The difference between true severity and the best approximation of the true severity is called:
Correct Answer: A
Explanation This question relates to fitting a distribution to the true severity of the operational risk loss we are trying to model. The quality of the fit, or the precision of the fit, has two elements to the difference between the severity as represented by our model and the true severity. To understand this, consider the three data points below: a. The true severity, b. The best approximation of the true severity in the model space, and c. The fit based on the dataset. - True severity is what we are trying to model. - The model space refers to the collection of analytical distributions (log-normal, burr etc) that we are considering to arrive at the estimate of the severity. - The 'best approximation of the true severity in the model space' is reached byestimating the parameters of the distribution that optimizes the risk functional. - The 'fit' is the actual parameter estimates we settle for with the distribution we have determined best fits the true estimate of our severity. When estimating parameters,we have various methods available for estimation - the least squares method, the maximum likelihood method, for example, and we can get different estimates depending upon the method we choose to use. Our severity model will be different from the true severity, and the total difference can be split into two types of errors: 1. Fitting error, represented by 'c - b' above: The difference between the fit based on the dataset and the best approximation of the true severity is called 'fitting error', ie, a measure of the extent to which we could have estimated the parameters better. 2. Approximation error, represented by 'b - a' above: Approximation error is the difference between the true severity, and the best approximation of the true severity that can be achieved within the model space is called 'approximation error'. One can reduce the approximation error by expanding the model space by adding more distributions. This will reduce the approximation error, but generally has the effect of increasing the fittingerror because the complexity of the model space increases, and there are more ways to fit to the true severity.
Question 2
The probability of default of a security over a 1 year period is 3%. What is the probability that it would have defaulted within 6 months?
Correct Answer: C
Explanation The question is asking for the probability of default over a 6 month period when the probability of annual default is known. If we let the 6 month probability of defaut be 'd', then the probability of survival at the end of 1 year would be (1 - d)^2. This we know is equal to 1 - 3% = 0.97. Therefore we can calculate 'd' to be equal to 1.51%. Choice 'c' is the correct answer, the others are incorrect. Note that an exam question may ask for probability of the security having survived after 6 months, in which case the answer might be 1 - 1.51%. Also note that such questions will always require you to use the probability of survival (1 - probability of default) for doing the calculations. That isbecause the probabilities of survival can be multiplied over periods of time, but not probabilities of default as the first default in any period is the 'game-over' event after which neither survival nor defaults mean anything. Therefore you generally always have to get the probability of survival till a point in time, and use that for any other calculations.
Question 3
When compared to a medium severity medium frequency risk, the operational risk capital requirement for a high severity very low frequency risk is likely to be:
Correct Answer: C
Explanation High frequency and low severity risks, for example the risks of fraud losses for a credit card issuer, may have high expected losses, but low unexpected losses. In other words, we can generally expect these losses to stay within a small expected and known range. The capital requirement will be the worst case losses at a given confidence level less expected losses, and in such cases this can be expected to below. On the other hand, medium severity medium frequency risks, such as the risks of unexpected legal claims, 'fat-finger' trading errors, will have low expected losses but a high level of unexpected losses. Thus the capital requirement for suchrisks will be high. It is also worthwhile mentioning high severity and low frequency risks - for example a rogue trader circumventing all controls and bringing the bank down, or a terrorist strike or natural disaster creating other losses - will probably have zero expected losses & high unexpected losses but only at very high levels of confidence. In other words, operational risk capital is unlikely to provide for such events and these would lie in the part of the tail that is not covered by most levels ofconfidence when calculating operational risk capital. Note that risk capital is required for only unexpected losses as expected losses are to be borne by P&L reserves. Therefore the operational risk capital requirements for a low severity high frequency risk is likely to be low when compared to other risks that are lower frequency but higher severity. Thus Choice 'c' is the correct answer.
Question 4
The cumulative probability of default for a security for 4 years is 11.47%. The marginal probability of default for the security for year 5 is 5% during year 5. What is the cumulative probability of default for the security for 5 years?
Correct Answer: C
Explanation The cumulative probability of default for the security for the 5 years is [1 - (1 - probability of default upto year 4)*(1 - probability of default in year 5)]. An easier way to think about this is that the Probability of survival till year 5 = (Probability of survival till year 4 * Probability of survival during year 5). Using the relationship that probability of default = 1 - probability of survival, we can calculate the required probability in all cases. In this case, the cumulative probability of default for the security for 5 years = 1 - (1 - 11.47%)*(1 - 5%) = 15.8695%, therefore Choice 'c' is the correct answer.
Question 5
Which of the following need to be assumed to convert a transition probability matrix for a given time period to the transition probability matrix for another length of time: I. Time invariance II. Markov property III. Normal distribution IV. Zero skewness
Correct Answer: C
Explanation Time invariance refers to all timeintervals being similar and identical, regardless of the effects of business cycles or other external events. The Markov property is the assumption that there is no ratings momentum, and that transition probabilities are dependent only upon where the rating currently is and where it is going to. Where it has come from, or what the past changes in ratings have been, have no effect on the transition probabilities. Rating agencies generally provide transition probability matrices for a given period of time, say a year. The risk analyst may need to convert these into matrices for say 6 months, 2 years or whatever time horizon he or she is interested in. Simplifying assumptions that allow him to do so using simple matrix multiplication include these two assumptions - time invariance and the Markov property. Thus Choice 'c' is the correct answer. The other choices (normal distribution and zero skewness) are non-sensical in this context.
