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. 006Write NPV as the product of two vectors. The cash flows are minus 100, 30, 40, 50 and 20 in years 0 to 4, and the discount rate is 10%. What is the NPV?Duration and ratesWarm upMoody'sNew York · 2018

    Try it first

    Which operation turns the two vectors into the NPV?

    Show the worked solution

    NPV is the dot product of the cash flow vector and the discount factor vector, and here it is about 11.56. The discount factors at 10% are 1, 0.909, 0.826, 0.751 and 0.683. Multiplying term by term gives minus 100, 27.27, 33.06, 37.57 and 13.66, which sum to 11.56. A positive NPV means the project earns more than 10%.

    Why is NPV a dot product at all?

    Think of a grocery bill. One list holds the quantity of each item, another holds each item's price, and the bill is quantity times price for each line, added up. NPV has the same shape: one vector holds the cash flows, the other holds what one rupee in each year is worth today, and the NPV is the sum of their products. Writing it this way separates the project, which is the cash flow vector, from the market, which is the discount factor vector. Change the rate and only the second vector changes.

    The relationship
    NPV=c⋅d=∑t=04ct dt,dt=1(1+r)t\text{NPV} = \mathbf{c} \cdot \mathbf{d} = \sum_{t=0}^{4} c_t\, d_t, \qquad d_t = \frac{1}{(1+r)^t}
    cthe cash flow vector, year 0 to year 4
    dthe discount factor vector, one entry per year
    rthe discount rate, 10%
    What it says in wordsMultiply each cash flow by the value today of one rupee in that year, and add the results.
    Multiply the two vectors term by term, then add the productsYear 0Year 1Year 2Year 3Year 4Cash flow-10030405020xDiscount factor1.0000.9090.8260.7510.683=Present value-100.0027.2733.0637.5713.660-100+27.27+33.06+37.57+13.66+11.56Year 0Year 1Year 2Year 3Year 4NPVRunning sum
    Multiplying the cash flows by the discount factors year by year gives present values of minus 100, 27.27, 33.06, 37.57 and 13.66, and adding them from minus 100 upward reaches an NPV of 11.56.

    What does the interviewer want to hear beyond the number?

    The thought process was part of the question, so say it in order. First build the discount factor vector from the rate, then take the dot product, then sanity check the sign and size. The undiscounted inflows are 140 against 100 out, so a positive but much smaller NPV is expected once four years of 10% are taken out. And in a spreadsheet the same idea is one SUMPRODUCT of two ranges, which is why the vector form is how a model is usually built.

    Then give the extension that shows range. With a term structure of rates, only the discount factor vector changes: each entry uses its own year's rate. With several scenarios, stack the cash flow vectors into a matrix and one matrix multiplication gives every scenario's NPV at once. The limitation is that the vector form assumes the cash flows are known; uncertain cash flows need expected values or scenarios first.

    Where candidates lose it

    Candidates reach for the NPV formula and start adding fractions, which gets the number but misses the question. The interviewer asked for two vectors precisely to see whether you can separate what the project pays from what time is worth.

    The other slip is discounting year 0. The first discount factor is 1; the minus 100 is already in today's money.

    What the interviewer asks next

    • How would you write the IRR condition using the same two vectors?
    • The rate for year 1 is 8% and for later years 10%. What changes in the vector form?
    • How would you compute the NPV for 1,000 cash flow scenarios in one operation?

    Asked at Moody's, Analytics, New York, 2018 (Wall Street Oasis): Construct an NPV formula using 2 vectors and show me your thought process.

  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?