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. 018Fund A returns 16% a year with 20% volatility. Fund B returns 12% with 10% volatility. Cash pays 6%. Which has the better Sharpe ratio, and which would you rather hold if you cannot borrow?Performance measurement and returnsCoreFund research and ratingsIndian AMCs

    Try it first

    Which fund has the higher Sharpe ratio?

    Show the worked solution

    B has the better Sharpe ratio, 0.6 against 0.5, but without borrowing a client who needs a 16% expected return can only get it from A. A earns 10 points over cash for 20 of volatility; B earns 6 for 10. With borrowing, B scaled to 20% volatility would offer 18%, beating A. Without it, B tops out at 12%.

    What does the Sharpe ratio measure?

    Think of two delivery riders. One earns Rs 1,000 a day riding 200 km; the other earns Rs 600 riding 100 km. The first earns more, the second earns more per kilometre. The Sharpe ratio is return above cash per unit of volatility, so it ranks how efficiently a fund turns risk into reward, not how much reward it delivers. A earns 10 points over cash for 20 points of volatility, 0.5; B earns 6 for 10, 0.6.

    The steeper line from cash is the better use of risk6%10%14%18%0%5%10%15%20%25%VolatilityReturnCash 6%Fund A: 16%, 20%Fund B: 12%, 10%B with borrowing: 18%Sharpe = slopeA: 10 / 200.5B: 6 / 100.6No borrowing allowed:B tops out at 12%;only A reaches 16%
    Fund B's line from cash rises more steeply than A's, a Sharpe ratio of 0.6 against 0.5, so with borrowing it would beat A at the same risk, but without borrowing it cannot go beyond its own 12% return.

    Why can the lower-Sharpe fund still be the one to hold?

    The Sharpe ratio assumes you can slide along the line from cash. Mix A with cash half and half and you get 11% at 10% volatility, worse than B's 12% at the same risk; that is B's Sharpe advantage at work. Going the other way, beyond B's own risk, requires borrowing, so an investor who cannot borrow and needs more than 12% expected return has to take the less efficient fund. With borrowing at the cash rate, B levered to 20% volatility would return 6% plus 0.6 times 20, which is 18%, beating A's 16%.

    The relationship
    S=R−Rfσ:SA=16−620=0.5,SB=12−610=0.6S = \frac{R - R_f}{\sigma}: \quad S_A = \frac{16 - 6}{20} = 0.5, \qquad S_B = \frac{12 - 6}{10} = 0.6
    Rthe fund's return
    R_fthe cash rate, 6%
    \sigmathe fund's volatility
    What it says in wordsSubtract what cash pays, then divide by the risk taken to earn the rest.

    Say the limits. Volatility treats upside and downside swings alike, and a fund with rare large losses can show a flattering Sharpe ratio until one arrives. Figures from a few years of history are noisy estimates, so a 0.5 against 0.6 difference may not be meaningful. And the answer to which to hold depends on the client's required return and tolerance for swings; the Sharpe ratio ranks the funds, it does not choose for the client.

    Where candidates lose it

    The common slip is picking A because it returns more, or because its excess return of 10 points beats B's 6. Both ignore the risk taken. The interviewer asked for a ratio and wants to see you divide.

    The opposite slip is saying B, full stop, to the second question. The Sharpe ranking assumes leverage is available; without it, the higher-return fund may be the only way to reach a client's target, and saying that is what the follow-up was set up to test.

    What the interviewer asks next

    • What mix of fund A and cash matches fund B's volatility, and what does it return?
    • Why might a fund with a high Sharpe ratio still lose a client a lot of money in one year?
    • What would the Sortino ratio change about this comparison?
  2. 055You buy an office REIT unit at Rs 300. It pays Rs 21 a year and you sell it for Rs 330 after five years. What are the equity multiple and the IRR, and why can two investments with the same multiple have very different IRRs?Performance measurement and returnsCoreInvescoNew York · 2025

    Try it first

    Which is closest to the IRR?

    Show the worked solution

    The equity multiple is 1.45x and the IRR is about 8.7%. You get back five payments of Rs 21 and Rs 330 on sale, Rs 435 in all, on Rs 300 in. The IRR is the rate at which those flows are worth exactly Rs 300 today. The multiple counts rupees and ignores time, so the same Rs 435 received in one lump at year 10 is still 1.45x but only about 3.8% a year.

    What does each measure actually count?

    Lend a friend Rs 300 and get Rs 435 back. Whether it came back in five years or fifteen, you can say you made 1.45 times your money, but you would not call the two loans equally good. The equity multipleTotal cash received divided by cash invested. It counts rupees and ignores when they arrive. counts how many rupees come back; the IRR counts how fast they come back. Here total cash is 5 x 21 plus 330, Rs 435, and 435 over 300 is 1.45x.

    Same money back, very different speedOffice REIT unitMultiple 1.45x | IRR 8.7%-30021212121351Same total, all at year 10Multiple 1.45x | IRR 3.8%-300435012345678910yearRs 21 a year, then Rs 330 on sale
    The REIT unit returns Rs 435 on Rs 300 through yearly payments and a sale at year 5, an IRR of 8.7%; the same Rs 435 in one payment at year 10 is still a 1.45x multiple but an IRR of only 3.8%.

    How do you get the IRR quickly in the room?

    Split it into income and growth. The payment of Rs 21 on Rs 300 is a 7% yield. The price rises from 300 to 330, 10% in five years, which compounds to just under 2% a year. Add them and you are near 9%; the exact answer is a little lower, 8.7%, because the price gain arrives only at the end. Then offer the check: at 8.7% the five payments and the sale discount back to Rs 300.

    The relationship
    300=∑t=1521(1+r)t+330(1+r)5  ⇒  r≈8.7%300 = \sum_{t=1}^{5} \frac{21}{(1+r)^t} + \frac{330}{(1+r)^5} \;\Rightarrow\; r \approx 8.7\%
    300the price paid for the unit
    21the yearly distribution
    330the sale price at year 5
    rthe IRR, the rate that makes both sides equal
    What it says in wordsThe IRR is the one discount rate at which everything you receive is worth exactly what you paid.

    Now the comparison the interviewer wants. The same Rs 435 in a single payment at year 5 is 7.7% a year, lower than 8.7% only because the distributions no longer arrive early. At year 10 it is 3.8%. The limit of IRR: it assumes the early cash can be reinvested at the same rate, and it says nothing about size, so a 20% IRR on Rs 1 lakh for one month is not better than 9% on Rs 1 crore for five years.

    Where candidates lose it

    The common slip is 9%: dividing the 45% total gain by five years. That treats the money as if it all came back evenly and ignores that the sale arrives last.

    The second loss is quoting only one of the two measures. Real estate and REIT desks ask for both because each hides what the other shows: the multiple hides time, the IRR hides size.

    What the interviewer asks next

    • The sale price is Rs 300 instead of Rs 330. What is the IRR?
    • Why do private real estate funds report both IRR and multiple to their investors?
    • The distribution is cut to Rs 15 in years 3 to 5. Which moves more, the multiple or the IRR?

    Asked at Invesco, Real Estate, New York, 2025 (Wall Street Oasis): Lots of basic questions asked about IRR, EM, Cap Rates etc

  3. 080A fund holds 30% in IT stocks against a 20% benchmark weight. The IT sector returned 5% while the whole benchmark returned 12%. Separately, its bank stocks, a 25% weight, beat the bank index by 3 points. Split the fund's active return into an allocation effect and a selection effect.Performance measurement and returnsCoreFund research and ratingsIndian AMCs

    Try it first

    What did the IT overweight do to relative performance?

    Show the worked solution

    Allocation cost 0.70% and selection added 0.75%, a net active return of about plus 0.05%. The IT overweight is 10 points in a sector that trailed the benchmark by 7 points: 0.10 x (5% minus 12%) is minus 0.7%. The bank stocks beat their index by 3 points on a 25% weight: 0.25 x 3% is plus 0.75%. Good stock picking almost exactly paid for a poor sector bet.

    What is the difference between allocation and selection?

    Picture a selector who picks four spinners for a pitch that suits pace, but whose four spinners bowl better than any other spinners in the country would have. Two separate decisions: how many of each kind, and which ones. Allocation measures the first decision, sector weights against the benchmark's weights; selection measures the second, how the stocks chosen inside a sector did against that sector. Splitting them tells a fund research team whether a manager's skill lies in calling sectors or in picking stocks, which matters more than the total when deciding what to trust next.

    The relationship
    A=(wp−wb)(Rs−Rb)S=wp (rs−Rs)A = (w_p - w_b)(R_s - R_b) \qquad S = w_p\,(r_s - R_s)
    w_p, w_bthe fund's and the benchmark's weight in the sector
    R_sthe sector index return
    R_bthe whole benchmark's return
    r_sthe return on the stocks the fund actually held in that sector
    What it says in wordsAllocation is the extra weight times how the sector did against the whole benchmark; selection is the weight held times how the chosen stocks did against their sector.
    Two decisions, two effects: how much in each sector, and which stocksAllocation: the sector bet(30% - 20%) x (5% - 12%)overweight x IT against the whole index= -0.70%Selection: the stock picks25% x (banks held - bank index)weight x 3 points of outperformance= +0.75%0-0.5%+0.5%-0.70%+0.75%+0.05%AllocationSelectionActive returnGood picking almost exactly paid for the sector bet
    The 10 point IT overweight cost 0.70% because IT trailed the benchmark by 7 points, and bank stocks that beat their index by 3 points on a 25% weight added 0.75%, leaving an active return of only plus 0.05%.

    Why is allocation measured against the whole benchmark rather than against zero?

    Because the extra 10% in IT had to come from somewhere, and the alternative was the benchmark itself. An overweight in a sector that makes money still costs you if that sector made less than everything else you could have held. Say the assumptions behind the split out loud: the bank weight matches the benchmark's, so banks carry no allocation effect; the IT stocks held matched the IT index, so IT carries no selection effect; and the 10 points taken from other sectors came from sectors that earned the benchmark's 12%.

    One detail an interviewer may probe. Using the fund's 25% weight in the selection term folds in what the BrinsonThe Brinson method, named after the authors who set it out in the 1980s, splits a fund active return into allocation, selection and an interaction term. framework calls the interaction effect; the textbook version uses the benchmark's weight and reports interaction separately. With the bank weights equal here, both give the same 0.75%.

    Where candidates lose it

    Candidates multiply the overweight by IT's own return, 10% x 5%, and call the IT bet a gain of 0.5%. That ignores what the money would have earned in the rest of the benchmark, and it turns a costly decision into a profitable-looking one.

    The other slip is netting everything into plus 0.05% and calling the manager roughly neutral. The split is the whole point: plus 0.75 on stocks and minus 0.70 on sectors describes a good picker whose sector calls are giving the gains away.

    What the interviewer asks next

    • What if the fund had been underweight IT by 10 points instead?
    • The bank index itself beat the benchmark. Where does that show up?
    • Over three years, which of the two effects would you trust more as evidence of skill, and why?