Question

Two risky portfolios exist for investing: one is a bond portfolio with a beta of 0.6 and an expected return of 7.1% and the other is an equity portfolio with a beta of 1.2 and an expected return of 15.8%. If these portfolios are the only two available assets for investing, what combination of these two assets will give the following investors their desired level of expected return? What are the betas of each investor's combination of the bond and equity portfolio?

a. Bart: Desired expected return 15%
b. Lisa: Desired expected return 13%
c. Maggie: Desired expected return 11%

Answer 1

To achieve their desired expected returns, Bart, Lisa, and Maggie must allocate 87%, 65%, and 43% of their portfolios to the equity portfolio, respectively. Their resulting portfolio betas would be approximately 1.11 for Bart, 0.96 for Lisa, and 0.86 for Maggie.

Step-by-step explanation:

To determine the combination of the two assets that will give the desired expected return for Bart, Lisa, and Maggie, we use the following formula to find the weight (w) of the equity portfolio in each combination:

Desired return = w * Return on equity + (1 - w) * Return on bonds

For Bart:
15% = w * 15.8% + (1 - w) * 7.1%
w = (15% - 7.1%) / (15.8% - 7.1%)
w ≈ 0.87 (87% in equity, 13% in bonds)

For Lisa:
13% = w * 15.8% + (1 - w) * 7.1%
w = (13% - 7.1%) / (15.8% - 7.1%)
w ≈ 0.65 (65% in equity, 35% in bonds)

For Maggie:
11% = w * 15.8% + (1 - w) * 7.1%
w = (11% - 7.1%) / (15.8% - 7.1%)
w ≈ 0.43 (43% in equity, 57% in bonds)

To find each investor's beta, we apply the formula:

Beta of combination = w * Beta of equity + (1 - w) * Beta of bond

Using the weights calculated above, we get:

• Bart's beta ≈ 0.87 * 1.2 + 0.13 * 0.6 ≈ 1.11
• Lisa's beta ≈ 0.65 * 1.2 + 0.35 * 0.6 ≈ 0.96
• Maggie's beta ≈ 0.43 * 1.2 + 0.57 * 0.6 ≈ 0.86

Therefore, each investor can adjust their portfolio allocation to achieve their desired expected return and associated beta.