Diversification Effect

Decompose the variance of an equally-weighted portfolio into idiosyncratic and systematic risk. What happens as $n\to\infty$ ? What does this mean?

Answer

Let $R_1,\dots,R_n$ be asset returns with $\mathrm{Var}(R_i)=\sigma^2$ and $\mathrm{Cov}(R_i,R_j)=\tau$ for $i\neq j$ . For the equally-weighted portfolio $R_p=\frac{1}{n}\sum_{i=1}^n R_i$ ,

\mathrm{Var}(R_p) =\frac{1}{n^2}\left(n\sigma^2+n(n-1)\tau\right) =\frac{\sigma^2}{n}+\frac{n-1}{n}\tau.

As $n\to\infty$ ,

\mathrm{Var}(R_p)\to \tau.

Meaning: As you diversify, the $\sigma^2/n$ term goes to $$0$$ (idiosyncratic risk is diversified away), while the $\frac{n-1}{n}\tau$ term tends to $\tau$ (systematic risk remains).