Let X₁, ...,X_n in Exercise 60 be independent, all expectations E{X_i} = m, but the variances be different, and Var{X_i} = σ²_i . How would you distribute the money between the assets adopting the variance as a measure of riskiness? Would it influence the expected return of your investment? Find the minimal variance if the total investment equals one.

Exercise 60

There are n assets with random returns X₁, ...,X_n. The term “return” means that, if you invest $1 in, say, the first asset, you will get an income of X₁ dollars. Note that a return may be less than one.

Assume that X₁, ...,X_n are i.i.d., E{X_i} = m, Var{X_i} = σ². Consider two strategies of investing n units of money: either investing the whole sum into one asset, for example, into the first, or distributing the investment sum equally between n assets. For both strategies, compute

The expected total return, i.e., the return per unit of money; and

The standard deviation of the return. Proceeding from what you got, compare the two strategies.