted margin is:

\[ M = 1{,}000P - 0.85(1{,}000P) - C = 150P - C. \]

The processor therefore benefits from rising copper prices, resembling a producer with 150 tonnes of unhedged output. Six short futures contracts provide an offsetting exposure of 150 tonnes. A consumer normally buys futures to hedge rising input prices, but that direction would increase this processor’s net exposure.

This hedge stabilizes the benchmark-price component of the planned margin. It does not protect against lower recovery, production interruptions, or other deviations from the planned physical output.

• A. Selling 40 contracts hedges the entire output exposure, overlooking the supplier invoice’s offsetting sensitivity to the same copper benchmark.
• B. Buying six contracts increases the processor’s positive copper-price exposure rather than offsetting the net exposure from its purchases and sales.
• C. The net positive exposure is 1,000 minus 850, or 150 tonnes; selling six 25-tonne contracts offsets that sensitivity.
• D. Buying 34 contracts hedges the gross input-cost exposure, but ignores the larger positive price exposure from refined copper sales.

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Question 52

Topic: Financial Markets and Products

A risk analyst evaluates an MBS backed by fixed-rate mortgages with a 6.00% coupon. The current refinancing rate is 6.00%, and borrowers face closing costs averaging 2% of outstanding principal.

The model shifts the discount curve and refinancing rate by the same amount, holding credit assumptions and discounting spreads fixed. Each scenario’s conditional prepayment rate (CPR) is applied as a constant annual rate over the remaining pool life. An option-free benchmark uses cash flows frozen at the MBS’s base-case projection. Prices are per $100 of outstanding principal.

Which interpretation of the MBS’s convexity and prepayment-driven duration response is best supported?

• A. Positive effective convexity over the 100 bp shocks, with prepayment-driven duration contraction present after the 25 bp decline.
• B. Positive effective convexity over the 100 bp shocks, with prepayment-driven duration contraction absent after the 25 bp decline.
• C. Negative effective convexity over the 100 bp shocks, with prepayment-driven duration contraction absent after the 25 bp decline.
• D. Negative effective convexity over the 100 bp shocks, with prepayment-driven duration contraction present after the 25 bp decline.

Best answer: C

Explanation: Mortgage borrowers hold an embedded prepayment option. Falling rates encourage refinancing when interest savings justify closing costs, but borrower frictions can delay this response. Here, the 25 bp decline leaves CPR unchanged, so there is no prepayment-driven shortening of cash flows.

The 100 bp decline raises CPR to 26%. Earlier principal repayment shortens duration and limits the MBS’s price appreciation relative to the option-free benchmark. Conversely, rising rates reduce CPR to 4%, extending cash flows and amplifying the price decline.

Under symmetric 100 bp shocks, the MBS prices average 98.50, below the base price of 100.00. This indicates negative effective convexity over that range. The benchmark prices average 100.25, above the base price, consistent with the positive convexity of fixed, option-free cash flows.

• A. Unchanged CPR contradicts contraction after the smaller decline, and the large-shock prices average below the base price, indicating negative effective convexity.
• B. Although prepayments remain unchanged after the smaller decline, the larger loss than gain under symmetric shocks indicates negative, not positive, effective convexity.
• C. The symmetric shocks produce a 3.00 gain and a 6.00 loss, while unchanged CPR after the smaller decline indicates no prepayment-driven contraction.
• D. The smaller decline leaves CPR at 8% and the MBS price equal to the fixed-cash-flow benchmark, providing no evidence of prepayment-driven contraction.

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Question 53

Topic: Foundations of Risk Management

A bank’s board has approved an enterprise-wide commercial-property concentration limit. Divisional limits were established separately and do not allocate the enterprise limit among business units.

Current practice:

• The CRO’s validated consolidated report shows an enterprise-limit breach for three consecutive months, although every division remains within its own limit.
• The executive committee closes each breach alert after confirming divisional compliance. No executive is responsible for reducing the aggregate exposure.
• Division-head bonuses depend on origination volume. The CRO’s reports reach the board only after CEO approval.

The board wants enterprise appetite breaches to trigger business decisions and credible independent challenge. Which change most directly addresses the bank’s ERM weakness?

• A. Hold division heads accountable for local-limit compliance and link their bonuses to results, with direct CRO compliance reporting to the board.
• B. Hold the CEO accountable for aggregate-limit remediation and link CEO bonuses to results, with independent CRO escalation to the board.
• C. Hold the CRO accountable for aggregate-limit remediation and link CRO bonuses to results, with executive-committee approval of escalation to the board.
• D. Hold division heads accountable for accurate exposure reporting and link their bonuses to results, with independent audit certification to the board.

Best answer: B

Explanation: Effective ERM connects enterprise risk information to accountable business decisions. Compliance with separately established divisional limits does not demonstrate compliance with an enterprise concentration limit. Here, measurement already identifies the breach; the failure is management’s response to that information.

The CEO can coordinate remediation across divisions and should be accountable for bringing aggregate exposure back within appetite. Linking incentives to remediation makes that responsibility consequential. The CRO should independently monitor and challenge management, with direct access to the board when breaches remain unresolved.

Risk culture is demonstrated by how leaders respond to unfavorable information. Closing accurate breach alerts because local limits remain satisfied shows that enterprise appetite is not governing actual decisions.

• A. Rewarding local-limit compliance preserves the existing mismatch: divisions can satisfy their limits while the bank continues to breach its enterprise appetite.
• B. CEO accountability connects enterprise exposure to business decisions and incentives, while independent CRO escalation prevents management from filtering unresolved breaches.
• C. This shifts business-risk remediation onto the independent control function and leaves escalation subject to the committee that already dismisses aggregate breaches.
• D. The consolidated exposure report is already validated; stronger reporting assurance does not establish ownership or incentives for reducing the enterprise concentration.

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Question 54

Topic: Quantitative Analysis

A risk analyst is preprocessing three monthly risk indicators, A, B and C, each measured in index points, using 120 observations per series. The goal is covariance-stationary inputs while avoiding unnecessary differencing.

Unit-root evidence: Each augmented Dickey-Fuller (ADF) test has a zero-frequency unit-root null. Assume the deterministic specifications and augmentation lags are appropriate, and no seasonal unit roots are present.

Additional results:

• Use 5% ADF critical values of -3.45 with a linear trend and -2.89 without a linear trend.
• OLS time-trend regressions report t-statistics of 5.6 for A and 7.4 for B.
• B’s first differences have an ADF statistic of -6.12 with a constant and no trend.
• C has stable calendar-month mean differences and no time trend.

At the 5% significance level, which transformation plan is best supported by the evidence?

• A. First-difference A; first-difference B; take C’s 12-month differences.
• B. First-difference A; remove B’s fitted linear trend; subtract C’s calendar-month means.
• C. Remove A’s fitted linear trend; first-difference B; subtract C’s calendar-month means.
• D. Remove A’s fitted linear trend; remove B’s fitted linear trend; take C’s 12-month differences.

Best answer: C

Explanation: An ADF rejection supports stationarity around the deterministic terms included in the test, not necessarily stationarity of unadjusted levels. For A, -4.32 is more negative than -3.45, supporting removal of its fitted linear trend.

For B, -2.11 does not cross the -3.45 rejection threshold, whereas its first-difference statistic, -6.12, crosses -2.89. First differencing is therefore supported. Failure to reject alone does not prove a unit root, but rejection for first differences reinforces this treatment. A stochastic trend, such as a random walk with drift, is addressed by differencing rather than subtracting a fitted time trend. Large ordinary regression t-statistics do not establish trend stationarity or validate conventional inference with nonstationary errors.

For C, -4.06 crosses -2.89 with calendar-month indicators included. Given the absence of seasonal unit roots, subtracting calendar-month means removes its deterministic periodic mean. Differencing A or taking 12-month differences of C would be unnecessary and could introduce additional moving-average dependence.

• A. Differencing A and seasonally differencing C are unnecessary because the evidence supports stationarity after removing A’s deterministic trend and C’s calendar-month means.
• B. A supports detrending rather than unnecessary differencing, while B’s large trend t-statistic does not override its unit-root evidence supporting first differencing.
• C. A rejects a unit root around its trend, B supports stationary first differences, and C supports stationarity after removing deterministic calendar-month effects.
• D. Detrending B does not address the supported stochastic trend, while C’s deterministic monthly pattern calls for mean removal rather than seasonal differencing.

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Question 55

Topic: Financial Markets and Products

A commodity fund owns 1,000 metric tons of copper, currently worth $9,000 per ton, and will sell the inventory in three months. Hedges mature or are closed out on the sale date. Assume no basis risk. Partial hedging is permitted.

Mandate:

• Net liquidation proceeds, including hedge settlements and deducting fees or premiums, must be at least $8.45 million if copper ends at $8,000 per ton and $9.45 million if it ends at $10,000 per ton.
• Before the sale, $550,000 is available for combined upfront hedge expenditure and margin or collateral. All margin and collateral calls must be funded before inventory sale proceeds become available.

Executable quotes: Linear-contract fees and initial deposits below are totals for a 500-ton hedge. Each futures contract covers 25 tons.

Ignore discounting and financing costs. Linear-contract reference prices move monotonically from $9,000 to each stressed terminal price. Initial deposits are refundable; collateral is credited against final obligations, and futures variation margin settles the futures gain or loss.

Which hedge minimizes nonrefundable upfront expenditure while meeting the mandate?

• A. Short a 500-ton forward at $9,000 per ton.
• B. Buy puts on 1,000 tons with a strike of $8,600 per ton.
• C. Enter a 500-ton receive-fixed, pay-spot swap at $9,000 per ton.
• D. Short 20 copper futures contracts at $9,000 per ton.

Best answer: C

Explanation: Partial linear hedging can preserve sufficient upside when protection is required at specified stress prices, rather than as a universal floor. For terminal copper price \(S_T\), in dollars per ton, the receive-fixed swap pays \(500(9,000-S_T)\). Combining this with the inventory proceeds and deducting the $4,000 fee gives $8.496 million in the lower stress and $9.496 million in the higher stress. Both exceed the required amounts.

When copper rises to $10,000, the swap has a $500,000 liability. Funding its collateral, initial deposit, and fee requires $529,000, within the $550,000 limit. The equivalent futures hedge has the same linear economic exposure, but its larger initial margin raises required cash to $577,000. Inventory appreciation offsets the derivative loss economically but cannot fund pre-sale margin calls.

The puts have the nonlinear payoff \(1,000\max(8,600-S_T,0)\), protecting downside while retaining upside. However, the premium is nonrefundable expenditure, unlike refundable initial deposits. The swap therefore provides the least-expensive feasible hedge under the stated criterion.

• A. The forward meets the stressed-value and liquidity limits, but its $8,000 nonrefundable fee exceeds the feasible swap’s $4,000 fee.
• B. The puts provide net stressed proceeds of $8.50 million and $9.90 million, but their $100,000 premium exceeds the feasible swap’s fee.
• C. The swap meets both stressed-value limits, requires peak cash of $529,000, and has the lowest nonrefundable expenditure among feasible hedges.
• D. The high-price stress requires $75,000 of initial margin, $500,000 of variation margin, and a $2,000 fee, exceeding available cash by $27,000.

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Question 56

Topic: Financial Markets and Products

On January 1, 2026, a treasurer must choose between paying an invoice immediately for $995,000 or using the following deferred payment schedule.

The zero rates are nominal annual rates with quarterly compounding, measured from January 1, 2026. Use a 30/360 day-count basis. Both schedules settle the same obligation, sufficient liquidity is available, and there are no taxes or transaction costs.

Which schedule minimizes the present value of the payment obligation, and what is its approximate advantage over the other schedule?

• A. The deferred payment schedule, by $98.
• B. The deferred payment schedule, by $6,301.
• C. The deferred payment schedule, by $1,554.
• D. The immediate payment schedule, by $8,532.

Best answer: C

Explanation: Each dated payment must be discounted using the zero rate for its own maturity and the stated compounding convention. Under 30/360, the payments occur in six months and one year, corresponding to two and four quarterly periods. The quarterly rates are 1.5% and 2%, respectively.

The deferred schedule therefore has present value:

\[ PV = \frac{500{,}000}{(1.015)^2} + \frac{550{,}000}{(1.02)^4} = 993{,}445.86. \]

The immediate payment requires no discounting and has present value $995,000. Choosing deferred payments reduces the present value of the obligation by $1,554.14, or approximately $1,554. A single rate should not replace the maturity-specific zero rates when valuing these separate cash flows.

• A. This result treats the quoted rates as effective annual rates, rather than nominal annual rates with quarterly compounding.
• B. This result applies the one-year 8% zero rate to both payments; the six-month payment must instead use its 6% zero rate.
• C. Discounting each payment at its maturity-specific quarterly-compounded zero rate gives a total present value of $993,446, approximately $1,554 below the immediate payment.
• D. This result applies the six-month 6% zero rate to both payments; the one-year payment must instead use its 8% zero rate.

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Question 57

Topic: Financial Markets and Products

A treasurer opens a long position in 10 crude oil futures contracts at $75.00 per barrel. Each contract represents 1,000 barrels.

Margin arrangements:

• Initial margin is $5,000 per contract; maintenance margin is $4,000 per contract.
• After each daily settlement, a balance below maintenance margin must be restored to initial margin using external cash.
• No excess margin is withdrawn. Ignore fees and interest.

Settlement prices:

The treasurer closes the entire position at the Day 3 settlement price, and the remaining margin balance is returned. Which pair gives the cash returned at closing and the cumulative futures trading P&L?

• A. Cash returned: $45,000; cumulative trading P&L: -$15,000.
• B. Cash returned: $45,000; cumulative trading P&L: -$5,000.
• C. Cash returned: $55,000; cumulative trading P&L: +$5,000.
• D. Cash returned: $55,000; cumulative trading P&L: -$15,000.

Best answer: D

Explanation: Daily marking to market credits or debits the long position by the settlement-price change multiplied by 10 contracts and 1,000 barrels per contract. The initial margin balance is $50,000, and maintenance margin is $40,000.

• Day 1: A $2.00 price decline creates a $20,000 loss. The balance falls to $30,000, triggering a $20,000 deposit to restore it to $50,000.
• Day 2: A $1.50 price increase creates a $15,000 gain, raising the balance to $65,000.
• Day 3: A $1.00 price decline creates a $10,000 loss, leaving $55,000 to be returned.

Cumulative trading P&L is -$20,000 + $15,000 - $10,000 = -$15,000. Equivalently, $55,000 returned minus $70,000 contributed equals -$15,000. Margin deposits fund the account; they do not create trading gains.

• A. Restoring the account to maintenance rather than initial margin understates the Day 1 deposit and the closing balance by $10,000.
• B. This calculation replenishes only to maintenance margin and excludes that additional contribution when measuring the trading loss.
• C. The $5,000 excess over the initial deposit is not trading profit because the treasurer also contributed $20,000 to meet a margin call.
• D. Total cash contributed is $70,000, including the $20,000 margin call, so returning $55,000 corresponds to a $15,000 trading loss.

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Question 58

Topic: Financial Markets and Products

A U.S. importer must pay EUR 1,000,000 today by purchasing euros with U.S. dollars at the quoted spot rate. The treasurer compares today’s USD outflow with what the same euro payment would have cost one year ago.

Market observations:

• Spot rate one year ago: USD 1.1000 per EUR 1.
• Spot rate today: USD 1.1550 per EUR 1.

Nominal rates are effective annual rates. Expected inflation over the coming year equals the past-year inflation shown. Use the initial spot rate as the base for a relative purchasing-power parity (PPP) benchmark.

Which conclusion about the USD payment cost and expected real interest rates is supported by these observations?

• A. The USD payment cost increased more than relative PPP implies, and the expected real interest rate is higher in the United States.
• B. The USD payment cost increased less than relative PPP implies, and the expected real interest rate is higher in the United States.
• C. The USD payment cost increased more than relative PPP implies, and the expected real interest rate is higher in the euro area.
• D. The USD payment cost increased less than relative PPP implies, and the expected real interest rate is higher in the euro area.

Best answer: C

Explanation: With spot rates quoted as USD per EUR, a rising rate means dollar depreciation and a higher USD cost of buying euros. The invoice’s USD cost increased from $1,100,000 to $1,155,000, or 5.0%.

Relative PPP adjusts the initial exchange rate by domestic inflation relative to foreign inflation:

\[ 1.1000 \times \frac{1.06}{1.02} \approx 1.1431 \text{ USD per EUR}. \]

This implies a 3.92% increase in the payment cost, less than the observed increase.

The Fisher relation gives an expected U.S. real rate of \(1.07/1.06-1 \approx 0.94\%\) and a euro-area real rate of \(1.045/1.02-1 \approx 2.45\%\). Higher U.S. inflation more than offsets its nominal-rate advantage.

The euro area’s higher expected real rate can support euro demand through capital-flow incentives. However, these observations do not establish the cause of the past spot movement, and relative PPP is an inflation-based benchmark rather than an exact short-run forecast.

• A. The higher U.S. nominal rate does not imply a higher real rate: inflation-adjusted rates are approximately 0.94% in the United States and 2.45% in the euro area.
• B. Relative PPP implies only a 3.92% cost increase, and Fisher-adjusted real rates are 0.94% in the United States and 2.45% in the euro area.
• C. The payment cost increased 5.0%, versus 3.92% under relative PPP, while the euro area’s expected real rate exceeds the U.S. rate.
• D. The observed USD payment cost increased 5.0%, exceeding the 3.92% increase implied by the two economies’ inflation rates under relative PPP.

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Question 59

Topic: Financial Markets and Products

A risk analyst is reconciling a projected monthly principal distribution from a mortgage pass-through. The servicing exhibit reports:

• Scheduled principal-and-interest payments for the month: $95,000.
• Projected annual conditional prepayment rate (CPR): 12%.

All loans are current. Monthly interest equals beginning principal multiplied by the annual coupon divided by 12. Assume no fees or defaults, and process prepayments immediately after scheduled payments. Retain full precision until final rounding.

What is the projected total principal distributed for the month, including scheduled principal and prepayments, rounded to the nearest dollar?

• A. A principal distribution of $151,975.
• B. A principal distribution of $150,486.
• C. A principal distribution of $158,462.
• D. A principal distribution of $157,906.

Best answer: D

Explanation: Mortgage principal distributions combine scheduled amortization and unscheduled prepayments. The scheduled payment includes interest, which must first be separated from principal.

The balance-weighted coupon is \(0.8 \times 4.5\% + 0.2 \times 7.5\% = 5.1\%\). Monthly interest on $10,000,000 is therefore $42,500. The $95,000 scheduled payment contains $52,500 of principal, leaving $9,947,500 outstanding.

CPR and single monthly mortality (SMM) are related through compounded survival:

\[ \mathrm{SMM} = 1-(1-\mathrm{CPR})^{1/12} = 1-0.88^{1/12} \approx 0.010596241. \]

Applying this monthly prepayment rate to the balance after scheduled principal produces $105,406.11 of prepayments. Adding $52,500 of scheduled principal gives $157,906.11, rounded to $157,906.

The $42,500 of interest is separate from the principal distribution. CPR/SMM measures unscheduled prepayments, not scheduled amortization, and CPR cannot simply be divided by 12.

• A. Using CPR divided by 12 gives 1% SMM and $99,475 of prepayments, ignoring compounding in the annual-to-monthly conversion.
• B. An unweighted average coupon of 6% overstates interest at $50,000 and understates scheduled principal; coupon weighting must reflect the unequal balances.
• C. Applying SMM to the $10,000,000 beginning balance overstates prepayments because $52,500 of scheduled principal is paid before the prepayment calculation.
• D. Coupon interest is $42,500, leaving $52,500 of scheduled principal; correctly converted SMM produces $105,406 of additional principal from prepayments.

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Question 60

Topic: Financial Markets and Products

An oil producer has two cash-settled crude oil forwards with the same dealer. Both settle in USD on 30 June 2026 using that day’s spot price. Both contracts remain outstanding until maturity, with no upfront payments or interim settlements.

The exhibit shows the dealer’s side of each trade. Prices in the trade rows are the agreed forward prices.

What is the producer’s net cash flow from settling both forwards on 30 June 2026?

• A. The producer receives $740,250.
• B. The producer pays $519,750.
• C. The producer receives $393,750.
• D. The producer receives $519,750.

Best answer: D

Explanation: Cash settlement depends on each contract’s agreed forward price and the settlement-date spot price. A long position receives \(Q(S_T-K)\), while a short receives \(Q(K-S_T)\), where \(Q\) is quantity, \(K\) is the agreed forward price, and \(S_T\) is settlement spot. A negative result represents a payment.

Because the records show the dealer’s positions, the producer is short 120,000 barrels under the first contract and long 45,000 barrels under the second:

• Short forward: \(120{,}000\times(83.40-78.15)=630{,}000\) USD.
• Long forward: \(45{,}000\times(78.15-80.60)=-110{,}250\) USD.

The producer’s net receipt is therefore $519,750. Opposite positions entered at different forward prices must be valued separately; applying one contract’s price to the net quantity discards the price difference locked in on the offsetting position.

• A. This adds both payoff magnitudes, treating the producer’s $110,250 loss on the second forward as a receipt rather than a payment.
• B. A $519,750 payment is the dealer’s net cash flow; the producer holds the opposite side of each contract.
• C. This applies the first forward’s price to the net 75,000 barrels, ignoring the different agreed price on the offsetting 45,000-barrel contract.
• D. The producer receives $630,000 on its short forward and pays $110,250 on its long forward, leaving a net receipt of $519,750.

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Question 61

Topic: Valuation and Risk Models

A bank’s risk analyst must report the one-day 99% historical VaR of a trading portfolio using 250 equally weighted daily loss observations. Positive values denote losses, and all observations are distinct.

The reporting policy specifies:

Use the smallest observed loss whose cumulative sample frequency reaches or exceeds the confidence level.

The losses are ranked in ascending order, with rank 1 representing the smallest loss. An excerpt is shown below.

Which amount should the analyst report?

• A. $3.90 million
• B. $3.10 million
• C. $3.25 million
• D. $3.40 million

Best answer: D

Explanation: Historical VaR at 99% confidence is the empirical 99th percentile of the loss distribution. Under the specified inverse empirical-CDF convention, select the first ranked loss whose cumulative frequency is at least 99%.

For 250 equally weighted observations, the required rank is \(\lceil 0.99 \times 250 \rceil = \lceil 247.5 \rceil = 248\). Rank 247 covers only 98.8% of the sample, while rank 248 covers 99.2%. Therefore, the reported one-day VaR is $3.40 million. The fractional target rank does not imply interpolation: the specified convention selects an actual observation.

• A. Rank 249 reaches 99.6% cumulative frequency, but rank 248 already satisfies the threshold at a smaller observed loss.
• B. Rank 247 includes only 247/250 = 98.8% of observations, which falls below the required 99% cumulative frequency.
• C. This amount interpolates halfway between ranks 247 and 248, whereas the policy requires selecting an observed loss.
• D. Rank 248 includes 248/250 = 99.2% of observations and is the first rank whose cumulative frequency reaches 99%.

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Question 62

Topic: Financial Markets and Products

A US insurer holds a €20 million fixed-rate bond paying a 3.0% annual coupon, with two years remaining until principal repayment. Treasury wants to convert all remaining bond cash flows into fixed USD cash flows.

An available currency swap has these remaining contractual terms:

• EUR leg: €20 million notional and a 3.0% annual coupon.
• USD leg: $22 million notional and a 4.0% annual coupon.
• Both legs pay coupons in exactly one and two years. Both notionals are exchanged with the final coupons; no principal exchange occurs today.

The spot exchange rate is $1.12 per €1. Ignore default risk, transaction costs, and any upfront settlement. Use these discount factors:

Which swap position achieves the objective, and what is its current USD value to the insurer, rounded to the nearest $1,000?

• A. Pay the EUR leg and receive the USD leg; current value is approximately +$911,000.
• B. Pay the EUR leg and receive the USD leg; current value is approximately -$911,000.
• C. Receive the EUR leg and pay the USD leg; current value is approximately -$911,000.
• D. Receive the EUR leg and pay the USD leg; current value is approximately +$911,000.

Best answer: B

Explanation: Transforming an asset requires paying away its existing cash flows and receiving the desired replacement cash flows. Paying the EUR leg cancels the bond’s coupons and principal repayment, leaving fixed USD receipts. The matching amounts and payment dates avoid residual EUR cash-flow or maturity mismatches.

Value each swap leg as a bond using its own currency’s discount factors. In millions of the respective currencies:

\( B_{EUR} = 0.60(0.9800) + 20.60(0.9500) = 20.158 \).

\( B_{USD} = 0.88(0.9600) + 22.88(0.9100) = 21.6656 \).

Convert the EUR present value into USD at the current spot rate, then subtract the paying leg from the receiving leg:

\( V_{USD} = 21.6656 - 1.12(20.158) = -0.91136 \) million.

The swap therefore has a current value of approximately -$911,000 before any upfront settlement. Its negative value does not alter which direction hedges the asset.

• A. The direction offsets the bond’s EUR receipts, but the USD leg’s present value is below the spot-converted EUR leg’s present value.
• B. Paying EUR offsets the bond receipts; the USD leg’s $21.6656 million value minus the converted EUR leg’s $22.57696 million value gives -$911,360.
• C. Receiving EUR increases the bond’s EUR exposure, and the value of receiving the EUR leg and paying the USD leg is positive rather than negative.
• D. The positive value is consistent with receiving EUR and paying USD, but this direction adds EUR receipts rather than converting the bond’s cash flows.

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Question 63

Topic: Foundations of Risk Management

During the 2007–2009 crisis, wholesale lenders decline to renew a bank’s funding. A central-bank facility approves a secured term cash loan against eligible securities, subject to a 20% haircut on their current fair value.

Bank data immediately before the intervention:

The bank pledges all eligible securities, draws the maximum loan, and repays today’s wholesale debt. It then recognizes the loan impairment. The securities were previously unencumbered, and pledging them does not transfer ownership. Ignore interest, taxes, and other balance-sheet changes.

Which closing cash and equity position is supported by these facts?

• A. Remaining cash is $10 million, and equity is -$3 million.
• B. Remaining cash is $4 million, and equity is -$3 million.
• C. Remaining cash is $4 million, and equity is $21 million.
• D. Remaining cash is $4 million, and equity is -$9 million.

Best answer: B

Explanation: Collateralized central-bank lending can relieve a rollover funding shortage by providing cash against securities rather than requiring their immediate sale. However, borrowing increases cash and liabilities equally; it does not directly increase equity. A collateral haircut limits the advance rather than creating an accounting impairment.

With amounts measured in USD millions, the maximum loan is \(30 \times (1-0.20)=24\). Closing cash is therefore \(4+24-24=4\). Repayment of existing wholesale debt reduces cash and liabilities equally and has no effect on equity. Recognizing the separate loan impairment leaves equity of \(6-9=-3\).

The intervention allows the bank to meet its immediate funding obligation, but the bank remains balance-sheet insolvent after recognizing the loss. Liquidity support does not automatically restore solvency or protect investors against losses.

• A. The 20% haircut limits borrowing against the $30 million collateral to $24 million, so the bank cannot obtain the proceeds needed for this cash balance.
• B. The $24 million advance leaves $4 million cash after repayment, while the $9 million impairment reduces the bank’s $6 million equity to -$3 million.
• C. The central-bank loan creates an equal liability, so treating its $24 million proceeds as additional equity overstates post-impairment capital by $24 million.
• D. The $6 million collateral haircut reduces borrowing capacity, not the securities’ carrying value, so it does not create an additional equity loss.

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Question 64

Topic: Foundations of Risk Management

A bank’s internal audit team compares two trading desks across three consecutive monthly reporting dates. Each desk’s risk manager reports functionally to the CRO, has direct access to the board risk committee, and uses the same system and criteria for escalating confirmed limit breaches.

Audit exhibit: Breach counts are per monthly reporting date. All counted breaches were material, verified, and unresolved when the committee pack was finalized.

Version histories show that Credit’s entries were removed during the desk head’s review at all three dates. The underlying calculations were unchanged, and the omitted breaches were not reported through another channel.

Which conclusion best identifies the primary governance weakness behind the reporting difference?

• A. The primary weakness is risk-appetite calibration, since recurring limit breaches show that the approved limits are unsuitable for the desks.
• B. The primary weakness is risk-data aggregation, since verified desk records are not being transferred consistently into the final committee packs.
• C. The primary weakness is effective risk independence, since desk management controls rewards and filters the reporting of adverse information.
• D. The primary weakness is escalation discipline, since the formal CRO reporting line and direct committee access preserve the risk managers’ independence.

Best answer: C

Explanation: Risk independence must operate in practice, not merely appear in an organizational chart. A functional reporting line to the CRO and direct committee access provide formal safeguards, but business management can undermine them when it controls risk staff appraisals, compensation, and the reporting of adverse findings.

Both desks identify the same number of verified, unresolved breaches and initially include them in their committee drafts. Only Credit repeatedly loses those entries during desk-head review. Combined with revenue-linked compensation and desk-controlled appraisal approval, this pattern supports a structural governance weakness that compromises credible challenge and escalation. The unchanged calculations and repeated omissions distinguish the problem from an isolated measurement or reporting error.

• A. Recurring breaches do not establish that limits are unsuitable, and equal breach counts do not explain why only Credit suppresses reporting.
• B. Both initial drafts contain the verified breaches, and Credit’s deletions occur during management review, indicating suppression rather than a data-transfer failure.
• C. Credit combines revenue-linked compensation with desk-head appraisal authority, while repeated management deletions demonstrate that nominal safeguards fail to protect independent escalation.
• D. Formal reporting lines and committee access do not preserve effective independence when desk management controls compensation and repeatedly removes adverse reports.

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Question 65

Topic: Financial Markets and Products

A treasury needs a one-year loan that starts one year from today and matures two years from today. It may lock its borrowing rate now or arrange financing when funding begins.

Current default-free curve:

Assume borrowing and lending occur at these rates without credit spreads or transaction costs. For its liquidity-preference forecast, the treasury estimates a positive term premium of 0.75 percentage points in the one-year forward rate. Ignore convexity adjustments and round rates to two decimal places.

Which comparison of the no-arbitrage forward borrowing rate and forecasts of the future one-year spot rate is correct?

• A. The forward rate is 6.01%; forecasts are 6.01% under pure expectations and 6.76% under liquidity preference. Market segmentation supplies no unique expected rate.
• B. The forward rate is 6.01%; forecasts are 6.01% under pure expectations and 5.26% under liquidity preference. Market segmentation supplies no unique expected rate.
• C. The forward rate is 5.00%; forecasts are 5.00% under pure expectations and 4.25% under liquidity preference. Market segmentation supplies no unique expected rate.
• D. The forward rate is 6.01%; forecasts are 6.01% under pure expectations and 5.26% under liquidity preference. Market segmentation also predicts a 6.01% future rate.

Best answer: B

Explanation: A no-arbitrage forward rate comes from current spot rates, not a guarantee about future rates. Let \(f_{1,2}\) denote the annual effective rate for borrowing from year 1 to year 2. Matching the compounded returns over two years gives:

\[ f_{1,2}=\frac{(1.05)^2}{1.04}-1\approx 0.060096. \]

The rate that can be locked today is therefore 6.01%. Ignoring convexity adjustments, pure expectations identifies this forward rate with the expected future one-year spot rate.

Under liquidity preference, the forward rate includes a positive premium above the expected future spot rate. Subtracting 0.75 percentage points gives a forecast of 5.26%.

Market segmentation instead emphasizes supply and demand within maturity segments and does not, by itself, produce a unique future short-rate forecast. The contractual forward rate can be fixed today, but neither model-based forecast guarantees the spot rate that will actually prevail.

• A. A positive liquidity premium places the forward rate above the expected future spot rate, so 0.75 percentage points must be subtracted, not added.
• B. No-arbitrage implies 6.01%, and subtracting the positive premium gives a 5.26% liquidity-preference forecast; market segmentation alone does not specify expected short rates.
• C. The 5.00% spot rate applies to two-year compounding; borrowing only during the second year requires the implied forward rate of 6.01%.
• D. Market segmentation links maturity-specific yields to supply and demand; it does not equate the implied forward rate with an expected future short rate.

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Question 66

Topic: Quantitative Analysis

A risk analyst must report tomorrow’s annualized realized-volatility forecast and a historical error estimate representative of real-time forecasting. Forecasts are produced at each trading day’s close.

Using information available at today’s close, the analyst estimates the model \(v_{s+1}=2.0+0.8v_s+\varepsilon_{s+1}\), where volatility is measured in percentage points and the innovation has conditional mean zero.

Available observations:

• Today’s annualized realized volatility: 18%.
• Yesterday’s annualized realized volatility: 15%.

Validation audit: Each procedure refits model coefficients on its training data. Expanding windows restrict estimation responses to observations available at each forecast origin. Random splitting can place dates later than a test forecast origin in the training sample. Centered smoothing replaces the predictor \(v_s\) with \((v_{s-1}+v_s+v_{s+1})/3\).

Which forecast and validation RMSE are supported for the report?

• A. Forecast 16.4% and report a validation RMSE of 1.5 percentage points.
• B. Forecast 16.4% and report a validation RMSE of 2.3 percentage points.
• C. Forecast 14.0% and report a validation RMSE of 2.3 percentage points.
• D. Forecast 16.4% and report a validation RMSE of 1.1 percentage points.

Best answer: B

Explanation: An AR(1) forecast conditions on the latest available observation. Because the innovation has conditional mean zero, tomorrow’s forecast is \(2.0+0.8\times18=16.4\%\). Using yesterday’s observation would ignore information already available at today’s close.

Historical validation should reproduce the information set available when each forecast would have been made. Expanding-window estimation with unsmoothed predictors satisfies this requirement, supporting the reported RMSE of 2.3 percentage points.

Random splitting can train a model on observations occurring after its test forecast dates. Chronological training alone is also insufficient if predictor construction leaks future information: the centered average contains \(v_{s+1}\), which is unknown at the close of day \(s\). The lower reported errors therefore do not establish an improvement achievable in real-time forecasting.

• A. The forecast uses today’s observation correctly, but random splitting allows future observations into training and does not reproduce historical information availability.
• B. The conditional forecast is 2.0 + 0.8 × 18 = 16.4%, and expanding-window validation with unsmoothed predictors preserves historical information availability.
• C. The validation procedure respects time ordering, but 14.0% uses yesterday’s 15% volatility rather than today’s available 18% observation.
• D. The forecast is valid, but the centered predictor includes next day’s realized volatility, which is unavailable at the historical forecast origin.

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Question 67

Topic: Foundations of Risk Management

A risk manager is validating a board summary used to decide whether desk-limit breaches require escalation. The exposure records and calculated sensitivities have been validated, and all reports use the same end-of-day snapshot. Position IDs support drill-down to the underlying exposure records.

Reference conditions:

• All positions are USD-denominated bonds. Positive signed notional denotes a long position; negative denotes a short position.
• Signed DV01 measures estimated loss for a one-basis-point parallel increase in yields. A negative value denotes a gain.
• Limits apply to the absolute value of net DV01 at each stated reporting level.

Position records and calculated sensitivities:

Limit report:

Published reporting:

• Exception report: No breaches listed.
• Management summary: Firm net DV01 of -$3,000 per bp; zero breached desk limits.

What should the management summary report for firm net DV01 and the number of breached desk limits?

• A. Report firm net DV01 of $13,000 per bp and one breached desk limit.
• B. Report firm net DV01 of -$3,000 per bp and one breached desk limit.
• C. Report firm net DV01 of -$3,000 per bp and zero breached desk limits.
• D. Report firm net DV01 of $13,000 per bp and zero breached desk limits.

Best answer: B

Explanation: Exposure-level records describe individual positions; DV01 is a calculated sensitivity rather than a notional amount. Aggregation must preserve the definition of the reported measure.

Rates has net DV01 of $12,000 - $7,000 = $5,000 per bp. Credit has $4,000 - $12,000 = -$8,000 per bp. Their signed sum gives firm net DV01 of -$3,000 per bp.

Limit monitoring uses absolute net DV01. Rates is within its $6,000 limit, while Credit exceeds its $7,000 limit. The firm remains within its $10,000 limit, but firm-level compliance does not eliminate a desk-level breach.

The exception report should identify Credit’s breach, and the management summary should report one breached desk limit. Traceability requires links from the summary to desk risk measures and limit comparisons, then to the underlying position records. Credit’s measure can be traced to C1 and C2 without reproducing every exposure record in the board summary.

• A. The $13,000 figure sums absolute desk sensitivities rather than signed desk sensitivities; the Credit desk breach is correctly recognized.
• B. Rates contributes $5,000 per bp and Credit contributes -$8,000 per bp, producing firm net DV01 of -$3,000 and one desk breach.
• C. The firm net DV01 is correct, but Credit’s absolute net DV01 of $8,000 per bp exceeds its $7,000 per bp limit.
• D. Grossing the desk sensitivities misstates the specified net measure, while zero breaches omits Credit’s absolute DV01 excess.

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Question 68

Topic: Valuation and Risk Models

A risk manager has sold 100 European call contracts on a non-dividend-paying stock. Each contract covers 100 shares. The manager will establish an initially delta-neutral stock hedge and hold it unchanged for one calendar day.

The desk provides these rounded Black-Scholes-Merton values for a long call. Monetary sensitivities are quoted per underlying share, and N denotes the standard normal cumulative distribution function.

Vega measures sensitivity to implied volatility. Rho measures sensitivity to the annual continuously compounded risk-free rate.

One-day changes:

• The stock price rises by $2.
• Implied volatility rises by 1 percentage point.
• The risk-free rate rises by 0.10 percentage points.

Use the supplied rounded values, second-order stock-price effects, and first-order volatility, elapsed-time, and rate effects. Ignore cross terms, higher-order effects, financing, and trading costs. Round P&L to the nearest dollar.

Which initial stock hedge and approximate one-day P&L for the combined position are appropriate?

• A. Sell 5,596 shares; approximate loss of $25,000.
• B. Buy 5,596 shares; approximate loss of $2,616.
• C. Buy 5,596 shares; approximate gain of $1,328.
• D. Buy 5,596 shares; approximate loss of $1,040.

Best answer: B

Explanation: A long European call on a non-dividend-paying stock has delta equal to \(N(d_1)\). The short position covers 10,000 shares, giving an initial delta of \(-10{,}000 \times 0.5596 = -5{,}596\). Buying 5,596 shares therefore establishes delta neutrality.

The fixed stock hedge cancels the initial first-order stock-price effect, but not the option’s other sensitivities. The remaining contributions are:

• Gamma: \(-10{,}000 \times \tfrac{1}{2} \times 0.0394 \times 2^2 = -788\) dollars.
• Vega: \(-10{,}000 \times 0.1972 \times 1 = -1{,}972\) dollars.
• Theta: \(-10{,}000 \times (-0.0273) \times 1 = 273\) dollars.
• Rho: \(-10{,}000 \times 0.1287 \times 0.10 = -128.70\) dollars.

These sum to a $2,615.70 loss, approximately $2,616. This is a local sensitivity approximation, not a full option revaluation.

• A. Selling shares reinforces the short calls’ negative delta; the stated loss corresponds to that non-neutral position rather than the required hedge.
• B. Buying 5,596 shares neutralizes the short calls’ initial delta; the remaining gamma, vega, theta, and rho effects produce a $2,615.70 loss.
• C. The $1,328 gain reverses the vega contribution: rising implied volatility creates a $1,972 loss for the short call position.
• D. The $1,040 loss reverses the gamma contribution: the short call position incurs a $788 gamma loss, not a gain.

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Question 69

Topic: Quantitative Analysis

A bank applies K-means to counterparty-review records to explore credit-risk patterns. No default or loss labels are used.

Preprocessing:

• Two numerical risk indicators have already been centered and standardized to unit variance. PCA retains only the first component, whose unit loading vector is \( (1/\sqrt{2}, 1/\sqrt{2}) \). This component explains 90% of the numerical variance.
• Review notes are represented by unigram word counts for the vocabulary bank, borrower, and owes, after lowercasing and removing punctuation.
• Each final clustering vector concatenates the retained component score and the three word counts, without additional scaling or weighting.

Records:

Which assessment of the Euclidean distance between these final clustering vectors and the information retained is supported?

• A. The distance is \(2\sqrt{2}\); an orthogonal PCA transformation preserves their numerical separation in the clustering input.
• B. The distance is \(\sqrt{2}\); unigram counts preserve debtor-creditor roles despite identical retained numerical scores.
• C. The distance is 0; retaining 90% of numerical variance establishes equivalent expected credit losses for the two records.
• D. The distance is 0; numerical contrast and debtor-creditor direction are absent from the clustering input.

Best answer: D

Explanation: PCA preserves directions of high input variance, not necessarily information most relevant to credit risk. Each record’s retained score is \(2/\sqrt{2}=\sqrt{2}\). Their original numerical difference lies entirely along the discarded direction \((1/\sqrt{2},-1/\sqrt{2})\).

Both notes contain bank, borrower, and owes once, so each produces the count vector \((1,1,1)\). Unigram counts retain word frequencies but lose word order and grammatical relationships, including who owes whom.

Consequently, both final vectors are \((\sqrt{2},1,1,1)\), and their Euclidean distance is zero. K-means operates on distances in this transformed feature space; it cannot recover distinctions that preprocessing removed. Retaining 90% of numerical variance does not guarantee preservation of risk-relevant information. Because neither PCA nor K-means uses loss labels, their outputs also do not independently establish expected-loss equivalence.

• A. Retaining all principal components preserves Euclidean distance, but retaining only the first discards the direction containing these records’ numerical separation.
• B. Unigram counts do not encode grammatical roles; both notes produce counts of (1, 1, 1), contributing zero distance.
• C. Explained input variance does not establish equivalent expected losses, especially when preprocessing removes economically meaningful distinctions and no loss labels are used.
• D. Both records project to the same component score and have identical unigram counts, eliminating their numerical contrast and the notes’ differing obligation directions.

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Question 70

Topic: Valuation and Risk Models

A bank compares credit capital for a portfolio of term loans and OTC foreign-exchange derivatives. It holds economic capital against unexpected one-year credit losses at 99.9% confidence and covers expected losses separately.

Reference conditions:

• The supplied supervisory rule sets regulatory credit capital at 8% of credit risk-weighted assets, which equal $350 million in both runs.
• Both runs use identical scheduled loan exposures, marginal counterparty default probabilities, recovery rates, and marginal distributions of future positive derivative exposures.
• The independent run simulates derivative exposures independently of counterparty default. The joint run links them: adverse exchange-rate scenarios increase the bank’s positive exposures while weakening the relevant counterparties.

Model estimates (USD millions):

Which conclusion about the capital comparison is supported by these results?

• A. Under the joint run, economic capital is $32 million and regulatory capital is $28 million; independent simulation can understate losses when exposure rises as counterparties deteriorate.
• B. Under the joint run, economic capital is $27 million and regulatory capital is $28 million; unchanged marginal default and exposure distributions preserve the unexpected-loss buffer.
• C. Under the joint run, economic capital is $32 million and regulatory capital is $32 million; the internal loss estimate adjusts the supervisory capital charge.
• D. Under the joint run, economic capital is $38 million and regulatory capital is $28 million; independent simulation can understate losses when exposure rises as counterparties deteriorate.

Best answer: A

Explanation: Economic credit capital covers unexpected loss, measured here as the one-year 99.9% credit-loss quantile less expected loss. The independent run produces $31 million less $4 million, or $27 million. The joint run produces $38 million less $6 million, or $32 million. Regulatory credit capital remains 8% of $350 million, or $28 million. Economic capital therefore moves from below to above regulatory capital without any change in the supervisory calculation.

The joint run incorporates wrong-way risk: market conditions that increase the bank’s positive derivative exposure also weaken its counterparties. Credit losses depend on exposure when default occurs, so matching marginal default and exposure distributions is insufficient to determine the joint loss tail. Derivative exposures are harder to measure than scheduled loan exposures because future values respond to market factors. Modeling exposure-default dependence addresses an independence limitation, but does not eliminate uncertainty in tail-loss estimates.

• A. Linking higher exposures to counterparty deterioration raises economic capital to $32 million, while the unchanged supervisory calculation produces $28 million.
• B. Identical marginal distributions do not fix the loss distribution: exposure-default dependence increases economic capital from $27 million to $32 million.
• C. The supervisory charge remains 8% of $350 million, or $28 million; the higher internal economic-capital estimate does not change that calculation.
• D. The $38 million quantile includes expected loss; subtracting the $6 million mean loss gives a $32 million unexpected-loss buffer.

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Question 71

Topic: Valuation and Risk Models

After a sharp fall in global commodity prices, a commodity-exporting sovereign updates its projections for the same upcoming 12-month period. The government receives export proceeds in USD, its USD debt service is contractually fixed, and it has no currency derivatives. Other fiscal revenue, usable foreign-exchange reserves, and committed financing are unchanged.

LCU denotes local currency units. Bond spreads are measured over maturity-matched US Treasury yields. The same bond issue and CDS contractual terms apply at both observations, and CDS market liquidity is unchanged.

Which interpretation of the change in sovereign credit risk is best supported by the exhibit?

• A. Repayment pressure is broadly unchanged; depreciation preserves local-currency export receipts and therefore maintains the resources available for USD debt service.
• B. Repayment pressure has increased; the bond-CDS widening difference measures the additional sovereign default risk associated with the longer bond maturity.
• C. Repayment pressure has increased; the bond-CDS widening difference is not a clean measure of additional sovereign default risk.
• D. Repayment pressure is broadly unchanged; fixed USD debt service and the unchanged rating provide stronger evidence of stability than the market spreads.

Best answer: C

Explanation: Sovereign foreign-currency repayment capacity depends on foreign-currency resources relative to foreign-currency obligations. USD export receipts fall 20%, while debt service remains $4 billion, reducing export-receipt coverage from 2.5 times to 2.0 times. Depreciation preserves local-currency export receipts at 40 billion LCU, but increases the local-currency cost of debt service from 16 billion to 20 billion LCU, a 25% rise.

The bond spread widens 160 bp, compared with 80 bp for CDS. Both movements are consistent with increased credit concerns, but the difference in widening is not a clean default-risk increment. The instruments reference different maturities and contractual claims, and the wider bond bid-ask spread indicates poorer liquidity. Market spreads also incorporate risk premiums rather than only expected default losses.

Ratings need not adjust simultaneously with market prices. The unchanged BBB rating therefore does not negate the deterioration in repayment resources or establish that spread changes are purely liquidity effects.

• A. Local-currency receipts remain 40 billion LCU, but the local-currency cost of debt service rises from 16 billion to 20 billion LCU.
• B. The 80 bp widening difference mixes different maturity exposures and deteriorating bond liquidity, so it cannot separately measure additional default risk.
• C. USD receipts decline against fixed USD obligations, while unequal maturities and poorer bond liquidity prevent attributing the relative spread widening entirely to default risk.
• D. Fixed payments do not imply stable repayment capacity when USD resources shrink, and an unchanged rating does not negate deteriorating cash flows.

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Question 72

Topic: Quantitative Analysis

A bank flags borrowers for review when their estimated one-year default probability is at least 10%. Its ensemble averages the leaf default probabilities from five classification trees, each fitted to a bootstrap sample of the same training data.

Current fitted ensemble: For a new borrower, the trees produce these probabilities.

Retraining comparison: The bank also compares a single tree with the five-tree ensemble across 100 independently drawn training samples of equal size from the same population. Each fit uses the same held-out validation set, and the new borrower’s features remain fixed.

Which interpretation of the review decision and model evidence is supported?

• A. Flag the borrower; the results support greater prediction stability for this borrower across repeated training samples.
• B. Do not flag the borrower; the results support variance reduction as the dominant source of the AUC improvement.
• C. Do not flag the borrower; the results support bias reduction as the dominant source of the AUC improvement.
• D. Do not flag the borrower; the results support greater prediction stability for this borrower across repeated training samples.

Best answer: D

Explanation: The ensemble’s estimated default probability is \( (2 + 4 + 11 + 13 + 15)/5 = 9\% \), below the 10% review threshold. Counting the three tree probabilities above 10% would substitute majority voting for the specified probability averaging.

The reduction in borrower-level prediction standard deviation from 6.0 to 3.0 percentage points demonstrates lower observed variability across retraining. The corresponding prediction variance is one-quarter as large. Higher mean validation AUC indicates better average ranking performance in the reported experiment.

These observations do not identify the dominant source of the AUC improvement. Systematic prediction bias has not been measured, and AUC is not squared-error loss with an additive bias-variance decomposition. Local prediction variability therefore cannot establish how much of the aggregate performance gain comes from bias or variance reduction.

• A. Three tree probabilities exceed 10%, but the ensemble averages probabilities; its 9% estimate is below the review threshold.
• B. Reduced prediction dispersion for one borrower does not quantify variance’s contribution to an aggregate ranking metric such as AUC.
• C. Neither validation AUC nor retraining standard deviation measures systematic prediction bias, so a dominant bias-reduction contribution is unsupported.
• D. The tree probabilities average 9%, and the lower retraining standard deviation directly supports greater prediction stability for this borrower.

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Question 73

Topic: Foundations of Risk Management

A risk committee reviews the following evidence from two historical cases.

Enron’s Raptor hedging structures:

• Legally separate entities entered contracts intended to offset losses on Enron’s merchant investments.
• Most of the entities’ capacity to meet those contracts rested on Enron shares or rights to acquire Enron shares, rather than independently funded assets.
• Investment values and Enron’s share price fell together, weakening the entities when protection payments were needed.

2016 Bangladesh Bank SWIFT incident:

• Investigators found malware in the bank’s local systems used to prepare and send SWIFT payment instructions.
• Fraudulent instructions passed message-authentication checks and were processed.
• Manipulation of local transaction records and payment confirmations delayed detection.

Which assessment of the principal failure mechanisms is best supported by this evidence?

• A. Enron retained economic exposure through dependent hedges; the SWIFT incident arose from a cyber-control failure at the member bank.
• B. Enron achieved independent risk transfer through separate entities; the SWIFT incident arose from a cyber-control failure at the member bank.
• C. Enron achieved independent risk transfer through separate entities; the SWIFT incident arose from a cyber-control failure in SWIFT’s central messaging network.
• D. Enron retained economic exposure through dependent hedges; the SWIFT incident arose from a cyber-control failure in SWIFT’s central messaging network.

Best answer: A

Explanation: Legal separation and authenticated messaging do not, by themselves, establish effective risk control. Enron’s investment exposure was apparently hedged with separate entities, but their capacity to pay depended largely on Enron shares. A joint decline in investment values and Enron’s share price increased the need for protection while weakening its source. Financial engineering created contractual hedges without reliable economic independence.

In the Bangladesh Bank case, the loss mechanism was fraudulent payment execution. Malware and record manipulation in the member bank’s local environment enabled and concealed the instructions. Successful message authentication did not make the payments legitimate or establish that SWIFT’s central network had been breached. The evidence therefore distinguishes a dependent risk-transfer arrangement from a member-bank cyber-control failure.

• A. Enron-linked backing weakened protection during joint declines, while compromised member-bank systems enabled fraudulent instructions that passed authentication checks.
• B. Legal separation did not establish independent protection because the entities’ payment capacity depended mainly on Enron’s own share value.
• C. Enron-linked backing undermined independent protection, and the cyber evidence identifies compromised local bank systems rather than compromised central messaging infrastructure.
• D. Malware and record manipulation in the bank’s local systems support an endpoint failure, not a compromise of SWIFT’s central messaging network.

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Question 74

Topic: Foundations of Risk Management

An investment committee is reviewing two active managers hired to outperform the same benchmark while controlling tracking error. Both supplement a diversified benchmark portfolio. The committee wants a mandate-consistent historical risk-adjusted ranking to inform its next capital-allocation review.

Valuation review: Manager A’s infrequently traded holdings were sometimes valued using stale prices. The risk team reconstructed A’s returns using timely independent valuations. Manager B’s returns required no revision. Neither manager’s mean return changed.

Reference conditions: All figures are annualized estimates from the same 36-month period. The benchmark’s mean return was 9%, and the risk-free return was 3%.

Which assessment is best supported for this review, rounding ratios to two decimal places?

• A. Rank Manager A higher, with an annualized information ratio of 0.60.
• B. Rank Manager B higher, with an annualized information ratio of 0.22.
• C. Rank Manager A higher, with an annualized information ratio of 1.50.
• D. Rank Manager B higher, with an annualized information ratio of 1.00.

Best answer: D

Explanation: For benchmark-relative active mandates, the information ratio measures active return divided by tracking error. It evaluates the reward earned for deviations from the benchmark, matching the managers’ assigned role.

Using timely valuations, Manager A’s information ratio is \( (12\% - 9\%) / 8\% = 0.375 \), rounded to 0.38. Manager B’s is \( (13\% - 9\%) / 4\% = 1.00 \). Manager B therefore provides stronger historical benchmark-relative risk-adjusted performance and should rank higher as an input to the allocation review.

Stale prices suppress A’s measured active risk and inflate its reported ratio. Using total portfolio volatility instead would measure a different risk exposure. Better historical measurement does not establish future performance: the ranking supports evaluation, but does not guarantee outperformance or independently determine the appropriate capital allocation.

• A. The 0.60 figure is A’s Sharpe ratio, using return above the risk-free rate and total volatility rather than active return and tracking error.
• B. The 0.22 calculation divides B’s 4% active return by 18% total volatility; the information ratio requires tracking error.
• C. The 1.50 calculation uses A’s stale-price tracking error of 2%, rather than the 8% tracking error obtained from timely valuations.
• D. B’s 4% active return divided by 4% timely-price tracking error produces 1.00, exceeding A’s corresponding information ratio of 0.38.

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Question 75

Topic: Valuation and Risk Models

A lender is forecasting realized credit losses on a corporate loan that is performing at time 0 (today). The table shows annualized, continuous-time hazard rates, constant within each one-year interval. Physical estimates are based on historical default data.

• Exposure at default remains $50 million throughout the three-year horizon.
• Recovery is 40% of exposure at default.
• Ignore prepayment and discounting.

Which statement correctly reports the unconditional probability of default during year 3 and the associated expected loss, assessed at time 0? Round percentages and losses in millions to two decimal places.

• A. The unconditional year-3 default probability is 7.95%, and the expected year-3 loss is $2.38 million.
• B. The unconditional year-3 default probability is 6.59%, and the expected year-3 loss is $1.98 million.
• C. The unconditional year-3 default probability is 6.37%, and the expected year-3 loss is $1.91 million.
• D. The unconditional year-3 default probability is 6.76%, and the expected year-3 loss is $2.03 million.

Best answer: C

Explanation: A continuous-time hazard rate is an intensity conditional on survival, not a one-year default probability. Forecasts of realized losses use physical probabilities. With piecewise-constant hazards, survival is the exponential of the negative accumulated hazard.

Here, survival through year 2 is \(S(2)=e^{-(0.02+0.04)}\approx0.9417645\), while survival through year 3 is \(S(3)=e^{-(0.02+0.04+0.07)}\approx0.8780954\). The unconditional probability of default during year 3 is therefore \(S(2)-S(3)\approx0.0636691\), or 6.37%. This incorporates both survival through year 2 and subsequent default during year 3.

Loss given default is 60%, so the expected year-3 loss is \(50\times0.60\times0.0636691\approx1.910073\) million dollars, rounded to $1.91 million. Risk-neutral hazards describe pricing probabilities, not the historical-frequency measure used for this forecast.

• A. These estimates use the risk-neutral hazards, which support valuation rather than the lender’s forecast of realized credit losses.
• B. This treats hazard rates as annual conditional default probabilities, using \(0.98\times0.96\times0.07\), instead of converting continuous-time intensities into survival probabilities.
• C. The physical default mass is \(e^{-0.06}-e^{-0.13}\); multiplying by $50 million and 60% loss given default yields $1.91 million.
• D. The 6.76% probability is conditional on survival through year 2; it omits the probability of surviving the first two years.

Questions 76-100

Question 76

Topic: Valuation and Risk Models

A credit-risk analyst must report one-year expected loss and unexpected loss for a homogeneous loan portfolio. For this report, unexpected loss is defined as the standard deviation of aggregate credit losses, not a capital quantile.

Portfolio assumptions:

• There are 100 loans, each with exposure at default of $1,000,000.
• Each loan has a one-year default probability of 4%.
• Recovery upon default is exactly 60% of exposure; a surviving loan produces no credit loss.
• Defaults are independent in the baseline calculation. Ignore discounting.

The analyst also considers a sensitivity case with positive pairwise default correlations, holding individual default probabilities, exposures, and recovery rates unchanged.

Which report correctly states the baseline measures, rounded to the nearest $0.01 million, and the effect of the sensitivity case?

• A. Expected loss: $1.60 million; unexpected loss: $0.78 million. Positive default correlation increases unexpected loss only.
• B. Expected loss: $4.00 million; unexpected loss: $1.96 million. Positive default correlation increases unexpected loss only.
• C. Expected loss: $1.60 million; unexpected loss: $0.78 million. Positive default correlation increases both expected and unexpected loss.
• D. Expected loss: $1.60 million; unexpected loss: $7.84 million. Positive default correlation increases unexpected loss only.

Best answer: A

Explanation: Expected loss is the mean credit loss; unexpected loss here measures dispersion around that mean. The fixed loss per default is $1,000,000 multiplied by 40%, or $400,000.

With independent defaults, the default count is binomial. For \(n\) loans, default probability \(p\), and fixed loss per default \(d\):

\[ E[L] = npd = 100 \times 0.04 \times 400{,}000 = 1{,}600{,}000. \]\[ \operatorname{SD}(L) = d\sqrt{np(1-p)} = 400{,}000\sqrt{100 \times 0.04 \times 0.96} \approx 783{,}837. \]

Thus expected loss is $1.60 million and unexpected loss is $0.78 million. Independence permits summing individual loss variances. Positive default correlations introduce positive covariance terms into portfolio variance, increasing unexpected loss. Expected loss remains unchanged because expectations add regardless of dependence when marginal default probabilities and losses upon default stay fixed.

• A. A $400,000 loss per default gives a $1.60 million mean and $0.78 million standard deviation; positive default covariance increases only dispersion.
• B. These amounts treat the entire exposure as lost upon default, ignoring the 60% recovery.
• C. With unchanged marginal default probabilities and losses upon default, dependence increases aggregate loss dispersion but does not change expected loss.
• D. Adding individual standard deviations produces $7.84 million; independent losses require adding variances and then taking the square root.

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Question 77

Topic: Valuation and Risk Models

A risk manager is assessing whether additional computing capacity will materially improve a bank’s one-day 99% VaR estimate. The current estimate is $18 million, based on 100,000 independent simulated losses from a normal return model with constant volatility. Parameters estimated from 250 historical observations remain fixed during simulation.

Diagnostic results:

• Across independently seeded runs, the standard deviation of the VaR estimates is $0.40 million.
• Bootstrap resampling and parameter refitting, with simulation noise made negligible, produces a 95% parameter-uncertainty interval for VaR of $14 million to $22 million.
• Historical validation data exhibit volatility clustering and heavier tails than the fitted model.

The proposed upgrade increases each run to 1,600,000 simulated losses, leaving the model, fitted parameters, and estimator unchanged. Assume standard large-sample Monte Carlo error scaling applies.

Which interpretation of the proposed upgrade is most justified?

• A. Sampling standard deviation should be about $0.10 million; parameter uncertainty should also fall by a factor of four, and distributional misspecification remains unresolved.
• B. Sampling standard deviation should be about $0.10 million; parameter uncertainty remains approximately unchanged, and distributional misspecification should diminish as tail coverage improves.
• C. Sampling standard deviation should be about $0.025 million; parameter uncertainty remains approximately unchanged, and distributional misspecification remains unresolved.
• D. Sampling standard deviation should be about $0.10 million; parameter uncertainty remains approximately unchanged, and distributional misspecification remains unresolved.

Best answer: D

Explanation: Monte Carlo sampling error measures numerical uncertainty conditional on the chosen model and fitted parameters. Under standard large-sample conditions, its standard deviation decreases in proportion to the inverse square root of the number of simulations. Here,

\[ 0.40\sqrt{\frac{100{,}000}{1{,}600{,}000}} = 0.10 \]

so the sampling standard deviation falls to $0.10 million.

Parameter uncertainty arises because the inputs were estimated from limited historical data. Increasing simulation size does not expand that dataset, so the $14 million to $22 million uncertainty interval remains approximately unchanged when numerical noise is already negligible.

Model misspecification concerns the assumed mechanism itself. A constant-volatility normal model does not capture the observed volatility clustering and heavier tails. More simulations improve precision within that model, not its correspondence to actual losses. A highly reproducible VaR estimate can therefore still be unreliable.

• A. Additional simulated losses do not add historical information for estimating parameters, so they do not narrow the parameter-uncertainty interval by the same factor.
• B. Denser sampling estimates the assumed normal tail more precisely; it does not introduce the heavier tails or volatility clustering observed in historical data.
• C. The number of simulations increases sixteenfold, so sampling standard deviation falls by a factor of four, not sixteen.
• D. Four times the sampling precision follows from sixteen times as many simulations, while neither historical parameter information nor the assumed return distribution changes.

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Question 78

Topic: Foundations of Risk Management

A risk analyst is evaluating how much of a manager’s portfolio performance is attributable to systematic factor exposures rather than estimated alpha. The analyst fits an OLS Fama-French three-factor regression with an intercept to monthly portfolio excess returns.

Over the same sample, the portfolio’s annualized average total return is 11.0%, and the annualized average risk-free return is 2.0%. All annualized averages equal monthly arithmetic averages multiplied by 12.

Factor conventions:

• Market: market return minus the risk-free return.
• SMB: small-cap stock returns minus large-cap stock returns.
• HML: high book-to-market stock returns minus low book-to-market stock returns.

Which interpretation correctly separates the portfolio’s factor exposures from its estimated annualized alpha?

• A. The portfolio has small-cap and growth tilts, a factor-attributed excess return of 7.6%, and estimated alpha of 1.4%.
• B. The portfolio has small-cap and growth tilts, a factor-attributed excess return of 7.6%, and estimated alpha of 3.4%.
• C. The portfolio has small-cap and growth tilts, a factor-attributed excess return of 6.6%, and estimated alpha of 2.4%.
• D. The portfolio has small-cap and value tilts, a factor-attributed excess return of 9.2%, and estimated alpha of -0.2%.

Best answer: A

Explanation: The Fama-French three-factor model attributes portfolio excess returns to market, size, and value exposures, with the intercept representing estimated alpha. Positive SMB indicates a small-cap tilt. Negative HML indicates a growth tilt because HML is defined as high minus low book-to-market returns.

The annualized factor contribution is:

\[ 1.10(6.0\%) + 0.60(3.0\%) - 0.40(2.0\%) = 7.6\%. \]

Average portfolio excess return is 11.0% minus 2.0%, or 9.0%. Because an OLS regression with an intercept has zero average residual, estimated annualized alpha is 9.0% minus 7.6%, or 1.4%.

Thus, most average excess return is attributed to factor exposures. The positive estimated alpha is performance unexplained by this model, not proof of persistent manager skill; statistical significance and model adequacy would require further assessment.

• A. Positive SMB and negative HML indicate small-cap and growth tilts; subtracting the 7.6% factor contribution from 9.0% excess return gives 1.4% alpha.
• B. Subtracting factor contributions from total return incorrectly includes the 2.0% risk-free return in alpha; the regression uses excess returns.
• C. The 6.6% contribution includes only the market factor; omitting the net positive size and value contributions overstates estimated alpha.
• D. The negative HML loading indicates a growth tilt and contributes -0.8%, rather than the +0.8% used in this calculation.

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Question 79

Topic: Valuation and Risk Models

A risk manager compares two sovereign-restructuring scenarios for a domestic bank. All bank losses are measured in local currency units (LCU).

Sovereign conditions:

• The government issues a floating local currency. Tax receipts, cash and available borrowing cannot cover its next local-currency debt payment.
• The government can legally finance that payment through central-bank money creation, but the cabinet rejects monetary financing because inflation is already high.
• Available dollars cannot cover the next dollar-denominated debt payment, and no additional dollars can be obtained before the payment date.

Bank exposures and stress assumptions:

• The bank holds local-currency sovereign bonds with principal of LCU 60 billion and dollar-denominated sovereign bonds with principal of USD 2 billion, both carried at par.
• The exchange rate is LCU 5 per USD 1.
• Each scenario separately imposes a 20% principal haircut on one debt class. Write-downs reduce bank capital one-for-one. Ignore taxes, hedges, exchange-rate changes and indirect effects.

Which assessment correctly compares the direct bank capital losses and the government’s repayment constraints?

• A. Local restructuring: LCU 12 billion; dollar restructuring: LCU 0.4 billion. Local repayment is constrained by willingness to monetize; dollar repayment is constrained by access to the payment currency.
• B. Local restructuring: LCU 12 billion; dollar restructuring: LCU 0.4 billion. Both repayment difficulties reflect insufficient nominal payment capacity rather than unwillingness to use available monetary powers.
• C. Local restructuring: LCU 12 billion; dollar restructuring: LCU 2 billion. Local repayment is constrained by willingness to monetize; dollar repayment is constrained by access to the payment currency.
• D. Local restructuring: LCU 12 billion; dollar restructuring: LCU 2 billion. Both repayment difficulties reflect insufficient nominal payment capacity rather than unwillingness to use available monetary powers.

Best answer: C

Explanation: Monetary sovereignty provides nominal local-currency payment capacity, but it does not guarantee willingness to use that capacity. Here, avoiding further inflation can motivate a local-currency default despite the government’s ability to create the payment currency. The dollar obligation presents a different problem: the government cannot create dollars and cannot obtain enough before payment is due.

The bank’s direct losses depend on its holdings, not on whether the sovereign issues the denomination currency:

• Local-currency haircut: \(60 \times 0.20 = 12\) billion LCU.
• Dollar haircut: \(2 \times 0.20 = 0.4\) billion USD, equivalent to \(0.4 \times 5 = 2\) billion LCU.

The bank therefore suffers greater direct capital damage from the local-currency restructuring. Monetary sovereignty does not eliminate the sovereign-credit risk embedded in domestic banks’ bond holdings.

• A. The dollar haircut is USD 0.4 billion, which must be converted at LCU 5 per dollar to obtain LCU 2 billion.
• B. The dollar loss is not converted into local currency, and rejection of available monetary financing is incorrectly classified as insufficient nominal payment capacity.
• C. The haircuts cost LCU 12 billion and LCU 2 billion; monetary financing is available but rejected locally, while dollar funding is unavailable.
• D. The losses are correctly converted, but the government retains nominal local-currency payment capacity and chooses not to exercise it.

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Question 80

Topic: Foundations of Risk Management

A GARP member at an asset manager prepares a liquidity-risk report for a prospective client considering an investment in the firm’s fund. The member’s bonus depends partly on securing the investment. The client wants an assessment covering both ordinary conditions and the firm’s approved stress scenario.

Verified findings:

• Liquid assets cover obligations under ordinary conditions.
• Liquid assets do not cover contractual margin calls under the approved stress scenario; the shortfall is material.
• An independent technical review has confirmed the analysis.

The sales director proposes this executive conclusion:

The fund has adequate liquidity to meet its obligations.

The complete stress results would appear in a technical appendix, and the member’s incentive would be disclosed in a conflicts section.

Which assessment of the proposed report is most consistent with the GARP Code of Conduct?

• A. Permissible, because disclosing the compensation arrangement and providing the complete stress results make the report transparent.
• B. Impermissible, because the member must eliminate investment-linked compensation before providing an assessment of the fund’s liquidity.
• C. Impermissible, because disclosing the incentive and stress results does not make a materially misleading executive conclusion acceptable.
• D. Permissible, because an adverse hypothetical stress result does not invalidate a favorable conclusion based on normal operating conditions.

Best answer: C

Explanation: Professional integrity and objectivity require risk communication that accurately conveys material findings despite commercial pressure. The report covers both ordinary conditions and an approved stress scenario. Its executive conclusion therefore must distinguish ordinary-condition adequacy from the verified stress shortfall rather than imply adequacy across the assessment.

Providing accurate calculations in an appendix does not cure a materially misleading summary. Similarly, disclosing investment-linked compensation alerts the client to a conflict but does not authorize inaccurate statements. The member should disclose the incentive and communicate the material stress limitation clearly. A stress scenario is hypothetical, but its findings remain relevant when stressed liquidity is explicitly part of the client’s requested assessment.

• A. Disclosure of the incentive and supporting analysis does not correct an executive conclusion that misrepresents a material finding.
• B. Investment-linked compensation creates a conflict requiring disclosure and objective conduct, not a categorical requirement to eliminate that compensation before performing the assessment.
• C. The unqualified adequacy conclusion contradicts the verified material shortfall within the assessment’s stated scope, regardless of disclosures elsewhere.
• D. The client requested coverage of ordinary and stressed conditions, so ordinary-condition adequacy cannot support an unqualified conclusion about the full assessment.

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Question 81

Topic: Financial Markets and Products

A bank reviews three outstanding bilateral interest rate swaps with the same counterparty. All exhibit amounts are in USD millions. Replacement values are measured from the bank’s perspective before collateral: positive values are owed to the bank, and negative values are owed by the bank.

Legal and collateral terms:

• Close-out netting is legally enforceable within each agreement, but not between agreements X and Y.
• The bank holds $1.2 million of eligible cash collateral restricted to agreement X and fully available against its close-out claim. Agreement Y has no collateral.

Assuming immediate close-out at these values, what is the bank’s aggregate current counterparty credit exposure after netting and collateral, before any default recovery?

• A. $6.3 million
• B. $3.5 million
• C. $0.0 million
• D. $2.3 million

Best answer: D

Explanation: Current counterparty exposure is the positive replacement value remaining after legally enforceable close-out netting and eligible collateral. Each separate netting agreement is evaluated independently, and negative exposure is floored at zero.

In USD millions:

• Agreement X: \(\max(7.5 - 4.0 - 1.2, 0) = 2.3\).
• Agreement Y: \(\max(-6.0, 0) = 0\).

The aggregate current exposure is therefore $2.3 million. A negative value under agreement Y represents an obligation of the bank, not an enforceable offset against its claim under agreement X.

The $450 million aggregate contractual notional determines swap cash flows; it is not current credit exposure. Potential future exposure concerns replacement values at future dates and depends on market movements and collateral arrangements, so it cannot be calculated from this valuation snapshot alone.

• A. Deducting collateral from the $7.5 million positive value ignores the enforceable $4.0 million offset within agreement X.
• B. Netting the swaps in agreement X gives $3.5 million, but the available cash collateral reduces the bank’s remaining exposure by $1.2 million.
• C. Combining all signed values would eliminate exposure, but agreement Y’s negative value cannot offset agreement X’s positive claim across these agreements.
• D. Agreement X leaves $2.3 million after netting and collateral; the separate negative-value agreement Y contributes no current credit exposure.

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Question 82

Topic: Valuation and Risk Models

A risk manager reviews three analyses of a bond and option portfolio:

• Distribution estimate: The modeled joint distribution of market-factor changes produces a one-trading-day 99% VaR of $8 million.
• Local rate bump: An immediate 1-basis-point parallel increase in risk-free yields, holding all other factors fixed, produces a $90,000 loss.
• Disruption scenario: Over ten trading days, risk-free yields rise 150 basis points, credit spreads widen 250 basis points, and implied volatility rises 12 percentage points. Full portfolio repricing at the end of the scenario produces a $30 million loss. No probability is assigned to the scenario.

Which interpretation of the $30 million result is supported?

• A. The $30 million measures loss under a specified ten-trading-day disruption and should be used to assess resilience to joint extreme factor moves.
• B. The $30 million measures marginal interest-rate exposure over ten trading days and should replace the one-basis-point result for routine hedging.
• C. The $30 million provides a lower bound for ten-trading-day 99% VaR and should be used to recalibrate the tail-loss estimate.
• D. The $30 million represents a ten-trading-day loss with an exceedance probability below 1% and should be used to quantify scenario likelihood.

Best answer: A

Explanation: Stress testing evaluates vulnerability to specified severe conditions. Here, full repricing under joint extreme factor movements produces a $30 million loss over ten trading days. That result supports resilience assessment, but it does not establish the scenario’s likelihood or a probabilistic loss quantile.

VaR instead summarizes a modeled loss distribution at a stated confidence level and horizon. The $8 million figure is a one-trading-day 99% loss quantile, not a maximum possible loss. It cannot directly determine the probability of the ten-day stress outcome.

Routine sensitivity analysis isolates exposure to a small change in one factor. The immediate one-basis-point rate bump measures local yield sensitivity while other factors remain fixed. It serves a different purpose from assessing losses under a large, multi-factor disruption.

• A. Joint extreme shocks and full repricing measure vulnerability to the specified disruption over ten trading days, rather than a probability-calibrated loss threshold.
• B. Joint large changes in yields, spreads, and volatility do not isolate marginal interest-rate exposure as the small, one-factor rate bump does.
• C. A specified scenario without probability calibration does not establish a lower bound for a loss quantile, even when its loss exceeds reported VaR.
• D. The one-day VaR cannot establish a ten-day exceedance probability, and the scenario itself supplies no probability information.

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Question 83

Topic: Foundations of Risk Management

A bank reconciles its lending and derivatives exposures daily. Its credit committee will decide tomorrow morning whether to renew a major counterparty’s credit facility.

Current assessment:

• The counterparty’s credit quality has materially deteriorated, and updated analysis estimates twice the stressed loss shown in the previous report.
• Current exposure remains unchanged at 70% of the approved counterparty limit.
• The risk team has validated the underlying data and the updated analysis.

Reporting arrangements: The committee receives monthly risk reports and immediate notifications of exposure-limit breaches. The next monthly report is due in ten days.

Which reporting change would most directly support the committee’s renewal decision?

• A. Send the committee a pre-approval limit-exception report showing breached thresholds, exposure owners, and required remediation actions.
• B. Send the committee a pre-approval risk supplement showing the credit deterioration, stressed-loss implications, and possible exposure-management responses.
• C. Send the committee a monthly stress-risk supplement showing loss drivers, scenario assumptions, and possible exposure-management responses.
• D. Send the committee a daily exposure dashboard showing limit utilization, reconciliation status, and links to transaction records.

Best answer: B

Explanation: Risk reporting should translate reliable risk information into timely oversight and action. Material deterioration can affect a credit decision even when exposure remains within an approved limit; limit compliance does not establish that the risk remains acceptable.

Here, the committee needs the validated change in credit quality and stressed loss before tomorrow’s renewal decision. A targeted supplement should explain the implications and available responses so the committee can assess whether to renew, amend terms, or reduce exposure. Accurate aggregation, reconciliation, and detailed monthly reports remain important, but they do not replace escalation based on materiality and the decision deadline.

• A. A breach-based report would not escalate this deterioration because exposure remains below its limit, despite the material increase in stressed loss.
• B. The supplement brings validated, material risk changes and potential responses to the responsible decision makers before they commit to renewal.
• C. The supplement provides decision-relevant analysis, but its monthly delivery occurs after the committee must decide whether to renew the facility.
• D. The dashboard supplies timely exposure data but does not communicate the material change in credit quality and stressed loss relevant to renewal.

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Question 84

Topic: Valuation and Risk Models

A bank analyzes a corporate bond position after its issuer is downgraded from BBB to BB. The issuer remains current on all payments, and no default has occurred.

Controlled model outputs: The bank changes only the rating input, holding other model inputs fixed.

The model’s default-state recovery value is $4,000,000. Forecast losses are measured from the applicable rating-state position value and exclude the immediate migration revaluation. VaR is the 99th percentile of these losses. The bank funds expected loss separately and defines economic capital as the unexpected-loss buffer above expected loss at that percentile.

Market observations: During the same announcement window, the issuer released better-than-expected earnings. Its stock price rose 6%, its bond price fell 3%, and its five-year CDS spread widened from 140 bp to 180 bp. These observations are separate from the controlled model outputs.

Which conclusion is best supported by the evidence?

• A. The modeled migration loss is $400,000 and economic capital rises by $600,000; the market reactions cannot isolate the downgrade’s effect.
• B. The modeled migration loss is $400,000 and economic capital rises by $470,000; the market reactions cannot isolate the downgrade’s effect.
• C. The modeled migration loss is $6,000,000 and economic capital rises by $470,000; the market reactions cannot isolate the downgrade’s effect.
• D. The modeled migration loss is $400,000 and economic capital rises by $470,000; the CDS widening measures the downgrade’s isolated effect.

Best answer: B

Explanation: Rating migration can generate a mark-to-market loss without an actual default. Holding nonrating inputs fixed, the BBB-to-BB revaluation reduces the position’s value from $10,000,000 to $9,600,000, producing a $400,000 migration loss. Relative to the original valuation, a $6,000,000 loss would instead correspond to the separate default state with $4,000,000 recovery.

Economic capital covers tail losses above separately funded expected loss. Subtracting expected loss from VaR gives $880,000 under BBB and $1,350,000 under BB, an increase of $470,000. This is the bank’s model-based buffer, not an automatic regulatory-capital requirement.

The bond-price decline and CDS widening are consistent with greater market-perceived credit risk. The equity rally can reflect favorable earnings news. Because both announcements occurred together, the observed market movements do not establish the downgrade’s separate causal effect.

• A. The $600,000 increase is the change in VaR alone; subtracting the $130,000 increase in expected loss leaves a $470,000 economic-capital increase.
• B. The rating-state valuation decline is $400,000, the unexpected-loss buffer increases by $470,000, and concurrent earnings news prevents separate causal attribution.
• C. The $6,000,000 loss compares the original value with default recovery, whereas the issuer migrated to BB without defaulting.
• D. CDS spreads also respond to concurrent information, so the widening cannot isolate the rating announcement’s effect when earnings news arrived in the same window.

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Question 85

Topic: Financial Markets and Products

A treasury operations analyst must fund settlement of two US Treasury purchases on the same date. Each security has a face amount of $1,000,000.

Settlement-date terms:

• Treasury bill: 90 actual days remaining to maturity, with an annual bank-discount quotation of 4.80% under the standard US Treasury-bill quotation convention.
• Treasury note: 4.00% annual coupon paid semiannually, with a clean-price quotation of 99-16 (points and thirty-seconds per $100 of face value).
• Note accrued interest: Actual/Actual convention, with 60 actual days accrued in a coupon period containing 182 actual days.

Ignoring transaction costs, how much total cash is required to settle both purchases, rounded to the nearest dollar?

• A. $1,989,667
• B. $1,989,758
• C. $1,989,593
• D. $1,983,000

Best answer: C

Explanation: A Treasury-bill bank-discount quotation applies the discount rate to face value using actual days over a 360-day year. The bill’s price is therefore:

\[ 1{,}000{,}000\left(1-0.048\times\frac{90}{360}\right)=988{,}000. \]

The note’s clean quotation converts to \(99+16/32=99.5\) per $100 of face value, giving a clean price of $995,000. Settlement requires the dirty price, which adds accrued interest to the clean price.

The semiannual coupon is $20,000. Under the stated Actual/Actual convention, accrued interest is \(20{,}000\times60/182\), or $6,593.41. Thus, the note requires $1,001,593.41 at settlement. Adding the bill price gives $1,989,593.41, which rounds to $1,989,593. The bill’s discount basis and the note’s accrued-interest basis must be applied separately.

• A. Using a fixed 180-day denominator for accrued interest overstates the note’s settlement price; the stated Actual/Actual coupon period contains 182 days.
• B. Using 365 rather than 360 days for the bill’s bank-discount quotation overstates its price and the total settlement cash.
• C. The bill costs $988,000, and the note’s dirty price is $1,001,593.41, producing total settlement cash that rounds to $1,989,593.
• D. This total combines the bill price with the note’s clean price, omitting the $6,593.41 of accrued interest payable at settlement.

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Question 86

Topic: Quantitative Analysis

A credit analyst is updating a borrower’s probability of default within the next year. The prior probability is 20%, and all reports concern the same borrower and horizon.

Evidence reviewed:

• A market-data vendor issues a warning.
• An internal dashboard displays that same warning, copied directly from the vendor with no additional analysis.
• A separate liquidity-monitoring service also issues a warning.

Warning likelihoods:

The market and liquidity warnings are conditionally independent both given default and given no default.

After considering all three reports, what posterior probability of default should the analyst use, rounded to one decimal place?

• A. 84.0%
• B. 36.8%
• C. 63.6%
• D. 42.9%

Best answer: C

Explanation: Bayesian updating changes a probability when new evidence has different likelihoods under the competing outcomes. Conditionally independent signals can be incorporated by multiplying their likelihood ratios. A copied report contributes no additional information once its original source is known.

The prior odds of default are \(0.20/0.80 = 0.25\). The market warning has a likelihood ratio of \(0.60/0.20 = 3\), while the liquidity warning has a likelihood ratio of \(0.70/0.30 = 7/3\). Therefore:

\[ \text{Posterior odds} = 0.25 \times 3 \times \frac{7}{3} = 1.75. \]

Converting odds to probability gives \(1.75/(1+1.75) = 63.6\%\). The dashboard republication must not be counted again; doing so would overstate the evidence supporting default.

• A. Treating the dashboard copy as an independent warning applies the market likelihood ratio twice, inflating posterior odds to 5.25 and probability to 84.0%.
• B. Updating the 20% prior using only the liquidity warning gives 36.8%, discarding the informative market warning.
• C. Counting the market warning once and incorporating the conditionally independent liquidity warning yields posterior odds of 1.75, equivalent to a 63.6% probability.
• D. The market warning alone produces 42.9%; retaining that estimate fails to incorporate the additional information supplied by the liquidity warning.

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Question 87

Topic: Valuation and Risk Models

A bank is assessing a corporate borrower’s default risk over the next year during an abrupt credit contraction.

Credit evidence:

• The borrower’s BBB rating is unchanged. The agency uses a through-the-cycle approach, and its most recent detailed published issuer review is nine months old.
• New financial disclosures show negative operating cash flow. A borrowing facility expires in three months, and lenders have withdrawn their renewal offers.
• A historical one-year rating transition matrix, pooled across several credit cycles, reports a 0.30% default frequency for BBB issuers.
• The bank’s point-in-time model, incorporating the new disclosures, estimates a one-year physical default probability of 1.80%.

Which interpretation of the difference between the two default estimates is most appropriate?

• A. The higher estimate reflects a shorter forecast horizon, because through-the-cycle ratings make the historical transition rate a multi-year rather than one-year default probability.
• B. The higher estimate reflects a probability-measure difference, because historical rating transitions provide risk-neutral probabilities while the bank’s model estimates physical probabilities.
• C. The higher estimate can coexist with the unchanged rating, because current-condition default risk can rise before a cycle-smoothed credit assessment changes.
• D. The higher estimate exceeds the appropriate stress benchmark, because a through-the-cycle cohort rate represents default probability conditional on adverse credit conditions.

Best answer: C

Explanation: Through-the-cycle ratings emphasize sustainable creditworthiness and may respond less quickly to cyclical developments than point-in-time default estimates. An unchanged rating therefore does not establish that current one-year default risk is unchanged.

The borrower’s negative operating cash flow and threatened refinancing provide new evidence that can increase near-term default risk. The 0.30% historical BBB default frequency averages outcomes across issuers and credit conditions; it does not fully condition on this borrower’s current circumstances. The 1.80% estimate can consequently be compatible with the unchanged rating, although the difference alone does not validate the model or guarantee a downgrade.

Both estimates concern physical default risk over one year. Neither through-the-cycle ratings nor pooled historical transitions turn that horizon into multiple years or make the historical average a stress-conditioned probability.

• A. Through-the-cycle describes the rating approach, not the transition horizon; both reported default estimates cover one year.
• B. Historical default frequencies estimate physical probabilities, not risk-neutral probabilities inferred from market prices.
• C. Recent cash-flow and refinancing deterioration can raise point-in-time default risk while a through-the-cycle rating remains unchanged.
• D. Pooling observations across credit cycles produces a historical average, not a default probability conditioned specifically on stressed conditions.

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Question 88

Topic: Quantitative Analysis

A risk analyst must choose an estimator of a loan portfolio’s population mean annual net return. The selection criterion is the lowest expected squared estimation error; unbiasedness is not a separate requirement.

Validation uses a fixed population whose mean return is 1.00%. Independent samples are drawn repeatedly with replacement:

• Estimator A averages 100 loans sampled from the full population.
• Estimator B averages 400 loans sampled only from records with no missed payments.

Treat these summaries as exact properties of the estimators’ repeated-sample distributions:

Which recommendation correctly applies the selection criterion? All mean squared errors below are expressed in squared percentage points.

• A. Choose estimator A, with mean squared error of 0.0016.
• B. Choose estimator B, with mean squared error of 0.0400.
• C. Choose estimator B, with mean squared error of 0.1300.
• D. Choose estimator A, with mean squared error of 0.1600.

Best answer: C

Explanation: Mean squared error combines sampling variability and estimator bias:

\[ \operatorname{MSE}(\hat{\mu}) = \operatorname{Var}(\hat{\mu}) + \operatorname{Bias}(\hat{\mu})^2. \]

Estimator A is unbiased because its repeated-sample mean equals the population mean. Its MSE is \(0.40^2 = 0.1600\). Estimator B has bias of \(1.30 - 1.00 = 0.30\) percentage points, giving MSE of \(0.20^2 + 0.30^2 = 0.1300\). Thus B meets the stated criterion despite being biased.

The reported standard deviations already measure uncertainty in the estimators, not dispersion of individual loan returns. Increasing sample size generally reduces sampling variability, but does not remove bias from excluding part of the target population. Similarly, a narrow confidence interval based only on standard error does not account for selection bias or establish accurate inference about the full portfolio.

• A. Dividing 0.40 squared by 100 again understates uncertainty because 0.40 already measures the dispersion of sample-mean estimates.
• B. The value 0.0400 includes only estimator B’s variance and omits its squared bias of 0.0900.
• C. Estimator B’s variance of 0.0400 plus squared bias of 0.0900 gives 0.1300, below estimator A’s 0.1600.
• D. Estimator A’s mean squared error is correctly calculated, but it exceeds estimator B’s 0.1300 despite A being unbiased.

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Question 89

Topic: Foundations of Risk Management

A bank holds a $20 million bond issued by a mortgage lender, purchased at par. It buys cash-settled, full-notional CDS protection on the bond from a financial institution heavily exposed to the same mortgage market. The CDS is bilateral and uncleared.

The bank’s risk report states:

Full-notional CDS protection transfers all credit losses on the bond away from the bank.

Housing-market stress:

• The bond issuer defaults, and the bond recovers 40% of face value.
• The protection seller also becomes insolvent. The contractual CDS payment equals the bond’s credit loss.
• The bank holds $2 million of cash collateral, which it can retain and apply against the CDS claim.
• The bank recovers 25% of the remaining unsecured CDS claim after applying collateral.

Ignore premiums, interest, discounting, and other cash flows. What net credit loss should the bank report for the bond and CDS together?

• A. A net credit loss of $7.5 million.
• B. A net credit loss of $7 million.
• C. A net credit loss of $9 million.
• D. A net credit loss of $0.

Best answer: A

Explanation: A CDS transfers credit risk only to the extent that its payment obligations are fulfilled. Here, the protection seller’s mortgage exposure makes its failure coincide with the reference issuer’s distress, illustrating wrong-way counterparty risk.

The bond loses 60% of its $20 million face value, or $12 million. Applying $2 million of collateral leaves a $10 million unsecured CDS claim. Recovery on that claim is $2.5 million, so total protection proceeds are $4.5 million and the bank retains a $7.5 million loss.

During the 2007–2009 crisis, correlated exposures and dependence on protection sellers undermined claims of complete risk transfer. Post-crisis collateral requirements reduced unsecured exposures, central clearing introduced multilateral default-management arrangements, and trade reporting improved transparency. These reforms address different weaknesses rather than guarantee payment: collateral can leave exposure gaps, clearing creates dependence on a central counterparty, and reporting reveals risk without absorbing losses.

• A. The $12 million bond loss is offset by $2 million of collateral and $2.5 million recovered on the remaining unsecured CDS claim.
• B. This applies the 25% recovery to the entire $12 million CDS claim rather than the $10 million unsecured balance after collateral.
• C. This applies a 75% loss rate to the full CDS claim, overlooking the cash collateral retained before unsecured recovery.
• D. Full-notional protection specifies the contractual payment, but the insolvent protection seller cannot satisfy that payment in full.

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Question 90

Topic: Valuation and Risk Models

A risk analyst is estimating the annual return volatility of a proposed long-short portfolio for risk budgeting. Position weights are measured relative to net asset value, and a negative weight denotes a short position. All volatilities and correlations describe the same annual return horizon.

Return correlations:

• Equity fund and bond fund: 0.40
• Equity fund and commodity fund: 0.60
• Bond fund and commodity fund: 0.20

What annual portfolio volatility should the analyst use, rounded to two decimal places?

• A. 14.32%
• B. 18.75%
• C. 12.43%
• D. 13.41%

Best answer: C

Explanation: Portfolio variance includes individual position variance contributions and pairwise covariance contributions:

\[ \sigma_p^2 = \sum_i w_i^2\sigma_i^2 + 2\sum_{i\lt j}w_iw_j\sigma_i\sigma_j\rho_{ij}. \]

Using decimal returns, the signed volatility exposures, \(w_i\sigma_i\), are 0.12, 0.05, and -0.06. Their squared values sum to 0.02050. The three pairwise covariance contributions, including the factor of two, are 0.00480, -0.00864, and -0.00120.

Therefore, portfolio variance is \(0.02050-0.00504=0.01546\), and portfolio volatility is \(\sqrt{0.01546}=0.12434\), or approximately 12.43%.

The short position’s own variance contribution remains positive because its weight is squared. Its covariance contributions with the long positions are negative because the weight products are negative and the return correlations are positive. Calculating consistently in decimal-return units prevents mixing percentage-squared and decimal-squared quantities.

• A. This estimate uses only the squared volatility exposures, producing variance of 0.02050, and omits the supplied covariance contributions.
• B. This estimate treats the commodity short weight as positive, reversing its covariance contributions and producing variance of 0.03514.
• C. Using signed weights and twice each pairwise covariance contribution gives portfolio variance of 0.01546, whose square root is approximately 12.43%.
• D. This estimate counts each pairwise covariance contribution once rather than twice, producing variance of 0.01798.

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Question 91

Topic: Quantitative Analysis

A risk analyst compares nested OLS regressions of portfolio returns using 100 independent observations. Returns are measured in percentage points. Both models contain an intercept, and the unrestricted model adds two factors with coefficients \(\beta_3\) and \(\beta_4\). Assume classical regression conditions with normally distributed, homoskedastic errors.

Regression output: Parameter counts include the intercept. Sums of squares are measured in squared percentage points.

• Corrected total sum of squares: 200.
• Ordinary 95% marginal confidence interval for \(\beta_3\): \([-0.10, 0.30]\).
• Ordinary 95% marginal confidence interval for \(\beta_4\): \([-0.30, 0.10]\).
• Applicable 5% critical value for the joint F-test: 3.09.

A reviewer writes:

The two reported marginal intervals form a region with guaranteed 95% simultaneous coverage for the two added coefficients.

Which assessment of the regression comparison and the reviewer’s statement is supported at the 5% significance level?

• A. The joint zero restriction is not rejected and adjusted R-squared rises; the marginal intervals do not guarantee 95% simultaneous coverage.
• B. The joint zero restriction is rejected and adjusted R-squared rises; the marginal intervals do not guarantee 95% simultaneous coverage.
• C. The joint zero restriction is not rejected and adjusted R-squared falls; the marginal intervals do not guarantee 95% simultaneous coverage.
• D. The joint zero restriction is not rejected and adjusted R-squared rises; the marginal intervals guarantee 95% simultaneous coverage.

Best answer: A

Explanation: The joint F-test compares the reduction in residual sum of squares with the unrestricted residual variance:

\[ F = \frac{(104-100)/(5-3)}{100/(100-5)} = 1.90. \]

Because 1.90 is below 3.09, the joint restriction \(\beta_3=\beta_4=0\) is not rejected at 5%. This does not establish that both coefficients are zero.

With \(k\) estimated parameters including the intercept, adjusted R-squared is \(\bar{R}^2 = 1-[RSS/(n-k)]/[TSS/(n-1)]\). Ordinary R-squared rises from 0.4800 to 0.5000, while adjusted R-squared rises from 0.4693 to 0.4789. This improvement is compatible with non-rejection: adjusted R-squared rises whenever the nested-model F-statistic exceeds 1, whereas the significance test uses the higher critical value.

Ordinary 95% marginal intervals do not automatically provide 95% simultaneous coverage. Joint inference accounts for covariance between coefficient estimates, so separate intervals or individual t-tests cannot replace the joint test. Neither improved fit measure establishes superior out-of-sample prediction.

• A. The F-statistic is below its critical value, adjusted R-squared increases, and ordinary marginal intervals do not establish the claimed simultaneous coverage.
• B. The joint F-statistic is 1.90, below 3.09, so the RSS reduction does not justify rejecting the joint zero restriction.
• C. After accounting for the additional parameters, adjusted R-squared rises from 0.4693 to 0.4789 rather than falling.
• D. Each interval has 95% marginal coverage, but their combined coverage for both true coefficients is not guaranteed to be 95%.

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Question 92

Topic: Financial Markets and Products

A risk analyst is evaluating a duration-only loss estimate for an instantaneous 200-basis-point increase in the yield of an option-free, fixed-rate bond position.

Position data:

• Current dirty market value: $100 million.
• Modified duration: 6.0 years.
• Convexity: 60 years squared, defined as \(C = \frac{1}{P}\frac{d^2P}{dy^2}\), where \(y\) is the annually compounded yield expressed as a decimal.
• Accrued interest and contractual cash flows remain unchanged during the instantaneous shock.

The desk treats an omitted second-order price adjustment greater than $0.5 million as material. Which estimated loss and assessment are supported by a duration-convexity approximation?

• A. A $10.8 million loss; the duration-only estimate materially overstates the loss.
• B. A $13.2 million loss; the duration-only estimate materially understates the loss.
• C. A $12.0 million loss; the duration-only estimate does not materially misstate the loss.
• D. A $9.6 million loss; the duration-only estimate materially overstates the loss.

Best answer: A

Explanation: Duration measures first-order price sensitivity, while convexity captures curvature in the price-yield relationship. Under the stated convexity convention,

\[ \frac{\Delta P}{P} \approx -D_{\text{mod}}\Delta y + \frac{1}{2}C(\Delta y)^2. \]

The yield increase is \(\Delta y = 0.02\). The duration-only estimate is \(-6(0.02) = -0.12\), implying a $12 million loss. The convexity correction is \(\frac{1}{2}(60)(0.02)^2 = 0.012\), adding $1.2 million to the estimated price change. The second-order estimate is therefore a 10.8% decline, or a $10.8 million loss.

The omitted correction exceeds the $0.5 million threshold. Its magnitude grows with the square of the yield move, making duration alone less reliable for larger shocks. This difference reflects price-yield curvature, not changed cash flows. The second-order estimate remains an approximation rather than an exact repricing.

• A. Duration implies a $12 million loss, while positive convexity offsets $1.2 million, exceeding the desk’s $0.5 million materiality threshold.
• B. Positive convexity reduces the estimated loss from a yield increase; subtracting its contribution incorrectly increases the loss to $13.2 million.
• C. The $12 million estimate ignores the convexity adjustment, which is $1.2 million and therefore material under the desk’s criterion.
• D. The $9.6 million estimate omits the one-half coefficient, doubling the convexity adjustment from $1.2 million to $2.4 million.

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Question 93

Topic: Financial Markets and Products

In September, a wheat merchant holds 50,000 bushels for sale in a local cash market in October. The proposed hedge uses ten December wheat futures contracts, each covering 5,000 bushels. The local market is not a futures delivery location.

The trader must establish the full short hedge by 09:02, but may activate it only when futures first trade at or below $6.20 per bushel.

Order-execution exhibit: An order is entered immediately before 09:00. Each displayed price is both a trade price and an executable bid with sufficient volume for the full hedge. These are the only prices during the order window. Marketable sell orders fill at the displayed bid; unfilled orders are canceled after 09:02.

Cash sale and hedge closure:

• Local cash price at 09:01: $6.16 per bushel.
• October cash sale price: $5.75 per bushel.
• December futures price when the hedge is closed at the cash sale: $5.86 per bushel.
• Ignore transaction costs and margin financing costs.

Which instruction meets the trading requirements and correctly assesses the eventual effective sale price and basis risk?

• A. Use a sell stop at $6.20; effective proceeds are $6.16 per bushel, with cash-futures basis risk eliminated upon execution.
• B. Use a sell limit at $6.18; effective proceeds are $6.13 per bushel, with cash-futures basis risk remaining.
• C. Use a sell stop-limit with a $6.20 stop and $6.18 limit; effective proceeds are $6.07 per bushel, with cash-futures basis risk remaining.
• D. Use a sell stop at $6.20; effective proceeds are $6.05 per bushel, with cash-futures basis risk remaining.

Best answer: D

Explanation: A sell stop becomes a market order when triggered. The first qualifying trade is $6.16, so the order fills there rather than at the $6.20 stop price. This four-cent gap is realized execution slippage, not the basis risk remaining after execution. A stop-limit order instead restricts the acceptable execution price and does not guarantee a fill.

The ten contracts match the cash quantity, but the local October sale and December futures are different exposures. Define basis as \(b = S - F\), where \(S\) is the local cash price and \(F\) is the futures price. Basis is zero at hedge entry but becomes \(5.75 - 5.86 = -0.11\) dollars per bushel at closure.

The short futures gain is \(6.16 - 5.86 = 0.30\) dollars per bushel. Effective proceeds are therefore \(5.75 + 0.30 = 6.05\) dollars per bushel. The eleven-cent basis decline explains why these proceeds fall below the futures entry price despite the quantity-matched hedge.

• A. Equal cash and futures prices at entry mean zero initial basis, not a guaranteed effective sale price; the basis subsequently falls to -$0.11 per bushel.
• B. A sell limit at $6.18 fills at the initial $6.24 bid, establishing the hedge before the $6.20 activation threshold is breached.
• C. The price gaps below $6.18 and never returns to that level during the order window, so the stop-limit activates but cannot fill.
• D. The stop order fills at $6.16; the $0.30 futures gain raises effective proceeds to $6.05 per bushel, leaving the changed basis unhedged.

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Question 94

Topic: Financial Markets and Products

A bank’s investment-banking division signs a binding agreement to purchase a manufacturer’s entire $100 million bond issue for $97 million and resell it to investors. All closing conditions have been satisfied. Investors have committed to purchase $80 million face value of the bonds.

The bank’s commercial-lending division expects its outstanding loan to the manufacturer to be repaid from the offering proceeds. Its head sends the securities research division this request:

Delay the scheduled report lowering the manufacturer’s earnings forecasts until the remaining bonds have been sold.

There is no factual dispute about the report or applicable publication restriction. Compliance will oversee the handling of the conflict.

Which interpretation of the financing obligation and allocation of report-timing authority is most appropriate?

• A. Treat the arrangement as firm commitment: purchase the full issue and place the report-timing decision with independent research management.
• B. Treat the arrangement as best efforts: purchase no unsold bonds and place the report-timing decision with independent research management.
• C. Treat the arrangement as firm commitment: purchase only the unsold bonds and place the report-timing decision with independent research management.
• D. Treat the arrangement as firm commitment: purchase the full issue and place the report-timing decision with the commercial-lending credit committee.

Best answer: A

Explanation: In firm-commitment underwriting, the bank purchases the securities from the issuer and bears the risk of distributing them. Here, its contractual payment is $97 million for the entire $100 million face-value issue. Investor commitments reduce the remaining distribution exposure to $20 million face value; they do not reduce the bank’s purchase obligation. Under best efforts, the intermediary instead undertakes to market the securities without committing to purchase the unsold balance.

The lending division’s expected repayment creates an incentive to suppress unfavorable research until distribution is complete. Underwriting also has a distribution interest. Research publication should therefore remain under independent research management, with compliance oversight and barriers against lending or underwriting influence. Necessary coordination among banking divisions does not justify allowing their financial interests to determine research conclusions or timing.

• A. The binding purchase agreement covers the entire issue, while independent research management avoids giving the financially interested lending division control over publication.
• B. Independent research control is appropriate, but the agreement to purchase the entire issue creates a firm-commitment obligation rather than a best-efforts mandate.
• C. The unsold bonds represent residual distribution exposure, but the bank’s contractual purchase obligation covers the full issue, not merely that residual.
• D. The purchase obligation is correctly identified, but commercial lending benefits from repayment and should not control research publication to protect that interest.

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Question 95

Topic: Valuation and Risk Models

A bank’s wire-transfer log identifies a treasury employee as the initiator of a fraudulent payment. A risk analyst must assign the resulting loss a primary operational-event classification.

Investigation findings:

• An outside attacker obtained the employee’s credentials through deception and used them to transfer funds to an attacker-controlled account.
• The employee did not intend the transfer or knowingly assist the theft.
• The payment platform validated the presented credentials and processed the instructions as designed, but existing controls did not detect the impersonation.

Which primary operational-event classification is supported by these findings?

• A. An external fraud event.
• B. A process failure event.
• C. An internal fraud event.
• D. A system failure event.

Best answer: A

Explanation: Operational-event classification follows the underlying loss mechanism, not merely the account name recorded in a transaction log. Here, an outside attacker deliberately obtained and misused credentials to steal funds. Because the employee was deceived rather than knowingly involved, the event is external fraud.

A control weakness can enable fraud without becoming the primary event classification. Likewise, a system can operate as designed while its authentication controls remain vulnerable to impersonation. Loss records should distinguish the fraudulent event from contributing control weaknesses. Classification also does not determine monetary severity: internal and external fraud can each produce small or substantial losses.

• A. An outside party deliberately stole funds through deception, with no knowing employee participation, supporting classification as external fraud.
• B. The controls failed to detect impersonation, but the loss-generating event was deliberate theft rather than an unintentional processing or execution error.
• C. Using an employee’s credentials does not establish internal fraud; the employee was deceived and did not knowingly participate in the theft.
• D. The platform processed instructions as designed; acceptance of stolen credentials indicates a control vulnerability rather than a system malfunction.

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Question 96

Topic: Foundations of Risk Management

A bank compares two loan portfolios with the following model-implied one-year aggregate loss distributions. Borrower defaults are independent in one portfolio and perfectly dependent through a common shock in the other.

Risk report: The 95% VaR is $5 million for the independent portfolio and $0 for the common-shock portfolio. An analyst interprets this as evidence that the common-shock portfolio needs less capital.

Committee objective: Expected loss is funded separately. The committee defines its additional buffer as 95% expected shortfall minus expected loss. For expected shortfall, include exactly the worst 5% probability mass, using part of the boundary probability if necessary.

Which pair of additional buffers is consistent with the committee’s objective?

• A. Independent portfolio: $9.60 million; common-shock portfolio: $9.60 million.
• B. Independent portfolio: $5.16 million; common-shock portfolio: $8.00 million.
• C. Independent portfolio: $5.00 million; common-shock portfolio: $0.00 million.
• D. Independent portfolio: $4.76 million; common-shock portfolio: $7.60 million.

Best answer: D

Explanation: Expected loss measures the average outcome, not the concentration of severe losses. Both portfolios have expected loss of $0.40 million, but dependence makes simultaneous defaults much more likely in the common-shock portfolio.

For the independent portfolio, the worst 5% consists of the entire 0.16% probability of a $10 million loss plus 4.84% probability of a $5 million loss. In millions of dollars:

\[ ES_{95\%} = \frac{0.0016(10) + 0.0484(5)}{0.05} = 5.16. \]

For the common-shock portfolio, the worst 5% contains the 4% probability of a $10 million loss plus 1% probability of zero loss, giving expected shortfall of $8.00 million. Deducting expected loss produces buffers of $4.76 million and $7.60 million, respectively.

The common-shock portfolio’s zero VaR reflects its 96% probability of no loss. It does not capture the severity beyond that quantile or guarantee protection against a large loss.

• A. These amounts subtract expected loss from the maximum modeled loss, rather than averaging losses over the worst 5% probability mass.
• B. These amounts are the portfolios’ expected shortfalls; they do not deduct the $0.40 million expected loss already funded for each portfolio.
• C. These are the reported VaR figures, which identify loss quantiles rather than average tail losses net of expected loss.
• D. Subtracting each portfolio’s $0.40 million expected loss from expected shortfalls of $5.16 million and $8.00 million gives the required buffers.

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Question 97

Topic: Financial Markets and Products

An analyst compares three fund transactions and their effects on existing investors who do not trade.

Reference conditions:

• Each fund initially has 1,000,000 shares outstanding.
• Public news at 15:30 raises the current fair value of each fund’s existing investments to $52,000,000.
• All instructions arrive at 15:50. The open-end fund’s order cutoff is 16:00, but its closing dealing NAV remains $50 because it uses earlier foreign-market closing prices.
• Ignore fees, taxes, transaction costs and subsequent changes in investment values.

Which conclusion about dilution of nontrading shareholders is supported by these records? Round the value transfer to the nearest $1,000.

• A. The open-end subscription transfers approximately $182,000 from nontrading shareholders to the subscriber.
• B. The closed-end trade transfers approximately $200,000 from nontrading shareholders to the secondary-market buyer.
• C. The ETF creation transfers approximately $100,000 from nontrading shareholders to the authorized participant.
• D. The open-end subscription transfers approximately $200,000 from nontrading shareholders to the subscriber.

Best answer: A

Explanation: Issuing open-end fund shares at a stale NAV can transfer underlying asset value from incumbent investors to new subscribers. Here, the subscription adds $5,000,000 cash and 100,000 shares. The resulting fair-value NAV, in dollars per share, is:

\[ \frac{52,000,000 + 5,000,000}{1,000,000 + 100,000} \approx 51.818182 \]

The subscriber’s stake is therefore worth approximately $5,181,818, exceeding its payment by $181,818, or $182,000 rounded. That gain equals the reduction in incumbent shareholders’ aggregate fair-value claim.

Exploiting predictable stale prices is market timing. It is distinct from late trading: this subscription arrived before the stated cutoff.

A closed-end secondary-market trade changes ownership without changing fund assets or share count. The ETF creation adds assets and shares at matching fair value, preserving its $52 fair-value NAV. Its exchange premium benefits the authorized participant at buyers’ expense, not through dilution of incumbent shareholders.

• A. The subscriber receives 100,000 of 1,100,000 shares backed by $57,000,000, making its stake worth $181,818 more than its payment.
• B. The discounted trade changes ownership of existing shares, not fund assets or shares outstanding, so nontrading shareholders are not diluted.
• C. The creation basket supplies full fair value for the new shares; the authorized participant’s $100,000 resale profit comes from exchange buyers.
• D. Multiplying 100,000 new shares by the initial $2 valuation gap ignores the change in fair-value NAV when subscription cash and new shares are added.

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Question 98

Topic: Financial Markets and Products

A risk analyst compares two hypothetical option-free bonds, each with $100 face value:

• Bond A: Six years to maturity, with no coupons.
• Bond B: Eight years to maturity, with a 10% annual coupon.

All remaining payments occur at integer numbers of years from valuation. The analyst values both bonds under two separate flat spot curves with annual compounding. Credit and liquidity effects are unchanged.

Valuation output: DV01 is the first-order dollar price decrease for a 1 basis point upward parallel shift, per $100 face value.

Using first-order sensitivity, which bond has the greater percentage price decrease at each reference yield?

• A. Bond B at 2% and Bond A at 10%.
• B. Bond A at 2% and Bond A at 10%.
• C. Bond B at 2% and Bond B at 10%.
• D. Bond A at 2% and Bond B at 10%.

Best answer: A

Explanation: DV01 measures a dollar change; percentage sensitivity requires dividing by price. For a 1 basis point increase, the percentage price decrease is approximately \(100 \times \text{DV01}/P\), where \(P\) is the bond price.

• At 2%: Bond A declines 0.0588%; Bond B declines 0.0621%.
• At 10%: Bond A declines 0.0545%; Bond B declines 0.0534%.

Within each flat spot curve, both bonds’ YTMs equal the reference yield. Bond B’s 10% coupon produces a premium price at 2% and a par price at 10%; coupon rate is not generally YTM.

Bond A’s Macaulay duration equals its six-year maturity. Bond B’s coupons shorten its present-value-weighted payment time relative to its eight-year maturity. At 2%, that time still exceeds six years. At 10%, heavier discounting shifts relative weight toward earlier coupons, bringing it below six years. Since modified duration divides Macaulay duration by the same annual yield adjustment within each scenario, the ranking reverses. Maturity alone cannot order sensitivity when coupons differ.

• A. Dividing DV01 by price gives greater proportional sensitivity for Bond B at 2% and Bond A at 10%.
• B. At the 2% reference yield, Bond A loses approximately 0.0588% per basis point, less than Bond B’s 0.0621%.
• C. At the 10% reference yield, Bond B loses approximately 0.0534% per basis point, less than Bond A’s 0.0545%, despite its larger dollar DV01.
• D. Both comparisons reverse the price-normalized sensitivities: the longer coupon-paying bond is more sensitive at 2%, while the shorter zero-coupon bond is more sensitive at 10%.

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Question 99

Topic: Financial Markets and Products

A risk analyst is reconciling a trading desk’s option portfolio. Both positions are cash-settled European options on the same stock and expire in six months. Each contract represents 100 shares.

The premiums were paid or received at inception. At expiration, the stock price is $57 per share. Assume zero interest rates and no transaction costs. Positive amounts represent receipts or gains to the desk; negative amounts represent payments or losses.

Which pair correctly reports the portfolio’s aggregate expiration payoff, excluding premiums, and total profit, including premiums?

• A. Expiration payoff: -$500; total profit: $480.
• B. Expiration payoff: $1,300; total profit: $2,280.
• C. Expiration payoff: -$100; total profit: $880.
• D. Expiration payoff: -$500; total profit: -$1,480.

Best answer: A

Explanation: Expiration payoff is intrinsic value with a sign determined by whether the position is long or short. European options can be exercised only at expiration.

• The long calls generate \(2 \times 100 \times \max(57 - 55, 0) = 400\) dollars.
• The short puts generate \(-3 \times 100 \times \max(60 - 57, 0) = -900\) dollars.

The aggregate expiration payoff is therefore -$500. At inception, the desk receives $1,620 for the puts and pays $640 for the calls, a net receipt of $980. With zero interest rates, total profit equals expiration payoff plus the net initial premium: \(-500 + 980 = 480\) dollars. A negative expiration payoff can therefore accompany a positive total profit.

• A. The calls pay $400 and the short puts cost $900; adding the $980 net premium receipt yields a $480 profit.
• B. The short puts require a $900 payment, not a receipt, so adding their intrinsic value to the long calls overstates payoff.
• C. The -$100 payoff counts only one contract per position; the portfolio holds two call contracts and three put contracts.
• D. The desk receives a net premium of $980; subtracting that receipt from the -$500 expiration payoff reverses its cash-flow sign.

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Question 100

Topic: Quantitative Analysis

A risk committee compares candidate hedges with an incumbent using daily losses measured in thousands of US dollars.

Study design and testing policy:

• Every hedge is evaluated on the same 36 trading days. Losses within each day are correlated, and marginal variances may differ. For each comparison, daily loss differences are independent across days and approximately normal.
• Before observing the data, the committee selected one candidate and specified the one-sided alternative of lower population mean loss. This single confirmatory comparison is assessed at 5% significance.
• An analyst also screened 12 additional hedges for lower mean loss and highlighted the smallest one-sided paired-test p-value. The committee requires Bonferroni control at a 5% familywise significance level for this separate exploratory family.

Reported results: All p-values test the alternative that the candidate has lower population mean loss than the incumbent.

Which conclusion is best supported under the committee’s testing policy?

• A. The evidence supports lower population mean loss for neither the prespecified hedge nor the selected exploratory hedge.
• B. The evidence supports lower population mean loss for the prespecified hedge, but not for the selected exploratory hedge.
• C. The evidence supports lower population mean loss for the selected exploratory hedge, but not for the prespecified hedge.
• D. The evidence supports lower population mean loss for both the prespecified hedge and the selected exploratory hedge.

Best answer: B

Explanation: A paired t-test evaluates the mean within-day loss difference, accounting for dependence between losses observed on the same day. Unequal marginal variances do not justify treating these matched observations as independent. The prespecified comparison therefore uses the paired-test p-value of 0.027 and rejects the null of no reduction at 5%.

Selecting the smallest p-value after screening multiple hedges creates additional opportunities for a false positive. Under the committee’s policy, the exploratory threshold is \(0.05/12 \approx 0.00417\). Equivalently, the selected hedge’s Bonferroni-adjusted p-value is \(12 \times 0.006 = 0.072\). Its result does not support a familywise-controlled claim of lower mean loss. Bonferroni control remains valid despite dependence among the exploratory comparisons. Failure to reject does not establish that the selected hedge has no benefit.

• A. The prespecified comparison meets its 5% threshold using the paired test; the separate exploratory family does not require adjusting this confirmatory comparison.
• B. The appropriate paired-test p-value of 0.027 meets the confirmatory threshold, while 0.006 exceeds the exploratory Bonferroni threshold of approximately 0.00417.
• C. This reverses the supported conclusions by using an independent-samples test for matched observations and overlooking the exploratory family’s multiplicity adjustment.
• D. The exploratory p-value must account for screening 12 hedges; its Bonferroni-adjusted value is 0.072, exceeding the required 5% level.

Review the reasoning

For each missed or guessed answer, record the topic, the decisive assumption and the step you need to revisit. Distinguish an arithmetic slip from choosing the wrong cash-flow date, distribution, valuation relationship or risk measure. Recalculate before reading the worked reasoning a second time.

Repeating this fixed set can reward answer recognition. A score is a description of this attempt, not a validated probability of passing. Use fresh app questions after reviewing weak areas; see practice-score guidance .

Report unclear wording or read practice updates .