Modern Portfolio Theory Optimization Problem is based on expected linear returns and covariances of linear returns.

That's said, variance and expected return of a portfolio based on linear returns r are computed as following:

$$\sigma^2_p = w'\Sigma w$$

$$E_p = \sum_{i=1}^{N}w_ir_i$$

Now, if using log-returns R instead of linear-returns r,

$$R = \log(1+r)$$

Then the expected log-return of a portfolio is computed as following:

$$E_p = \log\left( \sum_{i=1}^{N} w_i e^{R_i} \right)$$

I am struggling to find out the formula to compute the variance of the portfolio based on log-returns.

P.S. Actually, my goal is to compute the semi-variance of a portfolio semi-covariances matrix based on log-returns.


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