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. 005An active large cap fund charges 1.8% a year and an index fund on the same index charges 0.2%. How much must the active manager beat the index by, before costs, just to tie? And on Rs 10 lakh over 15 years with the index returning 12% a year, what does zero skill cost the investor?Costs and fee dragCoreFranklin TempletonSan Mateo · 2017

    Try it first

    Over 15 years, how big is the wealth gap if the active manager has no skill?

    Show the worked solution

    The manager must beat the index by 1.6 points a year before costs just to tie, and with zero skill the investor ends about Rs 10.4 lakh behind. Both funds earn the index's 12% before costs. Net, that is 11.8% against 10.2%. Rs 10 lakh compounds to Rs 53.3 lakh in the index fund and Rs 42.9 lakh in the active fund, a gap of about 19% of the final wealth.

    Why is the break-even the whole cost gap and not the active fee?

    Two taxis to the same station, one charging Rs 18 per km and one Rs 2. The expensive one only wins if it is a much shorter route. The investor's alternative is not zero cost, it is the index fund, so the active manager has to earn back the difference in costs, 1.6 points a year, before adding anything. Beating the index by 1% before costs sounds like skill and still leaves the investor 0.6 points a year behind the cheaper fund.

    A 1.6 point cost gap, compounded for 15 years1020304050Rs lakh051015YearsIndex fund: 53.3Active, no skill: 42.9Gap: Rs 10.4 lakhIndex return assumed 12% a year11.8% net against 10.2% net
    Rs 10 lakh compounding at 11.8% after costs reaches Rs 53.3 lakh in 15 years, against Rs 42.9 lakh at 10.2%, so a 1.6 point cost gap becomes a Rs 10.4 lakh gap, about a fifth of the final wealth.

    Why does 1.6 points a year become a fifth of the money?

    Because the fee is charged on the balance every year, and the rupees it takes would themselves have compounded. A cost gap compounds exactly like a return gap: the ratio of final wealth is (1.102 over 1.118) to the 15th power, about 0.806, so the active investor keeps about 81% of what the index investor has. The longer the horizon, the larger that share becomes; over 30 years it would be over a third.

    The relationship
    10(1.118)15−10(1.102)15=53.3−42.9=10.4 lakh10(1.118)^{15} - 10(1.102)^{15} = 53.3 - 42.9 = 10.4 \text{ lakh}
    1.118one plus the index fund's return after its 0.2% cost
    1.102one plus the active fund's return after its 1.8% cost, assuming no skill
    15years held
    What it says in wordsCompound the Rs 10 lakh at each net return and subtract; the gap is the price of paying for skill that did not show up.

    The limitation: the 12% index return is an assumption for the arithmetic, and some active managers do beat their index after costs. The question is not whether active management can win. It is how large the hurdle is, and 1.6 points a year, every year, is a high bar to clear consistently.

    Where candidates lose it

    The common wrong answer to the first part is 1.8%, the active fund's fee. The investor's real choice is the index fund, which also costs something, so the hurdle is the gap, 1.6 points.

    The common wrong answer to the second part is simple interest: 1.6% of Rs 10 lakh for 15 years, Rs 2.4 lakh. Fees come out of a growing balance and compound; say that sentence and then give the Rs 10.4 lakh.

    What the interviewer asks next

    • What gross alpha does the active manager need for the investor to end Rs 5 lakh ahead of the index fund?
    • The index fund also lags its index by 0.3% a year through tracking difference. How does that change the hurdle?
    • How would you explain this gap to a client in one sentence without recommending either fund?

    Asked at Franklin Templeton, Risk Management, San Mateo, 2017 (Wall Street Oasis): explain the difference between actively managed and passively managed mutual funds

  2. 044An ETF charges 0.05% a year, plus 0.03% brokerage each way and a 0.10% bid-ask spread. An index fund on the same index charges 0.20% a year with no trading cost. After how long a holding does the ETF become the cheaper choice?Costs and fee dragCoreVanguardMalvern · 2026

    Try it first

    Roughly how long must you hold before the ETF is cheaper?

    Show the worked solution

    After about 1.1 years, roughly 13 months. The ETF's trading costs are paid once: 0.03% brokerage on the way in and out, 0.06%, plus the 0.10% spread, 0.16% in all. Its fee is 0.15% a year below the index fund's. Divide the one-off cost by the yearly saving, 0.16 over 0.15, and the ETF catches up after 1.07 years. Hold longer and it wins; trade in and out and it loses.

    How do you compare a one-off cost with a yearly one?

    Think of buying a monthly rail pass against paying per ride. The pass costs more up front and less per trip, so it only pays if you ride enough. The ETF is the pass: it costs 0.16% to get in and out, then saves 0.15% a year against the index fund, so the break-even holding period is the one-off cost divided by the yearly saving. Count both sides of the trade: brokerage is paid when you buy and again when you sell, and crossing a 0.10% spread means buying a little above the middle price and selling a little below it.

    The relationship
    T∗=2b+sfIF−fETF=0.06%+0.10%0.20%−0.05%=0.160.15≈1.07 yearsT^{*} = \frac{2b + s}{f_{IF} - f_{ETF}} = \frac{0.06\% + 0.10\%}{0.20\% - 0.05\%} = \frac{0.16}{0.15} \approx 1.07 \text{ years}
    bbrokerage each way, 0.03%
    sthe bid-ask spread crossed over a round trip, 0.10%
    f_IF, f_ETFthe yearly fees of the index fund and the ETF
    What it says in wordsThe holding period at which the ETF catches up is its round-trip trading cost divided by the yearly fee it saves.
    One-off trading costs against a yearly fee: who wins depends on how long you hold0.2%0.4%0.6%0.8%1.0%0y1y2y3y4y5yYears heldCumulative cost, % of money invested1.07 yearsIndex fund, 0.20% a year: 1.00% after 5yETF: 0.16% up front + 0.05% a year: 0.41%brokerage 0.06 + spread 0.10
    The ETF starts 0.16% behind because of brokerage and spread but adds only 0.05% a year, while the index fund adds 0.20% a year from zero; the lines cross at about 1.1 years, and after five years the ETF has cost 0.41% against 1.00%.

    What does the gap look like in rupees?

    On Rs 10 lakh held for five years, the ETF costs about Rs 4,100 and the index fund about Rs 10,000, ignoring the small effect of compounding. Held for six months, the order flips: the ETF costs 0.185% against the index fund's 0.10%. ETFs win on long holds and lose on short ones, so the answer depends on the investor, not on the product.

    Name what the simple sum leaves out. Spreads on a thinly traded ETF can be much wider than 0.10%, and the price can sit at a premium or discount to NAV. Some investors pay yearly demat charges that matter on small balances. An index fund may carry an exit load in the early months, and both products have tracking differences that can outweigh a few basis points of fee. Each of these moves the break-even, so the useful answer is the method, with the inputs confirmed for the actual products.

    Where candidates lose it

    The common slip is answering from the fee alone: 0.05% is a quarter of 0.20%, so the ETF must be cheaper from day one. That ignores the costs you pay to trade it, which the index fund does not charge.

    The second slip is counting brokerage once, or the spread twice. Brokerage is paid on the buy and the sell, 0.06% in all; the 0.10% spread is crossed once over the round trip, half on each side.

    What the interviewer asks next

    • The investor adds Rs 10,000 every month through the ETF, paying brokerage each time. How does that change the answer?
    • At what spread would the ETF need a five-year hold to break even?
    • Why might an index fund still suit an investor who will hold for ten years?

    Asked at Vanguard, Generalist, Malvern, 2026 (Wall Street Oasis): the difference between a ETF and Mutual Fund