III AdvancedWeek 8 • Lesson 22Duration: 50 min

ES Expected Shortfall (CVaR)

When VaR breaks, how bad does it actually get?

Learning Objectives

•Understand Expected Shortfall and why it replaced VaR in Basel III
•Compute and interpret CVaR on real return data
•Know how ES informs practical risk thresholds
•See the difference between VaR and CVaR on the same distribution

Explain Like I'm 5

VaR says "you probably won't lose more than X." Expected Shortfall says "okay, but when you DO lose more than X, the average loss will be Y." It answers the follow-up question that VaR leaves hanging — how bad does the tail actually get?

Think of It This Way

If VaR is a speed limit sign, Expected Shortfall tells you the average speed of the cars that are breaking the speed limit. Knowing that some cars speed isn't enough — you need to know if they're doing 70 in a 65 zone or 120. ES gives you that picture.

1Why ES Fixes VaR's Biggest Problem

Expected Shortfall (also called CVaR or Conditional VaR) does what VaR can't: it tells you what happens in the tail. 95% ES = the average loss when you're in the worst 5% of outcomes. Two portfolios can have identical VaR but wildly different ES: • Portfolio A: VaR = 5%, ES = 6% — losses barely exceed VaR • Portfolio B: VaR = 5%, ES = 25% — losses can be catastrophic Same VaR. Completely different risk profiles. ES is also sub-additive — diversification always reduces portfolio ES. VaR doesn't guarantee this. In technical terms, ES is a "coherent risk measure" while VaR is not. That distinction pushed Basel III to switch from VaR to ES for bank capital requirements. Artzner, P. et al. (1999). "Coherent Measures of Risk." Mathematical Finance.

2Same VaR, Different Tail Risk

This is the chart that makes the concept click. Both portfolios have the exact same 95% VaR — the boundary of the tail is identical. But look at what happens inside the tail. Portfolio B has far more weight out in the extreme losses. Same VaR, completely different actual risk. If you only look at VaR, you miss the most important part of the picture.

VaR vs CVaR — Same VaR, Different Tail Risk

3ES in Practical Risk Management

Monte Carlo analysis produces ES at multiple confidence levels, and those numbers directly inform drawdown protection thresholds. • 95% ES tells you the average drawdown in the worst 5% of scenarios • 99% ES tells you the average drawdown in the worst 1% Drawdown-triggered risk scaling uses these estimates: at what drawdown level should you start getting worried about tail events? The answer comes directly from the Monte Carlo tail distribution. ES isn't just a theoretical upgrade over VaR. It's the actual input to real risk decisions. It tells you where to place the guardrails — and how high to make them.

4Sub-Additivity: Why the Math Matters

This sounds abstract, but it's practically important. Sub-additive means: risk of combined portfolio ≤ sum of individual risks. This is the mathematical way of saying diversification reduces risk. VaR is not sub-additive. You can genuinely have: • Asset A VaR = $5K • Asset B VaR = $5K • Portfolio (A + B) VaR = $12K That says diversification increased risk. That's broken. ES is always sub-additive. Portfolio ES ≤ sum of individual ES values. Guaranteed. This means you can use ES for portfolio construction, risk allocation, and budgeting without the math contradicting itself. This is why serious risk frameworks moved to ES. VaR doesn't play well with portfolios.

5ES/VaR Ratio — A Quick Diagnostic

The ES/VaR ratio tells you how heavy your tails are. For a normal distribution, the ratio is about 1.2× at 95% confidence. For real financial returns, it's typically 1.5–2.5×. If your ratio is above 2×, you have seriously fat tails and should be extra cautious with position sizing. This chart shows how the ratio varies across asset classes. Crypto has the fattest tails, followed by individual equities. Diversified portfolios have thinner tails — because diversification works.

ES/VaR Ratio by Asset Class (95% Confidence)

Key Formulas

Expected Shortfall

Average of all VaR levels beyond the confidence threshold. This captures the entire tail, not just the boundary. For discrete samples: average all returns worse than VaR.

Discrete ES

Sort returns worst to best. Average the worst (1-α)×n returns. That's your ES. Simple and model-free.

Hands-On Code

VaR vs Expected Shortfall

python

import numpy as np

def var_and_es(returns, confidence=0.95):
    """Compute both VaR and Expected Shortfall."""
    alpha = 1 - confidence
    sorted_returns = np.sort(returns)
    cutoff = int(len(returns) * alpha)
    
    var = -sorted_returns[cutoff]
    es = -sorted_returns[:cutoff].mean()
    
    print(f"{confidence:.0%} VaR: {var:.2%}")
    print(f"{confidence:.0%} ES:  {es:.2%}")
    print(f"ES/VaR ratio: {es/var:.2f}x")
    print(f"  Tail is {es/var:.2f}x worse than VaR boundary")
    
    return var, es

The ES/VaR ratio tells you about tail severity. Normal distributions give ~1.2×. Real financial data often runs 1.5–2×. That gap is why VaR alone is dangerous.

Knowledge Check

Q1.Two portfolios have 95% VaR = 5%. Portfolio A has ES = 6%, Portfolio B has ES = 20%. Which is riskier?

Assignment

Generate returns from both a normal distribution and a Student-t distribution (df=3). Compute VaR and ES for both. The t-distribution should show a much larger ES/VaR ratio — that's fat tails making VaR misleading.

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