Hedge Funds puzzles, solved step by step

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.

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.

The book's volatility is about Rs 23.1 crore a year, and the Rs 60 crore position carries about 74% of it on 60% of the capital. Stand-alone risks are Rs 18 crore and Rs 8 crore. Book variance is 18 squared plus 8 squared plus 2 x 0.5 x 18 x 8, which is 532, so volatility is Rs 23.07 crore. Each position's contribution is its covariance with the book over the book's volatility: Rs 17.17 crore and Rs 5.90 crore, which add back to the total.

Why is risk not shared out like capital?

Two friends share a taxi. One rides twice as far, straight through the traffic jam; the other gets off after a short hop. Splitting the fare by the number of bags each carries would be absurd. Risk belongs to a position in proportion to how much it moves and how much it moves with everything else, not to how much money sits in it. Here the first stock is larger, more volatile and positively correlated with the second, so it carries far more than its 60% of the capital.

Why do the contributions add up exactly to the total?

Split the variance. The cross term, 2 x 0.5 x 18 x 8 = 144, is shared equally, 72 to each position. So position A owns 324 + 72 = 396 of the 532 of variance and position B owns 64 + 72 = 136, and dividing each by the book's volatility of 23.07 gives rupee contributions that sum exactly to Rs 23.07 crore. The diversification benefit is the gap between the stand-alone total of Rs 26 crore and the book's Rs 23.07 crore.

What does a risk manager do with the split?

Cut where the risk is, not where the money is. Each rupee in position A carries 28.6 paise of marginal risk against 14.7 paise in position B, so trimming Rs 10 crore from A lowers book volatility by roughly Rs 2.9 crore. Recomputing exactly gives Rs 2.84 crore, close to the estimate. The limitation: the split is a snapshot at one correlation, and when correlations move, both the total and the split move with them.

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).

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.

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.

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.

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.

Let X be -1, 0 or 1 with equal chances and let Y = X squared. Y is fixed completely once you know X, yet the correlation is exactly zero. The covariance is E[XY] minus E[X]E[Y]; E[X] is 0 and E[XY] is the average of -1, 0 and 1, which is also 0. Correlation measures only straight-line co-movement, and a symmetric U shape has none.

What does correlation actually measure?

Think of a household's electricity bill against the outside temperature. It is high in the coldest months, when the heater runs, high in the hottest, when the air conditioner runs, and low in between. Correlation asks only whether one variable tends to rise along a straight line as the other rises, so a U-shaped link, however tight, can score zero. Over a year balanced around a mild middle, temperature explains the bill almost completely and a straight-line measure misses all of it.

How do you prove the covariance is zero?

Write out the three cases. X averages 0, Y averages 2/3, and XY takes the values -1, 0 and 1, which also average 0, so the covariance E[XY] - E[X]E[Y] is exactly 0. Symmetry does the work: every point to the right of the axis has a mirror image on the left with the same Y, so the upward slope on one side cancels the downward slope on the other. A continuous version works the same way: X normal with mean zero and Y equal to X squared.

Where does this bite on a desk?

Anywhere a payoff depends on the size of a move rather than its direction. A long straddleA call and a put bought at the same strike and expiry, which gains from a large move in either direction. gains from a big move either way, so over moves balanced around zero its return shows little correlation with the stock's return while being driven entirely by it. A risk report built only on correlations would call that position unrelated to the stock. Zero correlation means no straight-line link; only independence means no link at all.

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.

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.

About Rs 31.6 crore: Rs 10 crore times the square root of 10. If daily P&L is independent from day to day, with the same volatility and a mean near zero, variances add, so ten-day volatility is root 10 times daily volatility, and a normal quantile scales the same way. The rule also needs the positions held unchanged for ten days and a distribution that keeps its shape over the horizon.

Why not ten times the one-day number?

Picture ten friends each tossing a coin for Rs 100. The worst case is the group losing Rs 1,000, but the typical spread of the group's total is nowhere near ten times one person's, because some win while others lose. Independent daily moves partly cancel, so their variances add while their volatilities do not, and volatility grows with the square root of the number of days. Ten times would need every bad day to line up in the same direction, which is exactly what independence rules out. Value at riskThe loss a book should not exceed over a set horizon at a set confidence level, for example one day at 99%. inherits that square root when the distribution is normal.

What has to be true for the rule to hold?

List the assumptions, because that is the real question. Returns must be independent across days, volatility constant, the mean close to zero, the positions unchanged, and the distribution one that keeps its shape when summed, as the normal does. Break any one and the rule drifts. Positive autocorrelation, where bad days follow bad days, makes the true ten-day number larger. A book that is cut after losses makes it smaller. Fat tails make the one-day 99% quantile a poor guide to the ten-day one.

Which way does the error usually run?

In a calm market the rule is a fair approximation. In stress it tends to understate, because volatility rises and losses cluster just when the ten-day horizon matters. The square root of time is a scaling convenience, not a law, so a risk team checks it against ten-day P&L measured directly. The same assumption sits behind the desk habit of multiplying daily volatility by 16 to get an annual figure, 16 being roughly the square root of the trading days in a year; stretch it to 250 days here and you get Rs 158 crore, a number few would trust.

About 10% higher. The variance of a sum of 21 daily returns is 21 daily variances plus twice every covariance between pairs of days. With autocorrelation 0.1 at lag one, 0.01 at lag two and so on, the covariance terms add about 4.42 daily variances, lifting the total from 21 to about 25.4, a 21% rise. Volatility, the square root, rises by about 10%.

Why does a small autocorrelation matter over a month?

Think of a queue where each person who joins makes it slightly more likely that the next person joins too. Each nudge is small, but over a month of days they add up to longer queues than pure chance would give. The variance of a sum counts every pair of days, and there are 20 neighbouring pairs in a 21-day month, each counted twice, so a lag-one correlation of 0.1 alone adds 4 daily variances to the 21.

How do you do the sum?

Write the monthly variance as the daily variance times 21, plus twice the sum over every lag. Lag one appears in 20 pairs with correlation 0.1, lag two in 19 pairs with 0.01, lag three in 18 with 0.001: twice (2.0 + 0.19 + 0.018 and so on) is about 4.42. The total is 25.42 daily variances instead of 21, a ratio of 1.210, and the volatility ratio is its square root, 1.100. Over a long horizon the ratio tends to (1 + 0.1)/(1 - 0.1) = 1.222, so a month is already close to the limit.

What does this mean for risk and performance numbers?

Scaling daily volatility by the square root of time silently assumes zero autocorrelation. Positive autocorrelation, which shows up in trend-following returns and in portfolios of thinly traded assets priced from stale quotes, makes the square-root rule understate longer-horizon risk. Negative autocorrelation, typical of mean-reverting strategies, does the reverse: at minus 0.1 the monthly variance is only 0.83 times the rule's. The same effect is why correlations between two assets measured on daily returns can differ from those measured on monthly returns: prices that react to the same news on different days look less related day by day than month by month.