Which of the following statements are true: I. Heavy tailed parametricdistributions are a good choice for severity modeling in operational risk. II. Heavy tailed body-tail distributions are a good choice for severity modeling in operational risk. III. Log-likelihood is a means to estimate parameters for a distribution. IV. Body-tail distributions allow modeling small losses differently from large ones.
Correct Answer: D
Explanation When modeling for operational risk, we are generally concerned with tail losses - this isbecause the horizon for operational risk is 1 year at the 99.9th percentile. Since the 99.9th percentile is in the tail region, we would like to ensure that the tails are modeled as accurately as possible. Operational risk distributions are modeled usingheavy tailed distributions. Heavy tailed parametric distributions such as log-normal, pareto and others are therefore a good choice for modeling risk severity, therefore statement I is correct. Body-tail distributions are combinations of parametric distributions, with different types of distributions being used to model the body and the tail - this provides flexibility because small and medium losses upto a threshold can be modeled using one distribution, and losses beyond the threshold can be modeled usinga different distribution that is a better estimate of the tail. Statement II is therefore correct. A log-likelihood function simplifies the optimization of a regular likelihood function. We generally maximize (or minimize the risk functional) a likelihoodfunction with a view to estimating the parameters of the underlying distribution. If the likelihood function is complex, it may sometimes make it mathematically easier to optimize the log of the function - as that changes exponents and multiplications toadditions, while behaving in the same way as the underlying function. Therefore statement III is correct, log-likelihood is a means to estimate parameters for a distribution. Statement IV is correct as body-tail distributions allow modeling different partsof the distribution differently from each other.
Question 17
Financial institutions need to take volatility clustering into account: I. To avoid taking on an undesirable level of risk II. To know the right level of capital they need to hold III. To meet regulatory requirements IV. To account for mean reversion in returns
Correct Answer: B
Explanation Volatility clustering leads to levels of current volatility that can be significantly different from long run averages.When volatility is running high, institutions need to shed risk, and when it is running low, they can afford to increase returns by taking on more risk for a given amount of capital. An institution's response to changes in volatility can be either to adjust risk, or capital, or both. Accounting for volatility clustering helps institutions manage their risk and capital and therefore statements I and II are correct. Regulatory requirements do not require volatility clustering to be taken into account (at least not yet). Therefore statement III is not correct, and neither is IV which is completely unrelated to volatility clustering.
Question 18
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.
Question 19
If the default hazard rate for a company is 10%, and the spread on its bondsover the risk free rate is 800 bps, what is the expected recovery rate?
Correct Answer: B
Explanation The recovery rate, the default hazard rate (also called the average default intensity) and the spread on debt arelinked by the equation Hazard Rate = Spread/(1 - Recovery Rate). Therefore, the recovery rate implicit in the given data is = 1 - 8%/10% = 20%.
Question 20
Under the credit migration approach to assessing portfolio credit risk, which of the following are needed to generate adistribution of future portfolio values?
Correct Answer: D
Explanation The credit migration approach to assessing portfolio credit risk involves obtaining a distribution of future portfolio values from the ratings migration matrix. First, the frequencies in the matrix are used as probabilities, and expected future values of the securities belonging to each rating category are calculated. These are then discounted to the present using the discount rate appropriate to the 'future' rating category. This gives us a forward distribution of the value of each security in the portfolio. These are then combined using the default correlations between the issuers. The default correlation between the issuers is often proxied using asset returns, and recognizing that default occurs when asset values fall below a certain threshold. A distribution for the future value of the portfolio is generated using simulation, and from thisdistribution the Credit VaR can be calculated. Thus, we need the migration matrix, the risk horizon from which the present values need to be calculated, and the forward yield curve or the discount curve for each rating category for the risk horizon. Thus,Choice 'd' is the correct answer.
