Suppose your initial wealth is 100 and you have to allocate it to a portfolio of a stock and a bank deposit. The bank deposit pays an interest rate of 10%. The stock can go up or down in value. It can go up 50% or down 20%, each with probability 1 /2. This means that if you invest x [0, 100] percent of your portfolio into the stock, your final wealth will be 1.5x + 1.1(100 x) with probability 1/2 and 0.8x + 1.1(100 x) with probability 1/2. Suppose your utility of final wealth is given by u(w) = ln(w) (ln - natural logarithm).

1) What is your risk attitude given your utility function?

2) What % of your portfolio would you invest in the stock so as to maximize your expected utility? (Hint: the derivative of ln(w) is 1/w.)

3) Now change the setup so that if the stock goes up, it actually doubles in value. Using the same analysis as in part (b), what % of your portfolio would you invest in the stock so as to maximize your expected utility? Interpret your finding in case you can borrow from the bank at the interest rate of 10% to finance your stock investment (this is called margin borrowing). But if you cannot borrow, what fraction of your portfolio would you invest into the stock?

4) Now, relative to the original setup, change the setup so that if the stock goes down, it actually halves in value. Using the same analysis as in part (b), what percentage of your portfolio would you invest in the stock so as to maximize your expected utility? Interpret your finding in case you can borrow the stock and sell it, hoping to repurchase it later at a lower price (this is called shorting the stock). But if you cannot short the stock, what fraction of your portfolio would you invest into the stock?