# 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.

• We also know that the portfolio variance is a weighted combination of the individual assets' volatilities. I guess you forgot the covariances... Oh, you did not: More specifically, it's a quadratically weighted functional of the volatilities and covariances/correlations. OK, so you are in conflict with yourself there. Dec 4 '20 at 19:07

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?

• wish I could see a derivation of a portolio return distribution's characteristic function based on a convolution of Gaussin or t-distributions. let me know if there's a source demonstrating how this can be derived. In the same way that we have formulas for the portfolio expected return and portfolio variance being weighted combinations of their inputs, my question really was trying to draw out a similar formulation of a weighted-pdf/cdf combo Dec 4 '20 at 11:02
• For the multivariate normal, you can directly look it up (en.wikipedia.org/wiki/Multivariate_normal_distribution). The multivariate $t$ distribution is quite complicated (at least for me), but you might be able to find something online, e.g. here csam.or.kr/journal/… Dec 4 '20 at 11:10
• alot of the probability genres, like concentration of measure, and convolutions, tend to always assume the marginals are independent though, which is not the case for asset returns. your link included even says "The [characteristic functions] allow us to easily manipulate convolved probability density functions (pdfs) when they represent linear combinations of independent random variables." Dec 4 '20 at 11:13
• For normals, cou can de-correlate the variables using the cholesky matrix. Then, you have loadings on independent RVs. This 'trick' can be applied in various applications. Ultimatively, you always need to know how to disentangle your RVs... Dec 4 '20 at 12:08
• is it easy to extract the weight vector from a convolution? for example scipy convolve1d or numpy.convolve (the latter function is restricted to only 2 components) Dec 4 '20 at 12:15