What is the difference between these two optimization procedures?

Question

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 1

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.