Value at Risk and Expected Shortfall

Idea: Value at Risk is a quantile

How much could we lose tomorrow at a chosen confidence? The $95\%$ VaR is the smallest $x$ with $P(L\le x)\ge 95\%$.

• $L$ is tomorrow's loss, a gain counts as a negative loss
• Add chances from the best outcome up until you clear the level

Quick check: Read the VaR

Gain $3$ at $70\%$, lose $5$ at $20\%$, $12$ at $7\%$, $40$ at $2\%$, $100$ at $1\%$. What is the $95\%$ VaR?

Common mistake: Calling VaR the worst case

VaR is the best of the bad days, where the tail begins. Make the $100$ a $1000$ and it stays $12$.

Instead Expect about $2.5$ breaches of a $99\%$ VaR in $250$ days, and their size is not in the number.

Formula: Expected Shortfall

$\mathrm{ES}=\text{average}\text{loss}in\text{the}\text{worst}5\%$

The average of the tail beyond the VaR level. Cut the tail to its exact size.

Example: ES of the same book

Same book. What is the $95\%$ ES?

1. Fill the worst $5\%$ from the end: $1+2+2=5$
2. Only $2\%$ of the $7\%$ at $12$ is used
3. Tail sum: $1\times 100+2\times 40+2\times 12=204$
4. $\frac{204}{5}=40.8$

Idea: With a normal model

With mean $0$, both numbers are multiples of $\sigma$. Treat them as a first sketch.

• $99\%$: VaR about $2.326\sigma$, ES about $2.665\sigma$
• Over $4$ days scale by $\sqrt{4}=2$, if days are independent
• Heavy tails make normal multiples understate far losses

Quick check: Normal VaR

Daily returns are normal, mean $0$, $\sigma =1.5\%$. What is the $99\%$ VaR, in percent of the book?

Idea: Why desks prefer ES for limits

VaR is not subadditive: the VaR of a sum can exceed the sum of the VaRs.

• Each loan loses $10$ with $4\%$: alone, $95\%$ VaR $0$
• Together $1-0.96^2=0.0784$, above $5\%$, so VaR $10$
• ES never punishes diversifying: each $0.04\times \frac{10}{0.05}=8$, pair at most $16$

Idea: Say it in the room

State the level and the horizon first. VaR is a quantile, ES a tail average.

• Name the weak points: VaR ignores tail size
• A normal model understates heavy tails