There are n mutual funds and m asset categories. Let aij be the fraction of the capital of fund j invested in category i. We have initial capital C and we want to invest all or part of it in these funds. We denote by xj the amount invested in fund j = 1, . . . , n. No short selling is allowed, so xj has to be nonnegative.
(a) Describe the set X ⊂ Rn of all possible amounts invested in these funds (fund portfolios). What are its extreme points?
(b) Describe the set Y ⊂ Rm of all possible amounts invested in this way in the m asset categories (asset portfolios).
(c) Show that if a point y is an extreme point of Y , it has the form y = Ax, where x is an extreme point of X.
(d) Suppose y ∈ Y is an asset portfolio obtained by investing in some of the available funds. Prove that you can construct it by investing in no more than m + 1 funds.
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