VaR vs CVaR vs Expected Shortfall
Value at Risk (VaR) became the default market-risk metric in the 1990s. RiskMetrics, JP Morgan's 1994 framework, popularized it across banks and hedge funds. Today it is still in every risk dashboard, but it is no longer the headline regulatory measure. Basel III replaced 99% VaR with 97.5% Expected Shortfall in the Fundamental Review of the Trading Book (FRTB).
The CAIA Level II curriculum, the FRM Part I quantitative methods section, and the CFA Level III risk management readings all test these measures. The questions tend to focus less on the arithmetic (which is straightforward) and more on the interpretation (which is not).
VaR: The Definition
Value at Risk at confidence level \(\alpha\) over horizon \(T\) is the loss threshold such that the probability of exceeding it is \(1 - \alpha\):
\[ P(L > \text{VaR}_\alpha) = 1 - \alpha \]
A 99% 1-day VaR of $10M means there is a 1% chance of losing more than $10M tomorrow.
VaR has three appealing properties:
- Single number. The board, the trading desk, and the regulator all see the same dollar figure.
- Currency-denominated. Easier to understand than a Sharpe ratio.
- Sub-portfolio additivity is intuitive. Each desk has a VaR. They can be summed (with adjustments) to get firm-level VaR.
The problem is that the third property is also where VaR breaks.
VaR Is Not Coherent
In 1999, Artzner, Delbaen, Eber, and Heath introduced the four axioms a risk measure should satisfy: monotonicity, translation invariance, positive homogeneity, and sub-additivity.
Sub-additivity says: \(\rho(A + B) \leq \rho(A) + \rho(B)\). The risk of a combined portfolio is no greater than the sum of the risks of the parts. This is a formalization of diversification.
VaR violates sub-additivity for non-elliptical distributions. A combined portfolio of two short-volatility positions can have a higher VaR than the sum of the individual VaRs, because the rare tail event lines up between them. This means VaR can punish diversification, which contradicts the entire premise of risk management.
Standard exam stem: "A risk manager combines two strategies with VaRs of $1M and $2M and reports a combined VaR of $4M. Explain." The right answer cites VaR's failure of sub-additivity. The wrong answer is to claim a calculation error.
CVaR / Expected Shortfall
Conditional Value at Risk (CVaR), also called Expected Shortfall (ES), is the expected loss conditional on the loss exceeding VaR:
\[ CVaR\alpha = E[L \mid L \geq VaR\alpha] \]
In discrete form, CVaR is the average of the worst \((1-\alpha)\) percent of losses. For a 95% CVaR with 1,000 daily observations, you take the worst 50 days and average them.
Why ES Beats VaR
Expected Shortfall is coherent. It satisfies sub-additivity, so combining sub-portfolios cannot inflate the risk measure beyond the sum of the parts. This is why Basel III adopted ES.
ES also captures tail magnitude. VaR is silent about how bad losses get past the threshold. Two strategies can have the same 99% VaR but very different ES, because one has a fatter tail. ES surfaces the difference.
VaR tells you the cliff edge. ES tells you how far down you fall once you go over. For options-writing, credit-spread, and other tail-risk strategies, the ES gap matters. Reporting only VaR can hide losses that are 3x or 4x the VaR threshold in stress events.
Basel III: The 97.5% ES
The FRTB switched the regulatory measure from 99% VaR to 97.5% ES. For normal distributions, the two are roughly equivalent. The shift matters most under fat tails: under heavier-than-normal kurtosis, 97.5% ES is materially larger than 99% VaR, which forces banks to hold more capital against tail-risk exposures.
The formal Basel III definition uses scaled 10-day ES, with stress-period scaling and liquidity horizons by asset class. CAIA and FRM exam stems do not require the full FRTB calibration. The 97.5% ES vs 99% VaR concept is the load-bearing piece.
Three Estimation Methods
All three measures (VaR, CVaR, ES) can be estimated three ways:
1. Parametric (Variance-Covariance)
Assume returns are normally distributed with mean \(\mu\) and standard deviation \(\sigma\). Then:
\[ VaR\alpha = -(\mu + z\alpha \cdot \sigma) \]
Where \(z_\alpha\) is the standard-normal z-score (-1.645 for 95%, -2.326 for 99%).
For 99% 1-day VaR with daily mean of 0.05% and daily volatility of 1%:
\[ VaR = -(0.0005 - 2.326 \cdot 0.01) = 0.0228 = 2.28\% \]
Simple. Wrong for fat-tailed distributions. Use only when the normality assumption is defensible.
2. Historical Simulation
Take the last \(N\) daily returns, sort them, and read off the \((1-\alpha)\) quantile. For 99% VaR with 500 days of data, the 5th worst return is your VaR estimate. CVaR is the average of the worst 5.
No distributional assumption. Captures actual tail behavior in the sample. Limited by sample size and the fact that the worst event in the sample may be milder than the worst event the future holds.
3. Monte Carlo Simulation
Fit a distribution (or a multi-factor model) to the data, then simulate 10,000 (or 100,000) returns. Read off the quantile. This handles non-normal distributions and complex portfolio structures (options, credit derivatives, path-dependent payoffs).
Computationally heavy. Only as good as the input distribution. Garbage in, garbage out.
Practitioners typically run all three and compare. A large gap between parametric and historical VaR is a sign of fat tails. A large gap between historical and Monte Carlo is a sign of model error.
Modified VaR (Cornish-Fisher)
For strategies with material skew and kurtosis, the Cornish-Fisher expansion adjusts the parametric z-score:
\[ z'\alpha = z\alpha + (z\alpha^2 - 1)\frac{S}{6} + (z\alpha^3 - 3z_\alpha)\frac{K}{24} \]
Where \(S\) is skewness and \(K\) is excess kurtosis (kurtosis minus 3). Modified VaR is then:
\[ \text{mVaR}\alpha = -(\mu + z'\alpha \cdot \sigma) \]
For a strategy with skew of -1.5 and excess kurtosis of 5, the modified z-score at 99% confidence shifts from -2.326 to roughly -3.5, producing a VaR estimate roughly 50% larger than the parametric normal VaR.
This is the same expansion behind the Modified Sharpe ratio. It is a simple, defensible adjustment for moderate non-normality. For severe non-normality (heavy options-writing books, credit spreads in stress), even modified VaR can underestimate, and full Monte Carlo with a fat-tailed distribution is preferred.
VaR Backtesting
A 99% 1-day VaR predicts that losses will exceed the threshold roughly 1% of trading days, about 2-3 days per year. Kupiec's POF (proportion of failures) test compares the observed exceedance count to the expected count using a binomial test.
If a model produces 8 exceedances in a year when 2.5 were expected, the model is rejected. Basel III requires three months of daily backtesting and applies a multiplier penalty (3.0 to 4.0) on the VaR-based capital calculation when exceedances breach thresholds.
ES is harder to backtest because it conditions on tail events. The Acerbi-Szekely tests are the standard, but the literature is younger and the tests less powerful than VaR backtesting.
What FRM Part I Tests
FRM Part I covers VaR estimation methods, sub-additivity, and the basic concept of ES. CAIA Level II goes further into ES estimation, Cornish-Fisher modified VaR, and the application to hedge fund and structured-product portfolios.
For either exam, expect:
- Definition questions. Distinguish VaR from CVaR. State the four coherence axioms.
- Calculation questions. Parametric VaR given mean and sigma. Historical VaR given a sorted return series. Modified VaR given skew and kurtosis.
- Interpretation questions. Why did Basel III move to ES? What does sub-additivity mean? Why is VaR potentially misleading on a short-vol book?
- Backtesting questions. Apply Kupiec's test. Identify when a model should be rejected.
Common Mistakes
- Forgetting the negative sign. VaR is reported as a positive loss number. A z-score of -2.326 produces a positive VaR after the negation in the formula.
- Confusing 99% confidence with 1-day horizon. The two are independent parameters. A 95% 10-day VaR is different from a 99% 1-day VaR.
- Using parametric VaR for non-normal distributions. A short-vol book has a fat tail. Parametric VaR will underestimate.
- Treating VaR and ES as substitutes. They are complements. Report both.
How to Practice
Review the formulas (parametric, historical, Cornish-Fisher) on the CAIA Level II formula sheet, then drill the application questions.
The free CAIA Level II question bank has dozens of risk-measurement questions tagged by topic. Solutions walk through both the calculation and the choice of method (parametric vs historical vs Monte Carlo, VaR vs ES) so you build the judgment, not just the arithmetic.
The biggest mistake candidates make on these topics is reaching for parametric VaR by reflex. Train yourself to ask first: is the distribution normal enough to justify the assumption? If not, the right answer is almost always historical simulation, Monte Carlo, or modified VaR, and the exam stem usually telegraphs which one.