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  1. 013For a normally distributed P&L, compare the 99% VaR with the 97.5% expected shortfall. Why would switching from one to the other barely change capital for a normal book but raise it for a fat-tailed one?VaR and expected shortfallHardUBSZurich · 2021

    Try it first

    For a normal P&L, how do the 99% VaR and the 97.5% expected shortfall compare?

    Show the worked solution

    For a normal P&L they are almost identical: 2.33 against 2.34 standard deviations. On Rs 10 crore of daily volatility that is Rs 23.3 crore against Rs 23.4 crore. The 97.5% level was chosen so the switch would be neutral for a normal book. For a fat-tailed book with the same volatility, VaR is Rs 26.2 crore and expected shortfall Rs 29.1 crore, about 11% higher.

    What does each number actually measure?

    Think of a river's flood level. VaR is the line on the wall that the water passes one year in a hundred. Expected shortfallThe average loss on the days when losses exceed the VaR at a chosen confidence level. is how deep the water gets, on average, in the years it passes a lower line. VaR reads a single point in the tail; expected shortfall averages everything beyond its cut-off, so it responds to how far the tail stretches. For a normal distribution the 97.5% cut-off is 1.96 standard deviations, and the average of the losses beyond it is 2.338 standard deviations.

    The relationship
    ES97.5%=σ φ(1.96)0.025=2.338 σVaR99%=2.326 σ\text{ES}_{97.5\%} = \sigma\,\frac{\varphi(1.96)}{0.025} = 2.338\,\sigma \qquad \text{VaR}_{99\%} = 2.326\,\sigma
    sigmathe standard deviation of daily P&L
    phi(1.96)the height of the standard normal curve at the 97.5% cut-off
    0.025the probability of being beyond that cut-off
    What it says in wordsFor a normal book, the average loss beyond 1.96 standard deviations lands almost exactly on the 99% VaR.
    Same volatility, two tails: where VaR and expected shortfall landNormal tailVaR 99%: 23.3ES 97.5%: 23.412223242Daily loss, Rs croreFat tail (Student t, 3 degrees of freedom)VaR 99%: 26.2ES 97.5%: 29.112223242Daily loss, Rs croreGap between ES and VaR: 0.5%Gap between ES and VaR: 11.0%
    With the same Rs 10 crore volatility, a normal P&L puts the 99% VaR at Rs 23.3 crore and the 97.5% expected shortfall at Rs 23.4 crore, almost the same point, while a fat-tailed P&L puts them at Rs 26.2 crore and Rs 29.1 crore, 11% apart.

    Why does the fat-tailed book pay more under expected shortfall?

    Because its extreme losses are larger even though its everyday volatility is the same. Expected shortfall averages the tail, so a book that sells protection against crashes, whose losses are rare but very large, shows a much higher number under ES than under VaR. Using a Student t with three degrees of freedom and the same Rs 10 crore volatility, the 99% VaR rises only to Rs 26.2 crore, but the 97.5% expected shortfall reaches Rs 29.1 crore. That gap is exactly the risk VaR was criticised for ignoring.

    Give both sides, because the question asks for advantages and disadvantages. Expected shortfall sees the tail's depth and adds up sensibly across desks, since it is subadditive. But it is harder to backtest, because you are checking an average of rare events rather than a count of breaches, and it needs more data to estimate. VaR is easy to backtest and explain, and blind beyond its own line.

    Where candidates lose it

    The trap is assuming that a 97.5% measure must be smaller than a 99% measure. It compares the confidence levels and forgets that expected shortfall averages beyond its line while VaR stops at it.

    The second miss is stating that ES is always much larger than VaR. For a normal book it is not; say the 2.33 and 2.34 and explain that the gap only opens when the tail is fat.

    What the interviewer asks next

    • Why is VaR not subadditive, and can you build an example with two bonds?
    • How would you backtest an expected shortfall model?
    • A desk sells deep out-of-the-money puts. Which measure shows its risk better, and why?

    Asked at UBS, Risk Management, Zurich, 2021 (Wall Street Oasis): what are the advantages and disadvantages of ES compared to VaR?