3. An investor has utility of wealth function u(W ) = âˆ’ exp{âˆ’Î·W }. 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 Ìƒ which I assume is normally distributed with mean Î¼ and variance Ïƒ2. 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 Î¸ âˆˆ [0,1] of his/her wealth to the risk-free asset and the remaining fraction 1 âˆ’ Î¸ to the risky asset. Using calculus, determine a formula for the optimal fraction of the personâ€™s wealth, 1 âˆ’ Î¸âˆ— , to be allocated to the risky asset and describe how this fraction changes as a function of W, R, Î·, Î¼ and Ïƒ2. (Hint: if Z Ìƒ is a normally distributed random variable then the moment generating function, the expectation of exp{tZ Ìƒ} for any scalar t is given by:

ô°‚ Ìƒô°ƒZ+âˆž 1 22 22

E exp{tZ} = exp{tz}âˆš exp{âˆ’(zâˆ’Î¼) /2Ïƒ }dz = exp{tÎ¼+t Ïƒ /2}

âˆ’âˆž 2Ï€

Use this formula to compute the expected utility of a portfolio in which Î¸% of the personâ€™s wealth W is

invested in the riskless asset and 1 âˆ’ Î¸% is invested in the risky asset, i.e. use the formula above to find 1

ï¿¼ï¿¼

an expression for

Eô°‚u(W[Î¸R+(1âˆ’Î¸)Z Ìƒ])ô°ƒ=Eô°‚âˆ’exp{âˆ’Î·W[Î¸R+(1âˆ’Î¸)Z Ìƒ]}ô°ƒ

and then use calculus to find a formula for the optimal Î¸âˆ— and 1 âˆ’ Î¸âˆ— by taking the derivative of expected

utility with respect to Î¸ and setting it equal to 0 and solve for Î¸âˆ— or 1 âˆ’ Î¸âˆ—).

ô°‚ Ìƒô°ƒZ+âˆž 1 22 22

E exp{tZ} = exp{tz}âˆš exp{âˆ’(zâˆ’Î¼) /2Ïƒ }dz = exp{tÎ¼+t Ïƒ /2}

âˆ’âˆž 2Ï€

Use this formula to compute the expected utility of a portfolio in which Î¸% of the personâ€™s wealth W is

invested in the riskless asset and 1 âˆ’ Î¸% is invested in the risky asset, i.e. use the formula above to find 1

ï¿¼ï¿¼

an expression for

Eô°‚u(W[Î¸R+(1âˆ’Î¸)Z Ìƒ])ô°ƒ=Eô°‚âˆ’exp{âˆ’Î·W[Î¸R+(1âˆ’Î¸)Z Ìƒ]}ô°ƒ

and then use calculus to find a formula for the optimal Î¸âˆ— and 1 âˆ’ Î¸âˆ— by taking the derivative of expected

utility with respect to Î¸ and setting it equal to 0 and solve for Î¸âˆ— or 1 âˆ’ Î¸âˆ—).

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