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Finding Two-Asset Portfolio Weights That Maximize the Sharpe Ratio

Article Quant Q&A · Author: auhan
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Summary

The document outlines how to find the weights of two risky assets in a portfolio when their expected returns, standard deviations, and correlation are known. It expresses portfolio excess return as the weighted asset returns less the risk-free rate, and portfolio variance as the weighted variances plus a covariance term based on correlation. The Sharpe ratio is then portfolio expected excess return divided by portfolio volatility.

To identify weights that maximize this ratio, the answer proposes optimizing the Sharpe ratio subject to the weights summing to one. Thus, standard deviations and correlation alone are not enough to determine the preferred allocation: expected returns and the risk-free rate also enter the objective. The response gives the setup rather than carrying out the optimization or discussing constraints such as short selling, estimation error, or alternative portfolio objectives, so practical use requires those choices to be specified.

Key ideas

  • Portfolio variance depends on both asset volatilities and their correlation.
  • The Sharpe ratio compares expected excess return with portfolio volatility.
  • Maximizing the ratio requires an optimization over weights subject to a budget constraint.
  • Volatility and correlation alone do not establish the Sharpe-maximizing allocation.

Tags

Portfolio constructionStatisticsRisk management
Full text
# Do I calculate weights of assets correctly?


# Do I calculate weights of assets correctly?












I solved attached question but I am not sure whether I did part a and c correctly. Is there a way to calculate weights of A and B by just knowing their standard deviation and correlation's value?

## Answer by carbolymer (score 1)

https://quant.stackexchange.com/a/22454

To find the weights in the question (a) you should write your portfolio expected excess return and variance as: $$ E[R_p^e] = w_A R_A + w_b R_B - R_f \\ \sigma^2[R_p^e] = \sigma^2[w_A R_A + w_b R_B - R_f] = w_A^2\sigma_A^2 + w_B^2 \sigma_B^2 + 2 \rho_{AB}\sigma_A\sigma_B $$ The sharpe ratio is given by: $$ S(w_A,w_B) = \frac{E[R_p^e]}{\sigma[R_p^e]} $$ So, to find the weights which maximize Sharpe ratio, you should solve the equation: $$ \nabla S |_{w_A+w_B=1} = 0 $$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.