Your FREE Financial Risk Manager (FRM) Practice Test 2026 – 200+ Q&A

FRM Practice Questions

1. In the capital asset pricing model, what does the beta of a security measure?
   • The security's sensitivity to movements in the overall market portfolio
   • The total volatility of the security's returns including firm-specific risk
   • The security's expected return in excess of the risk-free rate
   • The correlation between the security and the risk-free asset

   Correct answer: The security's sensitivity to movements in the overall market portfolio

   Beta measures a security's sensitivity to movements in the overall market portfolio, capturing only systematic (non-diversifiable) risk. It is the covariance of the security's return with the market return divided by the variance of the market return. Total volatility describes standard deviation, not beta, and the excess return is the risk premium, a separate quantity.

2. A stock has a beta of 1.4, the risk-free rate is 3%, and the expected return on the market is 9%. Using the capital asset pricing model, what is the stock's expected return?
   • 12.6%
   • 11.4%
   • 9.0%
   • 15.6%

   Correct answer: 11.4%

   The expected return is 11.4%. CAPM gives expected return = risk-free rate + beta x (market return - risk-free rate) = $3\% + 1.4 \times (9\% - 3\%) = 3\% + 1.4 \times 6\% = 3\% + 8.4\% = 11.4\%$. The 12.6% figure mistakenly omits adding the risk-free rate to a different product, and 9.0% is simply the market return.

3. Under the capital asset pricing model, which type of risk is rewarded with a higher expected return?
   • Idiosyncratic risk specific to a single firm
   • Liquidity risk from thinly traded assets
   • Systematic risk that cannot be diversified away
   • Operational risk from internal process failures

   Correct answer: Systematic risk that cannot be diversified away

   Systematic risk that cannot be diversified away is the only risk rewarded under CAPM. Because investors can eliminate firm-specific (idiosyncratic) risk through diversification, the market does not compensate them for bearing it. Liquidity and operational risk are not the basis of CAPM's risk-return relationship.

4. What is the primary distinction between arbitrage pricing theory and the capital asset pricing model?
   • Arbitrage pricing theory ignores systematic risk entirely
   • Arbitrage pricing theory assumes investors are risk-seeking
   • Arbitrage pricing theory requires a normal distribution of returns
   • Arbitrage pricing theory uses multiple systematic risk factors rather than a single market factor

   Correct answer: Arbitrage pricing theory uses multiple systematic risk factors rather than a single market factor

   Arbitrage pricing theory uses multiple systematic risk factors rather than a single market factor. While CAPM explains expected return through one factor (the market), APT allows several macroeconomic or statistical factors to drive returns. APT still relies on systematic risk and does not require normally distributed returns.

5. In a multifactor arbitrage pricing theory model, a portfolio has a sensitivity of 0.8 to a factor with a risk premium of 4% and a sensitivity of 1.2 to a factor with a risk premium of 3%. With a risk-free rate of 2%, what is the portfolio's expected return?
   • 8.8%
   • 6.8%
   • 9.0%
   • 7.6%

   Correct answer: 8.8%

   The expected return is 8.8%. APT gives expected return = risk-free rate + sum of (factor sensitivity x factor premium) = $2\% + (0.8 \times 4\%) + (1.2 \times 3\%) = 2\% + 3.2\% + 3.6\% = 8.8\%$. Forgetting the risk-free rate yields 6.8%, which understates the result.

6. Arbitrage pricing theory assumes that any mispricing of an asset relative to its factor exposures will be:
   • Permanently sustained because markets are inefficient
   • Quickly eliminated by arbitrageurs who exploit risk-free profit opportunities
   • Corrected only by central bank intervention
   • Irrelevant because all assets share the same factor exposures

   Correct answer: Quickly eliminated by arbitrageurs who exploit risk-free profit opportunities

   Mispricing will be quickly eliminated by arbitrageurs who exploit risk-free profit opportunities. The no-arbitrage condition underpinning APT holds that investors will buy underpriced and sell overpriced assets until prices align with factor exposures. The theory does not assume permanent inefficiency or require central bank action.

7. In modern portfolio theory, the efficient frontier represents:
   • Portfolios that eliminate all systematic risk
   • The single portfolio with the lowest possible variance
   • Portfolios offering the highest expected return for each level of risk
   • Portfolios composed entirely of the risk-free asset

   Correct answer: Portfolios offering the highest expected return for each level of risk

   The efficient frontier represents portfolios offering the highest expected return for each level of risk. These are the optimal combinations that dominate all others; any portfolio below the frontier is inefficient. It is not limited to the minimum-variance portfolio, nor does it remove systematic risk.

8. Why does combining two imperfectly correlated assets reduce portfolio risk under modern portfolio theory?
   • Both assets must have negative expected returns
   • Diversification raises the correlation toward one
   • The risk-free rate falls as more assets are added
   • Their returns do not move in perfect lockstep, so part of each asset's volatility offsets the other

   Correct answer: Their returns do not move in perfect lockstep, so part of each asset's volatility offsets the other

   Risk falls because the returns do not move in perfect lockstep, so part of each asset's volatility offsets the other. Whenever the correlation is below one, the portfolio standard deviation is less than the weighted average of the individual standard deviations. Diversification lowers, not raises, effective correlation effects on risk.

9. A portfolio is split equally between two assets, each with a standard deviation of 20%, and their returns have a correlation of zero. What is the portfolio's standard deviation?
   • Approximately 14.1%
   • 20.0%
   • Approximately 28.3%
   • 0.0%

   Correct answer: Approximately 14.1%

   The portfolio standard deviation is approximately 14.1%. With equal weights of 0.5, each variance equal to $0.20^2 = 0.04$, and zero correlation, the cross term drops out: variance $= (0.5^2)(0.04) + (0.5^2)(0.04) = 0.01 + 0.01 = 0.02$, and $\sqrt{0.02} \approx 0.141$. The diversification benefit drops risk below the 20% of either asset held alone.

10. According to modern portfolio theory, what happens to the diversification benefit as the correlation between two assets approaches positive one?
    • The diversification benefit increases sharply
    • The diversification benefit disappears and portfolio risk approaches the weighted average of the individual risks
    • Portfolio risk falls to zero
    • The portfolio expected return doubles

    Correct answer: The diversification benefit disappears and portfolio risk approaches the weighted average of the individual risks

    As correlation approaches positive one, the diversification benefit disappears and portfolio risk approaches the weighted average of the individual risks. Perfectly correlated assets move together, so combining them provides no risk reduction. Only negative or low correlations deliver meaningful diversification.

11. Which statement best describes the purpose of enterprise risk management within a firm?
    • To eliminate all risk exposures the firm faces
    • To focus exclusively on market risk in the trading book
    • To manage risks across the entire organization in an integrated, top-down framework rather than in isolated silos
    • To replace the firm's internal audit function

    Correct answer: To manage risks across the entire organization in an integrated, top-down framework rather than in isolated silos

    Enterprise risk management aims to manage risks across the entire organization in an integrated, top-down framework rather than in isolated silos. It links risk appetite, strategy, and individual exposures so the firm sees aggregate and interacting risks. ERM neither eliminates all risk nor narrows focus to a single risk type.

12. A bank that previously managed credit, market, and operational risk in separate departments decides to adopt an enterprise risk management approach. What is the most likely benefit?
    • Complete removal of the need for risk capital
    • Guaranteed elimination of operational losses
    • Lower expected returns across all business lines
    • A consolidated view of aggregate risk that reveals concentrations and correlations across business lines

    Correct answer: A consolidated view of aggregate risk that reveals concentrations and correlations across business lines

    The most likely benefit is a consolidated view of aggregate risk that reveals concentrations and correlations across business lines. Siloed management can miss how exposures interact and accumulate; ERM aggregates them so leadership can manage total risk against appetite. ERM does not remove the need for capital or guarantee zero losses.

13. In an enterprise risk management framework, the firm's risk appetite is best described as:
    • The aggregate level and types of risk the firm is willing to accept in pursuit of its objectives
    • A regulatory minimum capital requirement set by supervisors
    • The maximum loss the firm experienced in the prior year
    • The forecasted return on the firm's investment portfolio

    Correct answer: The aggregate level and types of risk the firm is willing to accept in pursuit of its objectives

    Risk appetite is the aggregate level and types of risk the firm is willing to accept in pursuit of its objectives. It is set by the board and management and guides how much risk business lines may take. It is distinct from a regulatory capital floor or a backward-looking realized loss figure.

14. Which framework component is most central to enterprise risk management's integration of strategy and risk-taking?
    • Outsourcing all risk decisions to external consultants
    • Aligning the firm's risk appetite with its business strategy and capital allocation
    • Maximizing leverage to boost return on equity
    • Reporting only after losses exceed regulatory thresholds

    Correct answer: Aligning the firm's risk appetite with its business strategy and capital allocation

    The central component is aligning the firm's risk appetite with its business strategy and capital allocation. ERM connects how much risk the firm is willing to take with where it deploys capital and pursues opportunities. Outsourcing decisions and maximizing leverage run counter to integrated risk governance.

15. The Sharpe ratio measures a portfolio's:
    • Excess return per unit of systematic risk as measured by beta
    • Total return per unit of tracking error
    • Excess return per unit of total risk as measured by standard deviation
    • Return relative to a benchmark index only

    Correct answer: Excess return per unit of total risk as measured by standard deviation

    The Sharpe ratio measures excess return per unit of total risk as measured by standard deviation. It is computed as (portfolio return minus risk-free rate) divided by the portfolio's standard deviation. Using beta in the denominator describes the Treynor measure, not the Sharpe ratio.

16. A portfolio returns 11%, the risk-free rate is 3%, and the portfolio's standard deviation is 16%. What is its Sharpe ratio?
    • 0.69
    • 0.19
    • 1.38
    • 0.50

    Correct answer: 0.50

    The Sharpe ratio is 0.50. It equals (portfolio return - risk-free rate) / standard deviation = (11% - 3%) / 16% = 8% / 16% = 0.50. Dividing the full 11% return by the standard deviation, ignoring the risk-free subtraction, gives the incorrect 0.69.

17. Two managers report the same Sharpe ratio, but Manager A took on far more total volatility than Manager B. What can an analyst conclude?
    • Manager A generated proportionally higher excess return to compensate for the higher volatility
    • Manager A clearly outperformed on a risk-adjusted basis
    • Manager B took on more systematic risk
    • The two managers must have identical raw returns

    Correct answer: Manager A generated proportionally higher excess return to compensate for the higher volatility

    The analyst can conclude that Manager A generated proportionally higher excess return to compensate for the higher volatility. Because the Sharpe ratio holds excess return per unit of total risk constant, equal ratios with higher volatility imply higher absolute excess return. Equal ratios mean comparable risk-adjusted performance, not clear outperformance.

18. When comparing two portfolios using the Sharpe ratio, the portfolio with the higher Sharpe ratio is generally preferred because it:
    • Has a lower expected return
    • Delivers more excess return for each unit of total risk borne
    • Carries more idiosyncratic risk
    • Always has a higher beta

    Correct answer: Delivers more excess return for each unit of total risk borne

    The higher-Sharpe portfolio is preferred because it delivers more excess return for each unit of total risk borne. The ratio rewards efficient use of risk, so a higher value indicates better compensation per unit of volatility. A higher Sharpe ratio implies neither lower return nor a particular beta.

19. The Treynor measure differs from the Sharpe ratio primarily because it:
    • Ignores the risk-free rate entirely
    • Measures return relative to skewness
    • Divides excess return by beta rather than by total standard deviation
    • Uses gross return instead of excess return

    Correct answer: Divides excess return by beta rather than by total standard deviation

    The Treynor measure divides excess return by beta rather than by total standard deviation. It evaluates reward per unit of systematic risk, making it appropriate for well-diversified portfolios where idiosyncratic risk is negligible. Both Treynor and Sharpe use excess return over the risk-free rate.

20. Jensen's alpha for a portfolio is best interpreted as:
    • The portfolio's total return divided by its standard deviation
    • The portfolio's beta relative to the market
    • The risk-free rate earned on the portfolio
    • The portfolio's return above what the capital asset pricing model predicts for its beta

    Correct answer: The portfolio's return above what the capital asset pricing model predicts for its beta

    Jensen's alpha is the portfolio's return above what the capital asset pricing model predicts for its beta. A positive alpha indicates the manager earned more than the systematic-risk-implied return; a negative alpha indicates underperformance. Dividing return by standard deviation describes the Sharpe ratio instead.

21. A diversified portfolio earned 12% with a beta of 1.5. The risk-free rate is 2% and the market returned 8%. What is the portfolio's Jensen's alpha?
    • 1.0%
    • 2.0%
    • -1.0%
    • 4.0%

    Correct answer: 1.0%

    Jensen's alpha is 1.0%. The CAPM-required return is $2\% + 1.5 \times (8\% - 2\%) = 2\% + 9\% = 11\%$, and alpha equals the actual 12% minus the required 11%, giving 1.0%. Failing to subtract the required return from the actual return produces incorrect figures.

22. For evaluating a manager whose portfolio is only a small part of an investor's well-diversified holdings, which risk-adjusted measure is most appropriate?
    • The Sharpe ratio, because total risk fully matters
    • The Treynor measure, because only systematic risk is relevant in that context
    • The raw return with no risk adjustment
    • The portfolio's standard deviation alone

    Correct answer: The Treynor measure, because only systematic risk is relevant in that context

    The Treynor measure is most appropriate because only systematic risk is relevant in that context. When the portfolio is one piece of a larger diversified holding, idiosyncratic risk is already diversified away, so reward per unit of beta is the right gauge. The Sharpe ratio is better when the portfolio represents the investor's total wealth.

23. In risk management, expected loss is most accurately described as:
    • The worst-case loss in an extreme scenario
    • The loss that exceeds the firm's capital
    • The average loss anticipated over a period, typically covered by pricing and reserves
    • A loss that can never be predicted in advance

    Correct answer: The average loss anticipated over a period, typically covered by pricing and reserves

    Expected loss is the average loss anticipated over a period, typically covered by pricing and reserves. Because it is foreseeable, firms build it into product pricing and loan loss provisions as a cost of doing business. Worst-case and capital-exceeding losses describe unexpected loss instead.

24. Why do firms hold economic capital primarily against unexpected loss rather than expected loss?
    • Unexpected loss is always zero
    • Expected loss is larger than unexpected loss in every case
    • Regulators forbid holding capital against unexpected loss
    • Expected loss is anticipated and absorbed through pricing and reserves, while capital cushions deviations beyond the average

    Correct answer: Expected loss is anticipated and absorbed through pricing and reserves, while capital cushions deviations beyond the average

    Firms hold capital against unexpected loss because expected loss is anticipated and absorbed through pricing and reserves, while capital cushions deviations beyond the average. Expected loss is a predictable cost, so reserves cover it; capital protects solvency against unforeseen spikes. Unexpected loss is generally larger and material, not zero.

25. A lender prices a loan to recover the average historical default loss on similar loans. This pricing component corresponds to:
    • Expected loss
    • Unexpected loss
    • Economic capital
    • Operational risk capital

    Correct answer: Expected loss

    This component corresponds to expected loss. By recovering the average historical default loss through the loan's price, the lender treats expected loss as a routine cost embedded in the spread. Unexpected loss and economic capital address tail deviations beyond the average, not the average itself.

26. Unexpected loss is best characterized as:
    • The portion of loss recovered through collateral
    • The variability of losses around the expected loss, reflecting the uncertainty firms must hold capital against
    • A loss that is fully predictable from historical averages
    • The same quantity as expected loss

    Correct answer: The variability of losses around the expected loss, reflecting the uncertainty firms must hold capital against

    Unexpected loss is the variability of losses around the expected loss, reflecting the uncertainty firms must hold capital against. It captures how far actual losses can deviate from the average and drives economic capital needs. It is neither a recovery amount nor identical to expected loss.

27. The GARP Code of Conduct requires risk professionals to place which interest first?
    • Their employer's short-term profit above all else
    • Their own compensation above client outcomes
    • The integrity of the profession and the public interest above personal gain
    • The interests of regulators over those of clients

    Correct answer: The integrity of the profession and the public interest above personal gain

    The GARP Code of Conduct requires placing the integrity of the profession and the public interest above personal gain. Members must act with honesty, exercise due care, and avoid conduct that compromises the profession's reputation. Prioritizing personal compensation or short-term employer profit would violate these principles.

28. A risk manager discovers a material flaw in a model that overstates the firm's trading profits but is told to keep silent to protect a bonus pool. Under the GARP Code of Conduct, the manager should:
    • Conceal the flaw to preserve team morale
    • Disclose it only if a regulator specifically asks
    • Wait until after bonuses are paid to report it
    • Act with integrity and disclose the flaw, prioritizing professional standards over personal benefit

    Correct answer: Act with integrity and disclose the flaw, prioritizing professional standards over personal benefit

    The manager should act with integrity and disclose the flaw, prioritizing professional standards over personal benefit. The Code of Conduct demands honesty, professional integrity, and avoidance of conduct that misleads stakeholders. Concealing a material model flaw for personal or team gain breaches those duties.

29. Which behavior is consistent with the professional integrity principles of the GARP Code of Conduct?
    • Maintaining and improving professional competence and exercising independent judgment
    • Following any instruction from a superior without question
    • Sharing confidential client data to win new business
    • Accepting gifts that could impair objectivity

    Correct answer: Maintaining and improving professional competence and exercising independent judgment

    Maintaining and improving professional competence and exercising independent judgment is consistent with the Code of Conduct. Risk professionals are expected to stay current, act with diligence, and preserve objectivity. Blind obedience, breaching confidentiality, and accepting compromising gifts all conflict with the Code.

30. In modern portfolio theory, the capital market line plots the expected return and risk of:
    • Only portfolios of individual stocks with no risk-free asset
    • Efficient portfolios formed by combining the risk-free asset with the market portfolio
    • Portfolios that maximize idiosyncratic risk
    • The single global minimum-variance portfolio

    Correct answer: Efficient portfolios formed by combining the risk-free asset with the market portfolio

    The capital market line plots efficient portfolios formed by combining the risk-free asset with the market portfolio. Its slope is the market's reward-to-variability ratio, and every point represents an optimal mix of lending or borrowing at the risk-free rate with the market portfolio. It is not limited to stock-only portfolios.

31. Within the capital asset pricing model, the security market line describes the relationship between:
    • Expected return and total standard deviation
    • Beta and idiosyncratic risk
    • Expected return and beta for individual securities and portfolios
    • Price and trading volume

    Correct answer: Expected return and beta for individual securities and portfolios

    The security market line describes the relationship between expected return and beta for individual securities and portfolios. Any correctly priced asset plots on the line, with expected return rising linearly in beta. Plotting return against total standard deviation describes the capital market line, not the security market line.

32. A security plots above the security market line. According to the capital asset pricing model, this implies the security is:
    • Overvalued and should be sold
    • Earning exactly its required return
    • Free of systematic risk
    • Undervalued, offering more expected return than its beta justifies

    Correct answer: Undervalued, offering more expected return than its beta justifies

    A security plotting above the security market line is undervalued, offering more expected return than its beta justifies. Investors would buy it, pushing its price up and return down until it lies on the line. A security plotting below the line would be overvalued.

33. One key assumption of arbitrage pricing theory that distinguishes it from the capital asset pricing model is that APT:
    • Does not require identifying the true market portfolio
    • Assumes a single risk factor explains all returns
    • Requires investors to hold only the market portfolio
    • Ignores no-arbitrage conditions

    Correct answer: Does not require identifying the true market portfolio

    A key assumption is that APT does not require identifying the true market portfolio. CAPM depends on a theoretically all-inclusive market portfolio that is hard to observe, whereas APT relies on factor exposures and no-arbitrage. APT explicitly uses multiple factors and the no-arbitrage principle.

34. Which statement about the Sharpe ratio is most accurate when comparing a fund to its benchmark?
    • It rewards funds purely for raw return regardless of risk
    • It penalizes a fund for taking on higher total volatility even if returns are high
    • It uses beta to capture only market risk
    • It is unaffected by the risk-free rate

    Correct answer: It penalizes a fund for taking on higher total volatility even if returns are high

    The Sharpe ratio penalizes a fund for taking on higher total volatility even if returns are high. Because total standard deviation is the denominator, a high-return fund with very high volatility can have a modest Sharpe ratio. The measure depends directly on the risk-free rate and uses standard deviation, not beta.

35. In modern portfolio theory, an investor who can borrow and lend at the risk-free rate will choose a portfolio that:
    • Lies strictly inside the efficient frontier
    • Holds only the risk-free asset
    • Lies on the capital market line, combining the risk-free asset with the optimal risky portfolio
    • Maximizes total standard deviation

    Correct answer: Lies on the capital market line, combining the risk-free asset with the optimal risky portfolio

    The investor will choose a portfolio that lies on the capital market line, combining the risk-free asset with the optimal risky portfolio. This separation result shows that all investors hold the same risky tangency portfolio and adjust risk only through their risk-free allocation. Such portfolios dominate any point inside the efficient frontier.

36. A firm's economic capital is set to absorb losses up to a high confidence level. This capital is intended to cover:
    • Expected loss embedded in product pricing
    • Only routine operating expenses
    • Realized profits from prior years
    • Unexpected loss beyond the expected loss already provisioned

    Correct answer: Unexpected loss beyond the expected loss already provisioned

    Economic capital is intended to cover unexpected loss beyond the expected loss already provisioned. Expected loss is funded through pricing and reserves, while capital protects the firm against adverse deviations up to a chosen confidence level. It is not a provision for average losses or routine expenses.

37. Which scenario best illustrates a failure that effective enterprise risk management is designed to prevent?
    • Separate desks each taking large exposures to the same counterparty without any aggregate limit
    • A single trader hedging a position within an approved limit
    • A firm reporting its consolidated risk to the board quarterly
    • A risk committee setting a firm-wide risk appetite

    Correct answer: Separate desks each taking large exposures to the same counterparty without any aggregate limit

    The failure ERM is designed to prevent is separate desks each taking large exposures to the same counterparty without any aggregate limit. Without an integrated view, concentrated risk can build unnoticed across silos. The other choices describe sound practices that ERM promotes, not failures.

38. An analyst ranks two total-wealth portfolios and prefers the one with the higher Sharpe ratio. This choice is appropriate because the Sharpe ratio:
    • Accounts only for systematic risk
    • Accounts for total risk, which is the relevant measure when the portfolio is the investor's entire holding
    • Ignores the risk-free rate in the comparison
    • Measures return relative to a peer group median

    Correct answer: Accounts for total risk, which is the relevant measure when the portfolio is the investor's entire holding

    The choice is appropriate because the Sharpe ratio accounts for total risk, which is the relevant measure when the portfolio is the investor's entire holding. When no other assets diversify away idiosyncratic risk, total standard deviation captures all risk the investor bears. The Treynor measure, by contrast, focuses only on systematic risk.

39. Under the capital asset pricing model, if the market risk premium rises while a stock's beta stays constant, the stock's required return will:
    • Decrease, because beta offsets the premium
    • Stay unchanged, because beta is constant
    • Increase, because required return rises with the market risk premium scaled by beta
    • Fall to the risk-free rate

    Correct answer: Increase, because required return rises with the market risk premium scaled by beta

    The required return will increase, because required return rises with the market risk premium scaled by beta. CAPM adds beta times the market risk premium to the risk-free rate, so a larger premium raises required return for any positive beta. A constant beta does not freeze the required return when the premium changes.

40. A core insight of modern portfolio theory is that an asset should be judged by:
    • Its standalone standard deviation in isolation
    • Its historical price alone
    • Its idiosyncratic risk only
    • Its contribution to the risk and return of the overall portfolio, not its standalone risk

    Correct answer: Its contribution to the risk and return of the overall portfolio, not its standalone risk

    The core insight is that an asset should be judged by its contribution to the risk and return of the overall portfolio, not its standalone risk. Because correlations determine how an asset affects portfolio variance, a volatile asset can reduce total risk if it diversifies the portfolio. Standalone volatility alone is misleading.

41. A risk professional is offered a lucrative consulting role that would create a conflict with current client duties. Under the GARP Code of Conduct, the professional should first:
    • Disclose the potential conflict of interest to the affected parties
    • Accept the role and conceal it to avoid awkward conversations
    • Decline to inform anyone unless a loss occurs
    • Prioritize the higher-paying engagement automatically

    Correct answer: Disclose the potential conflict of interest to the affected parties

    The professional should first disclose the potential conflict of interest to the affected parties. The Code of Conduct requires transparency so that clients and employers can assess and manage conflicts. Concealing the conflict or automatically favoring the higher-paying role would breach the duty of integrity and fair dealing.

42. Which limitation of the capital asset pricing model does arbitrage pricing theory specifically address?
    • CAPM's assumption that returns are negatively skewed
    • CAPM's reliance on a single factor that may not capture all sources of systematic risk
    • CAPM's exclusion of the risk-free asset
    • CAPM's requirement that all betas equal one

    Correct answer: CAPM's reliance on a single factor that may not capture all sources of systematic risk

    APT addresses CAPM's reliance on a single factor that may not capture all sources of systematic risk. By permitting multiple macroeconomic or statistical factors, APT can explain return patterns that a one-factor model misses. CAPM does include a risk-free asset and does not assume all betas equal one.

43. A portfolio's Sharpe ratio falls after the manager adds a highly volatile position that did not raise the portfolio's return. The decline occurs because:
    • The risk-free rate rose sharply
    • Systematic risk fell while total risk was unchanged
    • Total risk in the denominator increased while excess return stayed flat
    • The portfolio's beta dropped to zero

    Correct answer: Total risk in the denominator increased while excess return stayed flat

    The decline occurs because total risk in the denominator increased while excess return stayed flat. The Sharpe ratio is excess return divided by standard deviation, so adding volatility without added return lowers the ratio. The change is driven by the denominator, not by a shift in the risk-free rate or beta.

44. In an enterprise risk management framework, who holds ultimate responsibility for approving the firm's risk appetite?
    • Individual trading desk heads
    • The external auditors
    • The most junior risk analyst
    • The board of directors

    Correct answer: The board of directors

    The board of directors holds ultimate responsibility for approving the firm's risk appetite. Sound ERM governance places risk-appetite oversight at the top so it aligns with strategy and is enforced firm-wide. Trading desks operate within the limits, and external auditors review controls rather than set appetite.

45. Two portfolios have identical expected returns, but Portfolio X has a lower standard deviation than Portfolio Y. According to modern portfolio theory, a risk-averse investor will:
    • Prefer Portfolio X because it offers the same return with less risk
    • Prefer Portfolio Y because higher risk guarantees higher return
    • Be indifferent because returns are equal
    • Prefer whichever has the higher beta

    Correct answer: Prefer Portfolio X because it offers the same return with less risk

    A risk-averse investor will prefer Portfolio X because it offers the same return with less risk. Modern portfolio theory holds that, for equal expected returns, lower variance is always preferred. Higher risk does not guarantee higher return, so Portfolio Y is dominated here.

46. A poorly diversified portfolio shows a strong Treynor measure but a weak Sharpe ratio. What does this divergence most likely indicate?
    • The portfolio has no systematic risk at all
    • The portfolio carries substantial idiosyncratic risk that total-risk measures penalize but beta-based measures ignore
    • The risk-free rate was applied inconsistently between the two measures
    • The portfolio's beta must be negative

    Correct answer: The portfolio carries substantial idiosyncratic risk that total-risk measures penalize but beta-based measures ignore

    The divergence most likely indicates the portfolio carries substantial idiosyncratic risk that total-risk measures penalize but beta-based measures ignore. The Sharpe ratio uses total standard deviation, so undiversified firm-specific risk drags it down, while the Treynor measure, based only on beta, is unaffected. This is why total-risk measures are preferred for undiversified, standalone portfolios.

47. A random variable has a probability density function. What does integrating that density over the entire range of possible values always equal?
    • One
    • Zero
    • The mean of the distribution
    • The variance of the distribution

    Correct answer: One

    The integral of a probability density function over its entire support equals one, because total probability must sum to certainty. The mean and variance are computed from weighted integrals of the variable and its squared deviations, not from the density alone, and zero would imply no probability mass exists.

48. For a continuous random variable, the cumulative distribution function F(x) gives which of the following?
    • The probability that the variable equals exactly x
    • The probability that the variable takes a value less than or equal to x
    • The rate of change of probability at the point x
    • The expected value of the variable above x

    Correct answer: The probability that the variable takes a value less than or equal to x

    The cumulative distribution function F(x) gives the probability that the variable is less than or equal to x, so it is non-decreasing and ranges from 0 to 1. The probability density function, not the CDF, describes the rate of change of probability, and for a continuous variable the probability of any exact value is zero.

49. The kurtosis of a normal distribution is 3. A return distribution with kurtosis well above 3 is best described as:
    • Platykurtic, with thinner tails than the normal
    • Perfectly symmetric with no excess risk in the tails
    • Leptokurtic, with fatter tails and a higher peak than the normal
    • Negatively skewed by definition

    Correct answer: Leptokurtic, with fatter tails and a higher peak than the normal

    A distribution with kurtosis above 3 is leptokurtic, meaning it has fatter tails and a more peaked center than the normal, which raises the chance of extreme outcomes. Platykurtic describes kurtosis below 3, and kurtosis measures tail weight rather than skewness, so it says nothing about symmetry by itself.

50. A risk analyst describes daily equity returns as having significant excess kurtosis. The most direct implication for risk measurement is that:
    • Average returns are systematically overstated
    • The mean and median must be equal
    • Volatility is constant through time
    • Extreme losses occur more often than a normal distribution would predict

    Correct answer: Extreme losses occur more often than a normal distribution would predict

    Significant excess kurtosis means the return distribution has fat tails, so extreme losses occur more frequently than a normal model predicts, leading normal-based risk estimates to understate tail risk. Kurtosis concerns tail thickness, not the location of the mean or whether volatility is constant.

51. In a Student's t-distribution, how does the shape compare to the standard normal distribution as the degrees of freedom decrease?
    • The tails become fatter, assigning more probability to extreme outcomes
    • The tails become thinner, concentrating probability near the mean
    • The distribution becomes increasingly skewed to the right
    • The mean shifts progressively above zero

    Correct answer: The tails become fatter, assigning more probability to extreme outcomes

    As the degrees of freedom of a Student's t-distribution decrease, the tails become fatter, assigning more probability to extreme outcomes, which makes it useful for modeling heavy-tailed financial returns. The t-distribution remains symmetric with a mean of zero, so it neither becomes skewed nor shifts its center.

52. An analyst uses a Student's t-distribution rather than the normal distribution to estimate value at risk because it:
    • Always produces a lower, more optimistic risk estimate
    • Better captures the fat tails observed in financial return data
    • Eliminates the need to estimate volatility
    • Guarantees the returns are independent over time

    Correct answer: Better captures the fat tails observed in financial return data

    The Student's t-distribution better captures the fat tails observed in financial returns, producing more conservative tail-risk estimates than the normal for the same volatility. It does not remove the need to estimate volatility, nor does it impose independence, and its heavier tails generally raise rather than lower risk estimates.

53. The chi-squared distribution is most commonly used in risk analysis to test:
    • The mean of two independent samples
    • The slope of a single-factor regression
    • Hypotheses about a population variance
    • Whether two assets are perfectly correlated

    Correct answer: Hypotheses about a population variance

    The chi-squared distribution is used to test hypotheses about a population variance, such as whether a model's variance equals a specified value. Tests on differences in means rely on the t-distribution, and regression slope significance uses a t-test, so those do not employ the chi-squared distribution.

54. The F-distribution arises when comparing which two quantities?
    • The difference between two sample means
    • A sample proportion against a hypothesized proportion
    • The covariance of two return series
    • The ratio of two independent sample variances

    Correct answer: The ratio of two independent sample variances

    The F-distribution arises from the ratio of two independent sample variances, which is why it underlies tests of equality of variances and the overall significance of a regression. Differences in means use the t-distribution and proportions use a z-test, so neither generates an F-statistic.

55. A confidence interval for a population mean becomes wider when which of the following increases, holding all else constant?
    • The desired confidence level
    • The sample size
    • The number of observations beyond the minimum
    • The precision of the estimate

    Correct answer: The desired confidence level

    Raising the desired confidence level widens the interval because greater certainty requires capturing a broader range of values. Increasing the sample size narrows the interval by improving precision, so larger samples and higher precision both work in the opposite direction of widening.

56. An analyst constructs a 95% confidence interval for a mean return and obtains 4% to 8%. The correct interpretation is that:
    • There is a 95% probability the next observed return falls between 4% and 8%
    • If the procedure were repeated many times, about 95% of such intervals would contain the true mean
    • The true mean has a 95% chance of being exactly 6%
    • Exactly 95% of historical returns lie between 4% and 8%

    Correct answer: If the procedure were repeated many times, about 95% of such intervals would contain the true mean

    The correct interpretation is that if the estimation procedure were repeated many times, about 95% of the resulting intervals would contain the true mean. The interval describes the location of the unknown parameter, not the probability of a single future return, and it makes no claim about exactly 6% or about the spread of historical data.

57. A point estimator is described as unbiased when:
    • It always produces the same value across samples
    • It has the smallest possible variance among all estimators
    • Its expected value equals the true population parameter
    • Its value converges to zero as the sample grows

    Correct answer: Its expected value equals the true population parameter

    An estimator is unbiased when its expected value equals the true population parameter, meaning it is correct on average across repeated samples. Minimum variance describes efficiency rather than unbiasedness, and an estimator producing identical or vanishing values would not generally track the parameter.

58. Among the desirable properties of an estimator, consistency means that the estimator:
    • Has zero bias in every finite sample
    • Is always normally distributed
    • Produces the lowest possible standard error in small samples
    • Converges toward the true parameter as the sample size grows large

    Correct answer: Converges toward the true parameter as the sample size grows large

    Consistency means the estimator converges toward the true parameter as the sample size grows large, so estimation error shrinks with more data. It does not require zero bias in finite samples, a normal sampling distribution, or the smallest small-sample standard error, which are separate properties.

59. In a quantile-quantile (QQ) plot used to assess whether returns are normally distributed, points that curve away from the straight reference line in the tails indicate:
    • Departures from normality such as fat tails
    • Perfect agreement with the normal distribution
    • A strong linear relationship between two assets
    • The presence of serial correlation in the data

    Correct answer: Departures from normality such as fat tails

    Points that curve away from the reference line in the tails of a QQ plot indicate departures from normality, such as fat tails, which is common in financial returns. Points lying on the line would indicate normality, while QQ plots assess distributional shape rather than correlation or serial dependence.

60. A risk team fits a probability distribution to historical loss data and compares competing fits. A graphical tool well suited to checking how well a chosen distribution matches the empirical data is:
    • A correlation matrix of asset returns
    • A QQ plot comparing empirical and theoretical quantiles
    • A control chart of daily volatility
    • A scatter plot of two regression residuals

    Correct answer: A QQ plot comparing empirical and theoretical quantiles

    A QQ plot comparing empirical and theoretical quantiles is well suited to checking distributional fit, because systematic deviation from the reference line reveals where the model and data disagree. A correlation matrix, volatility control chart, and residual scatter plot address relationships, variability, and regression diagnostics rather than overall distributional fit.

61. A discrete random variable representing the number of defaults in a fixed pool over a year is best modeled with which type of distribution?
    • A continuous distribution defined on all real numbers
    • A distribution that can take fractional default counts
    • A discrete distribution defined only on non-negative integers
    • A distribution with no upper or lower bound on outcomes

    Correct answer: A discrete distribution defined only on non-negative integers

    A count of defaults is a discrete random variable defined only on non-negative integers, so distributions such as the binomial or Poisson are appropriate. Continuous distributions, fractional outcomes, and unbounded real-valued ranges do not fit a quantity that can only be whole, non-negative counts.

62. The expected value of a discrete random variable is calculated as:
    • The simple average of the possible outcomes
    • The most likely single outcome
    • The range between the highest and lowest outcomes
    • The sum of each outcome multiplied by its probability

    Correct answer: The sum of each outcome multiplied by its probability

    The expected value of a discrete random variable is the probability-weighted sum of each outcome times its probability, which is the long-run average. A simple average ignores the probabilities, the most likely outcome is the mode, and the range describes dispersion rather than the central tendency.

63. The variance of a random variable is defined as the expected value of:
    • The squared deviations from the mean
    • The absolute deviations from the median
    • The outcomes themselves
    • The product of two correlated variables

    Correct answer: The squared deviations from the mean

    Variance is the expected value of the squared deviations from the mean, which is why it is always non-negative and emphasizes larger deviations. The expected absolute deviation is a different dispersion measure, the expected outcome is the mean, and the product of two variables relates to covariance.

64. When two random variables are independent, the variance of their sum equals:
    • The product of their individual variances
    • The sum of their individual variances
    • The sum of their variances plus twice their covariance
    • The larger of the two variances

    Correct answer: The sum of their individual variances

    For independent variables the covariance is zero, so the variance of the sum equals the sum of the individual variances. The general formula adds twice the covariance, but that term vanishes under independence, and neither the product nor the larger single variance gives the correct combined variance.

65. A portfolio holds two positions with equal weights, variances of 0.04 each, and a covariance of 0.02. The variance of the equally weighted portfolio return is:
    • 0.06
    • 0.02
    • 0.03
    • 0.10

    Correct answer: 0.03

    The portfolio variance is $(0.5^2)(0.04) + (0.5^2)(0.04) + 2(0.5)(0.5)(0.02) = 0.01 + 0.01 + 0.01 = 0.03$. The covariance term raises the variance above the simple weighted average of the individual variances, so 0.03 is correct rather than the larger or smaller alternatives.

66. A regression reports an R-squared of 0.64. The most accurate interpretation is that:
    • The model's slope coefficient equals 0.64
    • 64% of the predictions are exactly correct
    • The correlation between the variables is 0.64
    • 64% of the variation in the dependent variable is explained by the model

    Correct answer: 64% of the variation in the dependent variable is explained by the model

    An R-squared of 0.64 means 64% of the variation in the dependent variable is explained by the regression model. R-squared is not a slope coefficient or an accuracy rate, and in a simple regression the correlation would be the $\sqrt{R^2}$, which is 0.8 rather than 0.64.

67. Adjusted R-squared differs from ordinary R-squared because it:
    • Penalizes the addition of explanatory variables that do not improve the model
    • Always increases when any variable is added
    • Measures the statistical significance of each coefficient
    • Equals the F-statistic of the regression

    Correct answer: Penalizes the addition of explanatory variables that do not improve the model

    Adjusted R-squared penalizes the addition of explanatory variables that do not meaningfully improve fit, so it can fall when an unhelpful variable is added. Ordinary R-squared never decreases when variables are added, and coefficient significance and the F-statistic are separate diagnostics.

68. In time series analysis, an autoregressive AR(1) process expresses the current value as:
    • A function only of past error terms
    • A function of its own immediately prior value plus a random error
    • A constant with no dependence on prior values
    • The sum of two independent moving averages

    Correct answer: A function of its own immediately prior value plus a random error

    An AR(1) process models the current value as a function of its own immediately prior value plus a random error term, capturing first-order serial dependence. A model based only on past errors is a moving average, and a process with no dependence on prior values would be white noise.

69. In an AR(1) model written as $y_t = a + b\,y_{t-1} + e_t$, the process is covariance stationary only if:
    • The coefficient b equals exactly one
    • The constant a equals zero
    • The absolute value of the coefficient b is less than one
    • The error term has nonzero mean

    Correct answer: The absolute value of the coefficient b is less than one

    An AR(1) process is covariance stationary only if the absolute value of the slope coefficient b is less than one, which keeps shocks from accumulating without bound. A coefficient of exactly one produces a unit-root random walk, and stationarity does not require a zero constant or depend on the error mean being nonzero.

70. A seasonal pattern in a monthly time series is best handled by:
    • Ignoring it because it averages out over time
    • Increasing the regression's confidence level
    • Replacing the series with its first moment
    • Including seasonal dummy variables or seasonally adjusting the data

    Correct answer: Including seasonal dummy variables or seasonally adjusting the data

    Seasonality is best handled by including seasonal dummy variables or seasonally adjusting the data so the recurring pattern does not distort estimates. Seasonal effects do not simply average away, and changing a confidence level or reducing the series to its mean would not remove the systematic seasonal structure.

71. The mean reversion level of a stationary AR(1) process with intercept a and slope b is given by:
    • A divided by (1 minus b)
    • A multiplied by b
    • B divided by a
    • A plus b

    Correct answer: A divided by (1 minus b)

    The long-run mean, or mean reversion level, of a stationary AR(1) process equals a divided by (1 minus b), the value toward which the series tends over time. The other expressions do not yield the unconditional mean implied by the recursive structure of the model.

72. A measured time series that drifts persistently away from any fixed level and shows shocks that do not decay is most likely:
    • Strongly mean reverting
    • Non-stationary, exhibiting a unit root
    • White noise with zero autocorrelation
    • A stationary AR(1) with a small slope

    Correct answer: Non-stationary, exhibiting a unit root

    A series whose shocks persist and which drifts without returning to a fixed level is non-stationary and exhibits a unit root, the hallmark of a random walk. Mean-reverting, white-noise, and small-slope AR(1) series all return toward a stable level, which contradicts persistent drift.

73. In a linear regression, multicollinearity among explanatory variables tends to:
    • Bias the coefficient estimates toward zero
    • Lower the overall R-squared of the regression
    • Inflate the standard errors of the affected coefficients
    • Eliminate serial correlation in the residuals

    Correct answer: Inflate the standard errors of the affected coefficients

    Multicollinearity inflates the standard errors of the correlated coefficients, making individual estimates imprecise and often statistically insignificant even when the model fits well overall. It does not systematically bias the coefficients, does not reduce R-squared, and is unrelated to serial correlation.

74. A warning sign of multicollinearity in a multiple regression is:
    • A very low F-statistic with all coefficients significant
    • Residuals that are perfectly normally distributed
    • A slope coefficient exactly equal to one
    • A high overall R-squared but few individually significant coefficients

    Correct answer: A high overall R-squared but few individually significant coefficients

    A classic warning sign of multicollinearity is a high overall R-squared combined with few individually significant coefficients, because correlated regressors make it hard to isolate each variable's effect. Normally distributed residuals and a slope of one are unrelated, and multicollinearity does not typically produce a low F-statistic with all coefficients significant.

75. The standard error of the mean for a sample is calculated as:
    • The sample standard deviation divided by the $\sqrt{\text{sample size}}$
    • The sample variance divided by the sample size
    • The sample standard deviation multiplied by the sample size
    • The range of the sample divided by the sample size

    Correct answer: The sample standard deviation divided by the $\sqrt{\text{sample size}}$

    The standard error of the mean equals the sample standard deviation divided by the $\sqrt{\text{sample size}}$, so it shrinks as the sample grows. Dividing by the sample size, multiplying by it, or using the range would not yield the correct measure of the precision of the sample mean.

76. As the sample size used to estimate a mean increases fourfold, the standard error of the mean changes by a factor of:
    • One quarter
    • One half
    • Two
    • Four

    Correct answer: One half

    Because the standard error of the mean is inversely proportional to the $\sqrt{\text{sample size}}$, quadrupling the sample divides the standard error by $\sqrt{4}$, which is two, leaving it at one half. It does not fall to one quarter or rise, since larger samples reduce, not increase, sampling error.

77. A p-value in a hypothesis test represents the:
    • Probability that the null hypothesis is true
    • Probability of committing a Type II error
    • Probability of obtaining a result at least as extreme as observed, assuming the null hypothesis is true
    • Size of the estimated effect

    Correct answer: Probability of obtaining a result at least as extreme as observed, assuming the null hypothesis is true

    The p-value is the probability of obtaining a result at least as extreme as the one observed, assuming the null hypothesis is true; a small p-value casts doubt on the null. It is not the probability that the null is true, not the Type II error rate, and not a measure of effect size.

78. An analyst tests a coefficient and obtains a p-value of 0.02 against a significance level of 0.05. The correct decision is to:
    • Fail to reject the null hypothesis because the result is not extreme enough
    • Accept the null hypothesis as proven true
    • Conclude there is a 2% chance the alternative is correct
    • Reject the null hypothesis because the p-value is below the significance level

    Correct answer: Reject the null hypothesis because the p-value is below the significance level

    Because the p-value of 0.02 is below the 0.05 significance level, the analyst rejects the null hypothesis. A p-value below alpha is by definition extreme enough to reject, hypothesis testing never proves the null true, and the p-value does not give the probability that the alternative is correct.

79. A one-tailed hypothesis test differs from a two-tailed test in that it:
    • Places the entire rejection region in one tail of the distribution
    • Always uses a larger critical value
    • Tests for any difference regardless of direction
    • Requires a larger sample to be valid

    Correct answer: Places the entire rejection region in one tail of the distribution

    A one-tailed test places the entire rejection region in a single tail because the alternative specifies a direction, making it more powerful in that direction for a given significance level. A two-tailed test splits the rejection region across both tails to detect a difference in either direction, and sample size requirements are not what distinguishes the two.

80. An analyst wants to test whether a strategy's mean return is strictly greater than zero. The appropriate test is:
    • A two-tailed test with rejection regions in both tails
    • A one-tailed test with the rejection region in the upper tail
    • A test of variance using the chi-squared distribution
    • A test that the median equals the mean

    Correct answer: A one-tailed test with the rejection region in the upper tail

    Testing whether a mean return is strictly greater than zero is a directional claim, so a one-tailed test with the rejection region in the upper tail is appropriate. A two-tailed test addresses differences in either direction, and variance or median tests answer different questions entirely.

81. The law of large numbers states that as the number of independent observations increases, the sample average:
    • Converges to zero regardless of the distribution
    • Becomes increasingly volatile
    • Converges to the true population mean
    • Equals the population variance

    Correct answer: Converges to the true population mean

    The law of large numbers states that as the number of independent observations increases, the sample average converges to the true population mean. It does not drive the average to zero, increase its volatility, or relate the average to the population variance.

82. A simulation that draws a growing number of independent samples to approximate an expected payoff relies on which principle to justify that the average converges to the true value?
    • The principle of mean reversion
    • Bayes' theorem
    • The square-root-of-time rule
    • The law of large numbers

    Correct answer: The law of large numbers

    Such a simulation relies on the law of large numbers, which guarantees that the average of many independent draws converges to the true expected value. Mean reversion describes a different time-series property, Bayes' theorem governs conditional probability updating, and the square-root rule scales volatility over time.

83. Two events A and B are mutually exclusive. The probability that either A or B occurs equals:
    • The sum of their individual probabilities
    • The product of their individual probabilities
    • The larger of the two probabilities
    • Zero

    Correct answer: The sum of their individual probabilities

    For mutually exclusive events, which cannot both occur, the probability that either occurs equals the sum of their individual probabilities because there is no overlap to subtract. The product applies to the joint probability of independent events, not to a union, so it does not apply here.

84. Two events are statistically independent if and only if:
    • They cannot occur at the same time
    • The probability of their joint occurrence equals the product of their individual probabilities
    • Their probabilities sum to one
    • One event always causes the other

    Correct answer: The probability of their joint occurrence equals the product of their individual probabilities

    Two events are independent if and only if the probability of their joint occurrence equals the product of their individual probabilities, meaning one does not affect the other. Events that cannot occur together are mutually exclusive rather than independent, and summing to one or causation describe different relationships.

85. A skewness coefficient of zero for a return distribution indicates that the distribution is:
    • Heavily left-tailed
    • Heavily right-tailed
    • Symmetric about its mean
    • Necessarily normal

    Correct answer: Symmetric about its mean

    A skewness coefficient of zero indicates the distribution is symmetric about its mean, with the left and right sides mirroring each other. Negative skew indicates a longer left tail and positive skew a longer right tail, and symmetry alone does not guarantee the distribution is normal.

86. A Type I error in hypothesis testing occurs when an analyst:
    • Fails to reject a null hypothesis that is actually false
    • Chooses too small a significance level
    • Increases the sample size unnecessarily
    • Rejects a null hypothesis that is actually true

    Correct answer: Rejects a null hypothesis that is actually true

    A Type I error occurs when the analyst rejects a null hypothesis that is actually true, a false positive whose probability equals the significance level. Failing to reject a false null is a Type II error, and the choice of significance level or sample size are decisions rather than errors themselves.

87. A forward contract differs from a futures contract primarily because a forward is:
    • A private, customized agreement traded over the counter with no daily settlement
    • Standardized and traded on an exchange with daily margining
    • Always settled in cash rather than by delivery
    • Free of any counterparty credit risk

    Correct answer: A private, customized agreement traded over the counter with no daily settlement

    A forward contract is a private, customized over-the-counter agreement that is not marked to market daily, so gains and losses accumulate until maturity. Futures are exchange-traded, standardized, and settled daily through margin, and forwards carry counterparty credit risk precisely because there is no daily settlement or clearinghouse.

88. In futures trading, the daily process of crediting and debiting traders' margin accounts based on settlement price changes is called:
    • Novation
    • Marking to market
    • Securitization
    • Bootstrapping

    Correct answer: Marking to market

    Marking to market is the daily process of crediting and debiting margin accounts based on the change in the futures settlement price, which limits the buildup of credit exposure. Novation refers to the clearinghouse becoming the counterparty, while securitization and bootstrapping describe unrelated processes.

89. A trader's futures position triggers a margin call when the account balance falls below the:
    • Initial margin level
    • Contract's notional value
    • Maintenance margin level
    • Settlement price

    Correct answer: Maintenance margin level

    A margin call is triggered when the account balance falls below the maintenance margin level, at which point the trader must restore the balance up to the initial margin. The initial margin is the starting deposit, while notional value and settlement price are not the thresholds that trigger a call.

90. Under the cost-of-carry model, the forward price of a non-dividend-paying asset equals the spot price:
    • Discounted back at the risk-free rate
    • Reduced by the asset's expected return
    • Multiplied by its beta
    • Compounded forward at the risk-free rate

    Correct answer: Compounded forward at the risk-free rate

    For a non-dividend-paying asset, the forward price equals the spot price compounded forward at the risk-free rate, reflecting the financing cost of holding the asset. Discounting would understate the forward, and the asset's expected return or beta do not enter the no-arbitrage cost-of-carry relationship.

91. For a commodity that provides a convenience yield, the convenience yield acts in the cost-of-carry model like a:
    • Benefit of holding the physical asset that lowers the forward price
    • Storage cost that raises the forward price
    • Margin requirement on the futures position
    • Tax on delivery

    Correct answer: Benefit of holding the physical asset that lowers the forward price

    A convenience yield is a benefit of holding the physical commodity, so it behaves like a negative carrying cost and lowers the forward price relative to pure financing and storage. Storage costs raise the forward price, while margin requirements and taxes are not components of the convenience yield.

92. A market is said to be in contango when:
    • Futures prices are below the current spot price
    • Futures prices are above the current spot price
    • The spot price equals the futures price exactly
    • There are no futures contracts available

    Correct answer: Futures prices are above the current spot price

    A market is in contango when futures prices are above the current spot price, often reflecting positive net carrying costs. When futures trade below spot the market is in backwardation, so the opposite condition and the equal-price case do not describe contango.

93. A commodity market in backwardation typically reflects:
    • Very high storage costs with no convenience yield
    • An absence of any spot market
    • A high convenience yield or strong near-term demand for the physical commodity
    • A guaranteed arbitrage profit

    Correct answer: A high convenience yield or strong near-term demand for the physical commodity

    Backwardation, where futures trade below spot, typically reflects a high convenience yield or strong near-term demand for the physical commodity. High storage costs would push the market toward contango, and backwardation is a normal market state rather than evidence of a missing spot market or a riskless arbitrage.

94. In a plain vanilla interest rate swap, the two parties typically exchange:
    • The notional principal itself at initiation
    • Equity returns for bond returns
    • Two different currencies
    • Fixed-rate payments for floating-rate payments on a notional principal

    Correct answer: Fixed-rate payments for floating-rate payments on a notional principal

    In a plain vanilla interest rate swap, the parties exchange fixed-rate payments for floating-rate payments on a notional principal, and the notional is generally not exchanged. Exchanging principal or currencies describes a currency swap, and exchanging equity returns describes an equity swap.

95. At initiation, the value of a standard at-market interest rate swap to each party is:
    • Approximately zero
    • Equal to the notional principal
    • Equal to the present value of all fixed payments
    • Always positive to the fixed-rate payer

    Correct answer: Approximately zero

    At initiation an at-market swap is structured so the present values of the fixed and floating legs are equal, making the swap value approximately zero to each party. The notional is a reference amount, not the swap's value, and the swap's value to either side moves away from zero only as rates change afterward.

96. A currency swap differs from a typical interest rate swap because a currency swap usually involves:
    • Exchanging only fixed for floating payments in one currency
    • Exchanging principal amounts in two different currencies at initiation and maturity
    • No exchange of cash flows until maturity
    • A single payment at expiration

    Correct answer: Exchanging principal amounts in two different currencies at initiation and maturity

    A currency swap usually involves exchanging principal amounts in two different currencies at initiation and re-exchanging them at maturity, along with periodic interest payments in each currency. A single-currency fixed-for-floating exchange describes an interest rate swap, and currency swaps involve ongoing cash flows rather than one terminal payment.

97. The buyer of a call option has the right, but not the obligation, to:
    • Sell the underlying asset at the strike price
    • Receive a fixed dividend from the issuer
    • Buy the underlying asset at the strike price
    • Demand cash settlement of the spot price

    Correct answer: Buy the underlying asset at the strike price

    The buyer of a call option has the right, but not the obligation, to buy the underlying asset at the strike price. The right to sell at the strike belongs to a put holder, and options confer no dividend rights or automatic cash entitlements beyond their defined payoff.

98. At expiration, a long put option on a stock has a payoff equal to:
    • The greater of zero and the stock price minus the strike price
    • The stock price minus the option premium
    • The strike price regardless of the stock price
    • The greater of zero and the strike price minus the stock price

    Correct answer: The greater of zero and the strike price minus the stock price

    A long put pays the greater of zero and the strike price minus the stock price, since the holder can sell at the strike when the stock falls below it. The greater of zero and stock minus strike is the call payoff, and the premium and a fixed strike payoff do not describe the put's expiration value.

99. Put-call parity for European options on a non-dividend-paying stock links the call price, put price, stock price, and:
    • The present value of the strike price
    • The option's implied volatility
    • The stock's dividend yield
    • The risk-free rate of the underlying's issuer's bonds

    Correct answer: The present value of the strike price

    Put-call parity links the call and put prices with the stock price and the present value of the strike, expressed as call plus present value of strike equals put plus stock. Implied volatility and dividend yield are not direct terms in the basic non-dividend parity relationship.

100. If put-call parity is violated in the market, an investor can in principle:
     • Earn a higher expected return only by bearing more risk
     • Construct a risk-free arbitrage by trading the options, the stock, and a bond
     • Eliminate all volatility from the stock permanently
     • Force the company to pay a special dividend

     Correct answer: Construct a risk-free arbitrage by trading the options, the stock, and a bond

     A violation of put-call parity creates a risk-free arbitrage opportunity that an investor can exploit by simultaneously trading the options, the underlying stock, and a risk-free bond. The opportunity is riskless rather than a risk premium, and it does not affect the stock's volatility or corporate dividend policy.

101. A bond's yield to maturity is best described as:
     • The bond's annual coupon divided by its face value
     • The total dollar interest received over the bond's life
     • The single discount rate that sets the present value of the bond's cash flows equal to its price
     • The realized return guaranteed regardless of reinvestment

     Correct answer: The single discount rate that sets the present value of the bond's cash flows equal to its price

     Yield to maturity is the single discount rate that equates the present value of all the bond's cash flows to its current price. The coupon divided by face value is the coupon rate, total dollar interest ignores time value, and the realized return depends on reinvestment assumptions, so it is not guaranteed.

102. A bond trades at a price below its face value. This implies that its yield to maturity is:
     • Less than its coupon rate
     • Equal to its coupon rate
     • Equal to zero
     • Greater than its coupon rate

     Correct answer: Greater than its coupon rate

     A bond trading below face value is a discount bond, which occurs when its yield to maturity exceeds its coupon rate. A premium bond trades above par when the yield is below the coupon, and a par bond's yield equals its coupon, so only a higher yield explains the discount.

103. Bond duration measures the:
     • Sensitivity of a bond's price to changes in interest rates
     • Number of years until the bond matures, exactly
     • Probability that the bond will default
     • Bond's coupon rate adjusted for taxes

     Correct answer: Sensitivity of a bond's price to changes in interest rates

     Duration measures the sensitivity of a bond's price to changes in interest rates, approximating the percentage price change for a small yield change. Maturity is a separate concept, default probability relates to credit risk, and duration is unrelated to the bond's tax-adjusted coupon.

104. For a given change in yield, a bond with a longer duration will experience:
     • A smaller percentage change in price
     • A larger percentage change in price
     • No change in price
     • A change only if it is a zero-coupon bond

     Correct answer: A larger percentage change in price

     A bond with longer duration experiences a larger percentage price change for a given yield change, because duration scales interest-rate sensitivity. Shorter-duration bonds move less, and price sensitivity applies to all bonds, not only zero-coupon issues.

105. Convexity is used alongside duration because duration alone:
     • Overstates default risk
     • Ignores the bond's coupon entirely
     • Provides only a linear approximation that becomes less accurate for large yield changes
     • Applies only to floating-rate notes

     Correct answer: Provides only a linear approximation that becomes less accurate for large yield changes

     Convexity supplements duration because duration is a linear approximation that becomes less accurate for large yield changes, while convexity captures the curvature of the price-yield relationship. Duration concerns interest-rate risk rather than default risk, does account for coupons through cash-flow timing, and applies to fixed-rate bonds generally.

106. Because of positive convexity, when yields change by a large amount, the actual bond price is:
     • Lower than the price predicted by duration alone
     • Exactly equal to the duration estimate
     • Independent of the direction of the yield change
     • Higher than the price predicted by duration alone

     Correct answer: Higher than the price predicted by duration alone

     Positive convexity means the actual bond price is higher than the duration-only estimate for large yield moves in either direction, because the price-yield curve bows favorably. The duration estimate is a tangent line that underestimates price gains and overestimates price losses, so it is not exact and the benefit depends on convexity, not direction alone.

107. A repurchase agreement (repo) is economically equivalent to:
     • A collateralized short-term loan
     • An unsecured long-term bond
     • An equity ownership stake
     • A currency forward

     Correct answer: A collateralized short-term loan

     A repurchase agreement is economically a collateralized short-term loan, in which securities are sold and repurchased later at a higher price representing interest. It is not an unsecured or long-term instrument, an equity stake, or a currency forward.

108. In a repo transaction, the difference between the collateral's market value and the cash lent is known as the:
     • Coupon
     • Haircut
     • Basis
     • Tick size

     Correct answer: Haircut

     The haircut is the difference between the collateral's market value and the cash lent, providing the lender a cushion against a fall in collateral value. The coupon is a bond's interest payment, the basis is the spread between spot and futures, and tick size is the minimum price increment.

109. Eurodollars are best described as:
     • Euro-denominated deposits held in U.S. banks
     • A type of European stock index
     • U.S. dollar deposits held in banks outside the United States
     • Physical currency printed in Europe

     Correct answer: U.S. dollar deposits held in banks outside the United States

     Eurodollars are U.S. dollar-denominated deposits held in banks outside the United States, and Eurodollar futures historically referenced short-term dollar rates. They are not euro deposits, an equity index, or physical currency, despite the name.

110. Commercial paper is a money market instrument characterized by:
     • Long-term secured bonds with fixed coupons
     • Equity claims on a corporation
     • Government-guaranteed deposits
     • Short-term, unsecured corporate debt issued at a discount

     Correct answer: Short-term, unsecured corporate debt issued at a discount

     Commercial paper is short-term, unsecured corporate debt typically issued at a discount to face value to meet near-term funding needs. It is neither long-term secured debt, an equity claim, nor a government-guaranteed deposit.

111. The hedge ratio that minimizes the variance of a hedged position using futures is determined primarily by the:
     • Correlation and relative volatility between the spot and futures price changes
     • Number of contracts available on the exchange
     • Maturity of the longest available futures
     • Margin required per contract

     Correct answer: Correlation and relative volatility between the spot and futures price changes

     The minimum-variance hedge ratio is determined by the correlation between spot and futures price changes scaled by their relative volatilities, equivalent to the regression slope of spot on futures changes. Contract availability, futures maturity, and margin levels do not define the optimal hedge ratio.

112. Basis risk in a futures hedge arises because:
     • The clearinghouse can default
     • The spot and futures prices may not move perfectly together
     • Margin must be posted daily
     • Options and futures have different payoffs

     Correct answer: The spot and futures prices may not move perfectly together

     Basis risk arises because the spot and futures prices may not move perfectly together, so the basis can widen or narrow and leave the hedge imperfect. Clearinghouse default, daily margining, and differences between options and futures are separate considerations, not the source of basis risk.

113. A long hedger using futures is harmed when the basis, defined as spot minus futures, unexpectedly:
     • Goes to exactly zero at maturity
     • Equals the storage cost
     • Weakens (becomes more negative)
     • Becomes perfectly predictable

     Correct answer: Weakens (becomes more negative)

     A long hedger benefits when the basis strengthens and is harmed when it weakens, meaning spot minus futures becomes more negative than expected. Convergence to zero at maturity is the normal outcome, and a predictable basis or one equal to storage cost would not by itself create the loss.

114. A credit default swap (CDS) is a contract in which the protection buyer:
     • Receives a periodic premium for taking on default risk
     • Buys the underlying bond outright
     • Lends cash against equity collateral
     • Pays a periodic premium and receives compensation if a reference entity defaults

     Correct answer: Pays a periodic premium and receives compensation if a reference entity defaults

     In a credit default swap the protection buyer pays a periodic premium and receives compensation from the protection seller if the reference entity experiences a credit event such as default. The seller, not the buyer, receives the premium for taking on default risk, and a CDS is a derivative rather than an outright bond purchase or secured loan.

115. The periodic payment made by the protection buyer in a credit default swap is commonly called the:
     • CDS spread or premium
     • Recovery rate
     • Notional principal
     • Convenience yield

     Correct answer: CDS spread or premium

     The periodic payment by the protection buyer is the CDS spread or premium, which rises with the perceived credit risk of the reference entity. The recovery rate is the value recovered after default, the notional is the reference amount, and convenience yield is a commodity concept.

116. An exchange-traded fund (ETF) typically maintains a price close to its net asset value because:
     • A regulator fixes the price each day
     • Authorized participants can create and redeem shares to arbitrage away differences
     • The fund guarantees a minimum return
     • Trading is suspended whenever the price deviates

     Correct answer: Authorized participants can create and redeem shares to arbitrage away differences

     An ETF stays close to its net asset value because authorized participants can create and redeem shares, arbitraging away any meaningful premium or discount. The price is not fixed by a regulator, ETFs do not guarantee returns, and trading is not routinely suspended for deviations.

117. A key structural difference between a mutual fund and a typical ETF is that an ETF:
     • Can only be bought or sold at the end-of-day net asset value
     • Is always actively managed
     • Trades intraday on an exchange at market-determined prices
     • Cannot hold equities

     Correct answer: Trades intraday on an exchange at market-determined prices

     An ETF trades intraday on an exchange at market-determined prices, whereas a traditional mutual fund is bought or sold only at the end-of-day net asset value. ETFs may be passive or active and commonly hold equities, so those statements do not distinguish them.

118. A stock index futures contract is most commonly used by a portfolio manager to:
     • Collect dividends from the index components
     • Gain voting rights in index companies
     • Eliminate all idiosyncratic stock risk permanently
     • Quickly adjust market exposure without trading the underlying stocks

     Correct answer: Quickly adjust market exposure without trading the underlying stocks

     A portfolio manager uses stock index futures to quickly and cheaply adjust overall market exposure without buying or selling the underlying basket of stocks. Index futures do not convey dividends or voting rights, and while they hedge market risk, they do not permanently remove idiosyncratic risk.

119. To reduce the beta of an equity portfolio using stock index futures, a manager should:
     • Take a short position in the index futures
     • Buy additional index futures
     • Sell the entire portfolio for cash
     • Buy at-the-money call options on each stock

     Correct answer: Take a short position in the index futures

     Shorting stock index futures reduces a portfolio's beta by offsetting market exposure, since losses on the portfolio in a downturn are cushioned by gains on the short futures. Buying futures would raise beta, and liquidating the portfolio or buying calls are not the standard futures-overlay technique for lowering beta.

120. A foreign exchange forward rate that exceeds the current spot rate for a currency indicates the currency is trading at a:
     • Forward discount
     • Forward premium
     • Covered position
     • Negative basis

     Correct answer: Forward premium

     When a currency's forward rate exceeds its spot rate it is trading at a forward premium. A forward rate below spot would be a forward discount, while a covered position and a negative basis describe different concepts.

121. Covered interest rate parity states that the forward foreign exchange rate is determined by the spot rate and:
     • The expected inflation gap only
     • The relative equity market returns
     • The interest rate differential between the two currencies
     • The countries' trade balances

     Correct answer: The interest rate differential between the two currencies

     Covered interest rate parity states that the forward rate is determined by the spot rate adjusted for the interest rate differential between the two currencies, enforced by arbitrage. Inflation gaps, equity returns, and trade balances are not the direct inputs to the covered parity relationship.

122. A long position in a forward rate agreement (FRA) benefits when, by the settlement date, the reference floating rate:
     • Falls below the contract rate
     • Equals the contract rate exactly
     • Becomes negative
     • Rises above the contract rate

     Correct answer: Rises above the contract rate

     The long in an FRA, which effectively locks in a borrowing rate, benefits when the reference floating rate rises above the contracted rate, because the FRA compensates for the higher rate. If the rate falls below the contract rate the long loses, and a rate equal to the contract rate produces no payment.

123. The settlement amount on a forward rate agreement is paid at the start of the underlying loan period and is therefore:
     • Discounted back from the end of the period to a present value
     • Compounded forward to the end of the period
     • Equal to the full notional principal
     • Independent of the day count convention

     Correct answer: Discounted back from the end of the period to a present value

     Because an FRA settles at the start of the loan period rather than the end, the interest difference is discounted back to a present value at settlement. The notional is only a reference amount, not the payment, and the day count convention does affect the interest calculation.

124. The intrinsic value of a call option is:
     • The option premium minus transaction costs
     • The greater of zero and the stock price minus the strike price
     • The time remaining until expiration
     • The implied volatility multiplied by the strike

     Correct answer: The greater of zero and the stock price minus the strike price

     The intrinsic value of a call is the greater of zero and the stock price minus the strike, representing the immediate exercise value. The remainder of the premium is time value, while time to expiration and implied volatility influence the premium but are not the intrinsic value itself.

125. An option's time value is greatest when the option is:
     • Deep in the money near expiration
     • Deep out of the money at expiration
     • At the money with substantial time to expiration
     • Already exercised

     Correct answer: At the money with substantial time to expiration

     Time value is greatest for an at-the-money option with substantial time to expiration, where uncertainty about the final outcome is highest. Deep in- or out-of-the-money options near expiration have little time value, and an exercised option has none.

126. A protective put strategy combines a long stock position with a:
     • Short call option to generate income
     • Long call option to increase leverage
     • Short put option to collect premium
     • Long put option to limit downside losses

     Correct answer: Long put option to limit downside losses

     A protective put combines a long stock position with a long put, which sets a floor on losses while preserving upside. Selling a call would be a covered call, and the other combinations describe different strategies that do not provide the same downside floor.

127. A covered call strategy involves holding a stock and:
     • Writing a call option on that stock to earn premium income
     • Buying a put option on that stock for protection
     • Writing a put option on that stock
     • Shorting the same stock

     Correct answer: Writing a call option on that stock to earn premium income

     A covered call involves holding a stock and writing a call on it to earn premium income, capping upside in exchange for that premium. Buying a put is protective, writing a put is a separate income strategy, and shorting the same stock would offset the position entirely.

128. A floating-rate note's coupon resets periodically based on:
     • The issuer's stock price
     • A reference interest rate plus a fixed spread
     • A fixed rate set at issuance for the bond's life
     • The inflation rate alone, with no spread

     Correct answer: A reference interest rate plus a fixed spread

     A floating-rate note's coupon resets periodically to a reference interest rate plus a fixed spread, so its income rises and falls with short-term rates. A fixed coupon for life describes a conventional bond, and tying the coupon to stock price or inflation alone does not describe a standard floater.

129. Because its coupon resets to market rates, a floating-rate note generally has:
     • High duration similar to a long-term fixed-rate bond
     • No credit risk
     • Low interest rate (duration) risk
     • A guaranteed price above par

     Correct answer: Low interest rate (duration) risk

     A floating-rate note generally has low interest rate risk because its coupon resets to market rates, keeping its price close to par. It still carries credit risk tied to the issuer, has much lower duration than a long fixed-rate bond, and has no guarantee of trading above par.

130. In a securitization, a special purpose vehicle (SPV) is used primarily to:
     • Guarantee a fixed return to all investors
     • Provide voting rights over the underlying borrowers
     • Eliminate all credit risk in the structure
     • Hold the pooled assets separately from the originator to isolate them from the originator's bankruptcy

     Correct answer: Hold the pooled assets separately from the originator to isolate them from the originator's bankruptcy

     A special purpose vehicle holds the pooled assets separately from the originator so they are bankruptcy-remote and isolated from the originator's failure. The SPV does not guarantee returns, confer borrower voting rights, or eliminate credit risk, which is instead redistributed among tranches.

131. In a tranched securitization structure, the equity (most subordinated) tranche:
     • Absorbs the first losses from the underlying pool
     • Is repaid before all other tranches
     • Carries the lowest expected yield
     • Is fully government guaranteed

     Correct answer: Absorbs the first losses from the underlying pool

     The equity, or most subordinated, tranche absorbs the first losses from the underlying pool and therefore carries the highest expected yield to compensate for that risk. Senior tranches are repaid first, and equity tranches are not government guaranteed.

132. A swap can be decomposed for valuation purposes into a portfolio of:
     • Equity options
     • Forward contracts on the underlying rates or prices
     • Insurance contracts
     • Physical commodities

     Correct answer: Forward contracts on the underlying rates or prices

     A swap can be valued as a portfolio of forward contracts, since each exchange of payments is equivalent to a forward on the underlying rate or price. It is not naturally decomposed into equity options, insurance contracts, or physical commodities.

133. The value of an interest rate swap to the fixed-rate payer increases when:
     • Market interest rates fall after the swap is initiated
     • The notional principal is exchanged
     • Market interest rates rise after the swap is initiated
     • Volatility of equity markets declines

     Correct answer: Market interest rates rise after the swap is initiated

     The value of a swap to the fixed-rate payer increases when market rates rise after initiation, because the payer locked in a lower fixed rate while receiving the now-higher floating rate. Falling rates hurt the fixed payer, the notional is generally not exchanged in an interest rate swap, and equity volatility is irrelevant.

134. The delta of an option measures the:
     • Sensitivity of the option's price to the passage of time
     • Rate of change of the option's volatility
     • Probability that the option finishes at the money
     • Sensitivity of the option's price to a change in the underlying asset's price

     Correct answer: Sensitivity of the option's price to a change in the underlying asset's price

     Delta measures the sensitivity of an option's price to a small change in the underlying asset's price. Sensitivity to time is theta, sensitivity to volatility is vega, and delta is not a probability of finishing at the money.

135. Theta of a long option position is typically negative because:
     • The option loses time value as expiration approaches, all else equal
     • The underlying price always declines
     • Volatility rises with time
     • The strike price increases over time

     Correct answer: The option loses time value as expiration approaches, all else equal

     Theta of a long option is typically negative because the option loses time value as expiration approaches, all else equal, a phenomenon known as time decay. The sign of theta does not depend on the underlying always falling, on volatility, or on a changing strike, which is fixed.

136. A long straddle, consisting of a long call and a long put with the same strike and expiration, profits most when the underlying:
     • Stays exactly at the strike price
     • Makes a large move in either direction
     • Drifts slowly upward only
     • Has very low volatility

     Correct answer: Makes a large move in either direction

     A long straddle profits most when the underlying makes a large move in either direction, since one of the two long options gains substantially. A stable price near the strike causes both options to lose value, and low volatility works against the position.

137. A zero-coupon bond differs from a coupon bond in that it:
     • Pays a higher coupon than comparable bonds
     • Has no exposure to interest rate changes
     • Pays no periodic interest and is issued at a discount to face value
     • Always matures within one year

     Correct answer: Pays no periodic interest and is issued at a discount to face value

     A zero-coupon bond pays no periodic interest and is issued at a discount, with the investor's return coming from the gap between purchase price and face value at maturity. It still has interest rate risk, indeed high duration for long maturities, and zeros are not limited to one-year maturities.

138. The duration of a zero-coupon bond is:
     • Always less than one year
     • Equal to its coupon rate
     • Equal to its yield to maturity
     • Equal to its time to maturity

     Correct answer: Equal to its time to maturity

     The duration of a zero-coupon bond equals its time to maturity because all of its cash flow arrives at a single point. Its duration is therefore not limited to under a year and is unrelated to the coupon rate or yield level.

139. The clearinghouse in exchange-traded derivatives reduces counterparty credit risk by:
     • Acting as the buyer to every seller and the seller to every buyer
     • Guaranteeing a fixed profit to all members
     • Setting the market price each day
     • Eliminating the need for margin

     Correct answer: Acting as the buyer to every seller and the seller to every buyer

     The clearinghouse reduces counterparty credit risk by becoming the buyer to every seller and the seller to every buyer through novation, backed by margin and a default fund. It does not guarantee profits, set market prices, or remove the need for margin, which is central to its protection.

140. Compared with exchange-traded futures, over-the-counter forward contracts generally have:
     • Lower customization and lower credit risk
     • Greater customization but higher counterparty credit risk
     • Daily marking to market through a clearinghouse
     • Standardized contract sizes set by an exchange

     Correct answer: Greater customization but higher counterparty credit risk

     Over-the-counter forwards offer greater customization but carry higher counterparty credit risk because they lack a clearinghouse and daily settlement. Standardized sizes, daily marking to market, and clearing are features of exchange-traded futures, not forwards.

141. An interest rate cap protects a floating-rate borrower by:
     • Locking in a fixed rate for the entire loan
     • Paying off when the reference rate falls
     • Paying off when the reference rate rises above the cap's strike rate
     • Converting the loan to a fixed-rate bond

     Correct answer: Paying off when the reference rate rises above the cap's strike rate

     An interest rate cap protects a floating-rate borrower by paying off when the reference rate rises above the cap's strike rate, effectively limiting the borrower's interest cost. It does not lock in a single fixed rate or pay when rates fall, which would describe a floor.

142. An interest rate floor is most useful to a party that:
     • Pays floating-rate interest and fears rising rates
     • Wants to eliminate all credit risk on a loan
     • Seeks to increase the duration of a bond portfolio
     • Receives floating-rate income and wants to protect against falling rates

     Correct answer: Receives floating-rate income and wants to protect against falling rates

     An interest rate floor benefits a party receiving floating-rate income, because it pays off when rates fall below the floor's strike, protecting the investor's income. A borrower fearing rising rates would use a cap, and a floor addresses interest rate risk rather than credit risk or duration targeting.

143. An American-style option differs from a European-style option because the American option can be:
     • Exercised at any time up to and including expiration
     • Exercised only at expiration
     • Traded only in the United States
     • Settled only in cash

     Correct answer: Exercised at any time up to and including expiration

     An American-style option can be exercised at any time up to and including expiration, whereas a European-style option can be exercised only at expiration. The names refer to exercise style, not geography or settlement method.

144. Compared with a comparable European call, an American call on a non-dividend-paying stock is generally:
     • Always worth more because of early exercise
     • Worth the same because early exercise is not optimal
     • Worth less due to higher transaction costs
     • Worthless before expiration

     Correct answer: Worth the same because early exercise is not optimal

     An American call on a non-dividend-paying stock is generally worth the same as the comparable European call, because early exercise is not optimal when no dividends are paid. The early-exercise feature adds value only when it can be advantageous, such as for puts or dividend-paying stocks.

145. The minimum tick size of a futures contract refers to the:
     • Maximum daily price move permitted
     • Margin required per contract
     • Smallest allowable price increment for the contract
     • Number of contracts that can be traded

     Correct answer: Smallest allowable price increment for the contract

     The minimum tick size is the smallest allowable price increment by which a futures contract can move. A maximum daily move is a price limit, margin is the required deposit, and trading volume limits are separate concepts.

146. A bond's accrued interest is the interest that has:
     • Been paid in advance to the buyer
     • Been forgiven by the issuer
     • Accumulated over the bond's entire life
     • Accumulated since the last coupon payment but not yet been paid

     Correct answer: Accumulated since the last coupon payment but not yet been paid

     Accrued interest is the interest that has accumulated since the last coupon payment but has not yet been paid, and a bond buyer compensates the seller for it. It is neither prepaid, forgiven, nor measured over the bond's full life.

147. Value at risk (VaR) at a 99% confidence level over one day is best interpreted as:
     • The loss that will not be exceeded with 99% probability over one day
     • The maximum possible loss the portfolio can ever suffer
     • The average loss expected on a typical day
     • The loss that will occur exactly once per year

     Correct answer: The loss that will not be exceeded with 99% probability over one day

     A 99% one-day VaR is the loss threshold that will not be exceeded with 99% probability over one day, meaning a worse loss is expected only about 1% of the time. VaR does not bound the maximum possible loss, give an average loss, or guarantee a precise frequency of occurrence.

148. A primary limitation of value at risk as a risk measure is that it:
     • Always overstates the risk of fat-tailed portfolios
     • Says nothing about the magnitude of losses beyond the VaR threshold
     • Cannot be computed for equity portfolios
     • Is unaffected by the chosen confidence level

     Correct answer: Says nothing about the magnitude of losses beyond the VaR threshold

     A key limitation of VaR is that it identifies a loss threshold but says nothing about how severe losses can be once that threshold is breached. VaR does not systematically overstate fat-tailed risk, can be computed for equities, and clearly depends on the chosen confidence level.

149. Expected shortfall (conditional VaR) improves upon value at risk because it:
     • Ignores the tail of the loss distribution
     • Is always smaller than the corresponding VaR
     • Measures the average loss given that the loss exceeds the VaR threshold
     • Requires no assumptions about the loss distribution

     Correct answer: Measures the average loss given that the loss exceeds the VaR threshold

     Expected shortfall measures the average loss conditional on exceeding the VaR threshold, so it captures the severity of tail losses that VaR omits. It focuses on rather than ignores the tail, is generally larger than VaR, and still relies on assumptions about the loss distribution.

150. A risk measure is called coherent if it satisfies several properties, including subadditivity. Subadditivity means the risk of a combined portfolio is:
     • Always greater than the sum of the risks of its parts
     • Independent of diversification
     • Equal to the largest individual risk
     • No greater than the sum of the risks of its parts

     Correct answer: No greater than the sum of the risks of its parts

     Subadditivity requires that the risk of a combined portfolio be no greater than the sum of the individual risks, reflecting the benefit of diversification. A measure that can exceed the sum of its parts, such as VaR in some cases, violates subadditivity, while expected shortfall satisfies it.

151. Value at risk is sometimes criticized as not coherent because it can violate which coherence property?
     • Subadditivity
     • Monotonicity
     • Positive homogeneity
     • Translation invariance

     Correct answer: Subadditivity

     VaR can violate subadditivity, meaning the VaR of a combined portfolio may exceed the sum of the component VaRs, which is why it is not generally coherent. Monotonicity, positive homogeneity, and translation invariance are the other coherence properties that VaR typically does satisfy.

152. The parametric (variance-covariance) approach to value at risk assumes that:
     • Past returns will be replayed exactly into the future
     • Asset returns follow a specified distribution, often the normal distribution
     • Each scenario is generated by random simulation
     • Returns are completely unpredictable and have no distribution

     Correct answer: Asset returns follow a specified distribution, often the normal distribution

     The parametric, or variance-covariance, VaR approach assumes returns follow a specified distribution, commonly the normal, and computes VaR from estimated means and covariances. Replaying past returns describes historical simulation, random scenario generation describes Monte Carlo, and a distributional assumption is central rather than absent.

153. Under the parametric VaR method assuming normality, the one-day VaR for a position scales with which factor for the chosen confidence level?
     • The portfolio's expected return
     • The number of assets in the portfolio
     • The z-score (standard normal quantile) corresponding to that confidence level
     • The square of the volatility

     Correct answer: The z-score (standard normal quantile) corresponding to that confidence level

     Parametric normal VaR scales the portfolio's volatility by the z-score corresponding to the chosen confidence level, such as 1.65 for 95% or 2.33 for 99%. Expected return enters only as a small drift adjustment, the asset count is not the multiplier, and VaR scales with volatility, not its square.

154. Historical simulation value at risk estimates risk by:
     • Assuming returns are exactly normally distributed
     • Generating thousands of random scenarios from a model
     • Using only the single worst day ever recorded
     • Applying actual past return observations to the current portfolio and ranking the outcomes

     Correct answer: Applying actual past return observations to the current portfolio and ranking the outcomes

     Historical simulation applies the distribution of actual past return observations to the current portfolio and ranks the simulated outcomes to read off the VaR quantile. It makes no normality assumption, does not generate model-based random scenarios as Monte Carlo does, and uses the full ordered history rather than one worst day.

155. A drawback of historical simulation VaR is that it:
     • Assumes the future will resemble the historical sample period
     • Requires the returns to be normally distributed
     • Cannot capture nonlinear instruments at all
     • Ignores the actual portfolio holdings

     Correct answer: Assumes the future will resemble the historical sample period

     Historical simulation assumes the future will resemble the historical sample, so it can miss risks absent from that window, such as a regime not previously observed. It does not require normality, can in fact handle nonlinear instruments through full revaluation, and uses the actual holdings.

156. Monte Carlo simulation for value at risk differs from historical simulation because it:
     • Relies solely on the realized historical return path
     • Generates many scenarios from an assumed stochastic model of risk factors
     • Cannot value options or other derivatives
     • Always produces a lower VaR estimate

     Correct answer: Generates many scenarios from an assumed stochastic model of risk factors

     Monte Carlo VaR generates many scenarios from an assumed stochastic model of the risk factors and revalues the portfolio under each. It does not rely solely on the realized path as historical simulation does, can value derivatives through full revaluation, and does not inherently produce lower estimates.

157. Backtesting a VaR model involves:
     • Recalibrating the model to fit next year's forecast
     • Increasing the confidence level until no breaches occur
     • Comparing the number of actual losses exceeding VaR with the number predicted by the model
     • Replacing VaR with expected shortfall automatically

     Correct answer: Comparing the number of actual losses exceeding VaR with the number predicted by the model

     Backtesting compares the number of actual losses that exceed the VaR estimate (exceptions) with the number the model predicts at the chosen confidence level. It is a validation step, not a means of forcing zero breaches by raising confidence, recalibrating to a forecast, or swapping in expected shortfall.

158. In a 99% one-day VaR backtest over 250 trading days, the expected number of exceptions is approximately:
     • About 25
     • Zero
     • About 125
     • About 2 to 3

     Correct answer: About 2 to 3

     At 99% confidence the model expects breaches on roughly 1% of days, so over 250 trading days about 2 to 3 exceptions are expected. About 25 would correspond to a 90% level, zero would imply an overly conservative model, and 125 would imply a 50% level.

159. The Basel traffic light approach for VaR backtesting places a model in the red zone when:
     • The number of exceptions is large enough to strongly indicate the model understates risk
     • No exceptions occur at all
     • Exactly the expected number of exceptions occurs
     • The portfolio earns a profit every day

     Correct answer: The number of exceptions is large enough to strongly indicate the model understates risk

     In the Basel traffic light approach, the red zone is reached when the number of exceptions is high enough to strongly suggest the model understates risk, triggering supervisory action. Zero exceptions, the expected count, or daily profits do not place a model in the red zone.

160. Stress testing complements value at risk by:
     • Replacing the need to estimate volatility
     • Examining portfolio losses under severe but plausible scenarios outside normal conditions
     • Guaranteeing the portfolio cannot lose money
     • Measuring only average daily returns

     Correct answer: Examining portfolio losses under severe but plausible scenarios outside normal conditions

     Stress testing complements VaR by examining how a portfolio would perform under severe but plausible scenarios that lie outside normal market conditions and may not appear in historical data. It does not replace volatility estimation, prevent losses, or focus on average returns.

161. A reverse stress test starts from:
     • A normal market scenario and projects forward
     • The portfolio's expected return
     • A specified adverse outcome and works backward to identify scenarios that could cause it
     • The current VaR estimate only

     Correct answer: A specified adverse outcome and works backward to identify scenarios that could cause it

     A reverse stress test begins with a specified adverse outcome, such as a level of loss that would threaten the firm, and works backward to find the scenarios capable of producing it. This reverses the usual direction of starting from scenarios and projecting forward.

162. Expected loss on a credit exposure is commonly calculated as the product of:
     • Recovery rate, duration, and notional
     • Default probability and the risk-free rate
     • Volatility, beta, and exposure
     • Probability of default, loss given default, and exposure at default

     Correct answer: Probability of default, loss given default, and exposure at default

     Expected credit loss is the product of the probability of default, the loss given default, and the exposure at default. Recovery rate equals one minus loss given default and is not multiplied directly in this form, and the risk-free rate, volatility, and beta are not the components of expected loss.

163. Loss given default (LGD) is related to the recovery rate by which expression?
     • LGD equals one minus the recovery rate
     • LGD equals the recovery rate
     • LGD equals one plus the recovery rate
     • LGD equals the recovery rate squared

     Correct answer: LGD equals one minus the recovery rate

     Loss given default equals one minus the recovery rate, since whatever fraction of the exposure is recovered is not lost. It is not equal to, one plus, or the square of the recovery rate.

164. A bond's credit spread over a comparable risk-free rate primarily compensates investors for:
     • Interest rate risk only
     • Expected default loss and a premium for bearing credit risk
     • The bond's liquidity guarantee
     • Inflation alone

     Correct answer: Expected default loss and a premium for bearing credit risk

     A credit spread compensates investors for expected default losses plus a premium for bearing credit risk and related uncertainties. Interest rate risk is reflected in the underlying risk-free rate, and the spread is not a liquidity guarantee or purely an inflation adjustment.

165. As a bond approaches default, its credit spread typically:
     • Narrows toward zero
     • Remains constant by regulation
     • Widens significantly
     • Becomes negative

     Correct answer: Widens significantly

     As default risk rises, a bond's credit spread typically widens significantly to compensate investors for the elevated probability and severity of loss. Spreads do not narrow, stay fixed by rule, or turn negative as creditworthiness deteriorates.

166. In the binomial option pricing model, the risk-neutral probability is used to:
     • Forecast the actual probability of an up move
     • Estimate the option's real-world expected return
     • Determine the option's bid-ask spread
     • Discount expected payoffs at the risk-free rate to value the option

     Correct answer: Discount expected payoffs at the risk-free rate to value the option

     The binomial model uses the risk-neutral probability to compute an expected payoff that is then discounted at the risk-free rate, giving an arbitrage-free option value. The risk-neutral probability is a pricing device, not a forecast of the real-world probability, expected return, or trading spread.

167. In a one-step binomial tree, the value of a call option is found by:
     • Constructing a replicating portfolio of the underlying and a risk-free bond
     • Multiplying the strike by the expected return
     • Averaging the up and down stock prices
     • Discounting the strike at the dividend yield

     Correct answer: Constructing a replicating portfolio of the underlying and a risk-free bond

     A call in a one-step binomial tree is valued by constructing a replicating portfolio of the underlying asset and a risk-free bond that matches the option's payoffs, and pricing it by no-arbitrage. Simple averaging of prices or using expected returns and dividend yields does not produce the arbitrage-free value.

168. The Black-Scholes-Merton model assumes that the underlying asset price follows:
     • A mean-reverting process with jumps
     • A geometric Brownian motion with constant volatility
     • A discrete binomial process
     • A process with stochastic volatility

     Correct answer: A geometric Brownian motion with constant volatility

     The Black-Scholes-Merton model assumes the underlying follows a geometric Brownian motion with constant volatility and continuous price paths. Mean reversion with jumps, a discrete binomial process, and stochastic volatility are alternative assumptions that the basic model does not make.

169. A key limitation of the Black-Scholes-Merton model when applied to real markets is its assumption of:
     • Daily marking to market
     • Negative interest rates
     • Constant volatility, which conflicts with the observed volatility smile
     • Discrete trading intervals

     Correct answer: Constant volatility, which conflicts with the observed volatility smile

     A key limitation of Black-Scholes-Merton is its assumption of constant volatility, which conflicts with the observed volatility smile or skew in option markets. The model does not assume daily marking to market or discrete trading, and negative rates are not a core modeling flaw of the original framework.

170. The volatility smile or skew observed in option markets indicates that:
     • Implied volatility is identical for all options
     • Options cannot be priced by any model
     • Time value is always zero
     • Implied volatility varies across strike prices, contradicting constant-volatility assumptions

     Correct answer: Implied volatility varies across strike prices, contradicting constant-volatility assumptions

     The volatility smile or skew indicates that implied volatility varies across strike prices, which contradicts the constant-volatility assumption of Black-Scholes-Merton. It does not mean implied volatility is identical, that options are unpriceable, or that time value vanishes.

171. Implied volatility is best described as the volatility that, when input into an option pricing model:
     • Makes the model price equal the option's observed market price
     • Equals the historical volatility of the underlying
     • Minimizes the option's delta
     • Sets the option's intrinsic value to zero

     Correct answer: Makes the model price equal the option's observed market price

     Implied volatility is the volatility input that makes the model price equal the option's observed market price, effectively backing volatility out of the quoted price. It need not equal historical volatility and is unrelated to minimizing delta or zeroing intrinsic value.

172. The expected loss of a credit portfolio is generally covered by:
     • Economic capital held against extreme events
     • Provisions or pricing built into the product
     • Regulatory liquidity buffers
     • The risk-free rate alone

     Correct answer: Provisions or pricing built into the product

     Expected loss is generally covered by provisions or by being priced into the product, because it represents the average anticipated loss. Economic capital is held against unexpected loss beyond the expected level, while liquidity buffers and the risk-free rate address different concerns.

173. In external credit ratings, a downgrade from investment grade to below investment grade can force some institutional investors to sell because:
     • Below-investment-grade bonds pay no coupon
     • The bond is automatically converted to equity
     • Their mandates restrict holdings to investment-grade securities
     • The issuer must repurchase the bond immediately

     Correct answer: Their mandates restrict holdings to investment-grade securities

     A downgrade below investment grade can force selling because many institutional mandates restrict holdings to investment-grade securities, creating forced sales. Below-investment-grade bonds still pay coupons, are not automatically converted to equity, and downgrades do not trigger automatic issuer repurchase.

174. A through-the-cycle credit rating differs from a point-in-time estimate because it:
     • Updates continuously with daily market prices
     • Ignores the issuer's financial statements
     • Applies only to sovereign debt
     • Aims to be stable across the economic cycle rather than reflecting current conditions only

     Correct answer: Aims to be stable across the economic cycle rather than reflecting current conditions only

     A through-the-cycle rating aims to be stable across the economic cycle rather than reacting to current conditions, in contrast to point-in-time estimates that move with the current environment. It is not a continuously updating market-price measure, does consider issuer financials, and is not limited to sovereigns.

175. Country (sovereign) risk in valuation refers to the risk that:
     • A government may default on its obligations or take actions harming foreign investors
     • Equity markets always outperform bonds
     • Exchange rates are fixed permanently
     • Inflation is always zero in developed markets

     Correct answer: A government may default on its obligations or take actions harming foreign investors

     Country or sovereign risk is the risk that a government defaults on its obligations or takes actions, such as capital controls or expropriation, that harm foreign investors. It does not assert relative asset performance, fixed exchange rates, or zero inflation.

176. Operational risk under the Basel framework is defined as the risk of loss resulting from:
     • Adverse movements in market prices
     • Inadequate or failed internal processes, people, systems, or external events
     • Borrower default on a loan
     • Changes in interest rates only

     Correct answer: Inadequate or failed internal processes, people, systems, or external events

     Operational risk is defined as the risk of loss from inadequate or failed internal processes, people, and systems, or from external events. Market price movements are market risk, borrower default is credit risk, and interest rate changes alone fall under market risk.

177. A loss distribution approach to operational risk capital combines:
     • Only the largest single historical loss
     • The portfolio's market beta and volatility
     • The frequency and severity distributions of loss events
     • The bond's duration and convexity

     Correct answer: The frequency and severity distributions of loss events

     The loss distribution approach combines a frequency distribution of how often loss events occur with a severity distribution of how large each loss is, to produce an aggregate loss distribution. It does not rely on a single largest loss, market beta, or bond duration and convexity.

178. Liquidity risk in a portfolio includes the risk that:
     • The portfolio's expected return is too low
     • Interest rates will rise
     • Dividends will be cut
     • Positions cannot be sold quickly without a significant price concession

     Correct answer: Positions cannot be sold quickly without a significant price concession

     Liquidity risk includes the risk that positions cannot be sold quickly without accepting a significant price concession, which can amplify losses in stressed markets. Low expected return, rising rates, and dividend cuts relate to other risk categories rather than liquidity risk.

179. Funding liquidity risk is distinct from market (asset) liquidity risk because funding liquidity risk concerns:
     • A firm's ability to meet its cash obligations as they come due
     • The bid-ask spread of a single security
     • The volatility of equity prices
     • The credit rating of a counterparty

     Correct answer: A firm's ability to meet its cash obligations as they come due

     Funding liquidity risk concerns a firm's ability to meet its cash obligations as they come due, whereas market liquidity risk concerns the ability to sell assets without large price impact such as a wide bid-ask spread. Equity volatility and counterparty ratings address separate risks.

180. Marginal VaR measures the:
     • Total VaR of the whole portfolio
     • Change in portfolio VaR from a small increase in a position
     • Average VaR per asset
     • VaR contribution that always sums to less than total VaR

     Correct answer: Change in portfolio VaR from a small increase in a position

     Marginal VaR measures the change in total portfolio VaR resulting from a small increase in a particular position, indicating how that position affects overall risk. It is neither the total portfolio VaR nor a simple per-asset average, and component VaRs, not marginal VaRs, sum to the total.

181. Component VaR is useful because, unlike marginal VaR, the component VaRs of all positions:
     • Are always equal to each other
     • Ignore correlations between positions
     • Sum to the total portfolio VaR
     • Exceed the total portfolio VaR

     Correct answer: Sum to the total portfolio VaR

     Component VaR is useful because the component VaRs of all positions sum to the total portfolio VaR, allowing risk to be attributed across positions. The components are not equal, do account for correlations, and by construction add up to rather than exceed the total.

182. The square-root-of-time rule used to scale a one-day VaR to a ten-day VaR assumes that returns are:
     • Strongly serially correlated
     • Drawn from a fat-tailed distribution
     • Perfectly negatively correlated
     • Independent and identically distributed across days

     Correct answer: Independent and identically distributed across days

     The square-root-of-time rule, which multiplies one-day VaR by the $\sqrt{\text{horizon}}$, assumes returns are independent and identically distributed across days. Serial correlation, fat tails, or perfect negative correlation would violate the assumptions and distort the scaled estimate.

183. Using the square-root-of-time rule, a one-day VaR of $1 million scales to a four-day VaR of approximately:
     • $2 million
     • $4 million
     • $1 million
     • $16 million

     Correct answer: $2 million

     Scaling by the $\sqrt{\text{time}}$ multiplies the one-day VaR by $\sqrt{4}$, which is two, giving about $2 million. Multiplying by the horizon directly would wrongly give $4 million, and the other figures do not follow the rule.

184. Wrong-way risk in counterparty credit exposure occurs when:
     • Exposure falls whenever the counterparty weakens
     • Exposure to a counterparty increases as the counterparty's credit quality deteriorates
     • The counterparty is fully collateralized
     • Two trades perfectly offset each other

     Correct answer: Exposure to a counterparty increases as the counterparty's credit quality deteriorates

     Wrong-way risk occurs when exposure to a counterparty rises just as that counterparty's credit quality deteriorates, making the exposure and default risk positively related. Falling exposure with weakening credit is right-way risk, and full collateralization or perfectly offsetting trades reduce rather than create wrong-way risk.

185. A credit valuation adjustment (CVA) represents the:
     • Increase in value from rising interest rates
     • Tax benefit of holding the derivative
     • Reduction in a derivative's value to reflect counterparty default risk
     • Margin posted to the clearinghouse

     Correct answer: Reduction in a derivative's value to reflect counterparty default risk

     A credit valuation adjustment is the reduction in a derivative's value that reflects the expected loss from counterparty default risk. It is not an interest-rate-driven gain, a tax benefit, or the margin posted to a clearinghouse.

186. In the Merton structural model of default, a firm defaults when:
     • Its stock price declines on any single day
     • Its credit rating is downgraded
     • Interest rates rise above the coupon
     • The value of its assets falls below the face value of its debt at maturity

     Correct answer: The value of its assets falls below the face value of its debt at maturity

     In the Merton model, the firm's equity is treated as a call option on its assets, and default occurs when asset value falls below the face value of debt at maturity. A single-day stock decline, a rating downgrade, or rising rates do not by themselves define default in this structural framework.

187. In the Merton model, equity holders effectively hold a position equivalent to:
     • A call option on the firm's assets with a strike equal to the debt's face value
     • A risk-free bond
     • A put option on the firm's assets
     • A short position in the firm's debt

     Correct answer: A call option on the firm's assets with a strike equal to the debt's face value

     In the Merton model, equity is equivalent to a call option on the firm's assets with a strike equal to the face value of debt, since shareholders receive the residual only if assets exceed debt. Equity is not a risk-free bond, a put on the assets, or simply a short debt position.

188. A risk model exhibits model risk when:
     • It is backtested and validated regularly
     • Incorrect assumptions or implementation lead to mismeasured risk and flawed decisions
     • It uses observed market prices as inputs
     • It produces a single point estimate

     Correct answer: Incorrect assumptions or implementation lead to mismeasured risk and flawed decisions

     Model risk arises when incorrect assumptions, inputs, or implementation cause a model to mismeasure risk and lead to flawed decisions. Regular backtesting and validation mitigate model risk, and using market prices or producing a point estimate are not by themselves evidence of model risk.

189. A robust defense against model risk includes:
     • Using a single model with no review
     • Avoiding any documentation of assumptions
     • Independent validation and ongoing performance monitoring of the model
     • Removing all stress testing

     Correct answer: Independent validation and ongoing performance monitoring of the model

     A robust defense against model risk includes independent validation and ongoing monitoring of the model's performance to detect breakdowns. Relying on one unreviewed model, omitting documentation, and removing stress testing would increase rather than reduce model risk.

190. The expected exposure (EE) on a derivative counterparty position over time is:
     • Always equal to the notional principal
     • The maximum loss in the worst scenario
     • Independent of the time horizon
     • The average of the positive future exposure across simulated scenarios

     Correct answer: The average of the positive future exposure across simulated scenarios

     Expected exposure is the average of the positive future exposure across simulated scenarios at a point in time, reflecting the typical amount at risk. It is not the notional, not the worst-case exposure (which is potential future exposure), and it does vary with the time horizon.

191. Potential future exposure (PFE) differs from expected exposure because PFE represents:
     • A high-percentile (worst-case) estimate of future exposure
     • The average future exposure
     • The current mark-to-market value only
     • The notional amount of the trade

     Correct answer: A high-percentile (worst-case) estimate of future exposure

     Potential future exposure is a high-percentile, worst-case estimate of future exposure at a given confidence level, whereas expected exposure is the average. PFE is therefore larger than EE and is distinct from the current mark-to-market value or the notional.

192. The Basel III leverage ratio is designed to:
     • Replace all risk-weighted capital requirements
     • Constrain the buildup of leverage using a non-risk-based capital measure
     • Measure only market risk
     • Apply solely to insurance companies

     Correct answer: Constrain the buildup of leverage using a non-risk-based capital measure

     The Basel III leverage ratio constrains the buildup of leverage using a non-risk-based measure of capital to total exposure, serving as a backstop to risk-weighted requirements. It supplements rather than replaces risk-weighted capital, is not limited to market risk, and applies to banks.

193. Under Basel capital rules, risk-weighted assets are calculated to ensure that:
     • All assets receive the same capital charge
     • Only liquid assets require capital
     • Capital requirements are higher for riskier exposures
     • Capital is unrelated to asset risk

     Correct answer: Capital requirements are higher for riskier exposures

     Risk-weighted assets assign greater weight to riskier exposures so that capital requirements rise with risk. This explicitly ties capital to asset risk, in contrast to charging all assets equally or basing capital only on liquidity.

194. Extreme value theory (EVT) is applied in risk management to:
     • Forecast average daily returns
     • Estimate the most likely outcome
     • Eliminate the need for stress testing
     • Model the tails of a loss distribution more accurately than the normal distribution

     Correct answer: Model the tails of a loss distribution more accurately than the normal distribution

     Extreme value theory models the tails of a loss distribution more accurately than the normal distribution, improving estimates of rare, severe losses. It targets tail behavior rather than average returns or the most likely outcome and complements rather than replaces stress testing.

195. The peaks-over-threshold approach in extreme value theory models:
     • The distribution of losses that exceed a high threshold
     • The full distribution of all observed losses
     • Only losses below the median
     • The mean of the entire sample

     Correct answer: The distribution of losses that exceed a high threshold

     The peaks-over-threshold approach focuses on the distribution of losses that exceed a high threshold, typically fitted with a generalized Pareto distribution. It deliberately ignores the bulk of the data to characterize the tail rather than modeling the whole distribution or the central tendency.

196. For a portfolio of imperfectly correlated positions, the total VaR is generally:
     • Greater than the sum of the individual VaRs
     • Less than the sum of the individual position VaRs due to diversification
     • Exactly equal to the sum of the individual VaRs
     • Equal to the largest single position VaR

     Correct answer: Less than the sum of the individual position VaRs due to diversification

     Because of diversification among imperfectly correlated positions, total portfolio VaR is generally less than the sum of the individual VaRs. It would equal the sum only with perfect positive correlation, and it is not simply the largest individual VaR.

197. The undiversified VaR of a portfolio assumes that:
     • Positions are uncorrelated
     • Correlations are negative
     • All positions are perfectly positively correlated and move together
     • Volatility is zero

     Correct answer: All positions are perfectly positively correlated and move together

     Undiversified VaR assumes all positions are perfectly positively correlated and lose value together, which is the most conservative aggregation. Uncorrelated or negatively correlated positions would produce a smaller diversified VaR, and zero volatility would imply no risk.

198. A delta-normal VaR estimate for a portfolio containing options can be inaccurate because options have:
     • Zero exposure to the underlying
     • Constant value regardless of the underlying
     • No volatility sensitivity
     • Nonlinear payoffs that a linear (delta) approximation fails to capture

     Correct answer: Nonlinear payoffs that a linear (delta) approximation fails to capture

     Delta-normal VaR uses a linear approximation, which is inaccurate for options because their payoffs are nonlinear, requiring gamma and full revaluation methods for accuracy. Options clearly have exposure to the underlying, change in value with it, and are sensitive to volatility.

199. Incorporating gamma into a VaR calculation for an options portfolio improves accuracy by capturing:
     • The curvature of the option's value with respect to the underlying price
     • The option's sensitivity to interest rates
     • The bid-ask spread
     • The dividend payment schedule

     Correct answer: The curvature of the option's value with respect to the underlying price

     Including gamma captures the curvature of the option's value with respect to the underlying price, refining the linear delta estimate used in VaR. Interest rate sensitivity (rho), the bid-ask spread, and the dividend schedule are separate factors not addressed by gamma.

200. Mapping a complex fixed-income portfolio to a set of standard risk factors before computing VaR is done to:
     • Increase the number of inputs needed
     • Reduce dimensionality and use estimable factor volatilities and correlations
     • Eliminate the need for any historical data
     • Guarantee a normal distribution of returns

     Correct answer: Reduce dimensionality and use estimable factor volatilities and correlations

     Risk-factor mapping reduces the dimensionality of a complex portfolio so VaR can be computed using estimable volatilities and correlations of a manageable set of standard factors. It reduces rather than increases inputs, still relies on data, and does not guarantee normality.

201. In cash-flow mapping for fixed income, a bond's individual cash flows are assigned to:
     • A single average maturity ignoring timing
     • The equity risk factor
     • Standard maturity vertices (buckets) for risk measurement
     • The foreign exchange factor only

     Correct answer: Standard maturity vertices (buckets) for risk measurement

     Cash-flow mapping assigns a bond's individual cash flows to standard maturity vertices, or buckets, so interest rate risk can be measured against benchmark tenors. Collapsing to a single maturity loses timing information, and bond cash flows are not mapped to equity or solely to FX factors.

202. The recovery rate assumption in credit risk modeling is important because a higher assumed recovery rate:
     • Raises the probability of default
     • Has no effect on expected loss
     • Increases the exposure at default
     • Lowers the estimated loss given default and the expected loss

     Correct answer: Lowers the estimated loss given default and the expected loss

     A higher assumed recovery rate lowers loss given default, which directly reduces the estimated expected loss. The recovery rate does not change the probability of default or the exposure at default, and it clearly does affect expected loss.

203. Default correlation among borrowers in a credit portfolio matters because higher default correlation:
     • Increases the likelihood of many simultaneous defaults and fattens the loss distribution's tail
     • Reduces the portfolio's unexpected loss
     • Has no impact on tail risk
     • Always lowers the expected loss

     Correct answer: Increases the likelihood of many simultaneous defaults and fattens the loss distribution's tail

     Higher default correlation increases the chance of many borrowers defaulting together, fattening the tail of the loss distribution and raising unexpected loss and required economic capital. It does not reduce unexpected loss or leave tail risk unaffected, and it does not lower expected loss, which depends on individual default probabilities.

204. Incremental VaR measures the:
     • Average VaR of all positions
     • Change in total portfolio VaR from adding an entire new position
     • VaR computed assuming zero correlations
     • Loss in the single worst historical day

     Correct answer: Change in total portfolio VaR from adding an entire new position

     Incremental VaR measures the change in total portfolio VaR resulting from adding an entire new position, capturing that position's full effect including diversification. It is not an average across positions, a zero-correlation calculation, or a single worst-day loss.

205. A key reason firms hold economic capital against unexpected loss is that:
     • Expected loss cannot be estimated in advance
     • Unexpected loss is always smaller than expected loss
     • Unexpected loss represents the deviation of actual losses above the expected level
     • Regulators forbid provisioning for expected loss

     Correct answer: Unexpected loss represents the deviation of actual losses above the expected level

     Firms hold economic capital against unexpected loss because it represents the deviation of actual losses above the expected level, which provisions do not cover. Expected loss can be estimated and provisioned, unexpected loss can exceed it in the tail, and regulators do not forbid provisioning for expected loss.

206. Netting agreements between counterparties reduce credit exposure by:
     • Eliminating the need for any collateral
     • Guaranteeing both parties a profit
     • Converting all trades to fixed rates
     • Allowing offsetting positive and negative exposures to be combined into a single net amount

     Correct answer: Allowing offsetting positive and negative exposures to be combined into a single net amount

     Netting agreements reduce credit exposure by allowing offsetting positive and negative exposures with the same counterparty to be combined into a single net amount owed. They do not remove the need for collateral, guarantee profits, or convert trades to fixed rates.