Exercise: Building VaR's subadditivity counter-example

Consider two independent defaultable bonds $X$ and $Y$ with identical marginals:

\mathbb{P}(X = 5) = 0.98, \quad \mathbb{P}(X = -100) = 0.02, \quad (\text{same for } Y),

and $X, Y$ independent.

Tasks

Compute $\text{VaR}_{0.05}(X)$ and $\text{VaR}_{0.05}(Y)$ individually.
Enumerate the distribution of $X + Y$ (all four possible outcomes with probabilities).
Compute $\text{VaR}_{0.05}(X + Y)$ .
Verify numerically that $\text{VaR}_{0.05}(X + Y) > \text{VaR}_{0.05}(X) + \text{VaR}_{0.05}(Y)$ — VaR fails subadditivity.
For the same $X, Y$ , compute $\text{ES}_{0.05}$ on each individually and on the sum. Verify ES is subadditive here.
Interpretation. In plain English, explain why holding two independent defaultable bonds can look riskier under VaR than holding just one.

Hint

For part 3, you need to identify the quantile of a four-point distribution — list probabilities cumulatively and find the first value where cumulative probability exceeds $0.05$ .