Variance of an equal-weight portfolio of nine assets

Question

You spread money equally across 9 independent assets whose returns are identically distributed, each with variance 0.09 (standard deviation 0.3). The portfolio return is the equal-weight average of the 9 asset returns.

What are the variance and standard deviation of the portfolio return?

Accepted Answer

An equal-weight portfolio return is the average of the individual returns, so its spread follows the averaging rule for independent, identically distributed pieces: \operatorname{Var}(	ext{portfolio}) = \frac{\sigma^2}{n} = \frac{0.09}{9} = 0.01, \qquad 	ext{sd} = \frac{\sigma}{\sqrt{n}} = \frac{0.3}{3} = 0.1. You can also see it weight by weight: each asset has weight 1/9, contributing (1/9)^2 	imes 0.09, and 9 such terms give 9 	imes (0.09/81) = 0.01. Diversifying across independent assets shrinks variance by the factor 1/n. Takeaway. Equal weighting across 9 independent assets is just …

Variance of an equal-weight portfolio of nine assets

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