We consider 2 risky assets A and B whose mean returns and standard deviations are µA, µB, σA, and σB respectively. If µA ≥ µB and σA ≤ σB, and at least one inequality is strict, then any investor prefers A to B. If µA > µB and σA > σB, then the choice of the best risky asset depends on the investor’s risk aversion coefficient. In this case there is an investor (defined by a specific risk aversion coefficient) who is indifferent between A and B. a) Prove that in this case the risk aversion coefficient can be computed using the following formula A = 2 × µA − µB σ 2 A − σ 2 B b) Suppose that µA = 10%, µB = 5%, σA = 20%, and σB = 15%. Compute the risk aversion coefficient of the investor who is indifferent between A and B.
a) We can calculate the utility function based on below formula:
U = µ – 0.5 * σ 2* A
Where A is risk aversion coefficient.
So utility function in case of asset A will be:
UA = µA – 0.5 * σ2A * A
And utility function in case of asset B will be:
UB = µB – 0.5 * σ2B * A
Now for an investor to be indifferent between asset A and B, utility function will have to yield same result, so
µA – 0.5 * σ2A * A = µB – 0.5 * σ2B * A
µA – µB = 0.5 * A * (σ2A – σ2B)
A = 2 (µA – µB)/ (σ2A – σ2B)
b) From the above calculated formula we can now calculate “risk aversion coefficient”:
A = 2 * (.10 – .05)/((.2)2 – (.15)2) = 5.714