Let X 1 , ...,X n in Exercise 60 be independent, all expectations E{X i } =
Question:
Let X1, ...,Xn in Exercise 60 be independent, all expectations E{Xi} = m, but the variances be different, and Var{Xi} = σ2i . 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, ...,Xn. 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, ...,Xn are i.i.d., E{Xi} = m, Var{Xi} = 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.
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