Question

4) Consider an agent with initial wealth W to be allocated between a safe (risk-free) asset and a risky asset. Let xs denote the quantity (dollars) of wealth invested in the safe asset. Let Xr denote the quantity (dollars) of wealth invested in the risky asset. The safe asset has an interest rate of rs = 0. The risky asset has a probability n of earning an interest rate rr = -1 and a 1 – a probability of earning an interest rate rr = 1. Assuming that the agent has a utility function u(x) = log(x), answer the following: 4.1) Before solving for xs and X, using math, what do we know must be true if it > 1/2? What about if it < 1/2?

Answer 1

4.1) Here, we should determine first the significance of $\pi$ .

$\pi$ is the probability of earning negative returns (r_r=-1) . Therefore $\pi$ is the probability of loss.

Let us consider two ideal cases. If $\pi$ =1, the return of the risky asset is sure to be -1, which is less than the safe asset. Therefore, the risky asset should be avoided altogether. If $\pi$ =0, the return of the risky asset is sure to be +1, which is more than that of the safe asset. Therefore, all the wealth should be invested in the risky asset. Now consider the two cases given here:

If $\pi$ >1/2, there is more probability of a negative outcome. Therefore, the investment in the risky asset should be less than half to minimize risk and maximize returns.

Similarly, if $\pi$ <1/2, there is less probability of a negative outcome and more probability of positive outcome. Therefore, the investment in the risky asset should be more than half to minimize risk and maximize returns.