Mutual Fund Mastery puzzles, solved step by step

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

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017Two funds' daily returns are negatively correlated within any given month, yet their yearly returns are positively correlated. How can both be true?Statistics, correlation and diversificationHardSquarepoint CapitalMontreal · 2024
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Which explanation fits?
Show the worked solution
A shared driver that moves slowly lifts or sinks both funds together across years, while short-term noise pushes them in opposite directions day to day. Within a month the slow driver barely changes, so daily correlation reflects only the opposing noise. Across years it dominates. With daily noise of 0.8% at -0.5 correlation and a shared yearly drift of 20% standard deviation, yearly correlation comes out at about +0.57.
What kind of situation produces this?
Think of two ice cream stalls on the same beach. On any given day, a customer who buys from one does not buy from the other, so their daily sales move against each other. Across years, both do well in hot summers and badly in wet ones. Correlation is not one fixed number between two things; it depends on which driver dominates at the horizon you measure, and different drivers dominate at different horizons. For funds, the slow driver might be the economy's earnings cycle, which both portfolios share; the fast one might be money rotating between their two styles day to day.
In a simulated month the two funds move in opposite directions on 16 of 21 days, yet across years the shared drift adds 0.040 of covariance against 0.008 removed by the opposing noise, so yearly returns are positively correlated at about 0.57.
Can you show it with numbers?
Build each fund's yearly return from two parts. A shared drift, the same for both within a year but different from year to year, with a standard deviation of 20%. Plus daily noise of 0.8% a day for each fund, correlated at -0.5 between them, over 250 trading days. Within a month the drift is a constant, so it drops out of any correlation measured around the month's average, and the daily figure is the noise's -0.5. Across years, the drift contributes 0.2 squared, 0.040, to covariance, and the noise contributes -0.5 times 250 times 0.008 squared, -0.008, leaving +0.032.
The relationship
$\rho_{year} = \frac{\sigma_F^2 + \rho_d\, n\, \sigma_d^2}{\sigma_F^2 + n\,\sigma_d^2} = \frac{0.040 - 0.008}{0.040 + 0.016} = \frac{0.032}{0.056} \approx 0.57$
\sigma_F the standard deviation of the shared yearly drift, 20%
\rho_d the correlation of daily noise, -0.5
n trading days in a year, 250
\sigma_d each fund's daily noise, 0.8%
What it says in wordsYearly correlation is the shared drift's variance less the summed opposing noise, divided by each fund's total yearly variance.
Give the condition and a second mechanism. The sign flips only if the shared drift's variance is larger than the summed noise covariance; with a drift of 10% instead of 20%, covariance would be 0.010 minus 0.008 and correlation barely positive. A second route is timing: if one fund's holdings are priced with a lag, its daily moves look unrelated or even opposite to the other's, while over a year the lags wash out. Either way, the lesson for a fund analyst is that a diversification benefit measured on daily data may not exist at the horizon a client actually holds.
Where candidates lose it
The common failure is saying it is impossible, on the belief that correlation is a fixed property of two assets. The interviewer is checking whether you know correlation is horizon dependent and can name what drives each horizon.
The second failure is waving at small samples. Twelve monthly points are noisy, but noise is not an explanation; the strong answer builds a two-component model and states when the sign flips.
What the interviewer asks next
Using the same model, at what size of shared drift would yearly correlation be exactly zero?
Why might two funds that look like good diversifiers on daily data fail to diversify in a bear market?
How would stale prices in one fund distort its measured volatility as well as its correlation?
Asked at Squarepoint Capital, Hedge Fund, Montreal, 2024 (Wall Street Oasis): correlation can be negative intra-month but positive across a year, how?
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?
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?

\sigma_F	the standard deviation of the shared yearly drift, 20%
\rho_d	the correlation of daily noise, -0.5
n	trading days in a year, 250
\sigma_d	each fund's daily noise, 0.8%

\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

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

What kind of situation produces this?

Can you show it with numbers?

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 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