A corporate bond has a cumulative probability of default equal to 20% in the first year, and 45% in the second year. What is the monthly marginal probability of default for the bond in the second year, conditional on there beingno default in the first year?
Correct Answer: A
Explanation Note that marginal probabilities of default are the probabilities for default for a given period, conditional on survival till the end of the previous period. Cumulative probabilities of default are probabilities of default by a point in time, regardless of when the default occurs. If the marginal probabilities of default for periods 1, 2... n are p1, p2...pn, then cumulative probability of default can be calculated as Cn = 1 - (1 - p1)(1-p2)...(1-pn). For this question, we can calculate the marginal probability of default for year 2 by solving the equation [1 - (1 - 20%)(1 - P2) = 45%] for P2. Solving, we get the marginal probability of default during year 2 as 31.25%. Since this is the annual marginal probability of default, we will need to convert it to a monthly number, which we can do by solving the following equation where M1 is the monthly marginal probability of default. 1 - 31.25% = (1 - M1)^12, implying M1 = 3.07%
Question 2
In respect of operational risk capital calculations, the Basel II accord recommends a confidence leveland time horizon of:
Correct Answer: D
Explanation Choice 'd' represents the Basel II requirement, all other choices are incorrect.
Question 3
Which of the following statements are true: I. A high score according to Altman's Z-Score methodology indicates a lower default risk II. A high score according to theProbit or Logit models indicates a higher default risk III. A high score according to Altman's Z-Score methodology indicates a higher default risk IV. A high score according to the Probit or Logit models indicates a lower default risk
Correct Answer: D
Explanation A high score under the probit and logit models indicates a higher default risk, while under Altman's methodology it indicates a lower default risk. Therefore Choice 'd' is the correct answer.
Question 4
Credit exposure for derivatives is measured using
Correct Answer: C
Explanation Current replacement values are a very poor measure of the credit exposure from a derivative contract, because the future value of these instruments is unpredictable, ie is stochastic, and the range of values it can take increases the further ahead in the future we look. Therefore it is common for credit exposures for derivatives to be measured using forward looking exposure profiles, which are distributions of the expected value of the derivative at the time horizon for which credit risk is being measured. To be conservative, a high enough quintile of this distribution is taken as the 'loan equivalent value' of the derivative as the exposure.Choice 'c' is the correct answer. The notional value of derivative contracts generally tends to be quite high and unrelated to their economic value or the counterparty exposure. Therefore notional value is irrelevant.
Question 5
Calculate the 1-year 99% credit VaR of a portfolio of two bonds, each with a value of $1m, and the probability of default of 1% each over the next year. Assume the recovery rate to be zero, and the defaults of the two bonds to be uncorrelated to each other.
Correct Answer: C
Explanation This question requires the calculation of the credit VaR of the bonds - note that in the real exam the question may not refer to 'credit' VaR, but that canbe inferred from the context, ie because the probability of default is provided, it can only be asking for the credit VaR. (Note the difference from the market risk VaR which is driven by interest rate changes affecting the value of the bonds - there are other questions addressing that calculation). Credit VaR = Expected Value - Worst case portfolio value at the selected percentile (ie the confidence level) Thus if we know the distribution of the portfolio value in the future, we can find out the value at the required percentile (in this case 99%), and the VaR will be the difference between this value and the expected value of the portfolio. An important piece of information provided is that the defaults are independent, ie they are not correlated. This means joint probabilities of default or survival can be easily found by multiplying the relevant probabilities. The following outcomes are possible: 1. Both bonds default: Probability = 1% * 1% = 0.01%. Portfolio value = $0 (because both bonds have defaulted& there is zero recovery) 2. One bond defaults and the other survives: Probability = 2 * 1% * 99% = 1.98%. Portfolio value = $1m (because one bond survives with a value of $1m and the defaulted bond has a value of $0). (Note that because there are two waysin which this can happen, ie bond 1 defaults, bond 2 survives; and bond 1 survives, bond 2 defaults, we need to multiply the probability by 2). 3. Both bonds survive: Probability = 99% * 99% = 98.01%. Portfolio value = $2m. Expected value is therefore $1.98m (which is equal to 2 * $1m * (1 - 1%), or alternatively can also be obtained by multiplying the probabilities in the above three outcomes with the value associated with each). The future distribution of the value of the portfolio can be constructed from the three outcomes outlined above: a. Upto the 98.01th percentile the value of the portfolio is $2m, and the VaR is zero (being greater than the expected value, so there is nothing to lose) b. From the 98.01th percentile to the 99.99th percentile (98.01+the next 1.98%), the value of the portfolio is $1m. VaR in this range is $0.98m (=$1.98m - $1m) c. From the 99.99th to the 100th percentile the value of the portfolio is $0, and the VaR is $1.98m. Since the question is asking for VaR at the 99% confidencelevel, it lies in the range in 'b' above, and therefore the VaR is $0.98m. Therefore Choice 'c' is the correct answer and the rest are incorrect.
