Derivatives Foundation puzzles, solved step by step

The eigenvalues are 1 + 2 rho, once, and 1 - rho, twice; rho can go no lower than -1/2. The vector (1, 1, 1) is an eigenvector with eigenvalue 1 + 2 rho, and any vector whose entries sum to zero is an eigenvector with eigenvalue 1 - rho, which gives a two-dimensional space and hence a double root. A correlation matrix must have no negative eigenvalue, since each eigenvalue is a portfolio variance, so 1 + 2 rho is at least 0 and rho is at least -1/2.

Why can three assets not all be strongly negatively correlated?

Three friends cannot all sit opposite each other at a table: if A faces B and B faces C, then A and C are on the same side. Correlation has the same constraint. If asset A moves against B and B moves against C, A and C are pushed towards moving together, so there is a floor on how negative a common correlation can be, and for three assets that floor is -1/2. You can see the floor without any algebra by holding an equal-weight portfolio of the three, each with unit variance: its variance is (3 + 6 rho) / 9, which is (1 + 2 rho) / 3, and a variance cannot be negative. At rho = -1/2 the portfolio has zero variance; it is perfectly hedged, and nothing below that is possible.

How do you find the eigenvalues without expanding a determinant?

Write the matrix as (1 - rho) times the identity plus rho times the all-ones matrix J. The identity leaves every vector alone, so you only need the eigenvalues of J, and J is easy: it maps (1, 1, 1) to (3, 3, 3), eigenvalue 3, and it maps any vector whose entries sum to zero to the zero vector, eigenvalue 0, with a two-dimensional space of such vectors. Shifting and scaling by (1 - rho) turns those into 1 - rho + 3 rho = 1 + 2 rho for the market direction and 1 - rho for the two spread directions. Check with the trace: the eigenvalues add to 1 + 2 rho + 2(1 - rho) = 3, the sum of the diagonal, as they must. At rho = 0.3 they are 1.6, 0.7 and 0.7.

What does the structure tell a risk or trading desk?

The eigenvectors are the principal components. The (1, 1, 1) direction is the market factor, and its eigenvalue over the trace, (1 + 2 rho) / 3, is the share of total variance it explains: 53% at rho = 0.3 and 80% at rho = 0.7. The two spread directions carry the rest, equally. A long-short book that nets to zero across the three assets lives entirely in the 1 - rho directions, which is why pairs trades get calmer as correlation rises and why a correlation of 1 collapses them to nothing. For n assets the same argument gives eigenvalues 1 + (n - 1) rho and 1 - rho, so the floor is -1 / (n - 1): -1/3 for four assets and -1/9 for ten. Say the limitation as well: a historical correlation matrix estimated from more assets than observations is only barely positive semi-definite, and a hand-edited one, where a trader overrides a few pairs, can fail the test entirely, which is exactly the fault a risk system is built to catch.

Between about -0.75 and 0.95. The three correlations must form a positive semi-definite matrix, and for three variables that means its determinant 1 - a^2 - b^2 - c^2 + 2abc cannot be negative. Solving for c with a = 0.2 and b = 0.5 gives c = ab plus or minus sqrt((1 - a^2)(1 - b^2)) = 0.1 plus or minus sqrt(0.96 x 0.75) = 0.1 plus or minus 0.849.

Why can the third correlation not be anything it likes?

If Amit is tall when Bela is tall, and Bela is tall when Chitra is tall, then Amit and Chitra cannot be perfectly opposite; the middle person ties them together. Three pairwise correlations describe one joint set of three variables, and a joint set has a variance for every combination of them that cannot be negative, which is the positive semi-definite condition on the correlation matrix. With weak links, 0.2 and 0.5, the tie through Y is loose and the band is wide; with links of 0.9 and 0.9, X and Z would be forced to be strongly positively correlated.

How do you get the band without writing out eigenvalues?

Use the determinant. For a 3 x 3 correlation matrix with off-diagonal entries a, b and c, the determinant is 1 - a^2 - b^2 - c^2 + 2abc, and the matrix is valid exactly when that is at least zero. Treat it as a quadratic in the unknown c: c^2 - 2ab c + (a^2 + b^2 - 1) must be at most zero, so c lies between the two roots ab plus or minus sqrt((1 - a^2)(1 - b^2)). With a = 0.2 and b = 0.5 the centre is 0.1 and the half-width is sqrt(0.96 x 0.75) = sqrt(0.72), about 0.849. The geometric reading is the same thing: think of each variable as a unit vector, correlations as cosines of the angles between them, and the third angle is bounded by the sum and difference of the first two.

Where does this show up on a desk?

In any correlation matrix that was assembled by hand or from mismatched histories. A desk that marks the stock-to-index correlation at 0.9, the index-to-sector correlation at 0.9, and the stock-to-sector correlation at 0.3 has written down a matrix that no world can produce, and a basket option priced from it will give nonsense, often a negative variance for some combination. The fix is to project the matrix back to the nearest valid one before pricing. The limitation of the puzzle is that it covers only three variables; with more, every principal sub-matrix must pass, and the feasible set has no simple closed form.

About 0.17, exactly one sixth. For n equally weighted stocks with the same volatility and the same pairwise correlation rho, index variance is stock variance times (1/n + (1 - 1/n) rho). Index variance over stock variance is 0.15 squared over 0.30 squared = 0.25, so 0.25 = 0.1 + 0.9 rho and rho = 0.15/0.9 = 1/6. The large-n shortcut of 0.25 overstates it because it ignores the 1/n term.

Why does the index have less volatility than its members?

Ten shopkeepers in one market each have noisy daily takings, but the market's total takings are steadier, because one shop's bad day is often another's good day. Only the part of the noise they share, the weather or a festival, survives the averaging. Index variance has two parts: the stocks' own noise, which averages away as 1/n, and the common movement, which survives in proportion to the correlation. With ten stocks at 30% and no correlation at all the index would still have 9.5% volatility, and that floor is why the implied correlation is below the naive 0.25.

How do you derive the formula on the spot?

Write the index as the average of n returns and expand its variance. There are n variance terms, each sigma squared over n squared, and n(n - 1) covariance terms, each rho sigma squared over n squared, so index variance is sigma squared times (1/n + (n - 1) rho / n). Set that equal to 0.15 squared with sigma = 0.30 and n = 10: 0.0225 = 0.09 x (0.1 + 0.9 rho), so 0.25 = 0.1 + 0.9 rho and rho = 1/6. The derivation takes four lines, and the interviewer wants to hear the count of covariance terms, n(n - 1), said out loud.

What does a desk do with implied correlation?

It compares it with realised correlation and trades the gap. Selling index options and buying single-stock options is a short correlation position, called a dispersion trade, and it pays when the stocks move more independently than the 1/6 the market has priced in. The limitation of the puzzle is its symmetry: real indices have unequal weights and unequal volatilities, so the implied correlation is a weighted average that can differ from the simple formula, and the implied figure also moves with the skew of the index options, which the single number cannot show.

Give both assets a shared monthly drift and opposing daily noise. Let A's daily return be m/21 + e + u and B's be m/21 - e + v, where m is a monthly drift common to both, e is noise that hits them in opposite directions, and u and v are their own noise. Inside a month m is fixed, so daily returns co-move through -e alone: correlation -0.50 with equal variances. Over a month e sums to a modest total while m is large, so monthly returns correlate at +0.66.

Why can one pair of series carry two correlations?

Two shops on the same street both do better in a festival month and worse in a quiet one, so their monthly takings rise and fall together. But on any given day, the customer who buys lunch at one is not buying it at the other, so their daily takings move against each other. Nothing is paradoxical: the two effects live at different speeds. Correlation is not a property of two assets, it is a property of two assets at a horizon, because different components of the return dominate at different horizons. The slow shared component barely moves within a day; the fast opposing component barely survives a month.

How do you put numbers on the construction?

Take a drift m with a standard deviation of 12 points a month, opposing noise e of 1 point a day, and own noise u and v of 1 point a day. Within a month m is a constant, so it drops out of the daily covariance: cov = -var(e) = -1, and each variance is var(e) + var(u) = 2, so the daily correlation is -1/2. Over 21 days the sums of e, u and v each have variance 21, and the monthly covariance is var(m) - 21 var(e) = 144 - 21 = 123 against a variance of 144 + 21 + 21 = 186. The monthly correlation is 123/186 = 0.66, positive, because the shared drift's variance grows with the square of the month while the noise variance only grows in proportion to it.

Say where this shows up on a desk, because that is why the question is asked. Two stocks in one sector share the sector's slow repricing but trade against each other intraday as market makers hedge one with the other. A hedge ratio estimated from daily data will be wrong for a position held for months, and the reverse. The limitation of the example: a flat drift inside a month is a simplification, and in real data the slow component is itself noisy, so the sign flip is usually a fade from negative towards positive rather than a clean reversal.