Covariance and Correlation Inputs in Mean-Variance Portfolio Optimization

Summary

The discussion asks whether mean-variance optimizers produce the same portfolio when one accepts expected returns, volatilities, and correlations while another estimates a covariance matrix from historical prices. The answer explains that correlation together with each asset’s standard deviation contains the same information as covariance: covariance is recovered by scaling correlation by the two assets’ standard deviations. Thus, either representation can supply the same risk inputs when calculated consistently.

It also sketches two common formulations: minimizing portfolio variance subject to a minimum expected return, and maximizing expected return minus a risk-aversion-weighted variance penalty. These express related mean-variance tradeoffs, though their precise solutions depend on matching constraints, objectives, estimates, and implementation assumptions. The post’s central point concerns equivalent input representations; it does not address estimation error, constraints, transaction costs, or whether historical estimates are reliable forecasts.

Key ideas

• A covariance matrix can be reconstructed from asset correlations and standard deviations.
• Consistent covariance and correlation inputs encode the same portfolio risk information.
• Mean-variance optimization can minimize variance subject to a return target or penalize variance in a return objective.
• Equivalent information alone does not guarantee matching results if the estimators, constraints, or objectives differ.

Tags

# What is the difference between these two optimization procedures?


# What is the difference between these two optimization procedures?












In this portfolio optimization utility (and others), mean return, standard deviation and correlation among assets are required inputs.

http://finance.wharton.upenn.edu/~stambaugh/portopt.html

At the same time, I've seen other portfolio optimizers that start with historical price data and a covariance matrix is calculated as a step in optimizing.

http://investexcel.net/215/mean-variance-portfolio-optimization-with-excel/

If the same underlying data set is used, and the definition of the optimal portfolio is the same in both optimizers, will the results be the same?

## Answer by SRKX (score 4)

https://quant.stackexchange.com/a/3737

You know that the correlation between 2 assets is defined as

$$Corr(X,Y) = \frac{Cov(X,Y)}{\sigma_X \sigma_Y}$$.

So if you provide the algorithm with either the correlation matrix and the standard deviation of the components or with the covariance matrix alone, is has the same information.

The formula that will be used for the optimization algorithm will be something like:

$$\underset{w}{\arg \min} \quad w \Sigma w' \quad \text{s.t} \quad \mu w' \geq \bar{\mu}$$

Where $\Sigma$ is the covariance matrix and hence $w \Sigma w'$ is the variance of the portfolio with allocation $w$.

This problem is specified slightly differently in your examples, but yield equivalent results:

$$\underset{w}{\arg \max} \quad \mu w' - \rho w \Sigma w'\ $$

Where $\rho$ is the risk aversion coefficient.

Note that providing correlation as input means nothing... It simply was computed previously.

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