Why Portfolio Log Return Approximates the Weighted Sum of Asset Log Returns

Summary

The note gives an intuition for approximating a portfolio’s log return by the weighted sum of its assets’ log returns. For portfolio weights that sum to one, it expands each asset’s gross return, represented by the exponential of its log return, around zero. When individual returns are small, higher-order terms in that expansion can be neglected, leaving the portfolio’s gross return close to one plus the weighted sum of log returns.

Applying the approximation that the logarithm of one plus a small value is close to that value yields the stated relationship. The evidence is a short Taylor-series argument rather than a numerical example or empirical test. The result is approximate because it omits higher-order terms; it is most reasonable when returns are small. The note also distinguishes this approximation from the exact weighted-sum relationship for arithmetic returns, subject to weights summing to one.

Key ideas

• For weights summing to one, portfolio gross return is the weighted sum of asset gross returns.
• Expanding the exponential for small log returns makes each asset’s gross return approximately one plus its log return.
• The logarithm of one plus a small portfolio return is approximately that return itself.
• Combining these approximations gives the weighted-sum intuition for portfolio log return.
• The relationship is approximate, while the corresponding arithmetic-return aggregation is exact under the stated weight condition.

Tags

# Intuition behind log return of portfolio = weighted sum of log returns


# Intuition behind log return of portfolio = weighted sum of log returns












Suppose we have $n$ assets, each of which has weight $w_i$ in the portfolio. The log return of asset $i$ is denoted by $r_i$.

What's the intuition why this holds approximately:

$$ ln \left( \sum_i w_i e^{r_{i,t}}\right) \approx \sum_i w_i r_{i,t} $$

## Answer by Richi Wa (score 4, accepted)

https://quant.stackexchange.com/a/33939

The above relation really only approximately. If you consider arithmetic retunrs then it is exact.

For the approximation you just need to look at the Taylor series of the exponential: $$ e^x = 1 + x + \text{ terms of higher order}. $$ These terms of higher order ($x^2$ and $x^3$) become small if $x$ is much smaller than one - which holds true for returns. Thus $$ \sum_i w_i e^{r_i} \approx \sum_i w_i ( 1+ r_i) = 1 + \sum_i w_i r_i, $$ if $\sum_i w_i = 1$.

Then conside that $\ln(1+x) \approx x$ (you can check here) and you are done:

$$ \ln \left(1 + \sum_i w_i r_i \right) \approx \sum_i w_i r_i. $$

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