VaR and Expected Shortfall: why ES is the coherent risk measure
VaR is a quantile of your loss distribution. ES is the average of all worse-than-VaR losses. ES is *subadditive* (diversification reduces it); VaR isn't (you can construct portfolios where diversification penalises you under VaR).
Value-at-Risk (VaR) is the most widely quoted risk number on a trading floor and the most widely-misunderstood. It is just a quantile of the loss distribution: ‘we will not lose more than $X$ on 95% of days.’ Two famous failures of VaR drove the post-2008 regulatory move to Expected Shortfall (ES): VaR is blind to the size of losses past the cutoff, and it is *not subadditive* — you can build two portfolios where merging them increases VaR, violating the obvious principle that diversification should never hurt. This tutorial defines both, walks the canonical defaultable-bond example where VaR breaks, and shows ES restores the diversification axiom.
β Intro Β· expand
Try first (productive failure)
Before the worked example: spend 60 seconds taking your best shot at this.
A guess is fine β being briefly wrong about a problem makes the explanation
land harder when you read it. This appears once per tutorial; skip
if you already know the trick.
60s
β Try first Β· expand
Worked example
Two i.i.d. defaultable bonds each lose $100$ with probability $4\%$ and lose $0$ otherwise. Compute VaR$_{95\%}$ for one bond, for the two-bond portfolio, and the corresponding ES$_{95\%}$ values. Verify VaR is *not* subadditive while ES is.
β Worked example Β· expand
Practice 1 of 3Type a fraction, decimal, or expression β mathjs parses it.
β Practice Β· expand
Reflection
VaR is in the Basel III rules; ES replaced it for the trading-book in Basel 2.5 (FRTB). What was the political resistance to ES, and what made the post-Lehman 'tail-blindness' argument finally win?