Question

In an economy there are only two assets, the market portfolio and the risk free rate with the following characteristics:

E(r_m)=10%, r_f= 6%     

σ_m = 18%       

The expected utility function is given by: E(U) = E(r) – 0.5 A σ² .

a) Find the optimal allocation for an individual with A=1.0

b) Find the optimal allocation for an individual with A=2.0

Answer 1

Given,

E(r_m)=10%,

σ_m = 18%

r_f= 6%

a). for A = 1

optimal allocated weight of risky asset can be calculated using

Wm = (E(r_m) - r_f)/(A*σ²_m)

So, Wm = (0.10-0.06)/(1*0.18²) = 1.2346 or 123.46%

And weight of risk free asset = 1-1.2346 = -0.2346 or -23.46%

This means that an individual with A=1 will borrow 23.46% of his investment at risk free rate and invest 123.46% of his investment amount in risky asset

b). for A = 2

optimal allocated weight of risky asset can be calculated using

Wm = (E(r_m) - r_f)/(A*σ²_m)

So, Wm = (0.10-0.06)/(2*0.18²) = 0.6173 or 61.73%

And weight of risk free asset = 1-0.6173 = 0.3827 or 38.27%

This means that an individual with A=2 will invest 38.27% of his investment at risk free rate and invest 61.73% of his investment amount in risky asset.