Why Portfolio Return Density Is Not a Weighted Average of Asset Densities

Summary

The document asks whether the probability density of portfolio returns can be obtained by weighting and adding the return densities of its constituent assets. It starts from the familiar expression for portfolio return as a weighted sum of asset returns, with an additional residual term, then proposes a mixture-style density formula and asks whether it is correct.

The key distinction is between a weighted sum of random variables and a mixture that selects one asset's return distribution at random. A portfolio holds exposures to multiple assets at once, so its return distribution depends on their joint behavior, including dependence or covariance, as well as weights and any residual component. The prompt does not include an answer, corrected formula, data, or a specific portfolio construction. It therefore frames a probability-modeling question rather than demonstrating a particular density estimate; a simple weighted average of marginal densities is generally not enough to describe portfolio returns.

Key ideas

• Portfolio returns are formed by combining asset returns according to portfolio weights.
• A weighted sum of random variables is different from a mixture that selects one component distribution.
• The portfolio return density depends on the joint distribution of constituent returns, including their dependence.
• The proposed expression is not established by the linear formula for expected portfolio return alone.
• The document poses the question without giving a corrected formula or empirical example.

Tags

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


# 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?

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