How Diversification Reduces Idiosyncratic Portfolio Risk
Summary
The document asks why adding assets reduces portfolio-specific risk in an arbitrage pricing theory setting. With equal weights and uncorrelated residuals, the portfolio’s residual variance falls as the number of holdings grows. For unequal weights, the variance is expressed as the sum of each asset’s residual variance multiplied by its squared weight; the cross-covariance terms vanish under the assumption that idiosyncratic shocks are uncorrelated across assets.
The answer appeals to zero-mean residuals and the law of large numbers: as the portfolio includes more assets, weighted residual shocks can offset one another. This intuition requires conditions on how weights behave as the portfolio grows; the response acknowledges that unequal weights need restrictions but does not specify them. It also moves from a variance expression to convergence of a weighted sum of residuals, concepts related to but not identical to one another. The document therefore gives a useful diversification intuition, while leaving the precise assumptions needed for a formal result unstated.
Key ideas
- With equal weights and uncorrelated residuals, adding assets reduces the portfolio’s idiosyncratic variance.
- For unequal weights, residual variance depends on squared weights multiplied by individual residual variances.
- The law of large numbers motivates why zero-mean idiosyncratic shocks can offset across holdings.
- Diversification with unequal weights requires restrictions on the weights for the convergence argument to apply.
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Full text
# Mathematically: How does increasing the number of assets reduce idiosyncratic risk?
# Mathematically: How does increasing the number of assets reduce idiosyncratic risk?
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 by Christian S. (score 1, accepted)
https://quant.stackexchange.com/a/25756
- APT assumes that idiosyncratic risk is zero on average: $E[e_i]=0$.
- 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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.