Mutual Fund Mastery puzzles, solved step by step

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100
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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. 040Two retirees each start with Rs 60 lakh and withdraw Rs 6 lakh at the start of every year. Both see the same ten yearly returns, averaging 6.8%: one meets minus 20% and minus 10% in the first two years, the other in the last two. After ten years one has about Rs 2 lakh left and the other about Rs 39 lakh. Why?Withdrawals and after-tax arithmeticHardWealth and advisoryDistribution and sales

    Try it first

    What explains the gap of about Rs 37 lakh?

    Show the worked solution

    Because withdrawals turn the order of returns into a permanent difference. Without withdrawals both would end at the same Rs 108.1 lakh, since returns multiply in any order. But the first retiree takes Rs 6 lakh out of a pot that has just fallen, to Rs 33.5 lakh after two years, so later good years work on a small base and she ends with about Rs 2.0 lakh. The second takes withdrawals from a pot that has grown, and ends with about Rs 39.0 lakh.

    Why would order not matter without withdrawals?

    Returns multiply, and multiplication does not care about order. Rs 60 lakh left untouched through these ten years ends at Rs 108.1 lakh whichever year comes first. The withdrawals are what break the symmetry: a fixed rupee amount taken out of the pot is a bigger share of a small pot than of a large one.

    What exactly happens in the first two years?

    Think of a shopkeeper who must pay rent of Rs 6,000 on the first of every month by selling stock. If prices have just crashed, he sells many more items to raise the same Rs 6,000, and those items are gone when prices recover. A retiree withdrawing a fixed sum after a fall sells more units at low prices, and those units never take part in the recovery. The first retiree's Rs 54 lakh, after the first withdrawal, meets minus 20% and becomes Rs 43.2 lakh; Rs 37.2 lakh then meets minus 10% and becomes Rs 33.5 lakh. The second retiree's Rs 54 lakh meets plus 20% first and is Rs 64.8 lakh after year one.

    Same ten returns, same withdrawals, opposite order204060800012345678910Year; corpus in Rs lakh after each year's withdrawal and returnBad years last: ends at Rs 39.0 lakhBad years first: ends at Rs 2.0 lakhafter -20%, -10%: Rs 33.5 lakhRs 6 lakh out at the start of every year; both average 6.8%
    Both retirees withdraw Rs 6 lakh at the start of each year and see the same returns averaging 6.8%. The one who meets minus 20% and minus 10% first is down to Rs 33.5 lakh after two years and ends with Rs 2.0 lakh, while the one who meets them last ends with Rs 39.0 lakh.
    YearReturn, bad firstCorpus, Rs lakhReturn, bad lastCorpus, Rs lakh
    1-20%43.2+20%64.8
    2-10%33.5+13%66.4
    3+18%32.4+5%63.5
    4+15%30.4+7%61.5
    5+11%27.1+9%60.5
    6+9%23.0+11%60.5
    7+7%18.2+15%62.6
    8+5%12.8+18%66.8
    9+13%7.6-10%54.8
    10+20%2.0-20%39.0
    Year by year: Rs 6 lakh is withdrawn at the start of each year, then that year's return applies to what is left. The returns are the same set in reverse order.
    The relationship
    Vt=(Vt−1−W)(1+rt),V0=60,  W=6V_{t} = (V_{t-1} - W)(1 + r_t), \qquad V_0 = 60,\; W = 6
    V_tthe corpus at the end of year t, Rs lakh
    Wthe withdrawal at the start of each year, Rs 6 lakh
    r_tthe return in year t
    What it says in wordsEach year the withdrawal comes out first and the return applies only to what is left, so a fall before a withdrawal makes the withdrawal bite harder.

    What does an adviser do with this?

    The years just before and just after withdrawals begin carry the most weight; the risk is called sequence risk, and it is managed, not predicted. Common responses are holding two or three years of withdrawals in a steadier, short-term bucket so that a bad year does not force selling equity at a low, or trimming withdrawals after a fall. Note the mirror image of an SIP: for a saver adding money, bad years early are the kinder order; for a retiree taking money out, bad years early are the cruel one. The returns here are illustrative, and a 10% withdrawal rate is high by most planning conventions, which is what makes the gap so stark.

    Where candidates lose it

    The trap is answering that the average return decides the outcome, or insisting the order cannot matter because returns multiply. Both are true only for money that sits untouched. The question tells you withdrawals happen; build from there.

    The second loss is explaining it as bad luck in general. The interviewer wants the mechanism in one sentence: fixed withdrawals after a fall sell more units at low prices, and those units miss the recovery.

    What the interviewer asks next

    • How large a cash bucket would have let the first retiree avoid selling equity in years one and two?
    • If withdrawals were 4% of the current corpus instead of a fixed Rs 6 lakh, would the order still matter?
    • How would you explain sequence risk to a client in two sentences?
  2. 089A retiree has Rs 40 lakh invested at an assumed 8% a year and withdraws Rs 30,000 a month. How long does the money last, and what monthly withdrawal would last forever?Withdrawals and after-tax arithmeticHardWealth and advisoryDistribution and sales

    Try it first

    Roughly how long does Rs 40 lakh last at Rs 30,000 a month?

    Show the worked solution

    About 27.6 years, and about Rs 26,700 a month would last forever. At 8% a year, about 0.67% a month, Rs 40 lakh earns Rs 26,667 a month. Taking Rs 30,000 eats about Rs 3,333 of capital in month one, and the gap widens as the corpus shrinks, so the money runs out after about 331 months. Withdraw only the interest and the corpus never falls.

    Why does a small overdraw turn into a countdown?

    Think of a well that refills by 100 buckets a day. Draw 100 and it lasts forever; draw 110 and the level drops a little, so tomorrow it refills slightly less, and the drop speeds up until the well is dry. A corpus earning interest works the same way: withdraw no more than it earns and it lasts forever; withdraw a little more and each month's shortfall shrinks the base that earns next month's interest. Here she takes Rs 30,000 against Rs 26,667 of interest, only about Rs 3,333 too much at the start, and it still empties the account.

    Rs 40 lakh at 8%: how long each monthly withdrawal lasts1020304005101520253035Years of withdrawalsCorpus, Rs lakhRs 26,667 a month: lasts foreverRs 30,000: emptyat 27.6 yearsRs 35,000: 18.0 yearsRs 28,000: 38 yearsOn the Rs 30,000 path (dots):year 10: 33.9 lakh leftyear 20: 20.4 lakh left
    Withdrawing the Rs 26,667 a month the corpus earns keeps Rs 40 lakh intact forever, while Rs 30,000 a month empties it in about 27.6 years and Rs 35,000 in about 18.0, so a few thousand rupees a month decide whether the money is permanent or a countdown.

    How do you work out the number of months?

    Use the annuity formula and solve for the number of payments. With a monthly rate of 8% divided by 12, the corpus lasts as long as it takes the withdrawals to use up both the capital and the interest it earns along the way. The answer is about 331 months, or 27.6 years, and it is very sensitive to the withdrawal: Rs 28,000 a month lasts about 38 years and Rs 35,000 only about 18.0. The path is not a straight line: after ten years the corpus is still about Rs 33.9 lakh, and after twenty about Rs 20.4 lakh, because the fall speeds up near the end.

    The relationship
    n=−ln⁡ ⁣(1−P iW)ln⁡(1+i)=−ln⁡ ⁣(1−26,66730,000)ln⁡(1.00667)≈331 monthsn = \frac{-\ln\!\left(1 - \dfrac{P\,i}{W}\right)}{\ln(1+i)} = \frac{-\ln\!\left(1 - \dfrac{26{,}667}{30{,}000}\right)}{\ln(1.00667)} \approx 331 \text{ months}
    Pthe starting corpus, Rs 40 lakh
    ithe monthly rate, 8% divided by 12
    Wthe monthly withdrawal, Rs 30,000
    P ithe first month's interest, Rs 26,667
    What it says in wordsThe closer the withdrawal is to the interest earned, the closer the fraction gets to one and the longer the money lasts; at exactly the interest it lasts forever.

    What does this leave out?

    Three things, each of which shortens the runway. Rs 30,000 today buys less every year, so a real retiree needs a rising withdrawal, not a flat one. The 8% is an assumption, not a promise; a market-linked portfolio delivers it unevenly, and a bad few years early on, while withdrawals continue, does more damage than the same years late. And tax comes out of the return. Even the 'forever' figure only preserves Rs 40 lakh in rupees; after inflation, the capital it protects is shrinking. She is withdrawing 9% of the corpus a year while it earns 8%, and that one comparison tells you the plan has an end date.

    Where candidates lose it

    The quick answer divides Rs 40 lakh by Rs 3.6 lakh a year and says about 11 years. That treats the corpus as cash in a drawer and throws away the 8% it earns while it waits; the true runway is more than twice as long.

    The opposite slip is hearing 8% earned against 9% withdrawn and calling it roughly sustainable. Any withdrawal above the interest has an end date, and the gap compounds against her. Give the 27.6 years, then the Rs 26,700 that would never run out.

    What the interviewer asks next

    • She wants the money to last exactly 25 years. What monthly withdrawal allows that?
    • Inflation runs at 6%. How would you change the calculation?
    • Why does a bad market in her first three years of retirement matter more than one in her last three?