Quantitative Finance Stack Exchange is a question and answer site for finance professionals and academics. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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


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.


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?

share|improve this question
Your title is misleading. Your question seems to be about assuming a certain covariance matrix and estimating the covariance matrix from historical data. The contrast in title (Correlation vs Covariance) has little to do with it. – Ryogi Jul 5 '12 at 6:22

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.

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.