Exercise: Computing ES and VaR for a binary payoff

Consider the P&L $X$ of a single defaultable zero-coupon bond with face value $100$. With probability $0.97$ the bond pays $100$, and with probability $0.03$ the bondholder loses the entire principal (so $X = -100$).

Tasks

1. Compute $\text{VaR}_\alpha(X)$ for $\alpha \in \{0.01, 0.03, 0.05, 0.10\}$.

2. Compute $\text{ES}_\alpha(X)$ for the same four values of $\alpha$.

3. For which values of $\alpha$ is $\text{VaR}_\alpha = \text{ES}_\alpha$? For which is $\text{ES}_\alpha > \text{VaR}_\alpha$? Interpret each case.

4. Suppose the bond either pays $100$ (prob $0.97$) or only $50$ (prob $0.03$) — a haircut, not full default. How do VaR and ES at $\alpha = 0.05$ change?

Hint

Write out the CDF as a step function, then read off the quantiles and integrate.