Minimum-Variance Portfolios Versus Maximum-Sharpe Portfolios

Summary

The document distinguishes two portfolio optimization tasks using a dataset of stock prices for 15 companies over one year. The investor first estimates average returns, annualized volatilities, and correlations, then finds weights that minimize variance subject to a required return of 10%. The follow-up question asks what it means to optimize the portfolio’s risk-return tradeoff, rather than minimizing variance at a fixed target return.

The accepted response interprets the latter objective as finding the maximum-Sharpe, or tangency, portfolio. It describes the intuition in terms of balancing marginal contribution to expected return against marginal contribution to volatility, so small weight changes do not improve the portfolio’s return-to-volatility ratio. The document points to a derivation elsewhere but includes no calculations or performance evidence. The explanation is brief and leaves assumptions, such as return estimates and constraints on weights, unspecified; results would depend on those choices and the quality of the input estimates.

Key ideas

• Minimum variance with a required return and maximum Sharpe ratio are distinct optimization objectives.
• The stated data include average stock returns, volatilities, and correlations.
• The response identifies the risk-return optimization request as a maximum-Sharpe or tangency portfolio problem.
• Its intuition is to balance marginal return contribution against marginal volatility contribution.
• The document gives no computed weights and does not specify estimation or portfolio constraints.

Tags

# Optimize risk return of portfolio


# Optimize risk return of portfolio












I have gathered stock prices from 15 companies over a year. I calculated the averages, yearly volatilizes, and the correlation matrix.

I was asked to find the optimum weight for each stock with a required 10% return and I had to minimize variance. I solved this question.

However, I was then asked to find appropriate weights to optimize the risk-return of the portfolio.

As a student, I did not realize the difference in the questions. Thank you for your help.

## Answer by demully (score 1, accepted)

https://quant.stackexchange.com/a/51807

Your lecturer was almost certainly asking for the "max-Sharpe" portfolio.

See below for the formula. Essentially it equalises the marginal-contribution-to-return/marginal-contribution-to-vol, such that adding or subtracting from any weights won't change the expected-return/expected-vol of the portfolio.

Derivation of the tangency (maximum Sharpe Ratio) portfolio in Markowitz Portfolio Theory?

hope this helps, W

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