Why is Expected Shortfall considered superior to VaR, and what makes a risk measure 'coherent'?

The transition from VaR to Expected Shortfall as the primary regulatory risk measure (mandated by FRTB/Basel III.5) is a major theme in FRM Part II. Understanding why requires grasping the concept of coherent risk measures. **Value at Risk (VaR) — Definition:** VaR at confidence level α is the loss threshold such that losses exceed VaR only (1-α)% of the time. - Example: 99% 1-day VaR of $5.2 million means there's a 1% chance of losing more than $5.2M tomorrow. **Expected Shortfall (ES) — Definition:** ES at confidence level α is the expected (average) loss **given that losses exceed VaR**. - Example: 97.5% ES of $7.8 million means that on the worst 2.5% of days, the average loss is $7.8M. **The Four Properties of a Coherent Risk Measure:** A risk measure ρ is coherent if it satisfies: 1. **Monotonicity:** If portfolio X always produces worse outcomes than Y, then ρ(X) ≥ ρ(Y) - Makes sense: a riskier portfolio should have a higher risk measure 2. **Translation Invariance:** ρ(X + cash) = ρ(X) - cash - Adding $1M in risk-free cash reduces the risk measure by exactly $1M 3. **Positive Homogeneity:** ρ(λX) = λρ(X) for λ > 0 - Doubling the position doubles the risk 4. **Subadditivity:** ρ(X + Y) ≤ ρ(X) + ρ(Y) - **This is the critical property.** Diversification should not increase risk. The risk of a combined portfolio should be at most the sum of individual risks. **VaR Fails Subadditivity:** Here's a concrete example. Thornfield Asset Management has two portfolios: **Portfolio A:** 1% chance of losing $10M, 99% chance of gaining $0.5M **Portfolio B:** 1% chance of losing $10M, 99% chance of gaining $0.5M (A and B are independent) - VaR(99%) of A = $0 (99th percentile loss is actually a small gain) - VaR(99%) of B = $0 - VaR(99%) of A + B: - Joint probability of both losing: 0.01 × 0.01 = 0.01% → loss of $20M - Probability of exactly one losing: 2 × 0.01 × 0.99 = 1.98% → loss of ~$9.5M - These 1.99% worst cases exceed 1%, so VaR(99%) of A+B ≈ **$9.5M** **Result:** VaR(A+B) = $9.5M > VaR(A) + VaR(B) = $0 + $0 = $0 VaR says diversification *increased* risk! This violates subadditivity and can lead to absurd conclusions like telling desks to keep positions separate rather than netting them. **ES Satisfies All Four Properties:** Using the same example: - ES(99%) of A = $10M × (1%/1%) = $10M (the average loss in the worst 1%) - ES(99%) of B = $10M - ES(99%) of A+B: In the worst 1%, the average loss accounts for both the 0.01% chance of losing $20M and part of the 1.98% chance of losing $9.5M → ES ≈ $10.1M **Result:** ES(A+B) ≈ $10.1M ≤ ES(A) + ES(B) = $20M. Subadditivity holds. **Other VaR Shortcomings:** 1. **Tells you nothing about tail severity.** VaR says "1% chance of losing more than $5.2M" but doesn't say whether the average tail loss is $6M or $60M. ES explicitly captures this. 2. **Can be manipulated.** A trader can restructure positions to move losses just past the VaR threshold, keeping VaR constant while massively increasing tail risk. **The Backtesting Challenge:** ES is harder to backtest than VaR because: - VaR backtesting is binary: did the loss exceed VaR or not? Easy to count exceptions. - ES backtesting requires estimating the average of losses in the tail — you need many more observations of extreme events, which are rare by definition. - FRTB addresses this by using VaR for backtesting (the "traffic light" approach) while using ES for capital calculations. **Regulatory Implementation (FRTB):** - Basel replaced 99% VaR with 97.5% ES for internal models capital calculation - 97.5% ES ≈ 99% VaR for normal distributions, but captures tail risk better for fat-tailed distributions **Exam Tip:** Know the four coherence properties by heart and be able to construct the subadditivity counterexample for VaR. Also understand why ES is harder to backtest — this is a common exam question. Practice ES vs. VaR problems in our FRM Part II question bank.