What is Conditional VaR (CVaR / Expected Shortfall), and why did Basel III replace VaR with ES for market risk capital?

Question

AcadiFi · Accepted Answer

**Conditional VaR** (CVaR), also called **Expected Shortfall (ES)**, measures the average loss in the worst alpha-percent of scenarios. While VaR tells you the threshold loss at a given confidence level, ES tells you **how bad it gets beyond that threshold**.

**Formula (Continuous)**

**ES_alpha = E[Loss | Loss > VaR_alpha]**

For a discrete sample of n sorted losses (L_1 >= L_2 >= ... >= L_n) at alpha = 1%:

**ES_{1%} = (1 / floor(n x 0.01)) x SUM of the worst floor(n x 0.01) losses**

**Worked Example**

Dunbar Fixed Income Fund has 1,000 daily P&L observations. At the 99% confidence level:

- VaR_{99%} = $4.2 million (the 10th worst loss out of 1,000)
- The 10 worst daily losses (in $M): 9.1, 7.8, 6.5, 5.9, 5.3, 5.0, 4.8, 4.6, 4.4, 4.2

**ES_{99%} = (9.1 + 7.8 + 6.5 + 5.9 + 5.3 + 5.0 + 4.8 + 4.6 + 4.4 + 4.2) / 10 = $5.76M**

VaR says: "On 99% of days, losses won't exceed $4.2M."
ES says: "On the 1% of worst days, losses average $5.76M."

```mermaid
flowchart LR
    A["1,000 daily P&L observations"] --> B["Sort worst to best"]
    B --> C["VaR 99% = 10th worst = $4.2M"]
    B --> D["ES 99% = Average of 10 worst = $5.76M"]
    C --> E["Tells you the threshold"]
    D --> F["Tells you the expected severity beyond threshold"]
```

**Why ES Replaced VaR in Basel III (FRTB)**

| VaR Weakness | How ES Addresses It |
|---|---|
| Not subadditive — diversification can appear to increase risk | **ES is subadditive** — always rewards diversification |
| Ignores severity beyond the threshold | **ES averages the entire tail** |
| Can be gamed by shifting risk just beyond the VaR cutoff | **ES penalizes all tail scenarios** |
| No information about tail shape | **ES is sensitive to tail thickness** |

**Subadditivity Example**

Consider two portfolios X and Y:
- VaR(X) = $5M, VaR(Y) = $5M
- VaR(X + Y) could be $11M (superadditive!) — VaR says combining them increases risk
- ES(X) = $7M, ES(Y) = $7M
- ES(X + Y) will always be <= $14M — ES correctly reflects diversification benefit

**FRTB Specifics:**
- Basel moved from 99% VaR (10-day) to **97.5% ES** for internal models
- The confidence level was lowered because ES already captures tail severity
- Stressed ES is calibrated to the worst 12-month period in the bank's history

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What is Conditional VaR (CVaR / Expected Shortfall), and why did Basel III replace VaR with ES for market risk capital?

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Practice Questions

VaR Weakness	How ES Addresses It
Not subadditive — diversification can appear to increase risk	ES is subadditive — always rewards diversification
Ignores severity beyond the threshold	ES averages the entire tail
Can be gamed by shifting risk just beyond the VaR cutoff	ES penalizes all tail scenarios
No information about tail shape	ES is sensitive to tail thickness