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 can be 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 ways in 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% confidence level, 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.

Let the probability of default over a month be p. Therefore the probability of survival at the end of 12 months would be (1 - p)^12. Since the one year probability of default is 12%, we know that the probability of survival is 88%. Putting (1 - p)^12 = 88% and solving for p, we get p = 1.06%. Therefore Choice 'd' is the correct answer.

Generally, stress tests consider a shock or a severe scenario in order to determine the 'what-if' that circumstance were to materialize. They focus on the outcome based upon a set of shocks. In a reverse stress test, the outcome is assumed to be known (generally something as severe as bankruptcy, non-compliance with capital requirements etc), and the test is intended to work out what shocks or events would lead to such an outcome.

Reverse stress tests therefore start from a known stress test outcome (such as breaching regulatory captial ratios, illiquidity or insolvency) and then asking what events could lead to such an outcome for the bank. This can be quite a challenging task. Principle 9 laid out in the BCBS document on stress testing (May 2009) (which is part of the PRM syllabus effective March 1, 2010) lays down the expectations relating to reverse stress tests.

Therefore Choice 'a' is the correct answer. All the other choices are nonsensical.

Managing liquidity requires understanding and providing for two things: the amount of liquidity needed to pay for all current obligations (in both normal and stressed scenarios), and determining the time horizon over which this liquidity should be available. The first is essentially a function of the business of the firm, and the assets and liabilities resulting from operations. The second considers other factors such as the speed with which new cash can be borrowed (eg, from the repo markets), the consequences are of running out of liquidity (eg, maybe only an overdrawn fee as opposed to bankruptcy), etc. In other words, liquidity risk management answers two questions: how much, and for how long.

This question asks for identifying the factors that affect the 'how much' part. Choice 'd', Choice 'c' and Choice 'b' do affect the determination of the liquidity needs of the firm. Choice 'a' does not affect the liquidity needs, but the 'how long' part. Choice 'a' is therefore the correct answer.

The default correlation between two obligors goes up if the joint probability of default between them increases. This is intuitive. Also consider the formula for the default correlation between two obligors

Default correlation = [P(1,2) - P1 * P2] / √P1*(1-P1)*P2*(1-P2); where P(1,2) is the joint probability of default between the two and P1 and P2 are their individual probabilities of default. Obviously, an increase in P(1,2) will cause the default correlation to increase.

Stress tests can cover known events, but since the future is unknown, and new events may be entirely different from what has happened in the past, they provide no assurance that an institution's worst case losses would be covered. Hence II is false.

Stress testing is required to be performed as part of Basel II, and therefore III is false.

Historical events do not have sharply defined start and end dates. Often, even after a crises ends, its after effects may continue to affect the markets for a long time. In such cases, it may be difficult to define the start and end of the crises. In many cases, the crises may persist for months or even years, making it difficult for the risk manager to identify a time period that covers the essence of the crises, and yet is focused enough to constitute a plausible scenario. Therefore IV is false too. Only I is true, and the correct answer is Choice 'b'.

Component VaR measures the proportion of total VaR that can be allocated to each asset in the portfolio. It is based upon the covariance matrix multiplied by the weights, and each row represents the component VaR for the asset in question. Since the total of such a matrix is the total VaR, component VaR is additive. Component VaR is used to assess the contribution of each asset in the portfolio to total risk and has the useful property of being additive so some sense can be made of the contribution of each asset to total risk.

Incremental VaR, conditional VaR and VaR are sub-additive by definition, and therefore not the correct answer.

A bank's stress testing programme in relation to firm wide stress tests should document the type, frequency and the purpose of the programme, as well as methodologies for defining scenarios and the remedial actions envisaged. Choice 'b' is therefore a necessary attribute of a robust stress testing programme.

The programme should be supported by a robust systems infrastructure that allows the execution of periodic as well as ad-hoc stress tests at the right level (business unit, as well as firm-wide) at the right level of detail or granularity. Choice 'c' also therefore is a valid choice.

A related element is data quality - without which no stress tests can be be credible.

Therefore all the choices listed are correct and Choice 'd' is the correct answer.