Portfolio Management puzzles, solved step by step

Puzzles: 100
Traced to a firm: 31
Topics: 13
Hard: 30

Showing 1–4 of 4 · filtered from 100Clear filters

001A fund's returns have an R squared of 0.81 against its benchmark index. The fund's volatility is 20% a year and the index's is 18%. What are the correlation, the beta and the fund's residual volatility?Statistics and forecastingWarm upPerformance analysisAsset management

Try it first

Before you work it: how much of the fund's 20% volatility does the index fail to explain?

Show the worked solution

Correlation 0.9, beta 1.0 and residual volatility of about 8.7%. Correlation is the square root of R squared, so 0.9. Beta is correlation times the ratio of volatilities, 0.9 x 20 / 18, which is exactly 1.0. The unexplained 19% of the fund's variance of 400 is 76, and the square root of 76 is 8.7%: the fund's own risk, on top of what the index explains.

Why does 81% explained still leave so much unexplained?

Think of a household's monthly spending. If rent explains most of how the bill moves, the groceries, travel and surprises that make up the rest can still swing it by a lot. The fund is the same. R squared splits variance, and variance is volatility squared, so a small share of variance becomes a much larger share once you take the square root back. The fund's variance is 20 squared, 400. The index explains 81% of it, which is 324. The other 76 belongs to the fund alone, and the square root of 76 is 8.72%.

The index explains 324 of the fund's variance of 400 and leaves 76 unexplained; in volatility terms that is 18 points from the index and 8.7 points of the fund's own, which combine to 20 only because volatilities add in squares.

How do you get the correlation and the beta from R squared?

In a regression on a single index, R squared is simply the correlation squared, so the correlation is the square root of 0.81, which is 0.9. Beta is the correlation scaled by how volatile the fund is relative to the index: 0.9 times 20 over 18 is exactly 1.0. So the fund moves one for one with the index on average, and carries about 8.7 points of volatility the index does not explain. Mention the sign: the root could be minus 0.9, but a long-only equity fund with a positive slope takes the positive root.

The relationship

\rho=\sqrt{R^2}=0.9 \qquad \beta=\rho\,\frac{\sigma_f}{\sigma_i}=0.9\times\frac{20}{18}=1.0 \qquad \sigma_\varepsilon=\sigma_f\sqrt{1-R^2}=20\sqrt{0.19}\approx 8.7\%

R^2	the share of the fund's variance the index explains, 0.81
\rho	the correlation between fund and index returns
\sigma_f, \sigma_i	the volatilities of the fund, 20%, and the index, 18%
\sigma_\varepsilon	the residual volatility, the part of the fund's risk the index does not explain

What it says in wordsCorrelation is the root of R squared, beta rescales it by the volatility ratio, and the residual volatility is the fund's volatility times the root of the unexplained share.

What does the residual number tell a portfolio manager?

With a beta of 1.0, the residual volatility is the fund's tracking errorThe volatility of the difference between a fund's return and its benchmark's return. against the index. An R squared of 0.81 sounds index-like, but 8.7 points of tracking error is a genuinely active book: almost half as volatile as the market itself. Say the limitation too. The split assumes the relationship is linear and stable over the sample; a fund whose beta drifted during the period shows a lower R squared for reasons that have nothing to do with stock picking.

Where candidates lose it

The common slip is treating R squared as a share of volatility and answering 19% of 20%, which is 3.8%. The interviewer is checking whether you know that variances add and volatilities do not, which is the same fact that sits under every portfolio risk calculation.

The second slip is computing beta as 0.9 and stopping, forgetting that beta needs the volatility ratio. Say the three formulas in order and the numbers follow.

What the interviewer asks next

If the fund's beta were 1.2 with the same volatilities, what R squared would that imply?
How would you tell whether the 8.7 points are skill or just unintended sector bets?
Why might R squared against a style index be much higher than against the broad market?

014You backtest 20 independent trading strategies, none of which has any real edge, and test each one at the 5% significance level. What is the chance at least one looks significant, and what per-test threshold would hold that overall false alarm rate at 5%?Statistics and forecastingCoreQuantitative researchSystematic investing
Try it first
Chance that at least one of the 20 worthless strategies passes?
Show the worked solution
About 64%, and a per-test threshold of about 0.25%. Each worthless strategy passes by luck 5% of the time, so all 20 fail with chance 0.95 to the 20th, 35.8%, and at least one passes 64.2% of the time. To hold the overall rate at 5%, test each at 5% divided by 20, which is 0.25%; the exact version, 1 minus 0.95 to the power of one twentieth, is 0.256%.
Why does testing more ideas throw up a false winner?
Ask a room of 20 people to each flip a coin five times, and there is a fair chance someone gets five heads. Nobody in the room has a lucky hand; there were simply enough tries. A 5% test lets one worthless idea in twenty through by chance, so a researcher who tests twenty ideas should expect about one false winner, not be impressed by it. The expected number of false positives here is 20 x 0.05, exactly 1, and the chance of at least one is 64.2%.
Testing each worthless strategy at 5%, the chance that at least one looks significant reaches 64.2% at 20 strategies, while testing each at 0.25% holds it near 4.9%.
The relationship
$P(\ge 1 \text{ false}) = 1-(0.95)^{20}\approx 0.642 \qquad \alpha_{each}=\frac{0.05}{20}=0.25\%$
0.95 the chance a worthless strategy fails a 5% test
20 the number of independent strategies tested
\alpha_{each} the per-test threshold that caps the overall false alarm rate near 5%
What it says in wordsThe chance of at least one false winner is one minus the chance that every test correctly fails; dividing the level by the number of tests caps it.
What does a quant desk actually do about it?
Dividing the threshold by the number of tests is called the Bonferroni correctionA rule that divides the significance level by the number of tests run, so the chance of any false positive across all of them stays near the original level.. The real discipline is counting every test you ran, including the ones you dropped quietly, because the correction is only as honest as that count. A researcher who tried 200 variants and reports the best 20 has a far bigger multiple testing problem than the 20 suggest. Desks also hold out data the research never touched and demand a reason for the edge before the backtest. Say the limitation: the correction assumes independent tests, and for correlated strategies it is too strict, which costs real ideas.
Where candidates lose it
The fast wrong answers are 5%, which ignores that there are 20 tests, and 100%, which adds the chances. Say the complement and the answer arrives in one line.
The second trap is naming the fix without the cost. A tighter threshold throws away some genuine strategies too, and an interviewer on a systematic desk expects you to say that trade-off out loud.
What the interviewer asks next
If the 20 strategies are highly correlated, is the true chance of a false winner higher or lower than 64%?
Of 1,000 strategies, how many worthless ones pass at 5%?
Why is out-of-sample testing a better defence than a stricter threshold?
024A fund's measured alpha is 4% a year, with a standard error of 3%. Across all funds, true alphas average zero with a spread (standard deviation) of 1.5%. What is your best estimate of this fund's true alpha?Statistics and forecastingHardFund selectionQuantitative asset management
Try it first
Best estimate of the true alpha?
Show the worked solution
About 0.8% a year. Combine the fund's noisy measurement with what you know about funds in general, weighting each by its precision. The measurement's variance is 3 squared, 9; the spread of true alphas has variance 1.5 squared, 2.25. The measurement gets 2.25 / 11.25, a weight of 0.2, so the estimate is 0.2 x 4% + 0.8 x 0%, which is 0.8%. A noisy 4% deserves heavy shrinkage toward zero.
Why not take the 4% at face value?
A new restaurant with two five-star reviews is probably good, but you would not bet it beats every restaurant in town with thousands of reviews. When a measurement is noisy compared with how much things really differ, most of an extreme reading is luck, so the best estimate sits much closer to the average. Here true alphas across funds rarely stray far from zero, a spread of 1.5, while one fund's measured alpha can miss the truth by 3. A reading of 4% is far more likely to be a modest fund that got lucky than a fund with a true 4%.
The prior for all funds is centred on zero with a spread of 1.5 and the fund's measurement on 4% with a spread of 3, so the combined estimate lands at 0.8%, with a spread of 1.34, pulled 80% of the way back to zero.
The relationship
$\hat\alpha=\frac{\tau^2}{\tau^2+s^2}\,\alpha_{obs}=\frac{2.25}{2.25+9}\times 4\%=0.8\%$
\tau the spread of true alphas across funds, 1.5%
s the standard error of this fund's measured alpha, 3%
\alpha_{obs} the measured alpha, 4%
What it says in wordsThe estimate is the measured alpha scaled by how much of the total variance comes from real differences between funds.
How does a fund selector use this?
By refusing to rank funds on raw past alpha. Shrinkage keeps the ranking but compresses it, so a fund with a long, steady record keeps more of its measured alpha than one with a short, volatile record showing the same number. A fund whose standard error was 1% instead of 3% would keep 2.25 / 3.25, about 69% of its 4%. Say the limitations: the result depends on the assumed spread of true alphas, which itself is estimated, and on the average being zero; if the fund belongs to a peer group with a known positive or negative average, shrink toward that instead.
Where candidates lose it
Most candidates either take the 4% as measured, ignoring the noise, or answer zero, ignoring the evidence. The interviewer wants the weighted middle and the reason for the weights.
The arithmetic slip is weighting by standard deviations, 1.5 and 3, instead of variances, which gives one third and 1.3%. Precision is one over the variance, so square before you weight.
What the interviewer asks next
What if the fund had 20 years of data and a standard error of 1%?
How would you estimate the spread of true alphas across funds?
Why does this argument make top-quartile rankings unstable from one year to the next?
082Let X be the market's daily return, symmetric around zero, and let Y equal X squared, the shape of a long straddle's profit. What is the correlation between X and Y? Are they independent?Statistics and forecastingCoreQuantitative asset managementHedge funds
Try it first
What is the correlation?
Show the worked solution
The correlation is zero, but X and Y are completely dependent. Because X is symmetric, a 2% fall and a 2% rise both give Y of 4, so the up-slope and the down-slope cancel and the covariance is zero. Yet Y is fixed exactly by X. Zero correlation rules out a straight-line link only, not a link.
How can a perfect relationship show zero correlation?
An umbrella seller does well when it pours and an ice cream seller when it is scorching; a stall selling both does well on any extreme day and badly on mild ones. Its takings depend completely on the weather, but not in a straight line. Correlation asks only whether a straight line fits, so a U-shaped link, rising on both sides, averages out to zero.
Plotting Y equal to X squared for returns of minus 2% to plus 2% gives a U of 4, 1, 0, 1, 4, and the best-fit straight line through those points is flat at 2, so the correlation is zero although Y is fixed exactly by X.
What does the algebra say?
Covariance is the average of X times Y, less the product of the averages. With X symmetric around zero, the average of X is zero and the average of X cubed is zero, so the covariance, and with it the correlation, is exactly zero. Using five equally likely days of minus 2, minus 1, 0, 1 and 2 per cent gives the same answer by hand: the covariance works out to 0.
The relationship
$\text{Cov}(X, X^2) = E[X^3] - E[X]\,E[X^2] = 0 - 0 = 0$
E[X] the average return, zero by symmetry
E[X^3] the average cubed return, also zero by symmetry
What it says in wordsFor a symmetric variable, the covariance with its own square is zero.
Now the portfolio point. A long straddleBuying a call and a put at the same strike, so the position profits from a large move in either direction. profits roughly with the square of the move, so its returns can show near-zero correlation with the market while depending heavily on it. A risk model that reads zero correlation as no exposure will treat the straddle as a diversifier and miss that it is a pure bet on the size of market moves.
Where candidates lose it
The trap is equating zero correlation with independence. Independence implies zero correlation; the reverse fails, and this question is the standard counter-example.
The second trap is saying the correlation is high because Y is a function of X. Correlation measures a straight-line fit, and a symmetric U has no slope.
What the interviewer asks next
What if X is skewed, with bigger falls than rises? Is the correlation still zero?
Name a measure that would detect this dependence.
Why can hedge fund returns look uncorrelated with equities and still lose money in a crash?

0.95	the chance a worthless strategy fails a 5% test
20	the number of independent strategies tested
\alpha_{each}	the per-test threshold that caps the overall false alarm rate near 5%

\tau	the spread of true alphas across funds, 1.5%
s	the standard error of this fund's measured alpha, 3%
\alpha_{obs}	the measured alpha, 4%

E[X]	the average return, zero by symmetry
E[X^3]	the average cubed return, also zero by symmetry

Try it first

Why does 81% explained still leave so much unexplained?

How do you get the correlation and the beta from R squared?

What does the residual number tell a portfolio manager?

Where candidates lose it

What the interviewer asks next

Try it first

Why does testing more ideas throw up a false winner?

What does a quant desk actually do about it?

Where candidates lose it

What the interviewer asks next

Try it first

Why not take the 4% at face value?

How does a fund selector use this?

Where candidates lose it

What the interviewer asks next

Try it first

How can a perfect relationship show zero correlation?

What does the algebra say?

Where candidates lose it

What the interviewer asks next