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 X, 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 – 1 probability of earning an interest rate rr = 1. Assuming that the agent has a utility function u(x) = log(x), answer the following:

Answer 1

The agent needs to solve the following maximization problem to obtain the optimal investment portfolio:

Maximize π. u((1+r_{​​​​​​r})x_{​​​​​​r}+ x_{​​​​​​s} )+ (1-π).u((1+r_{​​​​​​r})x_{​​​​​​r}+ x_​​​s )

= π.ln(x_{​​​​​​s} ) + (1-π).ln(2x_r + x_{​​​​​​s} )

Now W= x_{​​​​​​s} + x_{​​​​​​r}

x_{​​​​​​r=} W - x_{​​​​​​s} . Now, substituting this in the agent's problem and taking first order derivative w.r.t x_{​​​​​​s} , we get:

π/x_{​​​​​​s} - (1-π)/(2W - x_{​​​​​​s} ) = 0

We get x_{​​​​​​s} = 2πW and x_{​​​​​​r} = W(1-2π)