Mutual Fund Mastery puzzles, solved step by step

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  1. 041In a category of 40 equity funds, the mean five-year return is 14% a year but the median is 11%. What does the gap tell you, and which figure should a client hear?Statistics, correlation and diversificationWarm upFund research and ratingsGlobal asset managers

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    Before you work it: what does a mean well above the median most likely mean here?

    Show the worked solution

    The gap says a few star funds are pulling the average up; the typical fund made about 11%. The median is the middle fund, so half the category earned 11% or less. A mean of 14% needs a small group far out on the right: here six funds between 24% and 42%, and only 9 of the 40 beat the mean. A client choosing a fund at random should hear 11%, together with the spread around it.

    Why can the average sit above most of the funds?

    Ten people sit in a tea stall, each earning about Rs 30,000 a month. A business owner earning Rs 30 lakh a month walks in. The average income in the room jumps above Rs 2.9 lakh; the median, the middle person, still earns about Rs 30,000. A mean is pulled by every extreme value, while a median only cares about the middle, so when a few values are very large the mean overstates what a typical member got. Fund returns behave the same way: a few funds that caught one theme early can drag the category average up.

    40 funds, one dot each: a few stars drag the mean away from the typical fund5%10%15%20%25%30%35%40%Five-year return, per cent a yearmedian 11%the typical fundmean 14%only 9 of 40 beat itsix star fundspull the mean right
    Thirty-four of the 40 funds returned between 5% and 17% a year and six star funds returned between 24% and 42%, so the median sits at 11% while the mean is pulled to 14% and only 9 funds beat it.

    How do you check that the stars explain the gap?

    Take them out and recompute. Without the six stars the other 34 funds average 10.9%, almost exactly the median. The six funds alone add about 3.1 percentage points to the category mean, which is the whole gap. A second check is to count: if the mean described a typical fund, about half the funds would sit above it. Here only 9 of 40 do.

    The relationship
    xˉ=140∑i=140xi=56040=14%,median=x(20)+x(21)2=11%\bar{x} = \frac{1}{40}\sum_{i=1}^{40} x_i = \frac{560}{40} = 14\%, \qquad \text{median} = \frac{x_{(20)} + x_{(21)}}{2} = 11\%
    x_ifund i's five-year annualised return
    x_(20), x_(21)the 20th and 21st returns after sorting, the middle pair of 40
    What it says in wordsThe mean adds every return, extremes included; the median takes the middle pair and ignores how far out the extremes are.

    Which figure should a client hear, and what else?

    The median, because it is the honest answer to what a typical fund in the category did. Then the spread, because 5% to 42% is the real range of outcomes, and the chance of picking a star in advance is small. Two further cautions belong in the same breath. Category figures usually include only funds that survived the five years, and funds that closed or merged were often the weak ones, so even the median flatters. And past five-year returns, mean or median, are not a forecast.

    Where candidates lose it

    The trap is quoting the mean because it is the number a factsheet or a sales deck usually leads with, or saying that the gap must be a data error. A mean above the median is the normal signature of a right-skewed set of returns.

    The second loss is stopping at the statistics. The interviewer asked which figure a client should hear. Say the median, say why, and add the spread and the survivorship caution.

    What the interviewer asks next

    • What would a mean below the median tell you about a category?
    • If you remove the top and bottom 10% of funds, what is that average called and why use it?
    • How does survivorship bias change both the mean and the median?