Maximum Sharpe Ratio Portfolio with Uncorrelated Assets

Worked Solution

How to Think About It: You are allocating between two uncorrelated bets. Each asset has its own Sharpe ratio -- you want to blend them to get the best risk-adjusted return. The key intuition: you should tilt more toward the asset with the higher Sharpe ratio per unit of risk capital, which means weighting proportional to $\mu_i / \sigma_i^2$. This is not the same as weighting by Sharpe ratio directly -- you also need to account for how much variance each position contributes. Since the assets are uncorrelated, there is no hedge benefit to consider, and the problem reduces to a clean optimization.

Quick Estimate: Take $\mu_1 = 0.10$, $\sigma_1 = 0.20$, $\mu_2 = 0.06$, $\sigma_2 = 0.15$. Then $\mu_1/\sigma_1^2 = 0.10/0.04 = 2.5$ and $\mu_2/\sigma_2^2 = 0.06/0.0225 \approx 2.67$. So $w_1 \approx 2.5/5.17 \approx 0.484$ and $w_2 \approx 2.67/5.17 \approx 0.516$. Despite asset 1 having the higher Sharpe ratio ($0.50$ vs $0.40$), asset 2 gets slightly more weight because its lower volatility makes each dollar of exposure more capital-efficient. The individual Sharpe ratios are $0.50$ and $0.40$, so the combined Sharpe ratio should be $\sqrt{0.25 + 0.16} = \sqrt{0.41} \approx 0.64$ -- strictly better than either alone.

Approach: Write the portfolio Sharpe ratio as a function of weight $w$, differentiate, and solve. Alternatively, recognize this as a special case of the tangency portfolio from mean-variance optimization.

Formal Solution:

The portfolio with weight $w$ in asset 1 and $(1-w)$ in asset 2 has:

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

Maximizing $\text{SR}$ is equivalent to maximizing $\text{SR}^2$. Set the derivative to zero. The numerator of $d(\text{SR}^2)/dw = 0$ gives:

$(w\mu_1 + (1-w)\mu_2)(w\sigma_1^2 - (1-w)\sigma_2^2) = (w^2\sigma_1^2 + (1-w)^2\sigma_2^2)\,(\mu_1 - \mu_2) \cdot \frac{1}{1}$

A cleaner route: from standard mean-variance theory, the tangency portfolio satisfies $\mathbf{w} \propto \Sigma^{-1}\boldsymbol{\mu}$. For uncorrelated assets, $\Sigma = \text{diag}(\sigma_1^2, \sigma_2^2)$, so:

$w_i \propto \frac{\mu_i}{\sigma_i^2}$

Normalizing so the weights sum to 1:

$w_1 = \frac{\mu_1/\sigma_1^2}{\mu_1/\sigma_1^2 + \mu_2/\sigma_2^2}, \quad w_2 = \frac{\mu_2/\sigma_2^2}{\mu_1/\sigma_1^2 + \mu_2/\sigma_2^2}$

For the maximum Sharpe ratio, substitute back. After algebra (or using the identity for uncorrelated tangency portfolios):

$\text{SR}_{\max}^2 = \frac{\mu_1^2}{\sigma_1^2} + \frac{\mu_2^2}{\sigma_2^2} = \text{SR}_1^2 + \text{SR}_2^2$

This generalizes immediately to $n$ uncorrelated assets:

$\text{SR}_{\max} = \sqrt{\sum_{i=1}^{n} \text{SR}_i^2}$

To verify: the optimal portfolio return is $\mu_p = \sum w_i \mu_i$ and variance is $\sigma_p^2 = \sum w_i^2 \sigma_i^2$. Substituting $w_i = c \cdot \mu_i/\sigma_i^2$ where $c = 1/\sum_j \mu_j/\sigma_j^2$:

$\text{SR}_p^2 = \frac{\mu_p^2}{\sigma_p^2} = \frac{\left(\sum c \mu_i^2/\sigma_i^2\right)^2}{\sum c^2 \mu_i^2/\sigma_i^4 \cdot \sigma_i^2} = \frac{c^2 \left(\sum \mu_i^2/\sigma_i^2\right)^2}{c^2 \sum \mu_i^2/\sigma_i^2} = \sum \frac{\mu_i^2}{\sigma_i^2}$

Answer: The optimal weights are $w_i \propto \mu_i / \sigma_i^2$, and the maximum portfolio Sharpe ratio satisfies:

$\text{SR}_{\max} = \sqrt{\text{SR}_1^2 + \text{SR}_2^2}$

For $n$ uncorrelated assets, individual Sharpe ratios combine in quadrature: $\text{SR}_{\max} = \sqrt{\sum_i \text{SR}_i^2}$.