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.
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