Mutual Fund Mastery puzzles, solved step by step

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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. 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?