Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 23 / 1 / a

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- Solution a

Solution

The no-borrowing portfolio constraint is

0 \leq x \leq T

. Net asset returns must be added to the original principal. Thus the two final portfolio wealths are

w_{g} = x (1 + r) + (T - x) (1 + g) = T (1 + g) - (g - r) x, w_{b} = x (1 + r) + (T - x) (1 + b) = T (1 + b) + (r - b) x .

The expected utility is

U (x) = p u (w_{g}) + (1 - p) u (w_{b}) .

The probabilities are those of the investor's beliefs; no risk-neutral measure is being used in this preference calculation.

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