Portfolio Variance and Semivariance from Log Returns

Summary

The document contrasts portfolio calculations using simple returns with those using log returns. It gives the familiar weighted covariance formula for variance under linear returns and expresses portfolio log return as the logarithm of the weighted sum of the assets’ gross returns. The author then asks how to derive portfolio variance when returns are represented logarithmically, with a particular interest in semivariance and a portfolio semicovariance matrix.

The text presents the modeling question and its mathematical setup, but does not provide a derivation, answer, or empirical evidence. It therefore serves mainly as a prompt to distinguish aggregating asset returns from transforming an already aggregated portfolio return. Any variance or downside-risk calculation would need to specify the return horizon, portfolio weights, and treatment of dependence; the document leaves these choices unresolved.

Key ideas

• Simple-return portfolio variance is expressed using the asset covariance matrix and portfolio weights.
• Portfolio log return is computed by aggregating gross asset returns before taking the logarithm.
• The document asks how to define portfolio variance and semivariance under log returns.
• No solution or evidence is supplied, so the risk calculation remains open.

Tags

# Variance of a portfolio based on log-returns


# Variance of a portfolio based on log-returns












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