Consider a risk averse investor with utility function U(W) = √W

Consider a risk averse investor with utility function U(W) = √W who is deciding how much of her initial wealth (Wo) to invest in a bond and how much to invest in a stock. The current prices of bond and stock are (Bo) and (So) respectively. Although neither security pays dividend or interest. Investor A expects to receive income from selling these securities at their end-of-period prices, which are B1 for the bond and S1 for the stock. Since the bond is riskless, its end-of-period prices is known with certainty to be B1 = Bo(1+r), where r is the riskless rate of interest. The price of stock at t =1 can be high or low; i.e., it will be So(1+s) with probability of 0.6 and it will be So(1-S) with probability of 0.4. Furthermore, assume that Wo= 100, r = .05 and s = 3.

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1. How much of the investor’s initial wealth should be invested in the stock, and how much in the bond?
2. What will be the investor’s expected wealth and standard deviation of wealth at t = 1 from this investment strategy?
3. Suppose that this investor starts out with initial wealth of $200 rather than $100. In this case, what proportion of her initial wealth should be invested in the stock, and how much in the bond?