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. 028A new fund offer is priced at Rs 10 a unit. An older fund holding exactly the same portfolio has an NAV of Rs 180. You put Rs 1 lakh into each, and the market then rises 20%. Which investment gains more?Behavioural trapsWarm upWealth and advisoryDistribution and sales

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    Pick before you calculate.

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    They gain exactly the same: Rs 20,000 each. Rs 1 lakh buys 10,000 units at Rs 10 or about 555.6 units at Rs 180. The same portfolio rising 20% lifts the NAVs to Rs 12 and Rs 216. Both holdings are worth Rs 1.2 lakh. A low NAV is not cheap and a high NAV is not expensive: what you earn is a percentage of the money you put in.

    Why does a Rs 10 unit feel cheaper than a Rs 180 unit?

    A pizza cut into twelve slices is not more pizza than the same pizza cut into four. Each slice is smaller, that is all. An NAV is the fund's portfolio value divided by the number of units in issue, so a lower NAV means the same money is cut into more, smaller slices. The instinct that Rs 10 is cheap comes from shares, where a low price is also misleading, and from shopping, where a lower price usually buys the same thing for less. Here it buys a smaller slice of the same thing.

    Rs 1 lakh into each fund, then the market rises 20%New fund, NFO at NAV 10Units bought10,000NAV before and after10 to 12Rs 1.00 lakhRs 1.20 lakhinvestedafter +20%+20%Older fund, NAV 180Units bought555.6NAV before and after180 to 216Rs 1.00 lakhRs 1.20 lakhinvestedafter +20%+20%
    Rs 1 lakh buys 10,000 units of the new fund at NAV 10 or about 555.6 units of the older fund at NAV 180. When the shared portfolio rises 20%, both NAVs rise 20%, and both holdings end at Rs 1.2 lakh, a gain of Rs 20,000 each.

    What actually decides the gain?

    Only the portfolio. A fund's return is the percentage change in its NAV, and the NAV moves by exactly the percentage the portfolio moves, less costs, whatever level it starts from. An NAV of 180 means the older fund has grown eighteen-fold since its own launch at 10; it tells you about the past and nothing about the next move. Two funds holding the same portfolio at the same cost will deliver the same return from here.

    The relationship
    value=Rs 1,00,000NAV0×NAV0(1+r)=Rs 1,00,000×(1+r)\text{value} = \frac{\text{Rs }1{,}00{,}000}{\text{NAV}_0} \times \text{NAV}_0 (1 + r) = \text{Rs }1{,}00{,}000 \times (1 + r)
    NAV_0the NAV you buy at, 10 or 180
    rthe portfolio's return, 20%
    What it says in wordsThe starting NAV cancels out: the money you end with depends only on the money you put in and the return.

    Is there any real difference between the two funds?

    Yes, but none of it is about the NAV level. A new fund has no track record, may take weeks to deploy the money it raises, and may carry a different expense ratio. An older fund may be larger, which matters in small caps. Those are real reasons to prefer one or the other. A lower NAV is not one of them, and saying so plainly is what the interviewer is listening for.

    Where candidates lose it

    The trap is answering that the NFO gains more because the investor holds more units, or because a Rs 10 price has more room to run. Both treat the unit count as if it were value. The interviewer is testing whether you can say, in one sentence, that returns are percentages.

    The quieter trap is the opposite: preferring the Rs 180 fund because a high NAV proves the manager is good. An NAV of 180 describes past growth since launch, not skill and not the future.

    What the interviewer asks next

    • The older fund charges 1.5% a year and the NFO 0.8%. Does that change your answer?
    • Why do fund houses launch new funds at Rs 10 when similar funds already exist?
    • How would you explain this to a client who insists the Rs 10 fund is cheaper?
  2. 076A market index has fallen five days in a row. Over its history it has risen on 52% of days, and each day's move is independent of the last. A client asks whether it is now due a rise. What is the chance the index rises tomorrow?Behavioural trapsWarm upWealth and advisoryDistribution and sales

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    Your instinct, before any arithmetic.

    Show the worked solution

    52%, the same as on any other day. If each day's move is independent, the five falls carry no information about tomorrow. A run of five falls was unlikely before it began, 0.48 to the fifth power or about 2.5%, but that probability belonged to days that are now finished. Believing a rise is due is the gambler's fallacy.

    Why does the streak not make a rise more likely?

    A captain who has lost five tosses in a row walks out for the sixth feeling owed a win. The coin has no record of the first five and no sense of fairness to restore; it is still 50:50. Independence means past outcomes do not enter the calculation for the next one, so tomorrow's chance of a rise is the 52% it always was. The feeling that things must even out is real, and it has a name: the gambler's fallacy. It is strongest exactly when a streak is long, which is when it does the most damage to decisions.

    Five falls behind you, and tomorrow's odds have not movedDay 1fellDay 2fellDay 3fellDay 4fellDay 5fellDay 6: tomorrow52%chance of an up dayThe odds each morning, before the day's move: green up, red down52 / 4852 / 4852 / 4852 / 4852 / 4852 / 48, the same barBefore Monday: five falls in a row = 0.48 x 0.48 x 0.48 x 0.48 x 0.48 = 2.5%After Friday: that rare thing has already happened, and tomorrow is still 52%
    Each of the five days began with the same 52% chance of a rise, and so does tomorrow: the 2.5% chance of five falls in a row applied before the streak started, not after it has happened.

    Where does the tiny probability come from, and why is it the wrong number?

    It answers a different question. Standing on Monday morning, the chance of five falls in a row was 0.48 multiplied by itself five times, about 2.5%, and the chance of six in a row was about 1.2%, roughly one in 82. Once five falls have happened, the only uncertainty left is tomorrow, and for independent days the chance of a rise given the streak equals the chance of a rise on any day. Six falls in a row is rare only when viewed from the start; viewed from Friday evening, it needs just one more fall.

    The relationship
    P(up6∣5 downs)=0.485×0.520.485=0.52P(\text{up}_6 \mid \text{5 downs}) = \frac{0.48^5 \times 0.52}{0.48^5} = 0.52
    0.48^5the chance of the five falls that have already happened
    0.52the chance of a rise on any single day
    |read as given that
    What it says in wordsDivide the chance of the whole sequence by the chance of the part already seen, and the streak cancels out, leaving 52%.

    Is the independence assumption true for real markets?

    Not exactly, and saying so earns credit. Day-to-day direction in a broad index is very hard to predict from the previous days, but the size of moves does cluster: a run of falls often comes with bigger swings, so tomorrow's move may be larger even if its direction is a near coin toss. For an adviser the danger is not the arithmetic but the conversation: a client who believes a bounce is due will add money for the wrong reason, or hold out for a rebound the odds do not promise. A streak on its own is not a reason to act; any case for investing has to rest on the client's plan and horizon, not on the last five days.

    Where candidates lose it

    The trap is answering with the streak's rarity. Candidates say six falls in a row is roughly a one in 82 event, so a rise is almost certain, and in doing so treat days that are already over as if they were still uncertain. The interviewer is checking whether you can separate the probability of a sequence from the probability of the next step.

    The quieter loss is stopping at 52% without naming the assumption. Say independent, then add one sentence on what real markets do differently, such as swings clustering, and the answer sounds like someone who has watched markets rather than memorised a rule.

    What the interviewer asks next

    • What is the chance of at least one up day in the next five?
    • If an up day followed an up day 55% of the time, how would your answer change?
    • A client wants to invest only after a 5% fall. How would you explain the cost of waiting?