Risk Management puzzles, solved step by step
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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?UBSZurich · 2021
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For a normal P&L, how do the 99% VaR and the 97.5% expected shortfall compare?
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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 relationshipsigma the standard deviation of daily P&L phi(1.96) the height of the standard normal curve at the 97.5% cut-off 0.025 the 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.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?
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?UBSRemote · 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.
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 relationship2.5 the number of observations in the worst 1% of 250 days 0.5 x 11 half 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
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?UBSAnonymous employee in · 2020
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What is the position's DV01, the loss for a one basis point rise in yield?
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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.
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 relationshipP position value, Rs 500 crore D modified duration, 6 z 2.33, the one-sided 99% point sigma_bp daily 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
099A desk's one-day 99% VaR is Rs 12 crore. What is the ten-day VaR under the usual scaling rule, and what has to be true about daily P&L for that rule to hold?BlackRockNew York · 2026
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What is the ten-day VaR under the usual rule?
Show the worked solution
About Rs 37.9 crore, Rs 12 crore times the square root of 10. The rule works because the variance of a sum of independent daily P&Ls is the sum of their variances, so volatility grows with the square root of time. It needs daily P&L to be independent, identically distributed with zero mean, and the position to stay unchanged for ten days.
Why the square root, not ten times?
Walk ten steps where each step is a coin toss, left or right. You rarely end ten steps away; the lefts and rights partly cancel, and the typical distance is about three steps, the square root of ten. Independent daily gains and losses partly offset each other, so the spread of the ten-day total grows with the square root of ten, not with ten. Multiplying by ten assumes all ten days are bad days in the same direction, which is the one path the independence assumption rules out.
The relationshipVaR_1 one-day 99% VaR, Rs 12 crore sqrt(10) the growth in volatility over ten independent days What it says in wordsMultiply the one-day VaR by the square root of the number of days, because variances of independent days add.From Rs 12 crore at one day, square-root scaling gives Rs 37.9 crore at ten days while simply adding days gives Rs 120 crore; with a daily autocorrelation of 0.2 the ten-day figure rises to Rs 45.5 crore. What breaks the rule, and in which direction?
Four things, and a market risk interviewer wants at least two. If losses cluster, with one bad day tending to follow another, the square root understates ten-day risk: an autocorrelation of 0.2 lifts the figure from Rs 37.9 crore to about Rs 45.5 crore. Volatility that rises after a shock, fat tails that do not shrink toward normal over a few days, and a position that cannot be cut or that the desk keeps adding to all push the same way. Mean reversion in P&L pushes the other way. The rule also assumes the portfolio is fixed for ten days, which a desk that trades daily does not satisfy.
Say where the rule is used: regulatory market risk capital has long been built on a ten-day horizon, and many banks produce it by scaling one-day VaR. The standards now also require horizons that differ by the liquidity of the risk, which is a direct admission that ten days of independence is not true for every position; confirm the current rules before quoting any detail.
Where candidates lose it
The fast wrong answer is Rs 120 crore, adding ten daily VaRs. It assumes perfect positive dependence across days, the opposite of the rule's premise.
The larger miss is giving Rs 37.9 crore without the conditions. The question asks what must be true: independence, stable distribution, and a static position, and which way the answer moves when they fail.
What the interviewer asks next
- What is the 250-day VaR under the same rule, and why would nobody trust it?
- How would you scale expected shortfall to ten days?
- Your desk's P&L shows positive autocorrelation. How do you adjust the ten-day figure?
Asked at BlackRock, Restructuring, New York, 2026 (Wall Street Oasis):
Techincal and behavioral (VAR, market views, stock valuation) why blackrock, python experience?
