Mutual Fund Mastery puzzles, solved step by step

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All topicsCompounding and time value9Statistics, correlation and diversification8Bond maths and duration10Performance measurement and returns8Costs and fee drag8Valuation riddles11Logic and numeracy brainteasers6Estimation and market sizing7Probability and expected value8NAV, units and fund mechanics7Risk, volatility and drawdown8Behavioural traps6Withdrawals and after-tax arithmetic4
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  1. 022A fund falls 35%. What gain does it need to get back to where it started, and at 12% a year, how long would that take?Risk, volatility and drawdownWarm upRisk and complianceIndian AMCs

    Try it first

    What gain recovers a 35% fall?

    Show the worked solution

    A gain of about 53.8%, which at 12% a year takes about 3.8 years. After the fall, 100 is 65. Getting back to 100 needs 35 more on a base of 65: 1 over 0.65, less 1, is 53.85%. At 12% a year, the time is the log of 1 over 0.65 divided by the log of 1.12, 3.80 years.

    Why is the recovery larger than the fall?

    Think of a shop that cuts a price by 50% in a sale, then raises it by 50% after. The shirt that was Rs 1,000 went to Rs 500 and comes back only to Rs 750. A loss is a percentage of the old, larger base, and the recovery is a percentage of the new, smaller base, so the same rupee gap is always a larger percentage on the way back up. A 35 point fall from 100 leaves 65, and 35 is a much bigger slice of 65 than of 100.

    The fall is measured on 100, the climb back on 65100Start65After the fall100Back to start-35+3535 on a base of 100 = 35%35 on a base of 65 = 53.8%Fall -> gain to recover-10%+11.1%-20%+25.0%-35%+53.8%-50%+100.0%at 12% a year: 3.8 years
    A fall of 35 from 100 is 35%, but the same 35 regained from a base of 65 is 53.8%, which takes about 3.8 years at 12% a year.

    How do you get the time without a calculator?

    Use the rule of 72 to bracket it. At 12% money doubles in about 6 years, and a 54% gain is about 0.62 of a doubling in log terms, so roughly 0.62 times 6, a little under 4 years. The exact figure is the log of 1.538 over the log of 1.12, 0.431 over 0.113, which is 3.80 years. A quick check: 1.12 to the third is 1.40 and to the fourth is 1.57, so the answer sits between three and four years, closer to four.

    The relationship
    g=11−d−1=10.65−1=53.8%,t=ln⁡(1/0.65)ln⁡1.12=0.4310.113≈3.8 yearsg = \frac{1}{1 - d} - 1 = \frac{1}{0.65} - 1 = 53.8\%, \qquad t = \frac{\ln(1/0.65)}{\ln 1.12} = \frac{0.431}{0.113} \approx 3.8 \text{ years}
    dthe drawdown, 35%
    gthe gain needed to recover
    tyears to recover at 12% a year
    What it says in wordsThe recovery needed is the inverse of what is left, less one, and the time is how many years of 12% growth it takes to multiply by that.
    FallGain needed to recover
    10%11.1%
    20%25.0%
    35%53.8%
    50%100.0%
    The gap between the fall and the recovery widens as the fall deepens; a halving needs a doubling.

    Say what it means and what it leaves out. Deep drawdowns cost time, not just money, which is why fund risk teams watch maximum drawdown alongside volatility. The 12% is an assumption for the arithmetic; returns after a fall can be faster or slower, and an investor who sells during the fall locks in the 35% and never earns the recovery at all.

    Where candidates lose it

    The common slip is answering 35%, as if gains and losses were symmetric. Candidates who say it in a risk interview have shown they do not see why drawdowns matter more than their headline size.

    The second slip is dividing 53.8 by 12 to get 4.5 years, which ignores compounding. Use logs, or step through 1.12 to the third and fourth powers, and say the answer is a little under four years.

    What the interviewer asks next

    • What fall needs a 100% gain to recover?
    • If the fund then earns 8% a year instead of 12%, how long does recovery take?
    • Why might a risk team set a limit on drawdown rather than on volatility?
  2. 048A fund's NAV at the end of each quarter runs 100, 130, 110, 140, 91, 120. What is its maximum drawdown, and why is it not measured from the starting NAV of 100?Risk, volatility and drawdownWarm upRisk and complianceIndian AMCs

    Try it first

    What is the maximum drawdown?

    Show the worked solution

    The maximum drawdown is 35%, from the peak of 140 to the trough of 91. Drawdown is always measured from the highest value reached before the fall, because that is the value an investor held and then lost. Measured from the start, 91 looks like a mild 9% dip, which hides how much was lost on the way. The fund later climbs to 120, still 14.3% below its peak.

    Why measure from the peak and not from the start?

    A house bought for Rs 1 crore, valued at Rs 1.4 crore at the top of a boom and then sold for Rs 91 lakh, did not lose 9%. Its owner watched Rs 49 lakh disappear. A drawdown measures the pain of falling from the best point reached, so it runs from the running peak, which keeps rising whenever the NAV sets a new high. Anyone who invested at 140, or held through it, lost 35%, and that is the risk the measure is built to show.

    Drawdown is measured from the highest point reached, not from the start80100120140t0t1t2t3t4t5running peak100130110140 peak91 trough120-35%maximum drawdown-15.4%from the start: only 9%
    The NAV path touches 140 and then falls to 91, a 35% drawdown from the running peak, while the earlier dip from 130 to 110 is only 15.4%; measured from the starting 100, the low of 91 would look like just a 9% fall.

    How do you compute it step by step?

    Walk along the path and keep two numbers: the highest NAV so far and the current NAV. At each point the drawdown is current over peak, less one. The maximum drawdown is the worst of those readings, here 91 over 140 less one, minus 35%. The dip from 130 to 110 gives minus 15.4%, a smaller drawdown. The final 120 against a peak of 140 is a drawdown of minus 14.3% still open.

    The relationship
    DDt=NAVtmax⁡s≤tNAVs−1,MDD=min⁡tDDt=91140−1=−35%DD_t = \frac{NAV_t}{\max_{s \le t} NAV_s} - 1, \qquad MDD = \min_t DD_t = \frac{91}{140} - 1 = -35\%
    NAV_tthe NAV at time t
    max NAV_sthe running peak up to time t
    MDDthe maximum drawdown, the worst reading
    What it says in wordsEach drawdown compares today's NAV with the best NAV so far; the maximum drawdown is the deepest of them.

    Add the recovery maths: from 91 the fund needs a 53.8% rise just to get back to 140. And the measure depends on how often you look. These are quarter-end NAVs; a daily series could show a deeper trough between the quarter ends, so a maximum drawdown should always be quoted with its data frequency and period.

    Where candidates lose it

    The trap is measuring from the starting NAV and answering 9%, because the question opens with 100. The interviewer is checking that you know the reference point moves up with every new high.

    The second slip is dividing by the trough, 49 over 91, and answering 54%. That is the gain needed to recover, not the drawdown. Say both numbers and label them.

    What the interviewer asks next

    • What gain does the fund need from 120 to set a new high?
    • Why might a daily NAV series show a larger maximum drawdown than quarter-end NAVs?
    • Two funds have the same volatility; one has a much larger maximum drawdown. What could explain it?
  3. 085A fund's monthly returns have a standard deviation of 5%. What is that as an annual volatility, and is a month of minus 12% a rare event?Risk, volatility and drawdownWarm upRisk and complianceIndian AMCs

    Try it first

    What is the fund's annual volatility?

    Show the worked solution

    About 17.3% a year, and a minus 12% month is a 2.4 standard deviation event: rare, not freakish. Annual volatility is the monthly figure times the square root of 12: 5% x 3.46 is 17.3%. A minus 12% month is 12 divided by 5, or 2.4 monthly standard deviations. A normal curve puts that at about 0.8% of months, roughly once a decade, and real markets produce such months more often.

    Why does volatility scale with the square root of time?

    Take ten steps where each one goes left or right on a coin toss. You rarely end ten steps from where you began; lefts and rights cancel, and a typical distance is only about three steps, the square root of ten. Monthly returns behave the same way: good and bad months partly cancel, so variance adds up over time and the standard deviation grows only with the square root of the number of months. That is why the conversion uses the square root of 12, about 3.46, and not 12.

    The relationship
    σyear=σmonth12=5%×3.464=17.3%z=−12%5%=−2.4\sigma_{year} = \sigma_{month}\sqrt{12} = 5\% \times 3.464 = 17.3\% \qquad z = \frac{-12\%}{5\%} = -2.4
    \sigma_{month}the standard deviation of monthly returns, 5%
    \sqrt{12}the square root of the number of months in a year
    zhow many monthly standard deviations the fall is from an average month
    What it says in wordsScale volatility up by the square root of time, and measure a single month against the monthly figure, not the annual one.
    Monthly returns with a 5% standard deviation, and where minus 12% sits-15%-3 sd-10%-2 sd-5%-1 sd0%+5%+1 sd+10%+2 sd+15%+3 sdminus 12%= -12 / 5 = -2.4 sdShaded tail:0.82% of months, 1 in 122Annual volatility5% x the square root of 12= 17.3% a yearMonthly return
    With a monthly standard deviation of 5%, a minus 12% month lies 2.4 deviations below the centre, and the normal curve puts only 0.82% of months beyond it, about one in 122, while the same fund's annual volatility is 17.3%.

    So how rare is a minus 12% month?

    Measure it in monthly standard deviations. A minus 12% month is 2.4 monthly standard deviations below an average month taken as zero, which a normal curve puts at about 0.8% of months, roughly one in 122, or once a decade. If the fund's average month is plus 1%, the fall is 2.6 deviations and a normal curve makes it a little rarer, 0.47%. Against the 17.3% annual figure the fall sounds small, and that is the confusion to avoid: judge a month on a monthly scale.

    Now the limitation. Equity returns have fatter tails than the normal curve: large falls cluster in crises and turn up more often than the bell shape allows. Treat the normal figure as a floor on how often such a month arrives, not as a promise that it will be rare. For a client the useful sentence is plain: a fund with annual volatility near 17% will, from time to time, lose more than a tenth of its value in a single month.

    Where candidates lose it

    The first loss is scaling the wrong way. Multiplying by 12 gives 60%, which describes a far wilder fund than this one, and it comes from treating risk as if it added up like returns. Say variance adds, then take the square root.

    The second is judging the minus 12% month against the annual 17.3% and calling it ordinary. Put the move and the volatility on the same time scale before comparing them, and the month turns out to be a 2.4 deviation event.

    What the interviewer asks next

    • What is this fund's volatility over a single week?
    • The fund's worst month in ten years was minus 20%. What does that tell you about the normal assumption?
    • Why might reported annual volatility understate the risk of a bad month?