FRM P2 Credit Risk: practice questions with answers

Question 1

A company's 3-year bonds trade at a credit spread of 180 basis points over the risk-free rate. The expected recovery rate is 40%. Assuming a constant hazard rate, what is the risk-neutral probability that the company defaults within 3 years?

1. A 5.26%
2. B 8.61%
3. C 9.00%
4. D 12.63%

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Answer: B. Hazard rate lambda = spread / (1 - R) = 0.018 / 0.60 = 3.0% per year.
Survival to year 3 = e^(-0.03 x 3) = e^(-0.09) = 0.9139.
Cumulative default probability = 1 - 0.9139 = 8.61%.

Question 2

Marlowe Bank has four OTC derivative trades with one counterparty, with current mark-to-market values to the bank (USD million) of +10, -6, +4 and -3. All trades are under a single enforceable ISDA master agreement with close-out netting and there is no collateral. By how much does netting reduce the bank's current exposure?

1. A USD 5 million
2. B USD 9 million
3. C USD 14 million
4. D USD 23 million

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Answer: B. Without netting, exposure is the sum of positive values only: 10 + 4 = 14 (the bank would still owe 6 + 3 on the negative trades).
With netting, exposure = max(sum of MTM, 0) = max(5, 0) = 5.
Reduction = 14 - 5 = USD 9 million.

Question 3

Wendover Bank has a large uncollateralised netting set with Calvert Airlines, consisting mainly of swaps on which the bank receives fixed and which have positive expected exposure to the bank. Calvert asks the bank to quote a new swap on which the bank would pay fixed. Which statement about the CVA charge for the new trade is most accurate?

1. A The new trade should be charged its stand-alone CVA, because CVA is additive across trades with the same counterparty under a netting agreement
2. B The new trade needs no CVA at all, because any trade that reduces existing exposure becomes free of counterparty risk once in the netting set
3. C The incremental CVA must exceed the stand-alone CVA, because adding trades to a netting set always raises the expected positive exposure
4. D Its incremental CVA, measured on the whole netting set, may be small or even negative, so the bank could quote better than its stand-alone CVA implies

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Answer: D. With an enforceable netting agreement, exposure is measured on the net value of all trades in the netting set, so a trade's CVA depends on what else is in the set.

- Stand-alone CVA treats the trade as if it were the only trade with the counterparty.
- Incremental CVA = CVA(netting set with the new trade) - CVA(netting set without it).

The existing receive-fixed swaps gain value to the bank when rates fall. A new pay-fixed swap gains when rates rise, so in many scenarios it offsets the existing exposure. Its incremental CVA can therefore be much lower than its stand-alone CVA, and may be negative, in which case the bank could in principle pay the counterparty for the trade. CVA pricing should be done incrementally at the netting-set level, which is why it is usually handled centrally, for example by a CVA desk.

Question 4

A logistic regression credit-scoring model gives z = -3.2 + 0.9 x (debt-to-income ratio) - 0.04 x (years at current address) + 1.1 x (missed payment flag, 1 or 0). An applicant has a debt-to-income ratio of 1.5, 20 years at the current address and one missed payment. What is the applicant's estimated probability of default?

1. A 6.1%
2. B 17.5%
3. C 21.2%
4. D 82.5%

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Answer: B. z = -3.2 + 0.9 x 1.5 - 0.04 x 20 + 1.1 x 1 = -3.2 + 1.35 - 0.80 + 1.10 = -1.55.
PD = 1 / (1 + e^(-z)) = 1 / (1 + e^1.55) = 1 / (1 + 4.711) = 17.5%.

Odds = e^(-1.55) = 0.212, and PD = odds / (1 + odds) = 0.175.

Question 5

Two borrowers have one-year default probabilities of 4% and 6%. The probability that both default within the year is 0.6%. What is the default correlation between the two borrowers?

1. A 0.0036
2. B 0.077
3. C 0.129
4. D 0.150

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Answer: B. Covariance of the default indicators = P(both) - P(A)P(B) = 0.006 - 0.0024 = 0.0036.

Standard deviations: sqrt(0.04 x 0.96) = 0.1960; sqrt(0.06 x 0.94) = 0.2375.

Default correlation = 0.0036 / (0.1960 x 0.2375) = 0.077.

Although a correlation of about 0.08 looks small, it makes the joint default probability 2.5 times what it would be under independence (0.24%).

Question 6

A retail credit scoring model is tested on 1,000 past applicants, of whom 80 later defaulted. At the proposed cut-off score the model would have rejected 300 applicants, 56 of whom defaulted. What is the false alarm rate, i.e. the proportion of non-defaulters that the model wrongly rejects?

1. A 26.5%
2. B 30.0%
3. C 73.5%
4. D 81.3%

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Answer: A. Non-defaulters = 1,000 - 80 = 920. Non-defaulters rejected = 300 - 56 = 244.

False alarm rate = 244 / 920 = 26.5%.

Hit rate = 56 / 80 = 70%. Plotting the hit rate against the false alarm rate for every possible cut-off traces out the ROC curve; moving the cut-off trades rejected good customers against accepted bad ones.