Question

You manage a risky portfolio with an expected rate of return of 17% and a standard deviation of 37%. The T-bill rate is 5%.

Suppose that your client prefers to invest in your fund a proportion y that maximizes the expected return on the complete portfolio subject to the constraint that the complete portfolio’s standard deviation will not exceed 17%.
a. What is the investment proportion, y?
b. what is the expected rate of return on the complete portfolio?

Answer 1

The investment proportion y is approximately 45.95%, and the expected rate of return on the complete portfolio is about 10.45%.

Step-by-step explanation:

To determine the investment proportion y for a client in a mixed portfolio of a risky asset and a risk-free asset, we can use the formula for the standard deviation of a two-asset portfolio: \[\sigma_{p} = y \sigma_{risky}\]

Given that the standard deviation of the complete portfolio should not exceed 17% and the standard deviation of the risky portfolio is 37%, set the equation: \[0.17 = y \times 0.37\]. Solving for y gives us: \[y = \frac{0.17}{0.37}\], which is approximately 0.4595 or 45.95%. Therefore, the investment proportion y is 45.95%.

To calculate the expected rate of return on the complete portfolio, we use the formula: \[ E(R_p) = y E(R_risky) + (1 - y) R_{T-bill} \]. Plugging in the given values gives us: \[ E(R_p) = 0.4595 \times 0.17 + (1 - 0.4595) \times 0.05 \], which evaluates to approximately 10.45%.

Answer 2

To constrain the portfolio's standard deviation to not exceed 17%, approximately 45.95% should be invested in the risky portfolio, and the rest in the risk-free asset. The expected rate of return of this complete portfolio is then approximately 10.11%.

Step-by-step explanation:

To solve this problem, we would typically use the theory of portfolio optimization, most commonly based on the Capital Asset Pricing Model (CAPM). However, the question provides us with some shortcuts for reaching our solution without having to dig deeply into the CAPM model. The expected rate of return for the risky portfolio is given as 17%, and the risk (as quantified by the standard deviation) is 37%. The risk-free rate, which in this case is the T-bill rate, is 5%.

When combining a risk-free asset and a risky portfolio, the combined standard deviation (σp) is simply y times the standard deviation of the risky portfolio because the risk-free asset has a standard deviation of 0. So we can set up the equation y × 37% = 17% to find out y, the investment proportion in the risky portfolio.

y = 17% / 37% = 0.4595

Thus, y, the investment proportion in the risky portfolio that will make the complete portfolio's standard deviation to not exceed 17%, is approximately 45.95%.

Next, to find the expected rate of return on the complete portfolio we use the following equation:

Expected return on complete portfolio = y × Expected return on risky portfolio + (1 - y) × Risk-free rate

Substitute y = 0.4595, Expected return on risky portfolio = 17%, Risk-free rate = 5%:

Expected return on complete portfolio = 0.4595 × 17% + (1 - 0.4595) × 5%

Expected return on complete portfolio ≈ 10.11%

The expected rate of return on the complete portfolio, when the investment proportion is optimized to limit the risk, is approximately 10.11%.