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. 038Two trading desks both report a 99% one-day VaR of Rs 5 crore. On their worst 1% of days, desk A lost Rs 6, 6.5 and 7 crore and desk B lost Rs 6, 12 and 30 crore. Compute the average tail loss for each desk and say which is riskier.VaR and expected shortfallWarm upBank market risk

    Try it first

    Before you average: which desk's tail average is larger, and by roughly how much?

    Show the worked solution

    Desk A's tail average is Rs 6.5 crore and desk B's is Rs 16.0 crore, so desk B is far riskier. Both desks have a 99% VaR of Rs 5 crore, which only says where the worst 1% of days begins. Averaging the losses beyond it, the expected shortfall, shows A's bad days cluster just past the line while B's run to Rs 30 crore.

    Why does the same VaR hide such different risk?

    Two rivers both have a flood mark at five metres that is crossed one year in a hundred. On one river, those floods reach six or seven metres; on the other, one of them reached thirty and washed the town away. The flood mark is the same; the town planner should care about the second river. VaR is the threshold of the bad days, not their size; two desks can share a threshold and have tails of completely different weight. Expected shortfallThe average loss on the days worse than VaR, so it measures the size of the tail rather than just where it starts. answers the question VaR leaves open: when it goes wrong, how wrong on average?

    Same VaR line, very different tails behind itDesk A6day 16.5day 27day 3VaR 5ES 6.5Desk B6day 112day 230day 3VaR 5ES 16.0Rs crore lost on each desk's three worst days in the sample
    Desk A and desk B share a 99% VaR of Rs 5 crore, but desk A's worst days average Rs 6.5 crore while desk B's average Rs 16.0 crore, because one of B's tail days lost Rs 30 crore.

    What does the ratio of expected shortfall to VaR tell you?

    Divide one by the other. A's ratio is 1.3 and B's is 3.2. For normally distributed returns at 99%, expected shortfall is only about 15% above VaR, so a ratio of 3.2 is a flag that something is behaving far from normal: an option position losing faster as the market moves, a concentrated name gapping, or a liquidity cliff. Desk A looks close to normal; desk B needs its positions read line by line.

    The relationship
    ESA=6+6.5+73=6.5,ESB=6+12+303=16ES_A = \frac{6 + 6.5 + 7}{3} = 6.5, \qquad ES_B = \frac{6 + 12 + 30}{3} = 16
    ESexpected shortfall, the average of losses beyond VaR
    3the number of days in the worst 1%, which implies about 300 days of history
    What it says in wordsAverage the losses on the days worse than VaR to see how heavy the tail is.

    State the limit honestly. Three observations make a fragile average: one more bad day for desk A, or one fewer outlier for desk B, would move the numbers a lot. Expected shortfall is also harder to backtest than VaR, because you are checking an average of rare events rather than a count of breaches. It is still the better answer to the question asked.

    Where candidates lose it

    The trap is saying the desks are equally risky because their VaR is equal, or ranking them on the single worst day without computing anything. The question hands you the tail precisely so you use it.

    The quieter miss is not naming the measure. Call the tail average expected shortfall and say one sentence about why regulators moved towards it: VaR is blind past its own line.

    What the interviewer asks next

    • Desk B's Rs 30 crore day came from one option position. What would you ask the desk?
    • Why is expected shortfall harder to backtest than VaR?
    • If you combined the two desks, could the combined VaR exceed the sum of the two? Could the expected shortfall?
  2. 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?VaR and expected shortfallWarm upBlackRockNew York · 2026

    Try it first

    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 relationship
    VaR10=VaR1×10=12×3.162=37.9\text{VaR}_{10} = \text{VaR}_1 \times \sqrt{10} = 12 \times 3.162 = 37.9
    VaR_1one-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.
    Risk grows with the square root of time, if days are independent2040608010012001 day: Rs 12 croreadding days: 120autocorrelated0.2: 45.5sqrt(10): 37.91246810Horizon, trading days99% VaR, Rs crore
    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?