Question

Assume that you manage a risky portfolio (can be perceived as a portfolio that maximizes

the Sharpe Ratio) with an expected rate of return of 22% and a standard deviation of 35%.
The T-bill rate is 6%.Your client’s degree of risk aversion is γ = 2.6, assuming a quadratic
utility function.
a. What proportion, y, of the total investment should be invested in your fund?
b. What is the expected value and standard deviation of the rate of return on your client’s
optimized portfolio?

Answer 1

Approximately 7.69% of the total investment should be invested in the risky portfolio. By substituting the calculated proportion and given values, it is determined that the expected value is approximately 1.69% and the standard deviation is approximately 2.69%.

Step-by-step explanation:

The proportion of the total investment, y, that should be invested in the risky portfolio can be determined using the formula:

y = (μ - Rf) / (γ · σp)

1. Given: Expected rate of return (μ) = 22%, Risk-Free rate (Rf) = 6%, Standard deviation (σp) = 35%, and Degree of risk aversion (γ) = 2.6
2. Substitute the values into the formula:
3. Therefore, approximately 7.69% of the total investment should be invested in the risky portfolio.

To find the expected value and standard deviation of the rate of return on the client's optimized portfolio, apply the following formulas:

Expected value: E(R) = y · μ

Standard deviation: σ(R) = y · σp

1. Substitute the values into the formulas:
   
2. Therefore, the expected value of the rate of return on the client's optimized portfolio is approximately 1.69% and the standard deviation is approximately 2.69%.

Answer 2

The optimal proportion of the total investment to be allocated to the risky portfolio is 50.2%.

The expected return of the client's optimized portfolio is 14.0%.

The standard deviation of the client's optimized portfolio is 17.6%.

How is that so?

Optimal Portfolio Allocation for Risk-Averse Investor

Given:

• Risky portfolio expected return: μ_f = 22%
• Risky portfolio standard deviation: σ_f = 35%
• Risk-free rate: μ_rf = 6%
• Client's risk aversion coefficient: γ = 2.6
• Utility function: Quadratic

To find:

a) Optimal Proportion (y):

In the case of a quadratic utility function and a risk-averse investor (γ > 0), the optimal proportion y of investment in the risky portfolio can be calculated using the following formula:

y = (μ_f - μ_rf) / (γ * σ_f²)

Substituting the given values:

y = (0.22 - 0.06) / (2.6 * 0.35²) ≈ 0.502

Therefore, the optimal proportion of the total investment that should be invested in the risky portfolio is approximately 50.2%.

b) Expected Return and Standard Deviation of Client Portfolio:

Once we have the optimal proportion y, we can calculate the expected return (μ_p) and standard deviation (σ_p) of the client's optimized portfolio using the following formulas:

Expected Return:

μ_p = μ_rf + y * (μ_f - μ_rf)

Substituting the values:

μ_p = 0.06 + 0.502 * (0.22 - 0.06) ≈ 0.140

Standard Deviation:

σ_p = σ_f * √(y)

Substituting the values:

σ_p = 0.35 * √(0.502) ≈ 0.176

Therefore, the expected return of the client's optimized portfolio is approximately 14.0%, and the standard deviation is approximately 17.6%.