Risk Management puzzles, solved step by step

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All topicsCapital and leverage6Compounding and drawdowns8Correlation and diversification8Counterparty exposure and collateral7Credit risk arithmetic10Duration and rates7Liquidity and balance sheet7Logic, estimation and brainteasers7Operational loss and fraud7Options and Greeks7Probability and base rates8Statistics and estimation10VaR and expected shortfall8
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  1. 063From 250 days of daily P&L, the five worst losses are Rs 18, 14, 11, 9.5 and 8 crore. What is the 99% one-day historical VaR, and what is the expected shortfall beyond it?VaR and expected shortfallCoreUBSRemote · 2020

    Try it first

    How many of the worst days sit beyond the 99% line in a 250-day sample?

    Show the worked solution

    99% VaR is about Rs 12.5 crore and expected shortfall about Rs 15 crore. One per cent of 250 days is 2.5 observations, so VaR sits between the second worst loss, 14, and the third, 11: Rs 12.5 crore by interpolation, or Rs 11 crore if the desk takes the third worst. Expected shortfall averages the worst 2.5 days: (18 + 14 + half of 11) / 2.5 = 15.

    Why is historical VaR a position in a list rather than a formula?

    If you rank a class of 250 students by exam score, the student at the 99th percentile is not computed, they are counted: roughly the second or third from the top. Historical VaR works the same way with losses. Rank the daily P&L from worst to best and count 1% of the sample in from the bad end; the loss you land on is the 99% VaR. With 250 days you land 2.5 places in, between two real days.

    99% historical VaR is a position in a ranked list: 2.5 losses deep0510152018#114#211#39.5#48#5Worst days, ranked (Rs crore of loss)2.5 obs deepES 15: average of the tail99% VaR12.5interpolated(11 if 3rd worst)ES15avg of worst 2.5
    In a 250-day sample the 99% line sits 2.5 observations from the worst end, between losses of Rs 14 crore and Rs 11 crore, so historical VaR is Rs 12.5 crore by interpolation or Rs 11 crore by the third-worst convention. Expected shortfall, the average of the worst 2.5 observations, is Rs 15 crore.

    Which convention is right, and does it matter?

    Neither is wrong; both are used, and the gap here is Rs 1.5 crore on a Rs 12.5 crore number. What is wrong is not saying which one you used, because two desks can report different VaR from identical data. Expected shortfall moves with the convention too: averaging only the two losses beyond Rs 11 crore gives Rs 16 crore rather than Rs 15 crore. Name the rule, then give the number.

    The relationship
    ES99%=18+14+0.5×112.5=37.52.5=15ES_{99\%} = \frac{18 + 14 + 0.5 \times 11}{2.5} = \frac{37.5}{2.5} = 15
    2.5the number of observations in the worst 1% of 250 days
    0.5 x 11half of the third worst loss, the fraction of it inside the tail
    What it says in wordsExpected shortfall is the average loss across the worst 1% of days, counting the third worst at half weight.

    The limitation is size. A 99% number from 250 days rests on two or three losses, so one bad day entering or leaving the window can move it by several crore. That is why regulators ask for backtesting and why expected shortfall, which uses the whole tail, is preferred where the tail is thin.

    Where candidates lose it

    The first trap is picking the worst loss, Rs 18 crore, as the 99% VaR. That is closer to a 99.6% number. Count 1% of the sample, 2.5 days, then read off the list.

    The second is giving a single number without the convention. Say 2.5 observations, say which rule you use, and show that expected shortfall is larger than VaR because it averages what lies beyond.

    What the interviewer asks next

    • What would the 97.5% expected shortfall be from the same list?
    • The worst day drops out of the window tomorrow. What happens to VaR and ES?
    • Why might a regulator prefer expected shortfall to VaR for setting capital?

    Asked at UBS, Risk Management, Remote, 2020 (Wall Street Oasis): Calculate VaR

  2. 088A treasury holds Rs 500 crore of government bonds with a modified duration of 6. Daily changes in yield have a standard deviation of 6 basis points. What is the one-day 99% VaR?VaR and expected shortfallCoreUBSAnonymous employee in · 2020

    Try it first

    What is the position's DV01, the loss for a one basis point rise in yield?

    Show the worked solution

    About Rs 4.19 crore. The position loses Rs 500 crore times 6 times 0.0001, Rs 30 lakh, for each basis point rise in yield. A 99% one-day rise is 2.33 times 6 basis points, about 14 basis points. Rs 30 lakh times 13.96 is Rs 4.19 crore. It assumes normal yield changes and a linear price response.

    Why start from DV01 instead of a price volatility?

    A taxi fare is a rate per kilometre times the distance. You would not guess the fare directly; you would multiply. For a bond, DV01 is the rupee rate per basis point and the yield move is the distance, so VaR is DV01 times the yield move at the chosen confidence. Yield volatility is what the market data gives you, and duration converts it to rupees. Guessing a price volatility for the bond skips the step the interviewer wants to see.

    Bond VaR = rupees per basis point x the 99% yield movePositionRs 500 cr, duration 6DV01Rs 30 lakh per bp99% yield move2.33 x 6 bp = 14.0 bpOne-day 99% VaRRs 4.19 crore500 x 6 x 0.0001 = 0.30 crore = Rs 30 lakh a bp0.30 x 13.96 = 4.19+14.0 bpworst 1% of days-18-12-60+6+12+18Daily change in yield, bp (standard deviation 6)yields up = bond price down,so the right tail is the loss
    A Rs 500 crore position with duration 6 loses Rs 30 lakh per basis point; a 99% daily yield rise is 2.33 times 6 basis points, 14.0 basis points, so the one-day 99% VaR is Rs 4.19 crore, the loss on the worst 1% of days.
    The relationship
    VaR99=P⋅D⋅0.0001⋅z99⋅σbp=500×6×0.0001×2.33×6=4.19\text{VaR}_{99} = P \cdot D \cdot 0.0001 \cdot z_{99} \cdot \sigma_{bp} = 500 \times 6 \times 0.0001 \times 2.33 \times 6 = 4.19
    Pposition value, Rs 500 crore
    Dmodified duration, 6
    z2.33, the one-sided 99% point
    sigma_bpdaily standard deviation of yield, 6 bp
    What it says in wordsMultiply rupees lost per basis point by the yield rise that is exceeded only one day in a hundred.

    What does this number leave out?

    Three things, and a treasury risk manager names them unprompted. The estimate assumes yield changes are normal, that the price responds in a straight line, and that every bond in the book moves with the same yield. Fat tails make a 14 basis point day more common than the normal says. Convexity makes the true loss slightly smaller than the linear figure. And a book spread along the curve has curve risk: if short yields rise and long yields do not, one DV01 figure misses it. Over ten days, the square-root rule would scale this to about Rs 13.2 crore, if daily moves are independent.

    Also say which way hurts. A holder of bonds loses when yields rise, so the one-sided 99% point on the upside of yields is the one that matters, which is why the figure uses 2.33 and not the two-sided 2.58.

    Where candidates lose it

    The usual slip is a units error: forgetting that a basis point is 0.0001 and producing a VaR a hundred times too large or too small. Say DV01 out loud first, Rs 30 lakh a basis point, and the rest follows.

    The other is using 2.58 because 99% sounds like a two-sided number. VaR is a one-sided loss measure, so the multiplier is 2.33.

    What the interviewer asks next

    • What is the 99% expected shortfall under the same normal assumption?
    • The book holds 2-year and 10-year bonds with the same total DV01. What risk does one number hide?
    • How would convexity change the VaR for a 300 basis point stress?

    Asked at UBS, Risk Management, Anonymous employee in, 2020 (Wall Street Oasis): Calculate VAR