Case 095Portfolio construction and optimisationHard

An optimiser sees two stocks with expected returns of 13% and 13.5%, both at 25% volatility, and a sample correlation of 0.95 from 36 months of data. Show how extreme the weights become, then shrink the correlation to 0.6.

The situation

Jaladhi Quant Equity Fund runs a mean-variance optimiser. Two private sector banks in its universe, call them A and B, have forecast returns of 13% and 13.5%. Both show 25% volatility, and 36 months of returns give a correlation of 0.95. Cash earns 6.5%.

The optimiser maximises the Sharpe ratio for a fully invested two-stock sleeve and allows short positions. A risk analyst proposes shrinking the correlation towards 0.5, the average pairwise correlation in the universe, which with the fund's shrinkage setting gives 0.6.

Your task

What weights does the optimiser choose at 0.95, how sensitive are they to the half-point return gap, and what does shrinkage change?

Quick check

At a correlation of 0.95, what does the optimiser do with a half-point return edge for B?

Worked solution

Try it on paper, then open one step at a time.

30-second answerThe answer to give first

At 0.95 the optimiser holds -22% in A and 122% in B, a 144% gross position built on half a point of forecast return. Swap the half point to A and the weights flip, a 144-point swing. That half point is about 0.11 standard errors of what 36 months can measure, so the optimiser is betting on noise. Shrinking the correlation to 0.6 gives 43/57, and the same swap moves the weights only 15 points.

Step 1Why does high correlation make the weights extreme?

Two petrol pumps across the road from each other, one a paisa cheaper: if they are truly identical, you would fill every car at the cheaper one. When two assets look almost identical, the difference between them looks almost riskless, and an optimiser will lever into any return gap between them. The spread between A and B has a volatility of only 7.9% at a correlation of 0.95, against 25% for each stock. A half point earned on so little risk looks like a bargain, so the optimiser sells A to buy more B.

The relationship
w∝Σ−1(μ−rf):wA∝6.5−0.95×7.0=−0.15,wB∝7.0−0.95×6.5=0.825w \propto \Sigma^{-1}(\mu - r_f): \quad w_A \propto 6.5 - 0.95 \times 7.0 = -0.15, \quad w_B \propto 7.0 - 0.95 \times 6.5 = 0.825
mu - r_fexcess returns: 6.5% for A and 7.0% for B
Sigma^-1the inverse covariance matrix; with equal volatilities it reduces to these terms
wweights, scaled so they sum to 100%
What it says in wordsEach stock's weight is its own excess return less the correlation times the other's; scaled to 100%, A is -22.2% and B 122.2%.
At correlation 0.95 a half-point guess flips the portfolio; at 0.6 it barely moves it0%100%A -22%B +122%rho 0.95, B +0.5A +122%B -22%rho 0.95, A +0.5A +43%B +57%rho 0.6, B +0.5A +57%B +43%rho 0.6, A +0.5sample correlation: weights swing 144 pointsshrunk correlation: weights swing 15 points
At Jaladhi's sample correlation of 0.95 the optimiser holds -22% A and 122% B, and moving the half point of expected return to A flips it to 122% and -22%; at a shrunk 0.6 the same swap moves the weights only from 43/57 to 57/43.
Step 2Is the half-point gap real?

Measure it against what the data can see. Thirty-six months estimate a stock's average return with a standard error of about 14.4% a year, and the gap between two stocks this correlated with an error of about 4.6%. A half-point gap is 0.11 of those standard errors: indistinguishable from zero. The optimiser is treating a number it cannot measure as a certainty and building a 144% gross position on it. This is why optimisers are sometimes called error maximisers: they bet hardest exactly where the inputs are least reliable.

Step 3What does shrinking the correlation change?

Shrinkage pulls a noisy sample estimate part of the way towards a sensible average, here 78% of the way from 0.95 to 0.5, landing at 0.6. At 0.6 the spread between A and B no longer looks riskless, so the optimiser holds 43% A and 57% B, both long, and a swap of the half point moves the weights only 15 points. The fund gives up a little theoretical efficiency and gains weights that do not lurch with every revision of a forecast.

As correlation rises, the optimiser leans ever harder on a tiny return gap50%100%150%200%00.20.40.60.80.95Correlation fed to the optimiserWeight in stock B0.6 shrunk: 57%0.95 sample: 122%
Jaladhi's optimal weight in B, given a half point of extra expected return, is 52% at zero correlation and 57% at 0.6, but 122% at 0.95 and rising steeply beyond, as the optimiser borrows more of A to buy B.
InputsWeight AWeight BGross exposure
Correlation 0.95, B expected +0.5-22.2%122.2%144%
Correlation 0.95, A expected +0.5122.2%-22.2%144%
Correlation 0.6, B expected +0.542.6%57.4%100%
Correlation 0.6, A expected +0.557.4%42.6%100%
A half-point change in which stock is expected to earn more swings Jaladhi's weights by 144 points at the sample correlation of 0.95 and by only 15 points once the correlation is shrunk to 0.6.

Say what else the fund should do, because shrinking one input is not the whole fix. Shrink the return forecasts towards each other as well, cap position sizes and gross exposure, and penalise turnover so the weights cannot chase each revision. The limit of shrinkage is that the target and intensity are choices; say how they were set and test the weights on data the optimiser did not see.

Where candidates lose it

The common loss is reading the extreme weights as a strong signal. The optimiser is not confident about B; it is exploiting a gap between two numbers that the data cannot tell apart.

The second is fixing it with a position cap alone. A cap hides the symptom, but the optimiser still flips its preference with every small change in forecasts; shrinking the inputs removes the cause.

What the interviewer asks next

  • How would you choose the shrinkage target and intensity?
  • What happens to the weights if you shrink the expected returns towards each other as well?
  • Why do long-only constraints partly act like shrinkage?

Company names and figures are illustrative.