# Density of a portfolio's returns is the weighted average of asset distributions?

The expected return of a portfolio can be formulated as a weighted average of the constituent assets' returns:

$$r_p = w_1 r_1 + w_2 r_2 + \dots + w_N r_N + \epsilon$$

Does it also follow that the empirical distribution, or density, of a portfolio's return series (a vector with observations $$x_{t=1}, x_{t=2},$$ etc, whose expected value is $$r_p$$ above) is just a weighted average of the constituent assets' return distributions?

$$f(r_p) = Pr(x_t) f(x_t|r_1) + Pr(x_t) f(x_t|r_2) + \dots + Pr(x_t) f(x_t|r_N)$$

where $$f(r_p)$$ is the portfolio's return density function such as a Gaussian, and $$Pr(x_t) f(x_t|r_N)$$ is the probability or likelihood that a datapoint came from asset $$n$$'s distribution.

If not, how should the probability formula be corrected?

• No, this is the domain of copulas – Chris Jul 15 '20 at 3:57
• what is the corresponding copula analogue of the firat linear equation shown above? – develarist Jul 15 '20 at 15:24