Finding the Distribution of Portfolio Returns
Summary
The document asks whether a portfolio’s return distribution can be formed by weighting the return distributions of its component assets. It distinguishes the straightforward weighted calculation of expected returns from portfolio variance, which also depends on cross-asset covariances, and focuses on the full probability distribution of a weighted sum.
The answer describes three ways to obtain or approximate that distribution: integrate the joint multivariate distribution, use the portfolio’s characteristic function and numerical inversion, or simulate returns and fit a flexible approximation such as a moment-matched distribution or a kernel estimate. These approaches address dependence among assets; marginal distributions alone are insufficient. The text offers no numerical example or performance comparison, and notes that both the required integrals and numerical inversion can be difficult. Simulation-based estimates also depend on the quality of the input distribution and approximation.
Key ideas
- A portfolio return distribution depends on the joint distribution of asset returns and their dependence.
- The exact distribution of a weighted sum can be derived by integrating the joint distribution.
- A characteristic function offers another route, followed by numerical integration to recover the distribution.
- Simulation can support moment matching or other approximations, including kernel methods.
- The suggested methods can be computationally difficult and are not compared with an example.
Tags
Full text
# Is the portfolio return distribution a weighted combination of individual asset return distributions?
# Is the portfolio return distribution a weighted combination of individual asset return distributions?
We know that the portfolio expected return is a weighted sum of the individual assets' expected returns (asset means). We also know that the portfolio variance is a weighted combination of the individual assets' volatilities. More specifically, it's a quadratically weighted functional of the volatilities and covariances/correlations.
Is the portfolio return distribution $\text{P}(Xw)$ likewise some sort of weighted combination of the individual assets' return distributions? where $X\in \mathbb{R}^{n\times k}$ is the multivariate asset returns matrix, and $w\in \mathbb{R}^k$ is the portfolio weight vector whose elements are fractions that sum to 1, making $Xw$ the portfolio's $n\times 1$-shaped return series vector. by "distribution", interested in both the pdf and cdf cases.
## Answer by Kermittfrog (score 3)
https://quant.stackexchange.com/a/59750
As @Martin has pointed out in his answer, of course it is.
Let $X=\sum_{i=1}^N w_ix_i$ denote the return of a portfolio of $N$ assets with multivariate distribution $f(x_1,x_2,\ldots,x_N)$.
The distribution of $X$ may be found by $(N-1)$-fold convolution of the $N$-dimensional distribution $f$. Unfortunately, the integrals are not that easily solved anymore.
Another way to solve this is to try and find the characteristic function of your portfolio return. This is usually a bit less complicated and you can then use complex numerical integration to find the distribution - which is nasty in itself as well.
A third way is to simply simulate from your distribution and moment-match a flexible distribution or use some other form of distribution approximation (e.g. kernel methods).
HTH?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.