Risk Management puzzles, solved step by step

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  1. 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

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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 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?