hard · Quantitative Finance
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 always eliminates risk in the limit
- mathrmVar(bar X)to 1-ρ, the smallest eigenvalue of the equicorrelation matrix
- mathrmVar(bar X)toρ/nto 0, the off-diagonal contribution vanishing faster than the diagonal
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