# Optimise the Sharpe ratio of a portfolio of uncorrelated assets

Given a portfolio of $$n$$ assets, mean returns vector $$\mu$$, covariance matrix $$K$$, one can calculate the portfolio weights $$w^*$$ that maximise the portfolio Sharpe ratio, by solving:

$$w^*=\text{argmax} \left[\frac {w^T \mu} {\sqrt {w^T K w}} \right]$$

Computing $$w^*$$ requires building the $$K$$ matrix and solving a system of quadratic equations, so it becomes computationally expensive when $$n$$ grows large.

However, if for some reason, we know that the asset returns are uncorrelated, the problem simplifies as: $$K=\begin{bmatrix} \sigma_1^2 & .. & 0 \\ .. & .. & .. \\ 0 & .. & \sigma_n^2 \\ \end{bmatrix}$$

In this case, I suspect we don't need to bother with the quadratic equations, there should be straightforward formula to compute $$w^*$$ by just plugging in the individual $$\mu_i$$ and $$\sigma_i$$ of each asset:

$$w_i^* = f(\mu_1,\sigma_1,\dots,\mu_n,\sigma_n)$$

I am unable to derive the formula analytically. HELP!

• The maximum Sharpe portfolio, also called the tangency portfolio in a Markowitz framework, has an analytical expression for the optimal weights $w^*$ under no short-selling constraints. See the answers to this post. Maybe that'll help you answer your question?
– Pleb
May 12, 2021 at 18:46
• In addition what Pleb said, the denominator becomes $\sqrt{(w_{1}^2 \sigma_{1}^2 + \ldots + w_{n}^2 \sigma^2_{n})}$ But I'm not sure if that helps with the maximization. May 13, 2021 at 3:29

Formally (and I mean it -- see below), the optimal weights are $$w=K^{-1}\mu$$. The portfolio pnl then has the mean $$Q=\sum_iw_i\mu_i$$ and the variance $$V=\sum_{ij}w_iw_jK_{ij}$$. The Sharpe is $$S=Q/\sqrt{V}$$. In the uncorrelated case the answer is given by the Pythagorean formula $$S=\sqrt{\sum_i\frac{\mu_i^2}{\sigma_i^2}}.$$
1. Two uncorrelated assets with Sharpe ratios $$S_1$$ and $$S_2$$. The optimally weighted two-asset book has the Sharpe $$\sqrt{S_1^2+S_2^2}$$.