Choosing Portfolio Weights from Sharpe Ratios and Correlation

Summary

The document explains how to combine two portfolios when their Sharpe ratios and correlation are known. Under the Markowitz framework, it gives a formula for the maximum achievable Sharpe ratio and a corresponding vector of portfolio weights, expressed using the two Sharpe ratios and their correlation. This provides a way to compare allocations across different correlation assumptions, including when diversification may improve the combined portfolio’s risk-adjusted return.

The answer gives formulas rather than worked allocations for the example correlations. Its result depends on the Markowitz assumptions and the inputs being suitable for that model; Sharpe ratios and correlation alone do not describe practical constraints such as leverage limits, costs, or estimation error. The formulas therefore offer a theoretical allocation rationale, not a complete implementation plan.

Key ideas

• The Markowitz framework can derive portfolio weights from individual Sharpe ratios and their correlation.
• The maximum combined Sharpe ratio depends on both portfolios’ Sharpe ratios and the correlation between them.
• Lower or negative correlation can increase the risk-adjusted return available through diversification.
• The formulas provide a theoretical allocation and do not account for practical constraints or estimation error.

Tags

# Portfolio Allocation given Sharpe Ratio


# Portfolio Allocation given Sharpe Ratio












If there are two portfolios with sharpe ratios of 1.2 and 0.5, what would be the allocation rationale.

If the correlation between portfolios is:

$a. 0 $

$b. 0.8 $

$c.-0.8 $

I see there is a diversification benefit in the c case, but is there a way to decide weights without more information ?

## Answer by steveo'america (score 2, accepted)

https://quant.stackexchange.com/a/39872

The optimal Sharpe you can achieve, by the Markowitz portfolio, is $$ \sqrt{\frac{1}{1-\rho^2} \left( 1.2^2 - 2 \rho (1.2) (0.5) + 0.5^2 \right)}. $$ The optimal portfolio is $$ \frac{1}{1-\rho^2} \begin{bmatrix} 1 & -\rho \\ -\rho & 1 \end{bmatrix} \begin{bmatrix} 1.2\\ 0.5 \end{bmatrix}, $$ where $\rho$ is the correlation of the assets.

You can do the rest of the math.

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