The key to pricing loans is to make sure that the prices cover expected losses. The correct measure of expected losses is the mean, and not the median. To the extent the median is different from the mean, the loans would be over or underpric ed.

The loss curve for credit defaults is a distribution skewed to the right. Therefore its mode is less than its median which is less than its mean. Since the median is less than the mean, the bank is pricing in fewer losses than the mean, which means over the long run it is underestimating risk and underpricing its loans. Therefore Choice 'd' is the correct answer.

If on the other hand for some reason the bank were overpricing risk, its loans would be more expensive than its competitors and it would lose market share. In this case however, this does not apply. Loan pricing decisions are driven by the rate of defaults, and not the other way round, therefore any pricing decisions will not reduce the rate of default.

Rating agencies target both accuracy and stability when they assign ratings. These two objectives can sometimes conflict, so a balance needs to be struck between the two. Rating agencies do not target any particular 'accuracy ratio' or rankings. Therefore Choice 'c' is the correct answer.

The minimum multiplication factor specified under Basel II is 3. Therefore the correct answer is Choice 'a'. The exact requirements are laid down below.

Each bank must meet, on a daily basis, a capital requirement expressed as the higher of (i) its previous day’s value-at-risk number measured according to the parameters specified in this section and (ii) an average of the daily value-at-risk measures on each of the preceding sixty business days, multiplied by a multiplication factor.

The multiplication factor will be set by individual supervisory authorities on the basis of their assessment of the quality of the bank’s risk management system, subject to an absolute minimum of 3. Banks will be required to add to this factor a “plus” directly related to the ex-post performance of the model, thereby introducing a built in positive incentive to maintain the predictive quality of the model. The plus will range from 0 to 1 based on the outcome of so-called “backtesting.”

The correct answer is Choice 'd'. The following is a brief description of the major approaches available to model credit risk, and the analysis that underlies them:

1. CreditMetrics: based on the credit migration framework. Considers the probability of migration to other credit ratings and the impact of such migrations on portfolio value.

2. CreditPortfolio View: similar to CreditMetrics, but adds the impact of the business cycle to the evaluation.

3. The contingent claims approach: uses option theory by considering a debt as a put option on the assets of the firm.

4. KMV's EDF (expected default frequency) based approach: relies on EDFs and distance to default as a measure of credit risk.

5. CreditRisk+: Also called the 'actuarial approach', considers default as a binary event that either happens or does not happen. This approach does not consider the loss of value from deterioration in credit quality (unless the deterioration implies default).

If the current level of the S & P 500 is 2000, and a single day volatility is 1%, and the delta (ie change in portfolio value from a one point change) is $100, then the 1 day volatility for the portfolio in dollars is 2000 * 1% * $100 = $2,000.

At the 99% confidence level, the value of the inverse cumulative density function for the normal distribution is 2.33 (=NORMSINV(99%), in Excel). Therefore the 1 day VaR will be 2.33 * $2000 = $4,660. Extending it to 10 days using the square root of time rule, we get the 10 day VaR as equal to SQRT(10)*4660 = $14,736.

Since this method of calculating VaR relies upon a delta approximation of a risk factor (in this case the S & P500), it is the parametric approach to calculating VaR (the other methods being historical simulation, and Monte Carlo simulation).

The 2015 Handbook provides an excellent example of parametric (and other) VaR calculations in Chapter 3 of Volume III of Book 3. The spreadsheet used for the illustration can be downloaded from http://www.prmia.org/prm-exam/handbook-resources.

Internal loss data is generally the highest quality as it is relevant, and is 'real' as it has occurred to the organization. External loss data suffers from a significant limitation that the risk profiles of the banks to which the data relates is generally not known due to anonymization, and may likely may not be applicable to the bank performing the calculations. Therefore, replacing external loss data with external loss data is not a good idea. Statement IV is therefore incorrect.

All other approach described are valid approaches for the risk analyst to consider and implement. Therefore statements I, II and III are correct and IV is not.

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%

It is important to note the difference between sensitivity analysis and stress testing. Sensitivity analysis applies to measuring the effect of changes on the outputs of a model by varying the inputs - generally one input at a time.

In scenario analysis, a number of variables may be changed at the same time to see the impact on the dependent variable. For example, a bank may measure the changes in the value of its mortgage portfolio by varying its assumptions on prepayment expectations, interest rates and other factors, using its modeling software or application. The changes in the inputs may or may not relate to integrated real world situations that may arise. Sensitivity analysis is purely a quantitative exercise, much like calculating the delta of a portfolio.

A stress test may include shocks or large changes to input parameters but it does so as part of a larger stress testing programme that generally considers the interaction of risk factors, past scenarios etc. At its simplest, a stress test may be no different from a sensitivity analysis exercise, but that is generally not what is considered a stress test at large financial institutions.

A stress test may consider multiple scenarios, for example one scenario may include the events witnessed during the Asian crisis, another may include the events of the recent credit crisis. Simulation generally refers to a Monte Carlo or historical simulation, and is often a more limited exercise.

The exercise described in the question is the closest to a scenario analysis, therefore Choice 'c' is the correct answer.

It is important to note that all of the choices referred to in this question are related to each other, and the boundaries between them tend to be fuzzy. At what point does a complex sensitivity analysis start resembling a scenario, or a stress test can always be debatable, but such a debate would be more about the symantics than be of any practical use.

The probability that only none of the three bonds will default is equal to the probability of all surviving. Since default correlation is zero, we can simply multiply the probabilities of survival. Therefore the correct answer is 94% = (1 - 1%) * (1 - 2%) * (1 - 3%)

Recovery rates vary a great deal from year to year, and are difficult to predict. Therefore statement III is true. Similarly, any attempt to predict these is hamstrung by a high standard error, which can be as high as the historical mean itself. The error does not cancel itself out due to the effect of the business cycle making the error directionally biased. Thus statement IV is false.

Statement II is true as these are all factors that make forecasting recovery rates for any credit risk model rather difficult. Statement I is false because recovery rates are difficult to predict and assumptions are not easy to make.