An investor lives for 2 periods and has the utility function u defined over the final wealth as u(w) The investor is born at t = 0 with wealth w0 and wants to choose a portfolio which maximizes her final wealth w1, which you will recognize is a random variable. There are two assets that the investor can invest in - (i) a bond which pays a sure return rf , and (ii) a stock which follows a Normal distribution N (, ?2 ).

1. Find the optimal portfolio allocation ? (the fraction of total wealth invested in stocks) when the utility function is given by u(w) = ?E(w1) ? ?^2/2*V (w1).
2. Next assume that the investor wants to minimize the value at risk for the portfolio. For example, suppose the portfolio follows a normal distribution N (, ?2 ). So, if she invests w0dollars today, the total wealth after 1 period follows the distribution w1 ? N (?w0( ? r) + rw0, ?^2w0^2?^2). The probability that the final wealth is a fraction (1 ? ?) of the initial wealth, that is w1 ? (1 ? ?)w0 is given by the Cumulative function ? [ (1 ? ?) ? ?( ? r) /?? ].Find the optimal portfolio which minimizes the value at risk.
3. Finally, assume that the investor must also eat something at time t = 0. Then she must set aside a certain consumption c0at time t = 0 and invests only w0 ?c0 in stocks or bonds. The total utility function that she maximizes now is u(w) = ?^2/2*c0^2 + ?* [?E(w1) ? ?^2/2*V (w1)]. Find the optimal consumption c0and the optimal portfolio allocation ? which maximizes the utility of the investor.

An investor lives for 2 periods and has the utility function a dened over the nal wealth as 11ij The investor is born at t = G with wealth mg and wants to choose a portfolio which maximizu her nal wealth ml, which you will recognize is a random variable. There are two assets that the investor can invest in - {i} a bond which pays a sure return ff, and [ii] a stock which follows a Normal distribution N (p, on). 1. Find the optimal portfolio allocation 6' {the fraction of total wealth invested in stocks) when the utility function is given by T2 aim) = crawl) 3?th Next assume that the investor wants to minimize the value at risk for the portfolio- For example, suppose the portfolio follows a normal distribution Nut, 0'2). So, if she invests wgdollars today, the total wealth after 1 period follows the distribution wl ~ N (3:000: r) + rum, zwgo'z] The probability that the nal wealth is a fraction (1 a) of the initial wealth, that is all 5 (1 (than is given by the Cumulative function 30' 45((1 -}-3[#-rll) Find the optimal portfolio which minimizes the value at risk. Finally, [this is the hard part) Assume that the investor must also eat something at time I = I]. Then she must set aside a certain consump- tion coat time I = {J and invests only wg an in stocks or bonds. The total utility function that she maximizes now is 2 2 aim) = gs: +13 7'13th %V(wl) Find the optimal consumption cuand the optimal portfolio allocation 6 which maximizes the utility of the investor