An equally weighted portfolio holds N assets, each with 30% annual volatility and a pairwise correlation of 0.3 with every other asset. What is the lowest volatility reachable as N grows without bound?

An equally weighted portfolio holds N assets, each with 30% annual volatility and a pairwise correlation of 0.3 with every other asset. What is the lowest volatility reachable as N grows without bound?

Approach: Write the equal weight variance as the average variance over N plus the average covariance times (1 - 1/N), then take N to infinity.

16.43%. The equal weight variance is sigma^2/N + (1 - 1/N)*rho*sigma^2, which tends to rho*sigma^2 = 0.3*0.09 = 0.027, so the floor is sqrt(0.027) = 16.43%. At N = 50 the variance is 0.09/50 + 0.98*0.027 = 0.0018 + 0.02646 = 0.02826 and the volatility is 16.81%, within 0.4 points of that limit, so the 51st name changes almost nothing. Only the idiosyncratic risk is removed by adding names. The common component implied by a pairwise correlation of 0.3 is systematic risk and no amount of diversification touches it, which is why the useful question about a book of 200 names is what the average correlation is rather than how many names it holds.

Follow-up: What average correlation would make a 50 name book as risky as a 10 name book at correlation 0.3?

Key concepts: systematic risk, idiosyncratic risk, pairwise correlation, diversification limit.