Mathematically: How does increasing the number of assets reduce idiosyncratic risk?

Question

As part of an Asset Pricing Module I'm currently taking, whilst looking at APT Ross (1974), we looked at how according to this model, risk originates from both systematic and idiosyncratic asset specific sources.

We first considered an N asset portfolio with equal weights to show how increasing N assets decreases the idiosyncratic (eP - here i am calling it the residual error term e of the portfolio P) residual variances:

Var(e) = (1/N)*(Average Sigma e)

It is clear to see that as N increases, the Variance of e decreases.

However, my question is for the case where N asset are held, but not in equal proportions. We end up with the following expression for the Var(eP):

Var(eP) = [(Summation from i=1 to N) (wi)^2 * (Sigma ei)] + All Covariance Terms

From our assumptions at the outset, idiosyncratic risks of say asset i don't affect asset j, so all the second terms from the above equation equal 0.

My question, in the below equation where wi is equal to the weight of asset i:

Var(eP) = [(Summation from i=1 to N) (wi)^2 * (Sigma ei)]

How can we see here that increasing N reduces idiosyncratic risk?

Thanks

Answer 1

1. APT assumes that idiosyncratic risk is zero on average: $E[e_i]=0$.
2. The law of large numbers.

From 1 and 2 it follows that as N increases, the weighted sum of idiosyncratic risks will converge to zero:

$\lim\limits_{N\to\infty}\sum\limits_{i=1}^N e_p=\lim\limits_{N\to\infty}\sum\limits_{i=1}^N w_ie_i=0$

Strictly speaking some restrictions on the weights would be needed in case the weights are unequal to $\frac{1}{N}$ (see this for example), but the above is what it comes down to.