medium · Quantitative Finance prob-stats
A portfolio's return is the sum S=sum_i=1^n X_i of n asset returns, each with unit variance and a common pairwise correlation ρ (an equicorrelation matrix).
As nto∞, what happens to the variance of the equally weighted average return bar X=S/n?
- mathrmVar(bar X)=ρ+(1-ρ)/ntoρ, so diversification cannot drive risk below the common-correlation floor ρ
- mathrmVar(bar X)=1/nto 0, since averaging n unit-variance returns eliminates all risk in the limit regardless of ρ
- mathrmVar(bar X)to 1-ρ, the smallest eigenvalue shared by the equicorrelation matrix's off-diagonal block
- mathrmVar(bar X)toρ/nto 0, since the off-diagonal covariance term always vanishes faster than the diagonal one
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