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Consider I have $2$ returns time series random variables $r_{i,t}=\ln{\left(\frac{p_{i,t+1}}{p_{i,t}} \right)}\stackrel{(i.d.)}{\sim}\mathcal{N}(\mu_i,\sigma_i)$.

Let's now compute a weighted average:

$$ R_t=\frac{\sum_{i=1}^{2}{w_{i,t}r_{i,t}}}{\sum_{i=1}^{2}{w_{i,t}}} $$

  • How would the distribution of $R_t$ look like if $w_{i,t}=p_{1,t}$ i.e. if we are doing a price-weighted average? Can we say something about $\mathrm{E}[r]$ and $\mathrm{Var}[r]$ for example?
  • And if instead $w_{i,t}=w=0.5$ i.e. we are compute a simple average?

Any insight is really appreciated.

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The numerator in your expression for $R_t$ is the logarithm of a weighted sum of lognormal random variables. The distribution of this quantity is complicated, but it has some known approximations (see e.g., Cobb and Rumi 2012). You could generate the distribution by simulation, or rely on one of the approximation methods in the linked paper. The logarithm of a weighted sum of lognormal random variables will be roughly (but not exactly) normally distributed. The moments can be found by simulation, or by the use of the characteristic function.

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