Mutual Fund Mastery puzzles, solved step by step

Puzzles: 100
Traced to a firm: 29
Topics: 13
Hard: 30

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002Fund A has 20% annual volatility and fund B has 12%. Their returns have a correlation of 0.3. What is the volatility of a portfolio that is 60% A and 40% B, and why is it below the 16.8% weighted average?Statistics, correlation and diversificationCoreFund research and ratingsGlobal asset managers
Try it first
Pick the portfolio's volatility before you work it.
Show the worked solution
About 14.2%, against a weighted average of 16.8%. Variance is 0.6 squared times 20% squared, plus 0.4 squared times 12% squared, plus twice 0.6 times 0.4 times 0.3 times 20% times 12%, which sums to 0.0202. Its square root is 14.2%. The 2.6 point gap exists only because the correlation is below 1.
Why is the mix not just the average of the two volatilities?
Two friends walking home on a windy night: if they stumble at exactly the same moments, holding hands does not steady them. If their stumbles come at different moments, each one's lean is partly caught by the other. Only the part of the two funds' swings that happens together adds up in full; the rest partly cancels, so the portfolio is calmer than the average of its parts. Correlation of 0.3 says most of their swings are not shared.
A 60/40 mix of a 20% and a 12% volatility fund has 14.2% volatility at a correlation of 0.3, below the 16.8% weighted average, and it would only reach 16.8% if the two funds moved in perfect step.
How do you work it quickly on paper?
Square the volatilities first, because variances are what add. Fund A's weighted variance is 0.36 times 0.04, which is 0.0144. Fund B's is 0.16 times 0.0144, which is 0.0023. The cross term is where correlation lives: 2 times 0.6 times 0.4 times 0.3 times 0.20 times 0.12 is 0.0035. The total is 0.0202, and the square root of 0.02 is about 0.141, so 14.2% is the answer.
The relationship
$\sigma_p = \sqrt{w_A^2\sigma_A^2 + w_B^2\sigma_B^2 + 2w_Aw_B\rho\,\sigma_A\sigma_B} = \sqrt{0.0202} \approx 14.2\%$
w_A, w_B the weights, 0.6 and 0.4
\sigma_A, \sigma_B the funds' volatilities, 20% and 12%
\rho the correlation between them, 0.3
What it says in wordsPortfolio variance is each fund's own variance, weighted, plus a shared term scaled by how closely the two move together.
Give the two edges to show you see the shape. At a correlation of 1 the formula collapses to the weighted average, 16.8%. At zero the cross term vanishes and the mix is 12.9%. At minus 1 it drops to 7.2%. The limitation is that correlations measured in calm years often rise in a selloff, so the benefit you computed can shrink exactly when it is wanted.
Where candidates lose it
Candidates answer 16.8% because averaging feels natural and the weights are right there. It is only true at a correlation of 1, and saying it tells the interviewer you do not see where diversification comes from.
The second loss is mixing units: adding volatilities in one term and variances in another, or forgetting the factor of 2 on the cross term. Write the formula once, square everything first, and take one square root at the end.
What the interviewer asks next
What correlation would make the 60/40 mix exactly as volatile as fund B on its own?
Which weight in A gives the lowest possible volatility at a correlation of 0.3?
Why might this calculation understate risk in a market crash?
029An active equity fund has an R-squared of 0.97 and a beta of 1.0 against its index, which has 16% volatility. Roughly how much active risk is the fund taking each year, and what does that suggest about its 1.8% expense ratio?Statistics, correlation and diversificationHardFund research and ratingsGlobal asset managers
Try it first
Before you work it: roughly how large is the fund's active risk, its tracking error?
Show the worked solution
About 2.8% a year of active risk, which leaves little room to earn back a 1.8% fee. An R-squared of 0.97 means 97% of the fund's variance moves with the index. With a beta of 1.0, the index part is 16% squared; total variance is that over 0.97, and the remaining 3% is the fund's own, about 2.8% a year as a volatility. To beat the index after a 1.8% fee, the manager needs an information ratio of about 0.64, which few sustain.
What does an R-squared of 0.97 actually say?
Picture a cover band that plays a famous album note for note, with one improvised solo per concert. Most of what you hear is the original. An R-squared of 0.97 says 97% of the fund's ups and downs are the index's ups and downs; only 3% of its variance is the manager doing something different. The fee, though, is charged on 100% of the money. That mismatch is what the interviewer wants you to find.
The step people miss is that R-squared is a share of variance, and variance is volatility squared. So you cannot take 3% of 16%. Work in squares, then come back with a square root. The fund's own volatility, measured against the index, is its tracking errorThe standard deviation of the gap between a fund return and its benchmark return; the size of the fund active bets, measured as a volatility., and when beta is 1.0 it equals this residual risk.
Of the fund's variance, 97% moves with the index and only 3% is its own. Converted back to volatility, the fund's total risk is about 16.2% a year and its own active risk about 2.8%, so a 1.8% fee needs an information ratio of about 0.64 just to break even.
The relationship
$\sigma_{\varepsilon} = \sqrt{\frac{\beta^2 \sigma_m^2}{R^2}\,(1 - R^2)} = \sqrt{\frac{0.0256}{0.97} \times 0.03} \approx 2.8\%$
\sigma_m the index volatility, 16%
\beta the fund's beta, 1.0
R^2 the share of the fund's variance explained by the index, 0.97
\sigma_{\varepsilon} the fund's own volatility, its active risk
What it says in wordsRebuild the fund's total variance from the index part, take the unexplained share, and square-root it back to a volatility.
Why does a 2.8% active risk make a 1.8% fee hard to earn back?
Think of the active risk as the size of the bets. A manager who deviates by about 2.8% a year and wants to beat the index after costs needs gross excess returns above 1.8% a year. That is an information ratio, excess return over active risk, of about 0.64, and managers who hold even 0.5 for a decade are rare. Put simply, the fund charges an active fee for a portfolio that is mostly the index, and the small active part has to work very hard to pay for all of it. Some analysts call the pattern closet indexing.
State the limits. R-squared and beta are estimated from past returns against one chosen index, and a different index can give a different R-squared. A high R-squared is also not proof of low skill; it only says the room for skill to show is small. The fair conclusion is a question for the fund, not a verdict on it.
Where candidates lose it
The common slip is taking 3% of the 16% index volatility and saying the fund's own risk is about 0.5%. R-squared divides variance, not volatility, and the square root turns a 3% share of variance into a much larger share of volatility, about 17% of the fund's total volatility.
The second loss is stopping at the number. The interviewer asked what it suggests about the fee: say that 2.8% of room against a 1.8% fee implies an information ratio of about 0.64 just to break even.
What the interviewer asks next
What R-squared would give the fund 6% of active risk with the same beta and index?
If the beta were 1.2 with the same R-squared, how would the active risk and its meaning change?
Why can a fund with a high R-squared still have a low correlation of its excess returns with the index?
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
Try it first
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.
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
$\bar{x} = \frac{1}{40}\sum_{i=1}^{40} x_i = \frac{560}{40} = 14\%, \qquad \text{median} = \frac{x_{(20)} + x_{(21)}}{2} = 11\%$
x_i fund 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?
053You backtest 20 fund-selection rules, none of which has any real skill. Each rule's measured alpha is pure noise with a standard deviation of 2% a year. What alpha does the best rule show in the backtest, and what should you expect from it out of sample?Statistics, correlation and diversificationHardFund research and ratingsGlobal asset managers
Try it first
Roughly what alpha does the best of the 20 rules show in the backtest?
Show the worked solution
The best rule shows about 3.7% of alpha in the backtest, and you should expect about zero from it afterwards. The largest of 20 normal draws sits about 1.87 standard deviations above the mean, and 2% noise times 1.87 is 3.7%. Picking the winner selects the luckiest draw, and luck does not repeat, so its expected alpha out of sample is the true alpha of every rule: zero.
Why does the best rule look good when none of them is?
Ask twenty people to toss a coin ten times and one of them will probably get eight heads. Nobody concludes that person has a talent for heads. When you choose the best of many attempts, you are choosing the luckiest draw, and the more attempts you make, the luckier the winner looks. One rule on its own shows more than 2% alpha only about one time in six. Among 20 rules, the best one shows more than 3% about 75% of the time.
Twenty rules with no skill scatter around zero with a 2% spread, the best of them shows about 3.7% in the backtest, and the same rule's expected alpha in the years after selection is zero.
The relationship
$E\left[\max_{i \le 20} \alpha_i\right] \approx 1.87 \times 2\% \approx 3.7\% \qquad E[\alpha_{\text{next}}] = 0$
alpha_i the backtest alpha of rule i, pure noise
1.87 how many standard deviations the largest of 20 normal draws sits above the mean, on average
2% the spread of the noise in each rule's alpha
What it says in wordsThe winner's backtest alpha measures how many rules you tried, not how good the winner is.
What should you expect out of sample, and how would you check a real rule?
Out of sample the noise is drawn again, fresh, and the rule has no skill, so its expected alpha is zero. The gap between 3.7% and zero is the price of searching, sometimes called selection biasThe distortion that comes from reporting the best of many tries as if it were the only try. The winner looks better than its true quality.. The honest checks follow from that. Hold back data the rules never saw and test only the winner on it. Raise the bar with the number of rules tried: a one-rule test might accept 2 standard deviations, but after 20 tries the best result is expected to reach 1.87 on luck alone. Ask whether the rule has an economic reason to work.
This is also why fund ranking tables are a weak guide on their own. A category with 20 funds and no skill still produces a fund that beat its peers by about 3.7% a year over the backtest window, and the marketing for that fund writes itself. The limit of the arithmetic: real rules are correlated, which shrinks the effective number of tries and the size of the winner's luck.
Where candidates lose it
The trap is answering zero to the first half. Every rule averages zero, but the question asks about the best one, and the maximum of 20 draws is far from the average draw. Candidates who say zero have not noticed the selection step.
The mirror trap is answering 3.7% to the second half and believing the winner will keep it. The interviewer wants both halves: a large number in the backtest and zero afterwards, with the reason.
What the interviewer asks next
How would the best backtest alpha change if you tested 200 rules instead of 20?
Your rules are strongly correlated with each other. Does the winner look more or less lucky?
How many years of out-of-sample data would you need to tell a true 2% alpha from zero at this noise level?
079Ten years ago a fund category had 100 schemes. Since then 30 were merged or closed after averaging 6% a year, and the 70 survivors averaged 12% a year. What was the true category average, and what does a database that shows only the survivors overstate?Statistics, correlation and diversificationCoreFund research and ratingsGlobal asset managers
Try it first
What was the average return across all 100 schemes?
Show the worked solution
The true average was 10.2% a year, and a survivor-only database overstates it by 1.8 points. Seventy funds at 12% and thirty at 6% average to 0.7 x 12% plus 0.3 x 6%, which is 10.2%. A database that drops merged and closed funds shows 12%, because the funds that disappeared were mostly the weak ones, and the gap compounds every year.
Why do dead funds disappear from the numbers?
Think of a coaching centre that advertises the average marks of students who stayed to the final exam. The ones who struggled and left are not in the average, so the centre looks better than its teaching. A fund database that lists only live schemes does the same: the funds that did badly were merged or shut, their records left the table, and the category average rose without anyone earning it. Fund houses merge weak schemes into stronger ones as a matter of routine, so the losers vanish quietly rather than with a headline. The name for this is survivorship bias.
Seventy surviving funds averaged 12% and thirty vanished funds averaged 6%, so the whole category earned 10.2%; a database that drops the dead funds reports the survivors' 12% and overstates the category by 1.8 points a year.
The relationship
$\bar r = \frac{70 \times 12\% + 30 \times 6\%}{100} = 10.2\%$
70, 30 the number of surviving and vanished funds
12%, 6% each group's average annual return
\bar r the true average across every fund that existed ten years ago
What it says in wordsWeight each group's return by how many funds were in it, including the ones no longer listed.
How much does the gap matter over ten years?
Compound it. Rs 1 lakh at 12% for ten years becomes about Rs 3.11 lakh; at 10.2% it becomes about Rs 2.64 lakh. A chart built on survivors shows roughly Rs 46,000 more per lakh than the category delivered to the average investor who chose a fund ten years ago. Compounding a category average is itself approximate, because each fund compounds on its own path, but the direction and the size of the gap are right.
The bias leaks into rankings too. A fund that looks top quartile among survivors may be only average once the vanished funds are put back, because the bottom of the table has been cut off. Before comparing a fund with its category, ask whether the category figure includes funds that no longer exist. Good research databases keep dead funds in; if yours does not, treat its averages as a ceiling rather than a middle.
Where candidates lose it
The first loss is quoting 12% because that is what the screen shows. The interviewer built the question to see whether you ask what is missing from the data, which is most of the skill in fund research.
The second is averaging 6% and 12% to get 9%. The groups are different sizes, so weight by count: seventy funds pull the average much closer to 12% than thirty pull it towards 6%, landing at 10.2%.
What the interviewer asks next
If the closed funds had been larger than the survivors, would you weight by count or by assets, and what changes?
How would survivorship bias affect a backtest of a rule that buys last year's top funds?
Where else in finance does the data quietly leave out the failures?
090Fund A's returns have a correlation of 0.9 with the index. Fund A's volatility is 24% a year and the index's is 16%. When the index moves 1%, how much does fund A typically move, and why is the correlation not the answer?Statistics, correlation and diversificationCoreFund research and ratingsGlobal asset managers
Try it first
When the index moves 1%, fund A typically moves about:
Show the worked solution
About 1.35%: beta is the correlation times the ratio of the two volatilities. Correlation of 0.9 says the fund's moves line up closely with the index; the ratio 24 over 16 says the fund's moves are 1.5 times as large. Together, 0.9 x 1.5 is 1.35. Correlation measures how tightly the points hug the line, beta measures how steep the line is, and a fund can have a high correlation with a beta below one.
What is the difference between how tight and how steep?
Two friends walking a dog on a lead: the lead's length sets how far the dog can stray, and the dog's pace sets how far it travels for each step you take. A short lead says nothing about whether the dog walks faster or slower than you. Correlation is the lead, how closely the fund's moves track the index's; beta is the pace, how far the fund moves for each 1% the index moves. They are related, but they answer different questions, and confusing them is the commonest slip in fund statistics.
The relationship
$\beta = \rho \times \frac{\sigma_{fund}}{\sigma_{index}} = 0.9 \times \frac{24\%}{16\%} = 1.35$
\beta the slope: the fund's typical move for a 1% index move
\rho correlation, how tightly fund and index move together, 0.9
\sigma annual volatility of the fund, 24%, and of the index, 16%
What it says in wordsBeta is correlation stretched by how much more the fund swings than the index.
Fund A, correlation 0.9, and fund B, correlation 0.6, share the same slope of 1.35 against the index, but fund A's points hug the line while fund B's scatter widely, which is the difference between beta and correlation.
Can two funds have the same beta and very different correlations?
Yes, and the figure shows it. Fund B has a correlation of only 0.6 but a volatility of 36%, so its beta is 0.6 x 36 / 16, also 1.35. Both funds move 1.35% for a 1% index move on average, but fund B's actual months scatter far from that line, because most of its swings come from things the index does not explain. The square of correlation measures the share explained: 81% of fund A's variance comes from the index against 36% of fund B's. For a fund researcher that changes the reading of beta itself: fund A's 1.35 is a reliable guide to a typical month, fund B's is an average around which a single month can land almost anywhere.
One more asymmetry worth having ready. Correlation is the same whichever way round you put the two series, but the slope is not: regressing the index on the fund gives 0.9 x 16 / 24, or 0.6. Beta always needs a direction, and for a fund it is the fund's return explained by the index. Both figures are estimates from past returns and shift with the window chosen, so quote them with the period they came from.
Where candidates lose it
The fastest wrong answer is 0.9%, read straight off the correlation. It sounds right because both numbers describe how fund and index relate, but correlation is capped at one and carries no information about size of moves, so it cannot be a slope.
The next slip is 1.5%, the volatility ratio, which forgets that 10% of the fund's movement has nothing to do with the index. Give the formula, then say in a sentence that correlation is tightness and beta is steepness.
What the interviewer asks next
What would fund A's beta be if its correlation with the index fell to 0.5?
Can a fund have a negative correlation and still lose money when the index falls?
Why might a fund's measured beta change when you switch from daily to monthly returns?

w_A, w_B	the weights, 0.6 and 0.4
\sigma_A, \sigma_B	the funds' volatilities, 20% and 12%
\rho	the correlation between them, 0.3

\sigma_m	the index volatility, 16%
\beta	the fund's beta, 1.0
R^2	the share of the fund's variance explained by the index, 0.97
\sigma_{\varepsilon}	the fund's own volatility, its active risk

x_i	fund i's five-year annualised return
x_(20), x_(21)	the 20th and 21st returns after sorting, the middle pair of 40

alpha_i	the backtest alpha of rule i, pure noise
1.87	how many standard deviations the largest of 20 normal draws sits above the mean, on average
2%	the spread of the noise in each rule's alpha

Try it first

Why is the mix not just the average of the two volatilities?

How do you work it quickly on paper?

Where candidates lose it

What the interviewer asks next

Try it first

What does an R-squared of 0.97 actually say?

Why does a 2.8% active risk make a 1.8% fee hard to earn back?

Where candidates lose it

What the interviewer asks next

Try it first

Why can the average sit above most of the funds?

How do you check that the stars explain the gap?

Which figure should a client hear, and what else?

Where candidates lose it

What the interviewer asks next

Try it first

Why does the best rule look good when none of them is?

What should you expect out of sample, and how would you check a real rule?

Where candidates lose it

What the interviewer asks next

Try it first

Why do dead funds disappear from the numbers?

How much does the gap matter over ten years?

Where candidates lose it

What the interviewer asks next

Try it first

What is the difference between how tight and how steep?

Can two funds have the same beta and very different correlations?

Where candidates lose it

What the interviewer asks next