Hedge Funds puzzles, solved step by step

Puzzles: 100
Traced to a firm: 38
Topics: 14
Hard: 30

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

010Every stock in a universe has 30% volatility and every pair has a correlation of 0.3. What is the volatility of an equal-weighted portfolio of 10 stocks, of 100 stocks, and of infinitely many?Portfolio and risk mathsCoreMulti-manager platformsQuant and systematic funds
Try it first
Where does the volatility end up with infinitely many stocks?
Show the worked solution
About 18.2% for 10 stocks, 16.6% for 100, and a floor of 16.4% for infinitely many. Portfolio variance is 30% squared times (0.3 + 0.7/n): the 0.7/n part is stock-specific noise that averages away, and the 0.3 part is shared movement that never does. The floor is 30% times root 0.3. Ten stocks capture most of the benefit; the next ninety add little.
Why does diversification stop working?
A choir of a hundred singers each slightly off key sounds more in tune than one singer, because the individual errors cancel. But if the whole choir takes its note from one badly tuned piano, no number of singers fixes it. Stock-specific risk is the individual error and averages away; the shared correlation is the piano, and it stays however many names you add. With every pair at 0.3, the shared part is 30% of each stock's variance.
Equal-weighted portfolio volatility falls from 30% for one stock to 18.2% for ten and 16.6% for a hundred, flattening onto a floor of 16.4% set by the 0.3 correlation that no amount of diversification removes.
The relationship
$\sigma_p^2 = \sigma^2\left(\rho + \frac{1-\rho}{n}\right) \qquad \sigma_\infty = \sigma\sqrt{\rho} = 30\% \times \sqrt{0.3} \approx 16.4\%$
\sigma each stock's volatility, 30%
\rho the correlation between every pair, 0.3
n the number of stocks, equally weighted
What it says in wordsPortfolio variance is a shared part that stays plus a specific part that shrinks with every name added.
How do the three numbers come out?
Plug in. Ten stocks: 0.09 x (0.3 + 0.07) = 0.0333, a volatility of 18.2%. One hundred: 0.09 x 0.307 = 0.0276, 16.6%. Infinitely many: 0.09 x 0.3 = 0.027, 16.4%. Going from one stock to ten cuts risk from 30% to 18.2%; going from ten to a hundred cuts only another 1.6 points. That is why a long book of 30 names, at 17.1%, is not as undiversified as it sounds, and why names added past a point buy almost nothing.
What does this mean for a hedge fund book?
The only way under the floor is to remove the shared factor itself, which is what a short leg or an index hedge does. If the correlation comes from the market, shorting the market against the long book strips out the shared piece and leaves stock-specific risk, which does diversify. The limitation is that correlations are not fixed. In a sell-off they rise, and the floor rises with them: at a correlation of 0.6 it is 23.2%, so a book that looked diversified at 0.3 starts behaving like a concentrated one.
Where candidates lose it
The common miss is saying volatility goes to zero with enough stocks. That holds only if the stocks are uncorrelated; any shared correlation leaves a floor, and the interviewer is testing whether you know it is there.
The second is computing the floor as 30% x 0.3 = 9%, which applies the correlation to volatility instead of variance. Variance floors at 0.3 times 0.09; take the square root at the end, not the start.
What the interviewer asks next
How many stocks do you need to get within one point of the floor?
If correlation rises to 0.6 in a crisis, where is the new floor?
How does a long-short book change this calculation?
035Three assets all have the same pairwise correlation, rho. What is the lowest value rho can take? What is the answer for n assets?Portfolio and risk mathsCoreMulti-manager platformsQuant and systematic funds
Try it first
Lowest possible common correlation for three assets:
Show the worked solution
Minus one half for three assets, and minus 1/(n minus 1) for n. Give each asset unit variance and add them up. The variance of the sum is 3 plus 6 rho, because there are three variances and six pairwise covariance terms. A variance cannot be negative, so rho is at least -1/2. With n assets the sum's variance is n plus n(n minus 1) rho, which gives rho at least -1/(n minus 1): -0.33 for four, -0.11 for ten.
Why can three assets not all be perfectly opposed?
Three friends cannot all disagree with each other on a yes-or-no question: if Ravi says yes and Meena says no, Arjun agrees with one of them. Perfect opposition is a relationship between two things; with three, two of them must lean the same way. The same limit holds for correlations. A set of numbers in a correlation matrix has to be internally consistent, and equal pairwise correlations become impossible well before -1 once there are three or more assets.
How do you find the exact floor?
Use the one fact that can never fail: a variance is zero or more. Add the three assets with unit variance; the sum has variance 3 from the three diagonal terms plus 6 rho from the six covariance terms, and 3 + 6 rho must be at least zero. That gives rho of at least -1/2. The picture is three arrows 120 degrees apart: each pair has a cosine of -1/2, and the three add to exactly zero, which is the boundary case. For n assets, n + n(n minus 1) rho at least zero gives -1/(n minus 1).
Three unit arrows 120 degrees apart add to zero, the arrangement where every pair has correlation minus one half, and the variance of the sum, 3 + 6 rho, turns negative below that value, which is impossible, so rho cannot fall below minus one half.
The relationship
$\operatorname{Var}\Big(\sum_{i=1}^{n} X_i\Big) = n + n(n-1)\rho \ge 0 \;\Rightarrow\; \rho \ge -\frac{1}{n-1}$
n the number of assets, each with variance 1
rho the common pairwise correlation
n(n-1) the number of ordered pairs, each contributing one covariance of rho
What it says in wordsThe variance of the equal-weighted basket must be non-negative, which caps how negative a shared correlation can be.
Say why a risk desk cares. A correlation matrix that breaks this rule is not a valid risk model: it implies some portfolio has negative variance, and an optimiser will pile into it. This happens in practice when correlations are estimated pair by pair from different data windows or overridden by hand in a stress test. The check is that the matrix is positive semi-definiteA matrix for which every weighted combination of the assets has a variance of zero or more., and the equal-correlation case is the cleanest example of the rule.
Where candidates lose it
The usual loss is answering -1, because that is the floor for any single pair. The interviewer is testing whether you see that the pairs constrain each other.
The second loss is getting -1/2 by intuition but having no proof. Say the variance of the sum in one line. It takes ten seconds, it generalises to n immediately, and it is the answer the follow-ups build on.
What the interviewer asks next
Two assets have correlation 0.9 with a third. What is the lowest possible correlation between the first two?
A stress test sets every pairwise correlation in a 20-asset book to -0.1. Is that a valid matrix?
How would you repair a correlation matrix that is not positive semi-definite?
046A fund has annual volatility of 12% and its benchmark index has 15%. The correlation between them is 0.9. What is the fund's tracking error?Portfolio and risk mathsCoreMulti-manager platformsQuant and systematic funds
Try it first
Your estimate of the tracking error:
Show the worked solution
About 6.7%. Tracking error is the volatility of the fund's return minus the benchmark's. The variance of a difference is the two variances minus twice the covariance: 0.12 squared plus 0.15 squared minus 2 x 0.9 x 0.12 x 0.15, which is 0.0144 + 0.0225 - 0.0324 = 0.0045. The square root is 6.71%. A correlation of 0.9 sounds tight but still leaves a sizeable gap.
What exactly is tracking error measuring?
Two friends walking to the same station take slightly different routes; tracking error is how far apart they typically are, not how fast either walks. Tracking errorThe standard deviation of the difference between a fund return and its benchmark return, usually quoted per year. is the volatility of the return difference, fund minus benchmark, so it depends on both volatilities and on how closely the two move together. The variance of a difference is Var(F) + Var(B) - 2 Cov(F, B), and the covariance is the correlation times the two volatilities.
Drawn as arrows of length 12 and 15 set 25.8 degrees apart, the angle whose cosine is 0.9, the fund and benchmark tips sit 6.7 apart, which is the tracking error; it would be 3.0% only at a correlation of 1 and rises to 13.7% at 0.5.
Why is the answer so much bigger than 3%?
Because the correlation is below 1, the gap is not just the difference in size. The variance of the difference is 144 + 225 - 324 = 45 in squared percent, and the square root of 45 is 6.7%: the 10% of correlation that is missing contributes more than the 3 point difference in volatility. The arrow picture shows it: two arrows 12 and 15 long, set about 26 degrees apart, have tips further apart than 3. That is just the law of cosines.
The relationship
$TE = \sqrt{\sigma_F^2 + \sigma_B^2 - 2\rho\,\sigma_F\sigma_B} = \sqrt{0.0144 + 0.0225 - 0.0324} \approx 6.7\%$
sigma_F, sigma_B the fund and benchmark volatilities, 12% and 15%
rho the correlation between their returns, 0.9
What it says in wordsTracking error is the volatility of the gap between fund and benchmark returns.
Say what it means for an allocator. A fund with 0.9 correlation to its index can still trail or beat it by 6 to 7 points in an ordinary year, so a single year of underperformance tells you very little. The limitation: the calculation assumes the correlation and volatilities are stable, and they tend to shift in stressed markets, which is when tracking error matters most.
Where candidates lose it
The fast wrong answer is 3%, the difference in volatilities, which is only true at a correlation of exactly one. The interviewer chose 0.9 precisely because it sounds close to one.
The second loss is forgetting the factor of 2 on the covariance term, which gives root(207), over 14%. Write the variance of a difference in full before plugging in numbers.
What the interviewer asks next
What correlation would give a tracking error of 5%?
If the fund had a beta of 0.72 to the index, what is its information ratio if it beats the index by 2% a year?
Why might a fund's realised tracking error jump in a sell-off?
070A portfolio is split equally between two assets, each with 15% annual volatility. Their correlation is 0.2 in normal markets and 0.8 in a crisis. What is the portfolio's volatility in each regime?Portfolio and risk mathsCoreMulti-manager platformsQuant and systematic funds
Try it first
What happens to the portfolio's volatility when correlation rises from 0.2 to 0.8?
Show the worked solution
About 11.6% in normal markets and 14.2% in a crisis. With equal weights and equal volatilities, portfolio variance is 15% squared times (1 + rho)/2. At a correlation of 0.2 that is 225 x 0.6 = 135, a volatility of 11.6%; at 0.8 it is 225 x 0.9 = 202.5, a volatility of 14.2%. About 77% of the diversification benefit disappears just when it is needed.
Where does correlation enter the arithmetic?
Picture an umbrella shop and an ice-cream stall owned by one family. On ordinary days one does well when the other is quiet, and the family income is smoother than either shop's. A city-wide power cut shuts both at once. Portfolio variance is each asset's own variance, weighted, plus a co-movement term that carries the correlation, so when correlation jumps the co-movement term grows and the smoothing shrinks. Here the own terms are 56.25 each and the co-movement term is 2 x 0.5 x 0.5 x rho x 225.
The relationship
$\sigma_p^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\rho\,\sigma_1\sigma_2 = 225 \times \frac{1+\rho}{2}$
w1, w2 the weights, 0.5 each
sigma1, sigma2 the volatilities, 15% each
rho the correlation between the two assets
What it says in wordsPortfolio variance is the weighted own variances plus a cross term that grows with correlation.
At a correlation of 0.2 the co-movement term adds 22.5 to a variance of 112.5, giving 11.6% volatility; at 0.8 it adds 90.0, giving 14.2%, so the benefit of holding both assets falls from 3.4 points to 0.8.
What are the two numbers?
Normal markets: 112.5 plus 2 x 0.25 x 0.2 x 225 = 22.5 gives 135, and the square root is 11.6%. Crisis: the co-movement term rises to 90, variance to 202.5, and volatility to 14.2%, a 22.5% jump in risk with no change in either asset's own volatility. The benefit of holding two assets instead of one falls from 3.4 points to 0.8. In a real crisis each asset's volatility usually rises too, so this is the milder case.
What would a risk manager do with this?
Stop sizing positions on correlations measured in calm markets. A risk limit set on a normal-market correlation understates crisis risk by about 22% here, before any rise in the assets' own volatility. Platforms that run many books run stressed-correlation scenarios for this reason, and ask each manager what the book looks like if everything moves together. The honest limitation: nobody knows the crisis correlation in advance; 0.8 is an assumption, and the answer should say so.
Where candidates lose it
The common loss is treating correlation as if it scaled volatility directly, or quoting 15% in both regimes because each asset's volatility has not changed. Correlation enters only through the cross term in the variance.
The second is computing both numbers and missing the point of the question: the benefit you were counting on shrinks exactly in the regime where you need it. Say that sentence.
What the interviewer asks next
At what correlation does the portfolio's volatility reach 13%?
With three equally weighted assets, what is the volatility at a correlation of 0.8?
Why might correlations between assets rise in a sell-off?

\sigma	each stock's volatility, 30%
\rho	the correlation between every pair, 0.3
n	the number of stocks, equally weighted

n	the number of assets, each with variance 1
rho	the common pairwise correlation
n(n-1)	the number of ordered pairs, each contributing one covariance of rho

sigma_F, sigma_B	the fund and benchmark volatilities, 12% and 15%
rho	the correlation between their returns, 0.9

w1, w2	the weights, 0.5 each
sigma1, sigma2	the volatilities, 15% each
rho	the correlation between the two assets

Try it first

Why does diversification stop working?

How do the three numbers come out?

What does this mean for a hedge fund book?

Where candidates lose it

What the interviewer asks next

Try it first

Why can three assets not all be perfectly opposed?

How do you find the exact floor?

Where candidates lose it

What the interviewer asks next

Try it first

What exactly is tracking error measuring?

Why is the answer so much bigger than 3%?

Where candidates lose it

What the interviewer asks next

Try it first

Where does correlation enter the arithmetic?

What are the two numbers?

What would a risk manager do with this?

Where candidates lose it

What the interviewer asks next