Risk Management puzzles, solved step by step

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100
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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. 024A desk's daily P&L has a standard deviation of Rs 2 crore. The head of the desk wants a loss number that is exceeded about one day a month. Which confidence level is that, and what is the VaR?VaR and expected shortfallCoreBank market riskTreasury and ALM

    Try it first

    Which confidence level matches one bad day a month?

    Show the worked solution

    About 95% confidence, and a VaR of about Rs 3.3 crore. A month has about 21 trading days, so one breach a month is a tail of 1 in 21, about 4.8%, a confidence level of about 95.2%. For a normal P&L that cut sits 1.67 standard deviations out, so the VaR is 1.67 times Rs 2 crore, about Rs 3.34 crore. At a round 95% it is Rs 3.29 crore.

    How do you turn a frequency into a confidence level?

    Think of a weather forecast that says a heavy rain day comes about once a month. That is the same statement as saying about 1 day in 30 is that wet. A VaR confidence level is just a frequency of bad days written as a percentage: one breach in 21 trading days is a 4.8% tail, which is a 95.2% confidence level. Counting trading days matters; using 30 calendar days would make the tail 3.3% and the level about 96.7%.

    One bad day a month is a 95% VaR: the shaded tail is 1 day in 21-6-4-20+2+4+6Daily P&L, Rs croreVaR: -3.341.67 x Rs 2 crore1 day in 21Translate1 day a month= 1 in 21 = 4.8%= 95.2% confidenceExact: 3.34 croreAt 95%: 3.29 croreabout 12 days a year
    For daily P&L with a Rs 2 crore standard deviation, one loss day in 21 sits beyond Rs 3.34 crore, 1.67 standard deviations out, which is close to the Rs 3.29 crore 95% VaR.
    The relationship
    VaR=zc σ=1.668×2≈3.34 crorez95%=1.645⇒3.29 crore\text{VaR} = z_{c}\,\sigma = 1.668 \times 2 \approx 3.34 \text{ crore} \qquad z_{95\%} = 1.645 \Rightarrow 3.29 \text{ crore}
    z_cthe number of standard deviations that leaves the chosen tail below it
    sigmathe daily P&L standard deviation, Rs 2 crore
    What it says in wordsMultiply the daily standard deviation by the number of standard deviations that matches the chosen frequency of bad days.

    Why is this translation useful on a desk?

    Because a desk head can act on a frequency and cannot act on a percentage. Saying we expect to lose more than Rs 3.3 crore about once a month, roughly 12 days a year, tells the head of the desk what to expect and when to worry. It also sets up the backtest: if losses beyond Rs 3.3 crore start happening three or four times a month, the model is understating risk, and that is visible within a quarter rather than a year.

    Add the limits. The VaR says nothing about how bad the bad day is; a loss of Rs 3.4 crore and one of Rs 10 crore both count as one breach. The normal assumption also understates fat tails, so the real one-in-21 loss may sit further out than Rs 3.3 crore. And the Rs 2 crore standard deviation is itself an estimate that moves with market conditions.

    Where candidates lose it

    The trap is picking 99% because it is the regulatory number people remember. A 99% VaR is breached about once in 100 trading days, closer to once every five months than once a month.

    The second miss is counting 30 calendar days. Losses happen on trading days, so the tail is 1 in 21, not 1 in 30.

    What the interviewer asks next

    • What loss would the desk expect to exceed once a year?
    • The desk had five breaches last month. What do you conclude, and what would you check first?
    • How would the answer change if the P&L had fat tails with the same standard deviation?
  2. 049A fund has an expected annual return of 12% and annual volatility of 20%, on Rs 100 crore. What is its one-year 95% VaR with and without the expected return, and when does the mean matter?VaR and expected shortfallCoreAsset manager risk

    Try it first

    Including the 12% expected return, what is the one-year 95% VaR?

    Show the worked solution

    Rs 32.9 crore ignoring the mean and Rs 20.9 crore including it. The 95% cut is 1.645 standard deviations below the mean: 1.645 x 20% is 32.9%, and a 12% expected return lifts the cut to minus 20.9%. At one year the mean cuts VaR by more than a third. At one day it barely matters: Rs 2.07 crore against Rs 2.02 crore.

    Why does the mean matter at one year but not at one day?

    On a short walk the path you take wanders more than your average direction moves you; on a long journey the direction wins. Expected return grows in proportion to time while volatility grows with its square root, so over short horizons the mean is noise and over long ones it is material. At one day the mean is 12% / 252, about 0.048%, against a daily volatility of 1.26%; the VaR figures differ by 2.3%. At one year the gap is 36%.

    At one year the mean moves the 5% cut by 12 points-60%-40%-20%0+20%+40%+60%One-year returnzero mean: -32.9%with 12% mean: -20.9%mean +12%95% VaR on Rs 100 croreone year: 32.9 vs 20.9one day: 2.07 vs 2.02Rs crore, zero mean vs with mean
    Over one year a 12% expected return shifts the whole distribution right, moving the 5% cut from minus 32.9% to minus 20.9%, whereas over one day the same mean moves VaR from Rs 2.07 crore only to Rs 2.02 crore.
    The relationship
    VaR95%=(1.645 σ−μ)×W=(32.9%−12%)×100=20.9\text{VaR}_{95\%} = (1.645\,\sigma - \mu) \times W = (32.9\% - 12\%) \times 100 = 20.9
    \sigmaannual volatility, 20%
    \muexpected annual return, 12%
    Wportfolio value, Rs 100 crore
    What it says in wordsVaR is how far the 5% worst outcome sits below zero: the volatility term pulls it down, the mean pushes it back up.

    Which figure should a risk report quote?

    Say which you are quoting, because both are used. Relative VaRVaR measured from the expected outcome rather than from zero, so it captures only the uncertainty and ignores the expected gain. ignores the mean and measures pure uncertainty; absolute VaR includes it and measures the loss from today's value. For a one-year horizon, many risk teams quote the zero-mean figure deliberately: a 12% expected return is an assumption, and counting it as a cushion lets an optimistic forecast shrink the risk number. Daily trading VaR usually ignores the mean for the simpler reason that it makes no difference.

    Limits to name: over a year, compounding and fat tails matter, so a normal model with a constant 20% volatility understates the chance of a large loss, and volatility itself changes over the year. The one-year figure is a rough guide to the size of a bad year, not a promise about it.

    Where candidates lose it

    The most common slip is adding the mean to the loss, giving 44.9, which moves the cut the wrong way. The second is ignoring the question's own hint and giving only 32.9.

    The quieter miss is not answering the when part. The square-root rule for volatility against the linear growth of the mean is the one sentence the interviewer is waiting for.

    What the interviewer asks next

    • Over what horizon does the mean cut VaR by half?
    • Why might a regulator prefer the zero-mean figure?
    • How would you compute one-year VaR if returns were lognormal?
  3. 074Under a normal distribution, 99% VaR should be about 1.41 times 95% VaR. A desk's historical VaR is Rs 5 crore at 95% and Rs 11 crore at 99%. What does the ratio tell you about the desk's P&L?VaR and expected shortfallCoreBank market riskModel validation

    Try it first

    The desk's ratio is 2.2 against a normal 1.41. What is the most likely reading?

    Show the worked solution

    The desk's loss tail is much fatter than a normal distribution. For a normal, 2.326 over 1.645 gives a ratio of 1.41, so a 95% VaR of Rs 5 crore implies about Rs 7.07 crore at 99%. The desk shows Rs 11 crore, a ratio of 2.2, fatter even than a Student t with 3 degrees of freedom at 1.93. Rare losses are far larger than volatility alone would suggest.

    Why does the ratio say anything about the shape of the tail?

    If a city's 1-in-20 rainy day brings 5 cm of rain and its 1-in-100 day brings 11 cm, the storms are of a different kind from the drizzle, not just more of it. Scaling up the whole distribution would move both numbers together. Volatility stretches every quantile by the same factor, so the ratio between two quantiles is a pure measure of shape, and a ratio well above 1.41 means the tail is fatter than normal.

    The gap between two VaR levels is a quick test of tail fatness0510Daily loss, Rs crore95%: 5normal 99%: 7.07desk 99%: 11Solid: normal. Dashed: fat-tailed, same 95% VaR.1.41Normal1.93t, 3 dof2.20Desk99% VaR / 95% VaR11 / 5 = 2.2 against 1.41
    A normal distribution and a fat-tailed one with the same 95% VaR of Rs 5 crore diverge in the tail: the normal's 99% VaR is Rs 7.07 crore and the fat-tailed one's is Rs 11 crore. The desk's ratio of 2.2 exceeds the 1.41 of a normal and the 1.93 of a Student t with 3 degrees of freedom.

    What would you do with that finding?

    Two things. First, distrust any risk number on this desk that is built from volatility and a normal multiplier, such as a parametric VaR or a stress scaled from the 95% figure, because it will understate the 99% loss by about a third. Second, look at the P&L on the worst days: fat tails on a desk usually come from positions that pay small amounts often and lose large amounts rarely, such as sold options or carry trades.

    The relationship
    VaR99VaR95∣normal=2.3261.645=1.41115=2.2\frac{\text{VaR}_{99}}{\text{VaR}_{95}}\Big|_{\text{normal}} = \frac{2.326}{1.645} = 1.41 \qquad \frac{11}{5} = 2.2
    2.326, 1.645the one-sided 99% and 95% points of a standard normal
    11, 5the desk's historical 99% and 95% VaR, Rs crore
    What it says in wordsFor a normal the ratio is fixed at 1.41 whatever the volatility, so a larger ratio signals a heavier tail.

    The limitation is sample size. A 99% historical VaR from a year of data rests on two or three observations, so one extreme day could produce the 2.2 by itself. Check the ratio across several windows before concluding the desk's business is fat-tailed rather than unlucky.

    Where candidates lose it

    The trap is reading the ratio as higher volatility. Volatility changes the size of both numbers but not their ratio; only the shape of the distribution changes the ratio.

    The second slip is concluding the desk is fine because the 95% number looks normal. A fat tail hides at the 95% level and shows up only further out, which is exactly why regulators moved capital towards expected shortfall.

    What the interviewer asks next

    • What would a ratio below 1.41 suggest?
    • How would you expect the desk's expected shortfall at 97.5% to compare with a normal's?
    • Which kinds of trading positions tend to produce fat left tails?