Free PRMIA 8008 Practice Exam with Questions & Answers | Set: 2

One way to think about this question is this: we are provided with two pieces of information: if the portfolio is worth $100 to start with, it will be worth $95 at the end of year 1 and $85 at the end of year 2. What it is asking for is the probability of default in year 2, for the debts that have survived year 1. This probability is $10/$95 = 10.53%. Choice 'b' is the correct answer.

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 probability of default for year 2 as [1 - (1 - 5%)(1 - 10.53%)] = 15%.

This question is different in that it uses terminology you will not find in the PRMIA handbook. Yet it is important to understand these as there may be a question based on this slightly different terminology. (It is only the terminology that is different, the concepts are the same.)

If you read the PRMIA handbook, there are three methods of calculating VaR: Analytical or parametric, historical simulation and Monte Carlo simulations. There is one more way of categorizing the methods of calculating VaR, and these are as follows:

1. Local valuation: This refers to analytical or parametric VaR. This relies upon a neat statistical formula to calculate VaR and assumes a normal distribution. It also relies upon a known covariance matrix between the different components of VaR. Local valuation based VaR is further subdivided into two types:

a. Linear VaR: Linear VaR is calculated assuming the portfolio is linear, and its value changes just based upon the delta of the portfolio. In such cases, once a change (eg, in stock values) is known, that change is multiplied by the delta alone to get the VaR. Second order effects, such as gamma or convexity are ignored.

b. Non-linear VaR: Non linear analytical VaR is calculated using both delta and the second derivative, ie gamma or the convexity. This is more accurate if the portfolio is non-linear.

The key thing about 'local revaluation' VaR is that it does not require us to reprice or completely value all instruments in the portfolio. All we have to know is the delta (or the gamma and convexity as well) and multiply that with the number of standard deviations of change in the risk factor that we are interested in. So if we are considering a bond, we don't have to recalculate the new value of the bond as we can just use the delta. This can be a significant computational advantage for a large financial institution where there may be a large number of positions.

2. Full revaluation: This refers to a VaR method where the asset in question is fully repriced based on the new value of the risk factor - and this includes both historical and Monte Carlo based VaR methods.

Local revaluation, or analytical method based VaR is computationally easier to calculate, specially if based on just the delta-normal method (ie ignoring second order effects from convexity or gamma). But it will give incorrect results if the portfolio includes substantial non-linearity or other complexities. The full revaluation methods will always give the correct results, but they can be computationally difficult to arrive at.

Statement I is completely inaccurate - local revaluation methods do take correlations into account through the correlation or covariance matrices. Statement IV is false too - the 'delta normal' VaR refers to Var calculations based upon just the delta and do not account for the convexity or optionality. Statements II and III are correct. Therefore Choice 'c' is the correct answer.

Choice 'b' is the correct answer. All other answers are incorrect.

Refer to the detailed loss event type classification under Basel II (see Annex 9 of the accord). You should know the exact names of all loss event types, and examples of each.

By assuming the assets to be independent, we are reducing the correlation from a positive number to zero. Reducing asset correlations reduces the combined standard deviation of the assets, and therefore reduces economic capital. Therefore Choice 'b' is the correct answer.

Note that this question could also be phrased in terms of the impact on VaR estimates, and the answer would still be the same. Both VaR and economic capital are a multiple of standard deviation, and if standard deviation goes down, both VaR and economic capital estimates will reduce.

An interesting property that distinguishes the three most used distributions for modeling event frequency is that for a given mean, their variances differ. The ratio of variance to mean (the variance-mean ratio, calculated as variance/mean) can then be used to decide the kind of distribution to use. Both the variance and the mean can be estimated from available data points from the internal or external loss databases, or the scenario exercise.

The variance-mean ratio reflects how dispersed a distribution is. (In the PRMIA handbook, the variance to mean ratio has been described as the "Q-Factor".)

The Poisson distribution has its mean equal to its variance, and therefore the variance to mean ratio is 1. For the negative binomial distribution, this ratio is always greater than 1, which means there is greater dispersion compared to the mean - or more intervals with low counts as well as more intervals with high counts. For the binomial distribution, the variance to mean ratio is less than one, which means it is less dispersed than the Poisson distribution with values closer to the mean.

In a situation where operational risk events are seen as clustering together, or dependent, the variance will be higher and it would be more appropriate to use the negative binomial distribution.

It is not generally sufficient to ensure the adequacy of liquidity across a group - ie it is not appropriate to just add up the sources and needs for liquidity across multiple entities in a group. This is because there can be restrictions on transferring liquidity between entities, particularly when the entities are located across borders. In cases where transfers of liquidity are permitted, there may be settlement delays in transferring funds from one entity to another. Therefore both statements I and II are incorrect.

Wholesale funding sources refers to the temporary interbank funding sources that need to be rolled over on very short intervals, often as short as overnight. These are not stable sources for long term funding. Statement III is therefore false.

Statement IV is correct as funding diversification refers to diversification of both funding sources and the duration for which the amounts are borrowed, ie tenor diversity.

Statement IV is the only correct statement and therefore Choice 'a' is the correct answer.

VaR calculations look at the lower part of the distribution of future portfolio values, for example, if the desired confidence level is 95%, the cut-off for the VaR calculation will be at the bottom 5%; similarly at 1% for a 99% confidence level. The number of observations that will end up in these bottom ranges will be few and sparse, and therefore their accuracy will generally be lower than, say, the average where observations are more likely to be concentratred. If the shape of the distribution of future portfolio values is not symmetrical and has a long tail to the left, then this problem gets further exacerbated as there may be even fewer and less reliable simulated numbers at the 5% or 1% quintiles. Thus the shape of the distribution will affect the accuracy of a VaR estimate. The distribution for a short option position, for example, will have a long tail to the left, and the VaR number will be quite significantly affected by a few simulations. On the other hand, for a long option position where the long tail is to the right, and we are interested in the left tail which is better defined and ends at zero we are more likely to get a better VaR estimate. Therefore Statement I is correct.

The number of simulations carried out directly affects the standard error, which is inversely proportional to the square root of the sample size (ie the number of simulations). THe accuracy of the VaR estimate can be increased by increasing the sample size (or reduced by reducing the sample size). Therefore Statement II is correct.

The confidence level selected for the VaR estimate also affects the accuracy of the estimate. To intuitively understand this, consider this extreme example where the desired confidence level is 99.9% and there are 1000 observations. Therefore the VaR will be determined by the last value in the sample, and will therefore be quite fickle and dependent upon what chance produces as the lowest value in the simulation. But if for the same sample the confidence level desired were to be 90%, there would be 100 observations beyond the 90% cut-off and this would be a much more stable and accurate number. Therefore the confidence level selected for the VaR estimate is also a determinant of the accuracy of the VaR estimate derived from the simulation. Statement III is correct.

Thus all statements are correct and Choice 'b' is the correct answer.

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.

In the case of mean reversion, the actual VaR would be lower than that estimated using the square root of time rule. This is because gains over a period would be followed by losses so that the price can revert to the mean. In such cases, the autocorrelation between subsequent periods is effectively negative. This means the combined VaR over the periods would be lower.

In the case of positive autocorrelation, the actual VaR would be higher than that estimated using the square root of time rule for exactly the opposite reason than that described for the mean-reverting case.

(Recall that Variance (A + B) = Variance(A) + Variance(B) + 2*Correlation*StdDev(A)*StdDev(B). In cases where correlation is zero, the variance can simply be added together (which is the case for iid observations). In cases where the correlation is negative, the combined variance (and therefore standard deviation and also VaR) will be lower; and where correlation is positive, the combined variance (and therefore standard deviation and also VaR) will be higher.)

Therefore statement II is correct, and so is statement III. Choice 'c' is the correct answer.

The probabilities of default of the two bonds are independent (as indicated by a zero default correlation). The various possible states of the portfolio are as follows:

First bond defaults, and the second does not: Probability * Loss = 0.03*0.92 * $50m = $1.38m

Second bond defaults, and the first does not: Probability * Loss = 0.97*0.08 * $50m = $3.88m

Both bonds default: Probability * Loss = 0.03*0.08 * $100m = $0.24m

Thus total expected loss on this portfolio = $5.5m. Since recovery rates are not provided, those should be assumed to be zero.

There is an easier way to solve this as well: default correlation does not affect expected losses, but their volatility. You can calculate the expected losses of the two bonds and add them up, ie, $50m*0.03 + $50m *0.08 = $5.5m