Why did Basel replace VaR with Expected Shortfall?

VaR has two key flaws: it provides no information about tail losses beyond the threshold, and it is not subadditive (can penalize diversification). Expected Shortfall addresses both issues by capturing the average tail loss and satisfying all properties of a coherent risk measure.

What is subadditivity and why does it matter?

Subadditivity means the risk of a combined portfolio should not exceed the sum of individual risks. It reflects the benefit of diversification. VaR can violate this property, potentially showing that merging two portfolios increases risk — which contradicts basic finance theory.

How do you calculate Expected Shortfall from historical data?

Sort daily losses from largest to smallest. At 97.5% confidence, ES is the average of the worst 2.5% of observations. For 1,000 observations, this means averaging the 25 largest losses. This captures not just the VaR threshold but the severity of losses beyond it.

Why is backtesting ES harder than backtesting VaR?

VaR backtesting simply counts exceptions (days where loss exceeded VaR) and compares to the expected frequency. ES backtesting must assess the magnitude of tail losses, requiring evaluation of how bad losses were when they occurred — a harder statistical problem with fewer data points.

What confidence level is used for ES under FRTB?

The FRTB uses 97.5% Expected Shortfall, compared to 99% VaR under the previous framework. Under normal distributions, 97.5% ES produces approximately the same capital requirement as 99% VaR, maintaining calibration continuity while gaining ES's theoretical advantages.

Expected Shortfall vs VaR: A Comprehensive Comparison for FRM Candidates

The debate between Value at Risk (VaR) and Expected Shortfall (ES, also known as Conditional VaR or CVaR) is one of the most important topics in risk management. The Basel Committee's decision to replace VaR with ES for market risk capital requirements under the Fundamental Review of the Trading Book (FRTB) makes this comparison essential for every FRM candidate.

Quick Recap: What Is VaR?

As we covered in our comprehensive VaR guide, Value at Risk answers the question: "What is the maximum loss over a given time period at a given confidence level?"

$$VaR_{\alpha} = -\inf{x : P(L \leq x) \geq \alpha}$$

For example, a 1-day 99% VaR of $10 million means there is a 99% probability that the portfolio will not lose more than $10 million in one day.

What Is Expected Shortfall?

Expected Shortfall answers a different question: "Given that losses exceed VaR, what is the average loss?"

$$ES_{\alpha} = E[L ;|; L > VaR_{\alpha}]$$

At the 97.5% confidence level, ES is the average of all losses in the worst 2.5% of scenarios. It captures the severity of tail losses, not just their threshold.

Head-to-Head Comparison

Property	VaR	Expected Shortfall
Question answered	How bad can it get (threshold)?	How bad is it when it gets bad (average)?
Tail information	None beyond threshold	Full tail captured
Coherent risk measure	No	Yes
Subadditivity	Not guaranteed	Always holds
Sensitivity to tail shape	Ignores tail shape	Sensitive to tail shape
Backtesting	Straightforward	More complex
Regulatory standard	Basel II.5 (legacy)	FRTB / Basel III.1
Confidence level	Typically 99% (1-day)	Typically 97.5% (1-day)

Why Is Subadditivity Important?

A risk measure ρ is subadditive if:

$$\rho(A + B) \leq \rho(A) + \rho(B)$$

Subadditivity means that combining portfolios should not increase total risk (diversification benefit). VaR can violate subadditivity, particularly for portfolios with heavy-tailed or skewed distributions. This means VaR can paradoxically show that a combined portfolio is riskier than the sum of its parts — penalizing diversification.

Expected Shortfall is always subadditive, making it a coherent risk measure (satisfying monotonicity, positive homogeneity, translation invariance, and subadditivity).

Example of VaR Subadditivity Failure

Consider two bonds, each with a 4% probability of default and 100% loss given default:

Bond A VaR (95%) = 0 (95% probability of no loss)
Bond B VaR (95%) = 0 (95% probability of no loss)
Portfolio VaR (95%) > 0 (probability of at least one default = 7.84%)

VaR shows the portfolio is riskier than the sum of individual VaRs — violating subadditivity. Expected Shortfall avoids this problem.

The FRTB Shift: VaR to ES

The Fundamental Review of the Trading Book (FRTB), finalized by the Basel Committee, replaces VaR with Expected Shortfall for market risk capital calculations:

Old standard (Basel II.5): 99% VaR, 10-day holding period
New standard (FRTB): 97.5% ES, variable liquidity horizons

Why 97.5% Instead of 99%?

Under normal distributions, 97.5% ES and 99% VaR produce approximately the same capital requirement. The 97.5% level was chosen to maintain calibration continuity while gaining the theoretical advantages of ES.

Calculating Expected Shortfall

Parametric (Normal Distribution)

Under normality:

$$ES_{\alpha} = \mu + \sigma \cdot \frac{\phi(z_{\alpha})}{1 - \alpha}$$

Where φ is the standard normal PDF and zα is the normal quantile.

Historical Simulation

Sort historical losses from largest to smallest. ES at 97.5% confidence is the average of the worst 2.5% of observations. If you have 1,000 daily returns, ES = average of the 25 worst days.

Monte Carlo

Run simulations, sort results, and average the losses beyond the VaR threshold. As we discussed in our Monte Carlo guide, this approach handles complex portfolios naturally.

Challenges with Expected Shortfall

Despite its theoretical superiority, ES has practical challenges:

Backtesting Difficulty

VaR backtesting is straightforward: count how many times losses exceeded VaR and compare to the expected frequency. ES backtesting is harder because it requires estimating the magnitude of tail losses, not just their frequency. The Acerbi-Szekely test is one approach, but no universally accepted standard exists.

Estimation Uncertainty

ES requires estimating the average of extreme losses, which inherently involves fewer data points than VaR (which only needs a single quantile). This makes ES estimates more volatile, especially at high confidence levels.

Model Sensitivity

ES is more sensitive to the assumed tail distribution. A change from normal to Student-t distribution affects ES much more than VaR, making model choice more consequential.

FRM Exam Focus Areas

Both VaR and ES appear extensively in FRM Part 1 and Part 2:

Coherent risk measure properties and why VaR fails subadditivity
Calculating ES from historical data, parametric assumptions, and Monte Carlo
FRTB rationale for switching from VaR to ES
Backtesting challenges for ES vs. VaR
Confidence level calibration (why 97.5% ES ≈ 99% VaR under normality)
Practical examples of subadditivity violations

Understanding this comparison is fundamental to mastering risk measurement for the FRM exam!