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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?

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  • 2
    $\begingroup$ No, this is the domain of copulas $\endgroup$ – Chris Jul 15 '20 at 3:57
  • $\begingroup$ what is the corresponding copula analogue of the firat linear equation shown above? $\endgroup$ – develarist Jul 15 '20 at 15:24

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