14.7 Combining strategies · M14 Portfolio and going live

Motivation

The cheapest Sharpe ratio you will ever get is from combining imperfectly correlated strategies. This section gives the exact gain.

Proposition 14.12Two strategies

Strategies 1 and 2 have Sharpe ratios $S_1, S_2$ (excess returns over volatility) and correlation $|\rho| < 1$ . The maximum Sharpe ratio of any combination is

S^* = \sqrt{\frac{S_1^2 + S_2^2 - 2\rho S_1S_2}{1 - \rho^2}}. \tag{14.7}

An equal-risk (each scaled to the same volatility, 50/50) combination has Sharpe $(S_1 + S_2)/\sqrt{2(1+\rho)}$ . $S^* > \max(S_1, S_2)$ unless $S_2 = \rho S_1$ (or $S_1 = \rho S_2$ ).

Proof

Scale both to unit volatility; then excess means are $s = (S_1, S_2)$ and $\Sigma = \begin{pmatrix}1&\rho\\\rho&1\end{pmatrix}$ , $\Sigma^{-1} = \frac1{1-\rho^2}\begin{pmatrix}1&-\rho\\-\rho&1\end{pmatrix}$ . By Corollary 14.4, $S^{*2} = s^\top\Sigma^{-1}s = (S_1^2 - 2\rho S_1S_2 + S_2^2)/(1-\rho^2)$ . For the 50/50 combination, mean $(S_1 + S_2)/2$ , variance $(1 + \rho)/2$ . For the last claim, $S^{*2} - S_1^2 = [S_2^2 - 2\rho S_1S_2 + \rho^2S_1^2]/(1-\rho^2) = (S_2 - \rho S_1)^2/(1-\rho^2) \ge 0$ , with equality iff $S_2 = \rho S_1$ .∎

Corollary 14.13 $N$ strategies

$N$ strategies with equal Sharpe $S$ , equal volatility and common pairwise correlation $\bar\rho$ , combined equally, have Sharpe $S\sqrt{N/(1 + (N-1)\bar\rho)}$ , which tends to $S/\sqrt{\bar\rho}$ as $N \to \infty$ .

Proof

Mean $S\sigma$ , variance $\sigma^2[1 + (N-1)\bar\rho]/N$ .∎

Numbers Two strategies with Sharpe 0.5 each: combined 0.71 at $\rho = 0$ and 0.58 at $\rho = 0.5$ . A Sharpe-0.8 strategy and a Sharpe-0.3 strategy with correlation $0.375 = 0.3/0.8$ : the combination is 0.80, no gain at all, because the second strategy's return is fully explained by its correlation with the first. Ten strategies of Sharpe 0.5: 1.58 if uncorrelated, 1.15 at $\bar\rho = 0.1$ , 0.82 at $\bar\rho = 0.3$ . Diversification across strategies is limited by their average correlation, which is why a new strategy is valuable in proportion to how different it is, not how good it looks alone. (This is also the content of M04's warning: ten variants of one idea are one strategy.)

Practitioner notes

Measure strategy correlations on daily P&L, and separately in drawdown periods (14.6). A swing book and an intraday book on the same index are often more correlated than they look, because both lose in regime changes.