Identifying and building a portfolio of uncorrelated trading strategies is the main aim of many quantitative hedge funds. Given that one would like to add a new strategy to an existing set of strategies, what is the marginal gain the can be expected over the status quo? In addition, how can one optimize the Sharpe ratio of this new set of strategies by allocating capital between different portfolio managers? We address these questions in a two dimensional setting below.

Let $X^i$ for $i=1,\ldots,m$ denote $m$ random variables that have a joint normal distribution $\mathcal{N}(\mu,\Sigma)$ which we interpret as the joint return distribution of a collection of $m$ distinct trading strategies. Let $X = w_1X^1+\cdots + w_nX^n$ denote a linear combination of these returns and $w=(w_1,\ldots,w_n)$ be their weight vector. Then $X$ is $\mathcal{N}(w^T\mu,w^T\Sigma w)$ distributed and the Sharpe ratio of the portfolio is given by $SR(m)=\mu^Tw/\sqrt{w^T\Sigma w}$.

In order to develop intuition for how the combined Sharpe ratio changes as a function of the means, vols, and correlation of the individual trading strategies, we restrict to the case of two random variables with variances $\sigma_1^2$, $\sigma_2^2$ and correlation $\rho$.   Then from the above we see that the aggregate Sharpe ratio is given by

$SR(2)=\frac{w_1\mu_1+w_2\mu_2}{\sqrt{\sigma_1^2w_1^2+\sigma_2^2w_2^2+2w_1w_2\rho\sigma_1\sigma_2}}$

In the case that two assets are uncorrelated and equally weighted, with the same mean and variance, then taking $\rho=0$, $\sigma_1=\sigma_2=\sigma$, $\mu_1=\mu_2=\mu$, and $w_1=w_2=1/2$ in the above we find that

$SR(2) = \frac{\mu}{\sqrt{\sigma^2/2}} = \sqrt{2} s,$

i.e. the Sharpe ratio of the equally weighted portfolio is $\sqrt{2}$ times greater than the Sharpe ratio of either individual portfolio. An example of this case would be a two asset risk parity strategy where their individual asset’s returns have very low correlation; in this case, combining the strategies improve the overall portfolio Sharpe ratio by an approximate factor of 1.4.

We now consider a question related to capital allocation. Namely, given two strategies for which we have estimates of their means, vols, and correlation, how should capital be allocated between these two strategies so that we maximize the Sharpe ratio of the aggregate portfolio? Consider the general case of two assets only supposing that $w_2=1-w_1$ so that the weights sum to $1$. In this case, maximizing the Sharpe ratio yields the optimal weights

$\hat{w}_1 = \frac{\mu_1\sigma_2^2-\mu_2\rho\sigma_1\sigma_2}{\mu_1\sigma_2^2+\mu_2\sigma_1^2- (\mu_1+\mu_2)\rho\sigma_1\sigma_2}, \quad \hat{w}_2 = 1-\hat{w}_1.$

The associated maximum Sharpe ratio value is given by

$SR(2) = \frac{1}{\sigma_1\sigma_2\sqrt{1 -\rho^2}}\sqrt{\mu_2^2\sigma_1^2+\mu_1^2\sigma_2^2-2\mu_1\mu_2\rho\sigma_1\sigma_2 }.$

Note in the case that the correlation tends to 1 that the numerator tends to 0 since the means and variances will become equal.

If we restrict to the case of equal means, zero correlation, and equal variances, then we see the Sharpe ratio is maximized with weights given by $w_1 = w_2 = 1/2$ with a value $\sqrt{2\mu^2\sigma^2}/\sigma^2=\sqrt{2}\mu/\sigma$, i.e. the original Sharpe ratio of one return series rescaled by $\sqrt{2}$ which is consistent with what we found before.

Variations of the $m$ dimensional extension of this argument is key to the capital allocation process of many quantitatively driven hedge funds.