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An investor has utility of wealth function u(W ) = ' exp{'W }. Suppose the investor can invest their wealth in either a risk-free asset (e.

3. An investor has utility of wealth function u(W ) = − exp{−ηW }. Suppose the investor can invest their wealth in either a risk-free asset (e.g. a government bond) paying a return R = (1 + r) or a risky asset (e.g. a mutual fund) paying a random return Z ̃ which I assume is normally distributed with mean μ and variance σ2. This consumer can invest all or part of his/her wealth in the risk-free asset, and the risky asset. Assume that the investor decides to allocate a fraction θ ∈ [0,1] of his/her wealth to the risk-free asset and the remaining fraction 1 − θ to the risky asset. Using calculus, determine a formula for the optimal fraction of the person’s wealth, 1 − θ∗ , to be allocated to the risky asset and describe how this fraction changes as a function of W, R, η, μ and σ2. (Hint: if Z ̃ is a normally distributed random variable then the moment generating function, the expectation of exp{tZ ̃} for any scalar t is given by:
􏰂 ̃􏰃Z+∞ 1 22 22
E exp{tZ} = exp{tz}√ exp{−(z−μ) /2σ }dz = exp{tμ+t σ /2}
−∞ 2π
Use this formula to compute the expected utility of a portfolio in which θ% of the person’s wealth W is
invested in the riskless asset and 1 − θ% is invested in the risky asset, i.e. use the formula above to find 1

an expression for
E􏰂u(W[θR+(1−θ)Z ̃])􏰃=E􏰂−exp{−ηW[θR+(1−θ)Z ̃]}􏰃
and then use calculus to find a formula for the optimal θ∗ and 1 − θ∗ by taking the derivative of expected
utility with respect to θ and setting it equal to 0 and solve for θ∗ or 1 − θ∗).
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