## Description

**Question:**

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.

**Answer:**

**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:

U_{A} = µ_{A} – 0.5 * σ^{2}_{A} * A

And utility function in case of asset B will be:

U_{B} = µ_{B} – 0.5 * σ^{2}_{B} * A

Now for an investor to be indifferent between asset A and B, utility function will have to yield same result, so

µ_{A} – 0.5 * σ^{2}_{A} * A = µ_{B} – 0.5 * σ^{2}_{B} * A

µ_{A –} µ_{B = 0.5 * A * (}σ^{2}_{A –} σ^{2}_{B)}

_{A = 2 (}µ_{A –} µ_{B)/ (}σ^{2}_{A –} σ^{2}_{B)}

_{Hence proved.}

**b)** From the above calculated formula we can now calculate “risk aversion coefficient”:

A = 2 * (.10 – .05)/((.2)^{2} – (.15)^{2}) = 5.714

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