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. 092A fund returned 60%, 2% and 3% in three successive years while the median peer returned 25%, 10% and 10%. The fund ranked in the bottom quartile in two of the three years, yet sits in the top quartile on its three-year return. How?Performance measurement and returnsHardFund research and ratingsIndian AMCs

    Try it first

    What are the fund's and the median peer's three-year returns, a year?

    Show the worked solution

    One outsized year carries the whole period: 18.9% a year against the peers' 14.8%. Compounded, 1.60 x 1.02 x 1.03 is 1.681 for the fund and 1.25 x 1.10 x 1.10 is 1.513 for the median peer. The 35-point lead in year 1 is larger than the 8 and 7 points given back later. A single period's rank says how much was earned, not how consistently, and here the latest two years run at 2.5% a year against 10.0%.

    How can losing two years out of three still win the period?

    A batter who scores 150 in one innings and 5 and 8 in the next two has a better series total than one who scores 40, 45 and 45, but nobody would call him the more reliable player. A multi-year return adds up the size of each year's result, not the number of years won, so one very large year can outweigh several small losses. Here the fund was 35 points ahead in year one, then 8 and 7 points behind. Compounding the three years keeps most of that first lead intact.

    Two bad years out of three, and still top over three years60%25%Year 12%10%Year 2bottom quartile3%10%Year 3bottom quartile18.9%14.8%3 years, a yeartop quartilethe fundpeer medianStart the clock one year later:fund 2.5% a year, peers 10.0%
    The fund trailed the median peer in two of three years, yet its 60% first year lifts its compounded three-year return to 18.9% a year against 14.8%, while the last two years alone show 2.5% against 10.0%.
    The relationship
    CAGR=(1.60×1.02×1.03)1/3−1=1.6811/3−1=18.9%\text{CAGR} = \left(1.60 \times 1.02 \times 1.03\right)^{1/3} - 1 = 1.681^{1/3} - 1 = 18.9\%
    1.60, 1.02, 1.03the fund's growth factor in each year
    1/3the cube root, which turns three years of growth into a yearly rate
    CAGRcompound annual growth rate
    What it says in wordsMultiply the yearly growth factors, then take the cube root to turn the three-year growth into an average yearly rate.

    What would a fund researcher look at instead?

    Consistency, measured several ways. Shift the start date by one year and the same fund drops from top to bottom: over the last two years it made 2.5% a year against the peers' 10.0%. That is the case for rolling returns, which compute the return over every possible window of a given length and show how often the fund beat its peers, not just whether it did over one window ending today. A researcher would also ask what produced the 60% year: one concentrated bet that paid off, a sector that ran, or something repeatable.

    One more point about averaging. The simple average of the fund's three years is 21.7%, well above the 18.9% it actually compounded at; volatile return streams always show a bigger gap between the two. Quote compounded returns for anything an investor actually experienced, and treat a period return as one data point, not a verdict. The quartile labels here are illustrative: they assume a peer group in which 2% and 3% fell in the bottom quarter and 18.9% in the top.

    Where candidates lose it

    The tempting wrong answer is that the numbers must be inconsistent, since a fund that trails in most years should not be able to come out ahead over the period. Candidates who say this have counted wins instead of adding up sizes, and missed that one 60% year outweighs two years of trailing by 7 or 8 points.

    The second loss is getting 18.9% right and stopping. The question invites the point that period returns hide consistency; say that rolling returns, and starting one year later, tell a very different story.

    What the interviewer asks next

    • In what order would the three years have to come for the fund to look worst on a three-year view?
    • How would you measure consistency across ten years of monthly data?
    • A fund house advertises its fund's five-year rank. What would you ask before believing it says anything about skill?
  2. 093Passive funds tracking an index hold an assumed Rs 50,000 crore. A stock is added to the index at a weight of 1.2%, and it trades about Rs 250 crore a day. How much must the passive funds buy, and how many days of the stock's trading is that?Estimation and market sizingHardPassive and index teamsIndian AMCs

    Try it first

    How many days of the stock's normal trading does the passive buying equal?

    Show the worked solution

    About Rs 600 crore, or 2.4 days of the stock's entire trading. Passive funds must hold the stock at its index weight: 1.2% of Rs 50,000 crore is Rs 600 crore. Against Rs 250 crore of daily trading, that is 2.4 days of every share traded, and they want to own it by the day the change takes effect. If they took only a quarter of each day's volume, the buying would take about 9.6 days.

    Why is index buying forced rather than optional?

    Picture a school that announces every student must own a particular textbook by Monday, and the only shop stocks a few dozen copies a day. The shopkeeper knows exactly how many will be needed and by when. An index fund has no view on the stock; its job is to hold every stock at its index weight, so when a stock enters the index, every passive fund tracking it must buy, in a known amount, by a known date. That predictability is what makes inclusion a trading event, not just a news item.

    Forced buying measured in days of the stock's tradingPassive buying needed1.2% x Rs 50,000 croreRs 600 croreDaily traded valueRs 250 crore a dayday 1day 20.4 day= 2.4 days of every share tradedPassive funds cannot take all the volume without moving the priceAt a quarter of each day's trading: Rs 250 x 25% = Rs 62.5 crore a dayRs 600 / Rs 62.5 = 9.6 days, but the funds want to own it on one day
    Passive funds holding Rs 50,000 crore must buy Rs 600 crore of a stock entering at a 1.2% weight, which is 2.4 days of all its trading at Rs 250 crore a day, and about 9.6 days if they take only a quarter of each day's volume.

    How do you size it, and why measure it in days?

    The rupee amount is just the weight times the passive money: 0.012 x Rs 50,000 crore is Rs 600 crore. On its own that number means little, because Rs 600 crore is trivial in one stock and enormous in another. Dividing by daily traded value turns it into a measure of difficulty: 2.4 days means the funds must buy every share traded for more than two full days, which nobody can do without pushing the price up. A desk usually assumes a buyer can take only a fraction of the day's volume quietly; at a quarter, the job takes about 9.6 days.

    The relationship
    Days=w×AUMADV=0.012×50,000250=2.4\text{Days} = \frac{w \times AUM}{ADV} = \frac{0.012 \times 50{,}000}{250} = 2.4
    wthe stock's weight in the index, 1.2%
    AUMmoney in funds tracking the index, Rs 50,000 crore, an assumption
    ADVaverage daily traded value of the stock, Rs 250 crore
    What it says in wordsForced buying in rupees, divided by what trades in a day, gives the number of days of all trading the funds need.

    Who positions for it, and what does the estimate leave out?

    Traders who expect the inclusion buy ahead of the effective date and sell to the index funds on it, so part of the price move usually happens before the passive funds trade at all. The passive funds also pay for the purchase by selling a slice of every other stock, about Rs 600 crore spread across the rest of the index, which barely registers in each. The estimate leaves out active funds that hug the same benchmark and buy for similar reasons, which makes the real demand larger, and it assumes the weight is set by the stock's full market value; index providers commonly weight by the shares freely available for trading, so check the actual weight before relying on the size.

    Where candidates lose it

    Candidates compute Rs 600 crore and stop. The rupee figure is the easy half; the interviewer wants it set against liquidity, because Rs 600 crore tells you nothing until you know whether the stock trades Rs 25 crore a day or Rs 2,500 crore.

    The second slip is assuming the funds can simply take all the volume. Say that a buyer can only take part of each day's trading without moving the price, and that traders front-run the known demand.

    What the interviewer asks next

    • The stock is removed from the index instead. Who is on the other side of the passive selling?
    • Passive assets double over five years. What happens to the days-of-volume figure for the same stock?
    • Why might a stock rise after the inclusion is announced but fall back after the effective date?