Recipe

Compute VaR and CVaR from a loss table

Use when: you have a discrete loss table (outcomes and their probabilities) and a risk level $\alpha$

Goal: the value-at-risk threshold and the conditional value-at-risk that follows it

Taught in Value-at-risk and conditional value-at-risk

Quick recipe

Find VaR: the smallest threshold covering $1-\alpha$ probability
Identify the tail beyond VaR
Average the tail to get CVaR

1
Sort outcomes from smallest to largest and accumulate probability until the running total first reaches $1-\alpha$ . That outcome is $\mathrm{VaR}_\alpha$ .

Example

Loss $100$ (prob $0.80$ ), $150$ (prob $0.15$ ), $600$ (prob $0.05$ ); $\alpha=0.05$ . At $t=100$ : $P(L\le t)=0.80$ , not enough. At $t=150$ : $P(L\le t)=0.95$ , clearing the bar exactly. $\mathrm{VaR}_{0.05}=150$ .

⚡ Shortcut

For a loss uniform on $[a,b]$ , skip the accumulation and use Section 8's closed form directly: $\mathrm{VaR}_\alpha=a+(1-\alpha)(b-a)$ . For $[100,200]$ at $\alpha=0.1$ : $\mathrm{VaR}_{0.1}=100+0.9(100)=190$ .

Why: this is exactly the $(1-\alpha)$ -quantile — the smallest cutoff whose cumulative probability first clears the confidence level.
2
Collect every outcome strictly greater than the VaR threshold, and confirm their combined probability equals $\alpha$ .

Example

Continuing the table above, the tail beyond $150$ is the single outcome $L=600$ , with probability exactly $0.05$ .
- the tail has exactly one outcome — $\mathrm{CVaR}_\alpha$ is that outcome itself.
- the tail has more than one outcome — $\mathrm{CVaR}_\alpha$ is a probability-weighted average over the whole tail, not just its largest value.
Why: $\mathrm{CVaR}$ is defined only over the tail beyond VaR — mixing in outcomes below the threshold would understate how bad the worst case actually gets.
3
Compute $E[L\mid L\ge\mathrm{VaR}_\alpha]$ : the probability-weighted average of the tail outcomes identified in Step 2.

Example

With the tail equal to the single outcome $600$ , the average is $600$ itself: $\mathrm{CVaR}_{0.05}=600$ .

Why: this is exactly the definition of conditional value-at-risk — the expected loss given landing in the worst $\alpha$ -tail.

Common mistakes

Reporting VaR alone and assuming it describes tail severity. Two tables can share the same VaR and have wildly different CVaR.
Averaging over ALL outcomes instead of only the tail beyond VaR.
Applying the uniform-distribution shortcut to a non-uniform loss table, where CVaR generally isn't a simple midpoint.
Forgetting to confirm the tail's total probability actually equals $\alpha$ before averaging it.