FRM P2 Market Risk: practice questions with answers

Question 1

Halvorsen Bank backtests its 99% one-day VaR model over the last 250 trading days and records 7 exceptions. Using Kupiec's proportion-of-failures (POF) likelihood ratio test at the 95% confidence level (chi-square critical value with one degree of freedom = 3.84), which statement is correct?

1. A LR is approximately 5.50, but the model is not rejected because 5.50 is below 6.63
2. B LR is approximately 3.01, so the model is not rejected
3. C LR is approximately 2.86, so the model is not rejected
4. D LR is approximately 5.50, so the model is rejected

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Answer: D. Under H0 the exception probability is p = 0.01; observed rate = 7/250 = 0.028.

LR = -2 ln[(1 - p)^(T-N) p^N] + 2 ln[(1 - N/T)^(T-N) (N/T)^N]
= -2 ln[0.99^243 x 0.01^7] + 2 ln[0.972^243 x 0.028^7] = 5.50.

LR is chi-square with 1 degree of freedom under H0. Since 5.50 > 3.84, the hypothesis that the model has correct (1%) coverage is rejected at the 95% confidence level. (Expected exceptions were only 2.5.)

Question 2

A trading desk's daily P&L is normally distributed with mean zero and standard deviation USD 2.0 million. What is the one-day expected shortfall at the 97.5% confidence level? (Use N^-1(0.975) = 1.960, N^-1(0.99) = 2.326, standard normal density at 1.960 = 0.05845 and at 2.326 = 0.02665.)

1. A USD 3.92 million
2. B USD 4.65 million
3. C USD 4.68 million
4. D USD 5.33 million

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Answer: C. For a normal distribution with mean zero, ES at confidence level c = sigma x phi(z_c) / (1 - c).

z = 1.960, phi(1.960) = 0.05845.
ES = 2.0 x 0.05845 / 0.025 = USD 4.68 million.

For comparison, 99% VaR = 2.326 x 2.0 = 4.65, which is why FRTB's switch from 99% VaR to 97.5% ES leaves capital similar for normal distributions but raises it for fat-tailed ones.

Question 3

A hedge fund buys a correlation swap on a three-stock basket with a fixed correlation strike of 0.45 and a notional of USD 5 million per unit of correlation. At maturity, the realized pairwise correlations are 0.62, 0.38 and 0.71. The buyer receives notional x (realized correlation - strike), where realized correlation is the average pairwise correlation. What is the settlement for the buyer?

1. A Receives USD 0.60 million
2. B Receives USD 1.32 million
3. C Pays USD 0.60 million
4. D Pays USD 0.83 million

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Answer: A. Realized correlation = 2 / (n^2 - n) x sum of rho_ij for i > j = 2 / 6 x (0.62 + 0.38 + 0.71) = 1.71 / 3 = 0.57.

Payoff to buyer = 5 million x (0.57 - 0.45) = USD 0.60 million received.

Question 4

Implied volatilities for options on a broad equity index decline steadily as the strike price rises (a volatility skew). Compared with a lognormal distribution, what does this imply about the risk-neutral distribution of the index?

1. A It has a fatter right tail and a thinner left tail
2. B It has fatter tails on both sides
3. C It is lognormal, because the skew only reflects bid-ask spreads
4. D It has a fatter left tail and a thinner right tail

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Answer: D. Out-of-the-money puts (low strikes) have high implied vols, so the market prices large declines as more likely than a lognormal model would. Out-of-the-money calls (high strikes) have lower implied vols, so large rises are seen as less likely. The implied distribution is therefore negatively skewed: fat left tail, thin right tail.

Explanations include the leverage effect (falling equity raises leverage and volatility) and crash-risk aversion.

Question 5

A risk manager uses the Hull-White volatility-weighted historical simulation approach. On one day in the sample the portfolio returned -2.4%, when the EWMA estimate of daily volatility for that day was 1.2%. Today's EWMA volatility estimate is 1.8%. What return is used for that day in the volatility-adjusted scenario set?

1. A -1.60%
2. B -2.94%
3. C -3.60%
4. D -5.40%

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Answer: C. Hull-White rescale each historical return by the ratio of current to then-current volatility:

r* = r x (sigma_now / sigma_then) = -2.4% x (1.8 / 1.2) = -3.60%.

Because volatility is higher now than on that day, the scenario is scaled up. The adjusted returns are then used as in ordinary historical simulation, so the VaR reflects current volatility and can exceed any loss in the raw sample.

Question 6

Ockley Partners is long USD 12 million of stock A (daily volatility 1.8%) and short USD 8 million of stock B (daily volatility 1.5%). The correlation between the two stocks' returns rises from 0.30 to 0.80. Using the delta-normal approach with zero expected returns, how does the portfolio's 99% one-day VaR change? (Use z = 2.326.)

1. A It falls by about USD 0.17 million
2. B It rises by about USD 0.10 million
3. C It rises by about USD 0.17 million
4. D It is unchanged at about USD 0.22 million, as only net exposure matters

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Answer: A. Dollar volatilities: A = 12 x 1.8% = 0.216; B = -8 x 1.5% = -0.120 (short).

Portfolio variance = 0.216^2 + 0.120^2 - 2 x rho x 0.216 x 0.120

- rho = 0.30: variance = 0.046656 + 0.0144 - 0.015552 = 0.045504; sigma = 0.2133; VaR = 2.326 x 0.2133 = 0.496
- rho = 0.80: variance = 0.046656 + 0.0144 - 0.041472 = 0.019584; sigma = 0.1399; VaR = 2.326 x 0.1399 = 0.326

Change = 0.326 - 0.496 = -USD 0.17 million.

Correlation risk works in opposite directions depending on the position: long-long portfolios lose diversification when correlation rises, whereas long-short (hedged) positions benefit. A correlation stress should therefore move correlations in the direction that hurts each portfolio, not simply raise them all.