Mutual Fund Mastery puzzles, solved step by step

Puzzles: 100
Traced to a firm: 29
Topics: 13
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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?
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

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

\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%

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

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