FRM Part 1 Sample Questions with Detailed Solutions

Correct answer: D. USD 285,000, and the combination is called wrong way risk

Expected loss changes when either probability of default or loss given default changes, so both values must be included in each calculation.

At origination:

EL = 10,000,000 × 0.30 × 0.015 = USD 45,000

In the downturn:

EL = 10,000,000 × 0.55 × 0.06 = USD 330,000

The rise is:

330,000 − 45,000 = USD 285,000

Expected loss increases by more than seven times even though neither input increases by that amount on its own. The reason is that probability of default and loss given default enter the calculation multiplicatively.

Probability of default increases fourfold, while loss given default rises from 30% to 55%. The product reflects both changes.

Wrong way risk occurs when probability of default rises while collateral values fall, so default becomes more likely at the same time that recovery becomes lower. Commercial real estate can produce this relationship because the same downturn can weaken borrowers and reduce the value of property securing their loans.

A similar effect can arise in derivatives when a contract gains value while the counterparty owing that value becomes more likely to default.

Why the other options are wrong

• A. USD 135,000, concentration risk incorrectly updates probability of default while leaving loss given default at its original 30%, giving 180,000 and a rise of 135,000. It also incorrectly identifies the effect as concentration risk.
• B. USD 135,000, wrong way risk identifies the interaction correctly but incorrectly leaves loss given default at its original level, even though deterioration in recovery is part of the interaction.
• C. USD 285,000, concentration risk gives the correct increase but incorrectly calls the interaction concentration risk. Concentration concerns exposure to a single counterparty, sector or country rather than probability of default and loss given default increasing together.

Module

Foundations of Risk Management

Chapter

FRM 1: The Building Blocks of Risk Management

Learning objective

Explain how risk factors can interact with each other and describe challenges in aggregating risk exposures

Correct answer: B. Mitigate, and the bank still carries the credit risk the collateral does not cover.

Requesting additional collateral is risk mitigation. The bank retains the exposure, but collateral reduces the loss that would occur if the borrower defaults.

Transfer is different because it moves part of the risk to an outside party at a financial cost through insurance, derivatives or securitization.

Collateral does not introduce a third party. The borrower pledges additional assets against the same exposure.

Residual risk remains because collateral may decline in value or become difficult to sell when the borrower defaults.

The risk is consequently reduced rather than eliminated. Avoidance would require the bank not to make the loan.

Residual risk is the exposure that remains after the risk management strategy has been applied. It needs to be included when the costs and benefits of a strategy are assessed.

Why the other options are wrong

• A. Mitigate, no residual correctly classifies collateral as mitigation but incorrectly states that no exposure remains. Collateral can decline in value and can be difficult to sell at the moment the borrower fails.
• C. Transfer, no residual incorrectly classifies the action as transfer and incorrectly states that the risk has been eliminated.
• D. Transfer, with residual correctly recognizes residual risk but incorrectly classifies collateral as transfer. No third party assumes the exposure.

Module

Foundations of Risk Management

Chapter

FRM 2: How Do Firms Manage Financial Risk?

Learning objective

Compare different strategies that a firm can use to manage its risk exposures and explain situations in which a firm would want to use each strategy

Correct answer: D. Risk tolerance is the tactical measure, describing the range of acceptable outcomes around a business objective, and risk appetite sits below risk capacity so that actual risk stays well below capacity.

Risk tolerance is the tactical measure.

It describes the acceptable range of outcomes associated with a particular business objective.

A tolerance is therefore linked to a specific objective and commonly expressed as a range around a target.

Risk appetite is broader.

It describes the aggregate amount and types of risk the firm is willing to accept.

Risk capacity is the maximum amount of risk the firm can bear.

Capacity depends on the firm's resources and regulatory constraints rather than on a discretionary board decision.

Risk appetite should be set below risk capacity.

The firm's actual risk profile should then remain within the relevant tolerance ranges.

The ordering runs from actual risk profile, through tolerance, to appetite, with capacity as the outer ceiling. Each gap in that ordering is a margin, and a firm that closes one is relying on nothing going wrong at the level above it.

A risk capacity below risk appetite would mean the firm intended to accept more risk than it could absorb.

Why the other options are wrong

• A. Appetite tactical, capacity below appetite incorrectly identifies appetite as tactical and reverses the capacity-appetite relationship.
• B. Appetite tactical, appetite below capacity correctly orders appetite below capacity but incorrectly identifies appetite as the tactical measure.
• C. Tolerance tactical, capacity below appetite correctly identifies tolerance as tactical but incorrectly places capacity below appetite.

Module

Foundations of Risk Management

Chapter

FRM 3: The Governance of Risk Management

Learning objective

Evaluate the relationship between a firm's risk appetite and its business strategy, including the role of incentives

Correct answer: B. Marking-to-market or margining, historically seen in the market for exchange-traded derivatives

Periodic revaluation followed by a transfer of the net change in value is marking to market, also called margining.

Margining keeps the net exposure between two counterparties small. Value that builds up on one side is transferred before the exposure can become larger.

The technique requires relatively sophisticated systems and has historically been used in the market for exchange-traded derivatives.

Netting is different. The bank calculates the difference between the asset and liability values with each counterparty. Documentation allows those exposures to be set against each other if the counterparty defaults.

Financial guarantees from monoline insurers were common in the municipal bond market. A monoline guarantee is insurance purchased from a third-party guarantor, so it is another separate credit risk mitigation technique.

Other traditional techniques include requiring collateral, using termination or put options triggered by agreed events, and reassigning a credit exposure to another party after a trigger such as a downgrade.

Why the other options are wrong

• A. Netting in exchange-traded derivatives incorrectly names the technique. Netting sets the asset and liability values with one counterparty against each other and does not involve periodic transfers of value.
• C. Margining in the municipal bond market correctly names the technique but incorrectly places it in the market where monoline guarantees were common.
• D. Netting in the municipal bond market incorrectly names the technique and incorrectly places it in the market where monoline guarantees were common.

Module

Foundations of Risk Management

Chapter

FRM 4: Credit Risk Transfer Mechanisms

Learning objective

Explain different traditional approaches or mechanisms that firms can use to help mitigate credit risk

Correct answer: A. 13.0%, and the same Sharpe ratio as the index

The fund's volatility is twice the index volatility of 15%, so the fund's standard deviation is 30%.

Beta equals correlation with the market multiplied by the ratio of the fund's standard deviation to the market standard deviation.

With correlation equal to 1.0:

The market risk premium equals 8.2% minus 3.4%, which is 4.8%.

The CAPM expected return is:

Doubling the market exposure doubles the excess return over the risk-free interest rate. It does not double the risk-free interest rate itself.

So the fund's expected return of 13.0% is less than twice the index's total expected return of 8.2%.

The index earns 4.8% of excess return for 15% standard deviation, while the fund earns 9.6% of excess return for 30% standard deviation.

Their Sharpe ratios are equal:

A fund with correlation of 1.0 with the market has no specific risk.

It lies on the capital market line with the market index, and every portfolio on that line has the same Sharpe ratio.

Why the other options are wrong

• B. 13.0%, higher Sharpe ratio correctly calculates the expected return of 13.0% but incorrectly increases the Sharpe ratio. Both excess return and standard deviation double.
• C. 16.4%, same Sharpe ratio incorrectly doubles the index's total expected return of 8.2%. This also doubles the risk-free interest rate component, which the CAPM does not do.
• D. 16.4%, higher Sharpe ratio incorrectly doubles the index's total expected return and then divides the resulting excess return of 13.0% by 30%, producing an incorrect Sharpe ratio of 0.43.

Module

Foundations of Risk Management

Chapter

FRM 5: Modern Portfolio Theory and the Capital Asset Pricing Model

Learning objective

Apply the CAPM in calculating the expected return on an asset

Correct answer: A. W offers 1.0 percentage point more than its APT value; buy W and short a mix with the same betas.

The factor premiums equal the expected returns on the factor portfolios minus the risk-free rate. Factor 1 has a premium of 7.0% minus 3.0%, or 4.0%. Factor 2 has a premium of 5.0% minus 3.0%, or 2.0%.

The Arbitrage Pricing Theory value of W is:

W offers 9.0%, which is 1.0 percentage point above the required 8.0%. W is underpriced.

A portfolio with the same betas as W holds USD 1.0 of factor portfolio 1 and USD 0.5 of factor portfolio 2 for each USD 1 of W. Those positions cost USD 1.5, so USD 0.5 must be borrowed at the risk-free rate to reduce the net investment to USD 1.

The matching portfolio has the same betas as W and an expected return of:

Buying USD 1 of W and shorting USD 1 of the matching portfolio requires no net investment, eliminates factor exposure and earns 1.0% per year.

Why the other options are wrong

• B. 1.0 point below, short W correctly measures the 1.0 percentage point difference but incorrectly reverses its direction. W is underpriced and should be bought.
• C. 4.0 points above, buy W correctly identifies the direction of the trade but incorrectly leaves the risk-free rate out of the required return, reducing the calculated value to 5.0%.
• D. 3.5 points below, short W incorrectly treats the 7.0% and 5.0% expected factor returns as factor premiums, producing an incorrect Arbitrage Pricing Theory value of 12.5%.

Module

Foundations of Risk Management

Chapter

FRM 6: The Arbitrage Pricing Theory and Multifactor Models of Risk and Return

Learning objective

Explain the Arbitrage Pricing Theory (APT), describe its assumptions, and compare the APT to the CAPM

Correct answer: C. Frequency

The failure concerns frequency. The board and senior management should determine how often risk reports are produced and distributed, and reporting frequency should increase during stress.

Frequency depends on the risk type and the purpose of each report. Fast-moving markets may require reports to be produced more frequently.

Some situations also require rapid risk analyses to support a decision. These situations should be planned for in advance to the extent possible.

The bank's systems could produce figures on request, which is consistent with adaptability. The aggregation capability was not the problem.

The failure is in the reporting plan. The monthly cycle remained unchanged during stress, and there was no plan for rapid analyses.

Why the other options are wrong

• A. Adaptability incorrectly places the failure in the aggregation capability. The systems could generate figures on request, which is consistent with adaptability.
• B. Review incorrectly assigns the failure to a supervisory principle. Review requires supervisors to evaluate a bank's compliance. It does not require the bank to use a particular reporting cycle.
• D. Remedial actions and supervisory measures incorrectly assigns the failure to another supervisory principle. It concerns the tools supervisors use to require corrective action.

Module

Foundations of Risk Management

Chapter

FRM 7: Principles for Effective Data Aggregation and Risk Reporting

Learning objective

Describe characteristics of effective data architecture, IT infrastructure, and risk-reporting practices

Correct answer: A. II only

Only statement II describes a disadvantage.

Statement II is a disadvantage. Firms may fail to develop sufficiently severe scenarios, so the scenarios can underestimate the impact of an extreme loss event or omit important risk exposures.

Statement I is incorrect as a description. Only a limited number of scenarios can be developed fully. The disadvantage is the uncertainty about whether the firm has selected the appropriate scenarios. A statement saying that an unlimited number of scenarios can be developed is factually incorrect and does not describe a genuine disadvantage.

Statement III is an advantage rather than a disadvantage. Outside regulator-defined programs, a firm can make scenario analysis as sophisticated or as straightforward as appropriate for its needs.

Event probabilities can also be difficult to gauge, and scenario analysis does not lead to the quantification of risk. Scenarios can become complex as they unfold and require many choices, and firms may focus too heavily on the most recent crisis when selecting scenarios.

Scenario quality, credibility, and assumptions can also vary and may be difficult to assess.

Why the other options are wrong

• B. I and III only incorrectly accepts false statement I and the advantage in statement III, while rejecting statement II.
• C. I and II only correctly accepts statement II but incorrectly accepts statement I, which reverses the limit on the number of scenarios that can be fully developed.
• D. III only incorrectly classifies an advantage as a disadvantage and rejects statement II.

Module

Foundations of Risk Management

Chapter

FRM 8: Enterprise Risk Management and Future Trends

Learning objective

Explain the role of scenario analysis in the implementation of an ERM program and describe its advantages and disadvantages

Correct answer: A. Leeson controlled both the trading of the positions and the Singapore back office that recorded and reported them.

Leeson had trading authority and also controlled the Singapore back office. This combination allowed him to conceal his losses.

The back office records, reconciles and reports trades. When the trader also controls those functions, there is no independent check that the reported positions and profits correspond to the actual trades.

The authorized strategy was an arbitrage between Nikkei futures traded on two exchanges. The strategy required offsetting trades in the two markets, so the net position should have remained small.

Leeson instead bought contracts in one market and continued to hold them. This created large speculative positions and substantial losses.

He concealed those losses in a reconciliation account and arranged for the account to be excluded from reports sent to London.

Barings' risk controllers questioned the size of the reported profits in late 1994 and raised concerns again in January 1995. Leeson's superiors rejected those concerns and referred to Barings' "unique ability to exploit this arbitrage."

When the losses were discovered, they exceeded what the bank could absorb. Barings was forced into liquidation, and ING bought it for GBP 1.

Why the other options are wrong

• B. Risk controllers never asked incorrectly says the reported profits went unquestioned. Risk controllers raised concerns twice, but Leeson's superiors rejected them.
• C. Arbitrage needed one-way positions incorrectly describes the authorized strategy. The arbitrage required offsetting positions in two markets. Leeson's one-way positions were a departure from that strategy.
• D. Account included in London reports incorrectly says head office received information about the reconciliation account. Leeson arranged for that account to be excluded from the reports.

Module

Foundations of Risk Management

Chapter

FRM 9: Learning from Financial Disasters

Learning objective

Analyze the following factors that contributed to the given case studies of financial disasters and examine the key lessons learned from these case studies: Rogue trading and misleading reporting, including the Barings case

Correct answer: B. R, S, P, Q

The order is R, S, P, Q. Each stage creates the conditions for the next one.

R comes first. As mortgage-backed securities lost value, the credit quality of the SIVs holding those securities deteriorated.

S follows from R. The asset-backed commercial paper issued by the SIVs was downgraded. Investors became skeptical about the pledged collateral, and many SIVs could no longer roll their paper.

P follows from S. Investors stopped rolling maturing asset-backed commercial paper. This forced banks to take SIV assets back onto their balance sheets and created uncertainty about how much additional funding the banks might need to provide. Banks responded by hoarding cash. In early August 2007, three-month Libor increased by more than 30 basis points.

Q follows from P. The reluctance to lend spread across the financial system, credit standards tightened, and the funding squeeze reached residential mortgages, commercial mortgages and business lending.

This final stage transmitted the financial crisis into the wider economy.

Why the other options are wrong

• A. S before R incorrectly places the asset-backed commercial paper downgrade before the decline in the asset values that caused the downgrade.
• C. P before S incorrectly has banks hoarding cash before the asset-backed commercial paper market froze. The uncertainty about possible backstop drawdowns that led banks to hoard cash followed the market freeze.
• D. Q before P incorrectly places the tightening of credit standards before the bank cash hoarding that contributed to it.

Module

Foundations of Risk Management

Chapter

FRM 10: Anatomy of the Great Financial Crisis of 2007-2009

Learning objective

Describe trends in the short-term wholesale funding markets that contributed to the financial crisis, including their impact on systemic risk

Correct answer: B. Varun only

The relevant prohibition concerns knowingly misrepresenting information relating to analysis, recommendations, actions or other professional activities. Varun knew that his description was misleading, while Nisha was unaware of the error when she issued her report.

Varun stated that the backtest covered the complete sample even though he knew that the worst-performing month had been removed. This makes the analysis appear stronger than it actually was and constitutes a knowing misrepresentation.

His description also conflicts with the requirement that professional communications contain factual data and avoid false information.

Nisha's report contained an error, but she did not know about it when the report was issued. After discovering the mistake, she corrected the report on the same day. On these facts, her original mistake is not a knowing misrepresentation.

Why the other options are wrong

• A. Nisha only incorrectly treats an unknown error as knowing misconduct while clearing Varun, who knew his description was misleading.
• C. Both correctly includes Varun but incorrectly includes Nisha, whose error was unknown when made and promptly corrected after discovery.
• D. Neither incorrectly clears Varun despite his knowledge that the supposedly full-sample analysis excluded the worst-performing month.

Module

Foundations of Risk Management

Chapter

FRM 11: GARP Code of Conduct

Learning objective

Describe the responsibility of each GARP Member with respect to professional integrity, ethical conduct, conflicts of interest, confidentiality of information, and adherence to generally accepted practices in risk management

Correct answer: A. Reversing the conditioning changes the denominator but not the intersection, so the two statements agree only when a breach and a downgrade are equally likely.

Reversing a conditional probability leaves the intersection in the numerator unchanged but changes the denominator.

So the two conditional probabilities are equal only when the two conditioning events have equal probabilities.

Both calculations use the same group in the numerator: exposures that both breached and were downgraded.

The first conditional divides that group by all exposures that breached.

The reversed conditional divides the same group by all exposures that were downgraded.

Consider a numerical example.

Suppose 20% of exposures breached, 12% were downgraded and 6% did both.

The numerator is 6% in both cases.

The denominators differ, so the conditional probabilities differ.

Independence does not make the two reversed conditionals equal.

Under independence, one conditional would equal the unconditional downgrade rate and the other would equal the unconditional breach rate.

Those two unconditional rates can still be different.

The two reversed conditionals are equal only when the two conditioning events are equally likely.

Why the other options are wrong

• B. Denominator, independent correctly identifies that the denominator changes but incorrectly names independence as the condition for equality.
• C. Intersection, equally likely correctly identifies the condition for equality but incorrectly claims that the intersection changes.
• D. Intersection, independent incorrectly identifies both the changing quantity and the condition.

Module

Quantitative Analysis

Chapter

QTA 1: Fundamentals of Probability

Learning objective

Define, describe, and calculate a conditional probability

Correct answer: B. 2.96%

The fixed fee is subtracted in every state, while the performance fee applies only to the portion of the gross return above the hurdle. This makes the net return a nonlinear function of the gross return, so it must be calculated separately in each state.

Here R is the gross return. The maximum function ensures that the performance fee is zero whenever the gross return does not exceed the 5% hurdle.

The expected net return is 2.96%.

The expected gross return is 5.10%, so the expected reduction from the fee structure is 2.14 percentage points. This exceeds the fixed 1.5% fee because the performance fee is also charged in the higher-return states.

Why the other options are wrong

• A. 2.34% incorrectly charges the 20% performance fee on the entire positive return rather than only on the portion above the 5% hurdle.
• C. 3.58% incorrectly applies the fee formula to the expected gross return of 5.10%. Because the performance fee is nonlinear, it must be calculated state by state before averaging.
• D. 4.46% calculates the performance fee correctly but omits the fixed 1.5% fee, leaving the expected net return exactly 1.5 percentage points too high.

Module

Quantitative Analysis

Chapter

QTA 2: Random Variables

Learning objective

Describe and apply the concept of a mathematical expectation of a random variable

Correct answer: B. Exactly one

Only statement III holds.

Statement I fails because the variance of a Student's t depends on its degrees of freedom:

Here v means degrees of freedom. This variance is finite only when v is greater than 2. At v equal to 2, the denominator equals zero, so a Student's t with 2 degrees of freedom does not have a finite variance. More generally, a moment of order m exists only when v is greater than m.

Statement II fails because a chi-squared variable cannot be negative. It is constructed as a sum of squared standard normal variables:

Each squared value is non-negative, so their sum is also non-negative. Lower degrees of freedom alter the shape and move the mode closer to zero, but they do not extend the support below zero.

Statement III holds. An exponential distribution has a mean equal to beta and a variance equal to beta squared. Its standard deviation is therefore the square root of beta squared, which equals beta.

So exactly one of the three statements is correct.

Why the other options are wrong

• A. None of the three incorrectly rejects statement III. For an exponential distribution, the mean and standard deviation both equal beta.
• C. Exactly two incorrectly accepts one of the two statements that fail. Accepting statement I would incorrectly use the Student's t variance formula without applying the condition that v must exceed 2.
• D. All three incorrectly accepts statements I and II as well as III, including the claim that a sum of squared values can be negative.

Module

Quantitative Analysis

Chapter

QTA 3: Common Univariate Random Variables

Learning objective

Illustrate the key properties and applications of the following distributions: Bernoulli distribution, binomial distribution, Poisson distribution, uniform distribution, normal distribution, lognormal distribution, Chi-squared distribution, Student's t distribution, F distribution, exponential distribution, and the Beta distribution

Correct answer: A. 11.95%

The weights must first be calculated from the two position sizes, and getting that step wrong makes everything after it wrong.

Note which fund carries the small weight. Fund A is the volatile one at 24%, and only a fifth of the money sits in it. So the combined volatility should be closer to 12% than 24%.

Next turn the correlation into a covariance:

Then the weighted sum:

The combined holding is less volatile than either fund on its own, which shows the benefit of combining the assets. Fund B alone runs at 12% and the pair runs at 11.95%, and the reason the gain is small is that 80% of the money is already in Fund B.

Why the other options are wrong

• B. 14.40% is 0.20 × 24% plus 0.80 × 12%, the weighted average of the two volatilities. That is the answer for a correlation of exactly 1, and the correlation is 0.30.
• C. 14.94% splits the money equally between the two funds instead of by their actual values. Equal weighting puts far too much into the 24% fund, and the volatility rises by three percentage points as a result.
• D. 20.05% reverses the two weights, putting 80% into Fund A and 20% into Fund B. It is the mirror image of the real portfolio and it runs about two thirds higher in volatility.

Module

Quantitative Analysis

Chapter

QTA 4: Multivariate Random Variables

Learning objective

Calculate the variance of a weighted sum of two random variables

Correct answer: A. None of the three statements

None of the three statements holds.

Statement I reverses the direction of a quantile. The 25% quantile is the point below which 25% of the observations lie. So 75% of the observations lie above it, not 25%.

Statement II incorrectly claims that an even-sized sample has no median. With an odd number of observations, the median is the central observation in the ordered sample. With an even number of observations, the median is the average of the two central observations.

Statement III reverses the relative sensitivity of the two dispersion measures. The interquartile range depends on quartile positions in the sorted sample and is relatively insensitive to an observation that moves far into a tail. Standard deviation uses squared deviations from the mean, so a distant outlier can have substantial influence on it.

Why the other options are wrong

• B. Exactly one of the three statements incorrectly accepts one false statement. Accepting statement I would reverse the definition of the 25% quantile by treating it as a point with 25% of observations above it.
• C. Exactly two of the three statements incorrectly accepts two false statements. Accepting statement III would treat the interquartile range as more sensitive to extreme observations than standard deviation, which reverses their actual behavior.
• D. All three statements incorrectly accepts the claim that an even sample has no median. For an even number of observations, the median is the average of the two central ordered values.

Module

Quantitative Analysis

Chapter

QTA 5: Sample Moments

Learning objective

Estimate quantiles, including the median, using sample data

Correct answer: C. 36.82%

The p-value is 36.82%, which is far above any size a risk team would use, so the difference is not significant.

The test statistic comes first:

For a two-sided alternative the p-value is the probability of a statistic at least as far from zero as this one, in either direction. The area to the right of 0.90 is one minus the cumulative probability. The area to the left of minus 0.90 is the same size by symmetry, so the two are added by doubling:

A p-value of 36.82% says that if the asset really did earn the benchmark return, a gap of this size or larger would show up in a little over a third of samples. That is ordinary, not surprising.

Why the other options are wrong

• A. 0.90% is the test statistic written as a percentage. The statistic is a pure number measured in standard errors, and it has no percentage attached to it.
• B. 18.41% is the one-tailed area, which is the p-value for a one-sided alternative. The question asks for a two-sided test, so both tails count. This is the most common mistake on this question.
• D. 81.59% is the cumulative probability itself, which is the area to the left of 0.90. That is the probability of a statistic smaller than the one observed, which is the complement of what a p-value measures.

Module

Quantitative Analysis

Chapter

QTA 6: Hypothesis Testing

Learning objective

Explain what the p-value of a hypothesis test measures

Correct answer: B. 1.153 to 3.207

A confidence interval reaches out from the estimate by the critical value multiplied by the standard error:

At a 95% confidence level the interval is two-sided and leaves 2.5% in each tail, so the critical value from the normal distribution is 1.96.

So the interval runs from 1.153 to 3.207.

The sample size of 50 is already reflected in the standard error. It has already done its work inside the standard error of 0.524, so it should not be applied again. A sample size does not enter the interval a second time.

Why the other options are wrong

• A. 0.833 to 3.527 uses 2.57, which is the two-sided critical value at a 99% confidence level. The interval is correctly built and set at the wrong confidence level.
• C. 1.318 to 3.042 uses 1.645, the two-sided critical value at a 90% confidence level. It is narrower than the correct interval, which is the direction less confidence moves it.
• D. 2.035 to 2.325 divides the standard error by the square root of 50 before building the interval. That step belongs to turning a standard deviation into a standard error, and it has already been taken inside the 0.524.

Module

Quantitative Analysis

Chapter

QTA 7: Linear Regression

Learning objective

Construct, apply, and interpret hypothesis tests and confidence intervals for a single regression coefficient in a regression

Correct answer: A. 25.0, and at least one of the two coefficients differs from zero

The F-test can be written with the two R-squared figures in place of the two residual sums of squares, which is the form to use when the question gives R-squared rather than sums of squares.

F = ((R-squared unrestricted − R-squared restricted) / q) / ((1 − R-squared unrestricted) / (n − k − 1))

Two coefficients are set to zero, so the number of restrictions q is 2. The unrestricted model carries three explanatory variables and an intercept, so the denominator degrees of freedom are 104 minus 3 minus 1, which is 100.

F = ((0.74 − 0.61) / 2) / ((1 − 0.74) / 100) = 0.065 / 0.0026 = 25.0

The statistic of 25.0 is well above the critical value of 3.09, so the null is rejected.

Now read what has been rejected. The null said both coefficients are zero. Rejecting it says that claim is false, and a claim that both are zero is false as soon as one of them is not. So the test establishes that at least one of the two coefficients differs from zero. It does not say which one, and it does not say both.

Why the other options are wrong

• B. 25.0, and both coefficients differ from zero has the arithmetic right and over-reads the conclusion. Most candidates who miss this question land here, because the null names both coefficients and it is natural to attach the rejection to both of them.
• C. 50.0, and at least one of the two coefficients differs from zero counts one restriction instead of two, giving 0.13 over 0.0026. Two coefficients were pinned to a value, so two restrictions were imposed.
• D. 50.0, and both coefficients differ from zero carries the restriction-count slip and the over-read conclusion together.

Module

Quantitative Analysis

Chapter

QTA 8: Regression with Multiple Explanatory Variables

Learning objective

Construct, apply, and interpret joint hypothesis tests and confidence intervals for multiple coefficients in a regression

Correct answer: C. Step III

The procedure removes one variable per round, not every insignificant one at once. The variable that goes is the one whose coefficient has the smallest absolute t-statistic among those that failed the test.

The reason for going one at a time is that removing a variable changes the significance of the ones left behind. Explanatory variables in financial data are correlated, so two variables can each look insignificant because they are competing to explain the same movement. Drop one and the other often becomes significant. A round that removes both throws away a variable that belonged in the model, and there is no step later in the procedure that puts it back.

Step III as Rohit has written it removes them together, so it is the step that departs from the procedure.

The other three steps are as they should be. Starting from a large model that includes everything of possible relevance is what working from the general to the specific means. A test size of 1% is a standard choice, with 0.1% the other common one, and these are tighter than the 5% used for a single hypothesis because the procedure runs many tests in sequence. Re-estimating and repeating until nothing insignificant is left is the stopping rule.

Why the other options are wrong

• A. Step I is the definition of the starting point. General-to-specific begins wide, with every variable that theory or intuition puts forward, and narrows from there.
• B. Step II names a test size inside the range the procedure normally uses. Choosing 1% rather than 5% is deliberate, because running a sequence of tests raises the chance that an irrelevant variable survives one of them.
• D. Step IV is the loop that makes the method work. Re-estimating after each removal is what step III failed to allow for, so a candidate who spots the problem and then blames step IV has the reason right and the label wrong.

Module

Quantitative Analysis

Chapter

QTA 9: Regression Diagnostics

Learning objective

Explain two model selection procedures and how these relate to the bias-variance tradeoff

Correct answer: C. 7.16, and the null of no autocorrelation is not rejected

Both joint tests in this chapter square the sample autocorrelations and add them up. The difference is what each squared term is multiplied by.

The Box-Pierce statistic multiplies the sum by the sample size and stops there.

The Ljung-Box statistic weights each squared autocorrelation by a factor that grows with the lag, which corrects the behavior of the test in samples that are not large.

Working the three terms:

The Ljung-Box weights are all larger than 1, so its statistic is always above the Box-Pierce statistic on the same data. Here the difference of 0.20 does not change the decision, and in a smaller sample it can.

The statistic of 7.16 is below the critical value of 7.815, so the null that all three autocorrelations are zero is not rejected. The residuals are consistent with white noise over these lags, which is the result a well specified model should produce.

Why the other options are wrong

• A. 6.96, not rejected is the Box-Pierce statistic, which is the uncorrected version of the same test. The conclusion happens to be the same, and the question asked for the corrected statistic.
• B. 6.96, rejected computes the Box-Pierce statistic and then compares it the wrong way round. A statistic below the critical value is what fails to reject.
• D. 7.16, rejected gets the arithmetic right and reverses the decision rule. Large values of the statistic indicate autocorrelation, so rejection needs the statistic above the critical value.

Module

Quantitative Analysis

Chapter

QTA 10: Stationary Time Series

Learning objective

Describe the Box-Pierce Q statistic and the Ljung-Box Q statistic

Correct answer: D. The lagged differences are included so that the residuals are white noise, which is what the Dickey-Fuller distribution assumes.

The word augmented refers to the added lagged differences. Without them the regression is a plain Dickey-Fuller test, with the difference regressed on the lagged level alone.

They are added because the critical values only hold when the residuals of the test regression are white noise. If the series carries short-run dynamics that the regression does not include, those dynamics end up in the residuals, the residuals are serially correlated, and the test statistic no longer follows the Dickey-Fuller distribution the critical values were built from. The lagged differences absorb the short-run dynamics so that what is left over is white noise.

Choosing how many to include is done by minimizing an information criterion, with the maximum lag length set by the frequency of the data, so around 24 for monthly data or 8 for quarterly. The Akaike criterion is preferred here even though it tends to select larger models than the Schwarz criterion does. The reason is asymmetric. Too few lags leaves serial correlation in the residuals and breaks the test, while too many costs some precision and breaks nothing. Including too many lags is the safer error.

Why the other options are wrong

• A. A normal distribution is not what the lags achieve, because the coefficient on the lagged level does not become normally distributed however many are added. That is why the Dickey-Fuller distribution exists and the reason its critical values are larger in absolute value than the normal ones.
• B. More degrees of freedom has it backward, because adding lags uses degrees of freedom rather than adding them, and the effect on power runs the other way. The lags are included because the test is invalid without them, not because they sharpen it.
• C. Dropping deterministic terms confuses two separate jobs. Which deterministic terms to include is settled by testing their significance, and no number of lagged differences makes that decision go away.

Module

Quantitative Analysis

Chapter

QTA 11: Non-Stationary Time Series

Learning objective

Describe how to test if a time series contains a unit root

Correct answer: C. 0.5000

Start with the daily correlation, which is the daily covariance divided by the two daily volatilities:

Now move to the annual horizon. Under independence the covariance scales linearly with the number of trading days, and so does each variance, so each volatility scales with the square root:

The factor of 250 appears above and below and cancels, leaving 0.5000. The correlation does not depend on the horizon at all.

The reason sits in what a correlation is. It is a covariance with the units divided out, so anything that scales the covariance by the same factor as the product of the two volatilities leaves it where it was. Lengthening the horizon does that under independence, and so does multiplying either series by any positive constant, which is what happens when a return is restated as a decimal rather than a percentage. A covariance moves under all of those and a correlation does not, which is why the correlation is the figure that gets compared across assets and across horizons.

Why the other options are wrong

• A. 0.0225 is the annual covariance, 250 multiplied by 0.000090, reported as though it were a correlation. It is still measured in squared return units and it has not been divided by anything.
• B. 0.0316 takes the daily correlation of 0.5000 and divides it by the square root of 250, applying a volatility scaling rule to a quantity that carries no units to scale.
• D. 0.7071 is the square root of 0.5000. A square root converts a variance into a volatility, and a correlation is not a squared quantity, so there is nothing here for it to undo.

Module

Quantitative Analysis

Chapter

QTA 12: Measuring Returns, Volatility, and Correlation

Learning objective

Define correlation and covariance and differentiate between correlation and dependence

Correct answer: A. Monte Carlo simulation draws from a data generating process that the user specifies, and bootstrapping draws from the observed data without specifying one.

The main difference between Monte Carlo simulation and bootstrapping is where the simulated data come from. A Monte Carlo simulation requires a complete data generating process, meaning a set of rules that specifies every random quantity in the model and the probability distribution from which it is drawn. Every simulated value is then generated from those rules. A bootstrap does not specify such a process. Instead, it repeatedly draws from the observed data with replacement. In other words, the same observation can be selected more than once.

After the data are obtained, the basic procedure is the same for both methods. Create a sample, calculate the statistic of interest, repeat the process many times, and average the results. This is why the two methods are often studied together. Their main difference is the source of the simulated observations.

Why the other options are wrong

• B. The reversed version swaps the two methods. Monte Carlo simulation specifies a data generating process. The bootstrap avoids having to specify one, which is one of its main advantages.
• C. Both methods use randomness. In bootstrapping, observations are selected randomly with replacement.
• D. A bootstrap does not require a distributional assumption, but this does not mean such an assumption is forbidden. The observed data still come from an underlying distribution even if the analyst does not specify it.

Module

Quantitative Analysis

Chapter

QTA 13: Simulation and Bootstrapping

Learning objective

Describe the bootstrapping method and its advantage over Monte Carlo simulation

Correct answer: D. 0.815

Temporal difference learning updates a Q-value before the full sequence of rewards has finished. It uses the reward received up to the next state together with an estimate of the value of that next state.

Begin with the current estimate. Action 3 in state 4 has a Q-value of 0.80.

Next find the value of state 3. The algorithm assumes the best currently known action will be chosen there, so the state's value is the largest available Q-value, not the average:

State 3 holds 0.40, 0.50 and 0.90, so its value is 0.90

Now apply the update using the reward of 0.20 and the next-state value of 0.90:

0.80 + 0.05 x (0.20 + 0.90 - 0.80)

= 0.80 + 0.05 × 0.30

= 0.80 + 0.015

= 0.815

The difference from the Monte Carlo approach is timing. Monte Carlo waits until the full episode is complete and uses the total reward that follows. Temporal difference learning updates earlier by combining the immediate reward with an estimate of future value.

Why the other options are wrong

• A. 0.770 leaves out the value of the next state.
• B. 0.800 uses the average of the three state-3 values, 0.60, causing the update term to become zero.
• C. 0.805 omits the reward of 0.20 received between the two states.

Module

Quantitative Analysis

Chapter

QTA 14: Machine-Learning Methods

Learning objective

Explain how reinforcement learning operates, and calculate Q-values utilized in the decision-making process

Correct answer: D. Below 0.50, because the bank is protecting against lending to a borrower who defaults.

Begin with the meaning of the threshold. It is the estimated probability above which the model classifies a loan as a default. Lowering the threshold causes more loans to be classified as defaults. This catches more borrowers who really will default, but it also wrongly rejects more borrowers who would have repaid. Raising the threshold has the opposite effect.

The relative costs of the two errors determine the preferred direction. Lending to a borrower who defaults may cost the lender the principal, while declining a borrower who would have repaid costs only the profit margin on that loan. Because the first error is much more expensive, the lender may accept more false alarms in order to catch more true defaults. This requires a threshold well below 0.50. In lending applications, a threshold as low as 0.05 may be used.

A threshold of 0.50 is not automatically correct. It is most natural when the two types of classification error have roughly equal costs. That is not the situation described for the loan portfolio.

The logic reverses if the error costs reverse. If turning away profitable business is much more costly than accepting a customer who fails to pay, the threshold would move above 0.50.

Why the other options are wrong

• A. Above 0.50 to protect against declining a good borrower describes what a higher threshold would tend to do, but that is not the lender's stated priority in this case.
• B. Above 0.50 to protect against lending to a defaulter identifies the expensive error but moves the threshold in the wrong direction. A higher threshold means fewer loans are classified as defaults.
• C. Below 0.50 to protect against declining a good borrower gives the correct direction for the wrong reason. Lowering the threshold causes more good borrowers to be rejected, not fewer.

Module

Quantitative Analysis

Chapter

QTA 15: Machine Learning and Prediction

Learning objective

Explain the role of linear regression and logistic regression in prediction

Correct answer: D. The Liquidity Coverage Ratio addresses the concern Rohan raises, and the Net Stable Funding Ratio addresses the concern Priya raises.

The two Basel III liquidity ratios address different time horizons.

Rohan is describing a short-term liquidity stress event: a downgrade, deposit outflows, and credit-line drawdowns occurring over a short period.

The Liquidity Coverage Ratio, or LCR, is designed to ensure that a bank has enough high-quality liquid assets to survive a severe 30-day liquidity stress.

Priya is describing a structural funding mismatch. Five-year assets funded with three-month liabilities create a continuing dependence on short-term refinancing.

The Net Stable Funding Ratio, or NSFR, is designed to encourage more stable funding over a longer horizon and reduce excessive maturity mismatch.

Why the other options are wrong

• A. The Liquidity Coverage Ratio addressing both ignores the separate long-term funding purpose of the NSFR.
• B. The Net Stable Funding Ratio addressing both ignores the LCR's specific short-term stress function.
• C. The two ratios swapped assigns each liquidity tool to the wrong problem.

Module

Financial Markets and Products

Chapter

FMP 1: Banks

Learning objective

Summarize the Basel committee regulations for regulatory capital and their motivations

Correct answer: D. All three

Statement I is correct. In a defined contribution plan, funds are invested for the employee, and the employee can usually choose among investment options. Some plans may also allow a lump-sum withdrawal at retirement.

Statement II is correct and describes a defining feature of a defined benefit plan. Actuaries calculate the present value of promised pension obligations each year and compare that value with the assets held by the plan. If there is a shortfall, the company must contribute enough to close it.

Statement III is also correct. A defined contribution plan creates much less financial risk for the employer because the employer does not promise a particular retirement value. However, the employer still has operational responsibilities. If it manages the employee funds badly, employees may sue. This creates residual operational risk.

So all three statements are correct.

Why the other options are wrong

• A. None of the three incorrectly rejects both employee investment choice and the employer's defined benefit shortfall obligation.
• B. Exactly one incorrectly accepts only one of the first two statements even though both describe core features of the plans.
• C. Exactly two incorrectly assumes a defined contribution plan leaves the employer with absolutely no remaining risk.

Module

Financial Markets and Products

Chapter

FMP 2: Insurance Companies and Pension Plans

Learning objective

Describe defined benefit plans and defined contribution plans and explain the differences between them

Correct answer: B. 1 in 16, and 16 funds

There are four independent years, and the probability of outperforming in each year is one half. So multiply the four probabilities.

0.5 × 0.5 × 0.5 × 0.5 = 1/16

A fund company operating sixteen funds therefore expects 16 × 1/16 = 1 of them to have a four-year winning record purely by chance, even if no manager has skill. That is the fund likely to appear in the advertisement.

The research behind the one-in-two probability is the important point. Funds that beat the market in one year beat it again in the following year about half the time. The same result applied to funds that had already beaten the market for two, three, four, five and six consecutive years. The length of the existing winning sequence does not increase the probability of another winning year. That is what no persistence means.

Why the other options are wrong

• A. 1 in 8, and 8 funds calculates one half to the third power, which counts only three years instead of four.
• C. 1 in 16, and 4 funds gets the probability correct but incorrectly uses the number of years as the number of funds.
• D. 1 in 32, and 32 funds calculates one half to the fifth power, which includes a fifth year that is not part of the record.

Module

Financial Markets and Products

Chapter

FMP 3: Fund Management

Learning objective

Describe characteristics of mutual fund and hedge fund performance and explain the effect of measurement biases on performance measurement

Correct answer: A. The two positions have the same payoff at every price above USD 100, and below USD 100 the forward loses money while the option payoff is zero.

Calculate the payoff from both positions at each asset price.

Above USD 100, the two payoffs are identical. The long forward pays the asset price minus USD 100, and an exercised long call also pays the asset price minus USD 100.

Below USD 100, the payoffs differ. The forward continues to pay the asset price minus USD 100, which becomes negative. The call holder simply lets the option expire, so the call payoff remains zero.

This shows the key difference between the two instruments. Above the strike, their payoffs are the same. Below the strike, the option holder has a right rather than an obligation, so the option payoff stops at zero while the forward payoff continues downward.

Why the other options are wrong

• B. Same above, both lose below correctly describes the upside but incorrectly gives the purchased call a negative payoff below the strike. A bought option cannot have a negative exercise payoff.
• C. Different above, forward loses below correctly describes the downside but incorrectly separates the two payoffs above USD 100, where both equal the asset price minus USD 100.
• D. Different above, both lose below incorrectly treats the two instruments as having different payoffs everywhere and may result from confusing payoff with profit.

Module

Financial Markets and Products

Chapter

FMP 4: Introduction to Derivatives

Learning objective

Identify and calculate option and forward contract payoffs

Correct answer: C. The member pays USD 4,250

Each contract is settled using its own price change, and the two positions move in opposite directions because one is long and the other is short.

The September position is long, so a falling futures price creates a loss:

The December position is short, so a falling futures price creates a gain:

The member pays USD 8,000 on one contract and receives USD 3,750 on the other. The two amounts are netted:

So the member pays USD 4,250 for the day.

Variation margin is netted across positions because gains on one contract can offset losses on another. In this type of position, a general price move often hurts one leg while helping the other.

Why the other options are wrong

• A. The member receives USD 4,250 correctly nets the amounts but reverses which side loses when prices fall.
• B. The member receives USD 11,750 adds the gain and loss instead of netting them and also puts the total on the wrong side.
• D. The member pays USD 11,750 adds the two amounts instead of offsetting them, incorrectly treating a falling price as a loss on both positions.

Module

Financial Markets and Products

Chapter

FMP 5: Exchanges and OTC Markets

Learning objective

Describe the implementation of a margining process, explain the determinants of and calculate initial and variation margin requirements

Correct answer: D. Uncleared derivatives between financial institutions must carry initial and variation margin.

A 2011 G-20 agreement introduced initial and variation margin requirements for uncleared derivatives, bringing their margin treatment closer to that of centrally cleared transactions.

The rules were phased in between 2016 and 2020.

They apply to uncleared transactions between two financial institutions and to transactions between a financial institution and a systemically important non-financial company.

This reduces the incentive to make a derivative slightly non-standard simply to keep it outside a central counterparty.

Before the rule, a dealer avoiding central clearing could also avoid posting initial margin.

After the rule, an uncleared transaction also requires initial margin. The uncleared requirement uses a ten-day close-out horizon rather than the five-day horizon typically used by a central counterparty.

The other three requirements arose from the Pittsburgh agreement.

Central clearing and electronic trading apply only to standardized transactions, so making a transaction non-standard can remove both requirements.

Reporting to a trade repository applies to all over-the-counter transactions, but reporting itself does not impose an additional cost on avoiding central clearing.

Why the other options are wrong

• A. Electronic platforms incorrectly selects a Pittsburgh requirement that applies only to standardized derivatives.
• B. Trade repository incorrectly selects a requirement that records transactions for regulators but does not make avoidance of central clearing more costly.
• C. Clearing by the end of 2012 incorrectly selects the requirement that making a transaction non-standard was intended to avoid.

Module

Financial Markets and Products

Chapter

FMP 6: Central Clearing

Learning objective

Explain regulatory initiatives for the OTC derivatives market and their impact on central clearing

Correct answer: A. 6

Investor A is already long 5 contracts after Day 1. So when Investor A sells 2 contracts on Day 3, that sale closes part of the existing long position rather than creating a new short position. The existing position of each trader must always be checked before deciding how a trade affects open interest.

Day 3 is the key step. Investor A sells 2 contracts, reducing the long position from 5 to 3. Investor E is new and takes a long position of 2. One trader is closing while the other is opening, so open interest stays at 7.

On Day 4, both sides reduce existing positions. Investor E sells 1, reducing the long position from 2 to 1. Investor C buys 1, reducing the short position from 7 to 6. Because both sides are closing, open interest falls by 1.

7 − 1 = 6

A final check confirms the answer. Total longs are 3 + 2 + 1 = 6, and Investor C holds a short position of 6.

Why the other options are wrong

• B. 7 incorrectly treats Day 4 as one trader opening and one closing. Both traders are actually closing positions.
• C. 8 incorrectly treats Investor E's new long on Day 3 as increasing open interest without recognizing that Investor A is closing on the other side.
• D. 10 incorrectly assumes every trade creates new open interest.

Module

Financial Markets and Products

Chapter

FMP 7: Futures Markets

Learning objective

Define and describe the key features and specifications of a futures contract, including the underlying asset, the contract price and size, trading volume, open interest, delivery, and limits

Correct answer: A. 0.64

The minimum variance hedge ratio equals the correlation multiplied by the standard deviation of the hedged asset's price changes divided by the standard deviation of the futures price changes.

The ratio of standard deviations adjusts for the different magnitudes of movement in the two prices. Here the futures price has the larger standard deviation, so less futures quantity is required for each unit of the hedged asset. The correlation then adjusts for the fact that the two price changes do not move together perfectly.

A hedge ratio of 0.64 means the minimum variance futures position is 0.64 units of futures exposure for each unit of spot exposure when the hedge is expressed in quantities. It does not mean that 36% of the economic risk is deliberately left unhedged. The ratio is the position size that minimizes the variance of the combined position given the covariance between spot and futures price changes.

Why the other options are wrong

• B. 0.80 incorrectly uses the correlation as the hedge ratio and omits the ratio of standard deviations. This would give the same result only if the two standard deviations were equal.
• C. 0.96 multiplies the correlation by the standard deviation of the spot price change, 0.80 times 1.20, and forgets to divide by the standard deviation of the futures price change of 1.50.
• D. 1.00 reverses the ratio of standard deviations, 0.80 times 1.50 divided by 1.20. Because the futures price is more volatile, its standard deviation belongs in the denominator. The result also looks like a one-for-one hedge, which ignores both the correlation and the difference in volatility.

Module

Financial Markets and Products

Chapter

FMP 8: Using Futures for Hedging

Learning objective

Define cross hedging and calculate and interpret the hedge ratio and hedge effectiveness

Correct answer: A. 0.7670, because the exchange treats a foreign currency as an asset and values it in US dollars, so the US dollar is the quote currency.

The forward rate is found by subtracting the negative forward points after dividing by 10,000.

The result is 1.30384 Canadian dollars per US dollar, which follows the USDCAD forward-market convention.

A US exchange quotes the Canadian dollar futures contract in US dollars per Canadian dollar, so the forward quote must be inverted.

Both rates describe the same exchange relationship. The difference is which currency is treated as the base currency.

Currency futures on a US exchange conventionally state the foreign currency's value in US dollars. The contracts are also specified on a fixed amount of foreign currency, including 100,000 Canadian dollars for the contract described.

Why the other options are wrong

• B. 0.7670 with the trading-country reason gives the correct numerical rate but uses an incorrect explanation. The quote arises because the foreign currency is priced in US dollars, not because the trading country's currency must be the base.
• C. 1.3038 with the dollar-as-quote-currency reason gives the correct convention in words but fails to invert the forward-market quote.
• D. 1.3038 with the trading-country reason incorrectly keeps the forward-market convention and gives a rule that does not determine the futures quote.

Module

Financial Markets and Products

Chapter

FMP 9: Foreign Exchange Markets

Learning objective

Explain and describe the mechanics of spot quotes, forward quotes, and futures quotes in the foreign exchange markets and distinguish between bid and ask exchange rates

Correct answer: D. The forward price will be higher than the futures price

A futures contract is settled daily, while a forward contract settles once at maturity. Daily settlement makes the timing of gains and losses relevant when interest rates are uncertain.

For a long futures position on an asset negatively correlated with interest rates, price increases tend to occur when interest rates are lower. Futures gains are then received immediately but reinvested at relatively low interest rates.

Price decreases tend to occur when interest rates are higher. Futures losses are paid immediately and therefore financed at relatively high interest rates.

These effects make the long futures position less attractive than the comparable long forward position. The futures price is therefore slightly below the forward price.

The expected direction of interest rates is not what determines this result. The relevant factor is the correlation between asset-price changes and interest rates.

Why the other options are wrong

• A. Prices the same applies when interest rates are constant or sufficiently predictable so that daily settlement has no differential reinvestment effect.
• B. Forward lower gives the relationship associated with positive correlation rather than negative correlation.
• C. Independent movement incorrectly treats forward and futures prices as unrelated. Their values remain closely linked through no-arbitrage relationships.

Module

Financial Markets and Products

Chapter

FMP 10: Pricing Financial Forwards and Futures

Learning objective

Explain the relationship between forward and futures prices

Correct answer: C. 4.50%

Convenience yield is the noncash benefit of holding the physical commodity rather than only a contract for future delivery. It is expressed as a rate so it can be incorporated into the cost-of-carry relationship.

Storage is a cost of holding the physical commodity, so its present value is added to the spot price. Convenience yield reduces the forward price in the same way that an income yield reduces the forward price of an asset.

Rearranging for convenience yield gives

The USD 2.00 storage cost is already stated as a present value, so it is added directly to the USD 128.00 spot price.

The convenience yield is 4.50% per annum.

The spot price plus storage equals the futures price before financing is considered. The convenience yield of 4.50% offsets the financing component in the stated relationship.

Why the other options are wrong

• A. -1.78% incorrectly omits the storage cost and uses 128 divided by 130 in the ratio.
• B. 1.13% incorrectly inserts the three-month interest amount in place of the annual interest rate even though the time adjustment is already handled by T.
• D. 10.75% incorrectly calculates an annualized carrying-cost measure rather than the convenience yield.

Module

Financial Markets and Products

Chapter

FMP 11: Commodity Forwards and Futures

Learning objective

Describe the cost of carry model and determine the impact of storage costs and convenience yields on commodity forward prices and no-arbitrage bounds

Correct answer: D. Option 1 is out of the money and option 3 is at the money.

Moneyness asks whether immediate exercise would produce a positive payoff. A positive payoff means the option is in the money. A zero payoff means it is at the money. A negative imagined payoff means it is out of the money. Time to expiration does not affect this classification.

A call is in the money when the asset price is above the strike price. A put is in the money when the asset price is below the strike price.

With the share price at USD 45:

Option 1 is out of the money and option 3 is at the money.

The same strike price can produce different moneyness classifications for a call and a put. The option type must be identified before the strike can be interpreted.

Why the other options are wrong

• A. Option 2 in and option 3 out correctly identifies option 2 but incorrectly treats a zero payoff as out of the money.
• B. Option 2 out and option 4 in reverses the put rule in both cases.
• C. Options 1 and 4 both in the money reverses the rule for both the call and the put.

Module

Financial Markets and Products

Chapter

FMP 12: Options Markets

Learning objective

Describe the various types and uses of options, and define moneyness

Correct answer: B. USD 0.737

Put-call parity links European call and put prices at the same maturity. With no dividends, a European call plus cash equal to the present value of the strike price has the same value as a European put plus one share.

Here c is the call price, p is the put price, S is the stock price, and PV(K) is the strike price discounted from maturity to today. Rearranging gives the call price minus the put price as the stock price minus the present value of the strike price.

The stock price remains unchanged when the maturity is extended. So the change in the difference between the call price and the put price comes from the change in PV(K).

The present value of the USD 40 strike price at six months is:

The present value of the same strike price at one year is:

The longer maturity lowers PV(K) by:

So the call price minus the put price increases by USD 0.5867. The call price rises by USD 1.324. For the difference between the two option prices to increase by only USD 0.5867, the put price must rise by the remaining amount.

The put price rises by USD 0.737.

Why the other options are wrong

• A. USD 0.587 incorrectly reports the change in the present value of the strike price instead of the change in the put price.
• C. USD 1.324 incorrectly assumes that the put price rises by the same amount as the call price.
• D. USD 1.911 incorrectly adds the change in the present value of the strike price to the call's increase instead of subtracting it.

Module

Financial Markets and Products

Chapter

FMP 13: Properties of Options

Learning objective

Explain put-call parity and apply it to the valuation of European and American stock options, with dividends and without dividends, and express it in terms of forward prices

Correct answer: C. Long the call with a strike price of USD 30 and short the call with a strike price of USD 40

A bull spread buys the lower strike option and sells the higher strike option, using options of the same type and maturity. With calls, the trader buys the USD 30 call and writes the USD 40 call.

Below USD 30, both calls expire worthless. Between USD 30 and USD 40, the lower-strike call has value while the higher-strike call does not. Above USD 40, both are in the money and their incremental payoffs offset, leaving a fixed spread payoff of USD 10.

The initial cost is USD 7.31.

The short call reduces the cost of the position but also caps the payoff above USD 40.

Why the other options are wrong

• A. Long call at USD 30, short put at USD 30 creates a synthetic long forward rather than a spread.
• B. Long call at USD 40, short call at USD 30 incorrectly reverses the strikes and creates a bear call spread.
• D. Long put at USD 40, short call at USD 40 combines a put and a call, so it is a combination rather than a spread.

Module

Financial Markets and Products

Chapter

FMP 14: Trading Strategies

Learning objective

Describe the use and calculate the payoffs of various spread strategies

Correct answer: B. USD 60.00

A knock-in option plus a matching knock-out option reproduces the corresponding plain vanilla option, provided the strike, barrier, maturity, and other relevant terms match. Either the barrier is reached, in which case the knock-in pays and the knock-out is dead, or it is not, in which case the knock-out pays and the knock-in never came alive, so exactly one of the two is live in every state.

The plain vanilla call and put values are

For European options on a non-income-paying asset, put-call parity is

where S sub 0 is the current stock price, K is the strike price, r is the risk-free rate and T is the time to maturity. Substituting:

The strike price is USD 60.00.

The barrier levels are no longer needed after the knock-in and knock-out values have been recombined into the plain vanilla call and put.

Why the other options are wrong

• A. USD 58.23 incorrectly reports the present value of the strike rather than the strike itself.
• C. USD 62.79 incorrectly uses only the knock-out option values rather than reconstructing the full call and put.
• D. USD 66.62 incorrectly reverses the put-call parity rearrangement.

Module

Financial Markets and Products

Chapter

FMP 15: Exotic Options

Learning objective

Identify and describe the characteristics and payoff structures of the following exotic options: gap, forward start, compound, chooser, barrier, binary, lookback, Asian, exchange, and basket options

Correct answer: D. SOFR is free of credit sensitivity and it is backward looking.

LIBOR represented unsecured bank borrowing and therefore contained a bank credit component.

SOFR is based on secured overnight Treasury repo transactions, so it carries no bank credit component, unlike an unsecured term bank borrowing rate. The spread between three-month USD LIBOR and a three-month rate built from overnight rates was normally around ten basis points, and it reached an all-time high of 364 basis points in October 2008.

A compounded three-month SOFR rate is also backward looking under the standard in-arrears construction because it is calculated from overnight observations accumulated through the period. Traditional three-month LIBOR was known near the beginning of the interest period.

Alternative credit-sensitive benchmarks have been proposed to capture bank funding conditions more directly. The Bloomberg Short-Term Bank Yield Index measures the average yields at which large global banks access unsecured wholesale funding, and AMERIBOR reflects the actual borrowing costs of community and regional banks in the United States.

Why the other options are wrong

• A. Credit sensitive and forward looking describes characteristics associated more closely with traditional LIBOR.
• B. Credit sensitive and backward looking correctly identifies the usual compounded-SOFR timing but incorrectly assigns material bank credit sensitivity to the secured reference rate.
• C. Free of credit sensitivity and forward looking correctly identifies the low-credit-risk nature but incorrectly describes the standard compounded-in-arrears timing.

Module

Financial Markets and Products

Chapter

FMP 16: Properties of Interest Rates

Learning objective

Describe Treasury rates, LIBOR, Secured Overnight Financing Rate (SOFR), and repo rates, and explain what is meant by the "risk-free" rate

Correct answer: B. USD 1,049.76

A make-whole call provision sets the call price as the present value of the remaining promised cash flows, discounted at the risk-free rate plus a stated spread.

The discount rate is the three-year Treasury rate of 4.00% plus 20 basis points, or 4.20%. The remaining cash flows are USD 60 in one year, USD 60 in two years, and USD 1,060 in three years.

BA II Plus: use the cash flow worksheet. CF0 is 0, C01 is 60 with F01 of 2, and C02 is 1060 with F02 of 1. Press NPV, enter I as 4.2, then compute NPV to get 1,049.76.

The 20-basis point spread matters because it raises the discount rate and lowers the call price relative to discounting at the Treasury rate alone. A make-whole provision is attractive to issuers because it imposes little economic loss on bondholders. Since bondholders are paid close to the present value of what they would otherwise receive, they demand less compensation for the call feature.

A fixed-price call uses a schedule in the indenture and is valuable to the issuer, so bondholders charge for it. A make-whole call is recalculated at the call date and pays bondholders close to fair value.

Why the other options are wrong

• A. USD 1,000.00 is only principal. The make-whole price is the present value of remaining coupons and principal.
• C. USD 1,055.50 discounts at the Treasury rate and omits the 20-basis point spread.
• D. USD 1,180.00 adds the remaining cash flows without discounting.

Module

Financial Markets and Products

Chapter

FMP 17: Corporate Bonds

Learning objective

Describe the mechanisms by which corporate bonds can be retired before maturity

Correct answer: C. Exactly two

Statement I is correct. Agencies charge fees for guaranteeing and servicing mortgage pools. Those fees are deducted before cash flows are passed to investors.

Statement II is incorrect. The guarantee protects against default, not prepayment. When a mortgage prepays, investors receive principal early and must reinvest it.

Statement III is correct. GNMA is a US government agency. FNMA and FHLMC are government-sponsored enterprises. Their obligations do not carry an explicit US government guarantee in the way GNMA securities do, although most market participants treat an implicit guarantee as existing, and both enterprises received substantial government support during the financial crisis.

Two statements are correct.

The three agencies exist so that banks can make long-term loans without keeping them on their books. Selling the mortgages returns the funds, which can then be lent to new home buyers.

Why the other options are wrong

• A. None of the three incorrectly rejects statements I and III.
• B. Exactly one incorrectly rejects one of the two correct statements.
• D. All three incorrectly accepts statement II. Agency guarantees do not remove prepayment risk.

Module

Financial Markets and Products

Chapter

FMP 18: Mortgages and Mortgage-Backed Securities

Learning objective

Summarize the securitization process of mortgage-backed securities (MBS), particularly the formation of mortgage pools, including specific pools and to-be-announced (TBAs)

Correct answer: A. 0.967438

Two-year and five-year note contracts round maturity down to the nearest month. Here the bond has 3 years and 8 months remaining, so no rounding adjustment is needed.

The next coupon is two months away. At that coupon date, the bond will have 3 years and 6 months remaining, or 7 semiannual periods. The semiannual coupon is 2.5 and the semiannual yield used for conversion-factor purposes is 3%.

On the BA II Plus, use N = 7, I/Y = 3, PMT = 2.5, and FV = 100. The value is 96.8849.

Adding the coupon about to be paid:

Discount back two months, which is one third of a semiannual period:

Four months have accrued since the last coupon:

Subtract accrued interest and divide by 100:

The conversion factor is 0.967438.

Why the other options are wrong

• B. 0.968849 incorrectly uses the Treasury bond contract's three-month rounding rule.
• C. 0.977182 incorrectly handles accrued interest at the forward coupon date rather than at the current valuation date.
• D. 0.984104 is the dirty price divided by 100 and omits accrued interest.

Module

Financial Markets and Products

Chapter

FMP 19: Interest Rate Futures

Learning objective

Explain and calculate a U.S. Treasury bond futures contract conversion factor

Correct answer: B. Kavya is correct, because the spread over the floating reference rate is only set for the current interest period and is reset at the market's view of the company's credit quality

A borrowing cost of Floating + 0.90% contains two separate components.

The swap can offset the floating reference rate, but it does not offset the 0.90% credit spread charged by the lender.

If the borrower's credit quality deteriorates, the spread charged when the floating borrowing is rolled can widen. The company may then face a higher total effective fixed cost than the original comparative-advantage calculation suggested.

This is the main criticism of the argument. The fixed borrowing quote may be locked for the full swap term, while the floating borrowing spread can be reset over time.

So Kavya is correct. The swap creates a fixed-rate liability only if the borrower can continue financing at the same credit spread.

Why the other options are wrong

• A. Rohan, both fixed for five years incorrectly assumes the swap fixes the borrower's credit spread.
• C. Rohan, floating offsets exactly correctly notes the reference-rate offset but ignores the remaining credit spread.
• D. Kavya, floating payments may fall short uses the wrong mechanism. The two floating reference-rate amounts offset; the issue is the changing borrowing spread.

Module

Financial Markets and Products

Chapter

FMP 20: Swaps

Learning objective

Describe the comparative advantage argument for the existence of interest rate swaps and evaluate some of the criticisms of this argument

Correct answer: B. 15.06%

First convert the investment ratio into portfolio weights.

If the fraction invested in A is x, the fraction invested in B is one quarter of x.

So 80% is invested in A and 20% in B.

The volatility of a two-asset portfolio is:

Here the w terms are portfolio weights, the sigma terms are asset volatilities, and rho is the correlation between the two returns.

Substitute the values.

The portfolio volatility is 15.06%.

The expected returns are not used in this calculation.

The result is below the weighted average volatility of 16.40% because the correlation is below 1, which creates diversification.

Why the other options are wrong

• A. 12.78% incorrectly omits the covariance term, the third piece under the square root.
• C. 15.20% incorrectly interprets B as one quarter of the total portfolio and uses weights of 0.75 and 0.25.
• D. 16.40% is the weighted average of the two volatilities and would apply only with perfect positive correlation.

Module

Valuation and Risk Models

Chapter

VRM 1: Measures of Financial Risk

Learning objective

Describe the mean-variance framework and the efficient frontier

Correct answer: C. Exactly two

Statements I and III are correct. Statement II is incorrect.

Statement I is correct because Monte Carlo simulation can fully revalue a portfolio under each simulated scenario.

So it can handle both linear and non-linear portfolios.

Statement II is incorrect.

Monte Carlo simulation is not restricted to normal risk factor distributions.

A multivariate normal distribution may be used as an intermediate step to impose the correlations, after which each sampled percentile is transformed into whatever distribution is actually wanted for that risk factor.

Statement III is correct.

Full portfolio revaluation over thousands of scenarios can be computationally expensive.

Using a delta-gamma approximation instead of full revaluation can reduce that cost, an approach sometimes called partial simulation.

So exactly two of the three statements are correct.

Why the other options are wrong

• A. None of the three incorrectly rejects both valid statements.
• B. Exactly one incorrectly rejects one of statements I and III, most often statement III on the view that the delta-gamma approximation belongs to the delta-normal model alone.
• D. All three incorrectly accepts the claim that risk factors themselves must be normally distributed.

Module

Valuation and Risk Models

Chapter

VRM 2: Calculating and Applying VaR

Learning objective

Explain the Monte Carlo simulation method for calculating VaR and ES and identify its strengths and weaknesses

Correct answer: C. 1.08%

The simplified estimator is:

Convert the percentage returns into decimals, so 1.1% becomes 0.011 and minus 0.6% becomes minus 0.006. Squaring removes every sign, so the four negative days contribute as the positive days do.

The squared returns are 0.000121, 0.000036, 0.000225, 0.000400, 0.000009, 0.000016, 0.000324, 0.000009, 0.000009 and 0.000016. They add to 0.001165.

So:

Take the square root:

The estimated daily volatility is 1.08%.

Day 4 is doing a large share of the work. Its squared return of 0.000400 is more than a third of the total, from a day whose return was only about twice the typical size.

Why the other options are wrong

• A. 0.01% reports the variance rate rather than its square root, and a variance rate is not a volatility.
• B. 0.87% averages the absolute returns instead of the squared returns. Averaging absolute returns is a real alternative and is considered more robust when the distribution is non-normal, but it is a different estimator and gives a different number.
• D. 1.14% divides by m minus 1 instead of the specified m.

Module

Valuation and Risk Models

Chapter

VRM 3: Measuring and Monitoring Volatility

Learning objective

Compare and contrast different approaches for estimating conditional volatility

Correct answer: C. A structured product's rating depended almost entirely on a model whose correlation inputs proved too optimistic, and because the agencies were open about their models the creators could design products to reach a target rating.

Corporate bond ratings combine quantitative analysis with judgment. The process uses historical and projected financial information, industry conditions, peer comparisons, financing plans, qualitative factors, and discussions with management.

Structured product ratings depend much more heavily on quantitative models.

Problems arose when key model inputs, including assumptions about correlations among mortgage defaults, proved too optimistic.

The issue became more severe for structured products built from other structured products.

The agencies also disclosed the models they used. This allowed product creators to work backward from the rating methodology, design products to achieve a target rating, obtain preliminary feedback, and modify structures until they met the desired threshold.

Why the other options are wrong

• A. A meeting with management describes part of the corporate bond rating process rather than the structured product process.
• B. Both halves wrong describes the corporate bond process and also reverses the transparency of the structured-product models.
• D. Confidential models correctly identifies the model-based process but incorrectly states that the models were not disclosed.

Module

Valuation and Risk Models

Chapter

VRM 4: External and Internal Credit Ratings

Learning objective

Explain historical failures and potential challenges to the use of credit ratings in making investment decisions

Correct answer: C. 79.74%

The rating must remain BBB through both the first and second years. Under the stated independence assumption, the two annual survival probabilities are multiplied.

The probability that the foreign currency rating remains BBB after two years equals 79.74 percent.

A one-year probability of 89.3 percent becomes less than 80 percent over two independent years. Under this calculation, roughly one BBB sovereign in five would move away from the rating within two years.

Why the other options are wrong

• A. 75.86% incorrectly squares the 87.1 percent local currency probability rather than the 89.3 percent foreign currency probability.
• B. 78.60% incorrectly subtracts the 10.7 percent annual probability of leaving the rating twice from 100 percent instead of multiplying the two survival probabilities.
• D. 89.30% incorrectly uses only the one-year probability and leaves out the second year.

Module

Valuation and Risk Models

Chapter

VRM 5: Country Risk: Determinants, Measures, and Implications

Learning objective

Describe factors that influence the level of sovereign default risk; explain and assess how rating agencies measure sovereign default risks

Correct answer: D. 4.36%

Expected loss equals the probability of default multiplied by the loss given default, meaning the proportion of the exposure lost if default occurs.

The expected loss rate is 1.26%. This is the average credit loss the bank expects and must recover through the interest rate rather than through capital.

The average interest rate must cover the expected loss, the expenses of managing the loans and the cost of funding them. The deposits funding the loans cost 1.7%.

The bank also requires an additional 20 basis points. A basis point means one hundredth of one percent, so 20 basis points equals 0.20%.

Why the other options are wrong

• A. 1.26% equals the expected loss alone and incorrectly stops before expenses, funding cost and the additional 20 basis points are included.
• B. 2.66% incorrectly omits the 1.7% funding cost. It includes the expected loss, expenses and 20 basis points only.
• C. 4.16% equals the average interest rate before the additional 20 basis points and incorrectly stops one step early.

Module

Valuation and Risk Models

Chapter

VRM 6: Measuring Credit Risk

Learning objective

Define and calculate expected loss (EL)

Correct answer: B. USD 100.0 million

Under the basic indicator approach, operational risk capital equals 15% of the three-year average annual gross income.

Gross income equals interest earned minus interest paid plus non-interest income.

Interest paid is subtracted, so year 1 gross income is:

For the other two years, gross income equals 900 minus 470 plus 240, or 670, and 960 minus 500 plus 270, or 730.

The three-year average is:

Applying the 15% rate gives:

The bank's operational risk capital requirement equals USD 100.0 million.

The basic indicator approach applies one percentage to total gross income. It does not distinguish among business lines or incorporate the bank's own operational loss history.

Why the other options are wrong

• A. USD 80.0 million incorrectly applies 12%, a business line percentage used under the standardized approach, to the correct gross income average.
• C. USD 109.5 million incorrectly uses only the most recent year's gross income rather than the three-year average.
• D. USD 170.0 million incorrectly omits interest paid from the gross income calculation.

Module

Valuation and Risk Models

Chapter

VRM 7: Operational Risk

Learning objective

Compare the basic indicator approach, the standardized approach, and the advanced measurement approach for calculating operational risk regulatory capital

Correct answer: A. The measure is conditional on a repeat of the stressed period, so it does not move when current market conditions change.

Using a fixed stressed window makes the measure conservative because the inputs come from a period of severe market conditions.

The same design also makes the measure less responsive to current market conditions.

If the stressed historical window remains unchanged, the number changes mainly because the current portfolio changes. A traditional measure can change because the portfolio changes and because the rolling data window incorporates recent market movements.

This creates a trade-off. A measure anchored to a crisis provides protection against severe conditions but may not indicate that today's market environment is changing.

For this reason, stressed measures complement traditional measures rather than replacing them.

The traditional measure reflects more recent market behavior, while the stressed measure reflects losses associated with a previously observed crisis period.

Why the other options are wrong

• B. A smaller capital number incorrectly reverses the effect of stressed inputs, which generally produce larger losses at the same confidence level.
• C. Positions not held in the stressed period incorrectly assumes the bank must have held the same portfolio historically. Historical risk-factor movements are applied to today's portfolio.
• D. The average loss beyond the threshold describes expected shortfall rather than a disadvantage created by stressed inputs.

Module

Valuation and Risk Models

Chapter

VRM 8: Stress Testing

Learning objective

Describe stressed VaR and stressed ES, including their advantages and disadvantages, and compare the process of determining stressed VaR and ES to that of traditional VaR and ES

Correct answer: C. USD 105.01

The law of one price says that two portfolios with identical cash flows on identical dates must have the same price.

The third bond can therefore be priced by constructing a portfolio of the first two bonds that reproduces its cash flows.

Per USD 100 of face value, the first two bonds pay semiannual coupons of USD 2.00 and USD 3.125. The third bond pays USD 4.00.

Let the weight on the first bond be w and the weight on the second bond be 1 - w. Because the weights add to 1, the portfolio repays USD 100 at maturity. The coupon condition is:

The negative weight means shorting USD 77.78 of face value of the 4% bond and buying USD 177.78 of face value of the 6.25% bond.

The third bond has a coupon higher than either available bond. Producing that higher coupon requires a short position in the lower-coupon bond and an overweight position in the higher-coupon bond.

The third bond should be priced at USD 105.01 per USD 100 of face value.

Why the other options are wrong

• A. USD 90.39 incorrectly removes the negative sign and uses weights of 0.7778 and 0.2222, which produces a semiannual coupon of only USD 2.25.
• B. USD 100.00 incorrectly assumes that bonds sharing a maturity date must trade at par, even though their coupon cash flows differ.
• D. USD 107.10 incorrectly adds the full USD 9.40 price difference between the first two bonds instead of applying the required portfolio weights.

Module

Valuation and Risk Models

Chapter

VRM 9: Pricing Conventions, Discounting, and Arbitrage

Learning objective

Define the "law of one price," explain it using an arbitrage argument, and describe how it can be applied to bond pricing

Correct answer: A. I and II only

A term structure flattens when the difference between long-maturity and short-maturity interest rates becomes smaller.

Description I has both interest rates falling, with the long-maturity rate falling by more. The difference narrows, so this is a bull flattener.

Description II has both interest rates rising, with the short-maturity rate rising by more. The difference also narrows, so this is a bear flattener.

Description III has both interest rates falling, but the short-maturity rate falls by more. The long-maturity rate therefore ends farther above the short-maturity rate, so the curve steepens.

Descriptions I and II are flattenings.

The terms flattening and steepening describe changes in the difference between maturities rather than the overall direction of interest rates. A curve can flatten while interest rates rise or while they fall, and the same holds for a steepening.

Why the other options are wrong

• B. I and III only incorrectly accepts description III even though the short-maturity interest rate falls by more and the difference widens.
• C. II and III only incorrectly accepts description III and incorrectly rejects the bull flattener in description I.
• D. I, II and III incorrectly classifies description III as a flattening instead of a steepening.

Module

Valuation and Risk Models

Chapter

VRM 10: Interest Rates

Learning objective

Define the "flattening" and "steepening" of rate curves and describe a trade to reflect expectations that a curve will flatten or steepen

Correct answer: C. USD 1.40

Under an unchanged term structure, interest rates for each remaining maturity are assumed to remain the same.

Six months later, the bond has eighteen months remaining. Its three remaining discounting periods use the same 1.00%, 1.60%, and 2.20% annual rates currently associated with those maturities.

Build the discount factors using half of each annual rate.

The bond has two coupon payments of USD 1.50 and a final payment of USD 101.50.

Carry roll-down equals the price change plus the coupon received.

The carry roll-down is USD 1.40 per USD 100 of face value.

Why the other options are wrong

• A. -USD 0.10 is the price change alone and incorrectly omits the coupon.
• B. USD 0.51 uses the forward-rates-realized assumption instead of the unchanged term structure.
• D. USD 1.50 includes only the coupon and incorrectly assumes no price change.

Module

Valuation and Risk Models

Chapter

VRM 11: Bond Yields and Return Calculations

Learning objective

Describe the common assumptions made about interest rates when calculating carry roll-down, and calculate carry roll-down under these assumptions

Correct answer: D. The barbell has the higher convexity, because convexity rises more than in proportion to duration as maturity lengthens, so the twenty-year bond adds more convexity than the five-year bond takes away.

The three bonds have effective durations of 4.68, 8.18 and 12.62, but effective convexities of 24.82, 78.90 and 212.46. For bonds like these, convexity rises much more than in proportion to duration as maturity lengthens.

The duration of a portfolio is the value-weighted average of the durations of its bonds, and its convexity is the value-weighted average of their convexities.

The barbell matches the bullet's duration of 8.18 with about 56% of its value in the five-year bond and 44% in the twenty-year bond. Its convexity with those weights is

So the barbell's convexity of 107.38 is well above the bullet's 78.90 at the same duration.

Measured against the bullet, the twenty-year bond's convexity is 133.56 higher and the five-year bond's is 54.08 lower. Weighted by value, the twenty-year bond adds 0.44 × 133.56 = 58.77 and the five-year bond takes away 0.56 × 54.08 = 30.28, which leaves the barbell about 28.5 above the bullet.

The higher convexity helps the barbell when the whole curve moves up or down by the same amount. The bullet can still do better when the curve changes shape, so the barbell does not beat the bullet for every change in the curve.

Why the other options are wrong

• A. Barbell higher, convexities added gets the direction right but incorrectly adds the two bonds' convexities, as the DV01s of a portfolio are added, when a portfolio's convexity is the value-weighted average of its bonds' convexities.
• B. Bullet higher, one bond gets the direction wrong, because the five-year bond's shortfall against the bullet is more than made up by the twenty-year bond's much larger excess.
• C. Bullet higher, convexity rises with maturity starts from the correct relationship but incorrectly compares the bullet with the barbell's five-year bond only, leaving out the twenty-year bond, which carries the most convexity.

Module

Valuation and Risk Models

Chapter

VRM 12: Applying Duration, Convexity, and DV01

Learning objective

Construct a barbell portfolio to match the cost and duration of a given bullet investment and explain the advantages and disadvantages of bullet and barbell portfolios

Correct answer: C. 0.6037

The bond first loses 0.9056 when its own two-year interest rate rises by one basis point. That sensitivity must then be allocated between the neighboring key rate shifts.

The zero-coupon bond pays USD 5,000 in two years. At 4.00% with semiannual compounding, there are four periods at 2.00%:

Raising the two-year interest rate to 4.01% gives four periods at 2.005%:

The 0.9056 is the bond's sensitivity to a one-basis-point increase in its two-year spot rate. Because two years is not itself a key rate, that sensitivity is split between the one-year and four-year key rate shifts according to their interpolation weights.

The two-year maturity lies between the one-year and four-year key rates. At two years, the one-year key rate shift contributes:

The four-year shift contributes the remaining one third, which equals 0.3019. The two KR01s sum to 0.9056.

Why the other options are wrong

• A. 0.3019 is the four-year KR01, corresponding to the complementary interpolation weight rather than the one-year shift.
• B. 0.4528 incorrectly divides the sensitivity equally between the two key rates even though two years is not the midpoint between one and four years.
• D. 0.9056 is the full sensitivity to the bond's own two-year spot rate before that sensitivity is allocated between the neighboring key rates.

Module

Valuation and Risk Models

Chapter

VRM 13: Modeling Non-Parallel Term Structure Shifts and Hedging

Learning objective

Calculate the KR01s of a portfolio given a set of key rates

Correct answer: A. Early exercise is optimal at the lower node at six months, where the payoff of USD 8.04 beats the roll-back value of USD 7.39.

At every pre-maturity node, the American put value is the larger of its immediate exercise payoff and its continuation value.

At the lower six-month node, the stock price is USD 24.962. Immediate exercise pays 33.00 - 24.962 = 8.038. The continuation value is 7.385. Immediate exercise gives the larger value, so the node value becomes USD 8.038.

At the upper six-month node, the stock price is USD 36.055, above the USD 33.00 strike. Immediate exercise pays zero, so the continuation value of USD 1.444 is retained.

Today, the put has intrinsic value of USD 3.00, while its continuation value is USD 4.591. Holding the option has the larger value, so immediate exercise is not optimal today.

Why the other options are wrong

• B. Exercise today at USD 3.00 incorrectly compares the immediate exercise payoff with zero instead of with the continuation value of USD 4.591.
• C. Exercise at both six-month nodes incorrectly treats the upper node as in the money even though USD 36.055 is above the USD 33.00 strike.
• D. No early exercise anywhere on the tree is correct at the upper node and today but incorrect at the lower six-month node, where 8.038 exceeds 7.385.

Module

Valuation and Risk Models

Chapter

VRM 14: Binomial Trees

Learning objective

Calculate the value of an American and a European call or put option using a one-step and two-step binomial model

Correct answer: B. USD 27.56 million

The cost of a new warrant issue to existing shareholders is the warrant value multiplied by the proportion of the enlarged company that the existing shareholders continue to own.

There are 40 million existing shares and 5 million warrants.

So the enlarged share count is 45 million.

The cost per warrant to existing shareholders is USD 5.511.

Multiply by the number of warrants.

The total cost to existing shareholders is approximately USD 27.56 million.

The dilution fraction reflects that existing shareholders own 40 out of 45 parts of the enlarged company after exercise.

Why the other options are wrong

• A. USD 3.44 million incorrectly uses the warrant holders' share of the enlarged company rather than the existing shareholders' share.
• C. USD 31.00 million incorrectly charges existing shareholders the full warrant value without applying the dilution fraction.
• D. USD 34.88 million incorrectly inverts the dilution fraction.

Module

Valuation and Risk Models

Chapter

VRM 15: The Black-Scholes-Merton Model

Learning objective

Describe warrants, calculate the value of a warrant, and calculate the dilution cost of the warrant to existing shareholders

Correct answer: D. The naked position loses if the stock price rises well above USD 105, and the covered position loses if the stock price falls well below USD 100

A naked written call is exposed to a large rise in the stock price. Below the USD 105 strike, the option expires worthless and the dealer keeps the USD 10,000,000 premium. Above the strike, the option payoff increases dollar for dollar with the stock price.

At a stock price of USD 125, the option payoff is USD 20 per share, or USD 20,000,000 across 1,000,000 options. After the USD 10,000,000 premium, the dealer has a net loss of USD 10,000,000.

A covered written call includes ownership of 1,000,000 shares purchased at USD 100. A rise above the strike is covered by the stock holding because the shares can be delivered at USD 105. A large fall in the stock price creates the main loss because the dealer continues to own the shares while the option expires worthless.

At USD 75, the stock position has lost USD 25,000,000 relative to its USD 100 purchase price. The USD 10,000,000 option premium offsets part of that loss.

So the naked call is exposed to rising stock prices, while the covered call is exposed to falling stock prices.

Why the other options are wrong

• A. Naked loses on a fall, covered loses on a rise incorrectly reverses the main risk direction of both positions.
• B. Naked loses on a rise, covered loses on a rise gets the naked position correct but incorrectly assigns the covered-call loss to the upside.
• C. Naked loses on a fall, covered loses on a fall gets the covered position correct but incorrectly assigns downside loss to the naked written call.

Module

Valuation and Risk Models

Chapter

VRM 16: Option Sensitivity Measures: The Greeks

Learning objective

Describe and assess the risks associated with naked and covered option positions