Question

Using CAPM, a stock has a beta of 1.13 and expected return of 12.1%. Risk-free asset currently earns 3.6%.

(a) what is the expected return on a portfolio that is equally invested in the two assets?

(b)   A portfolio of two assets has a beta of 0.50, what are portfolio weights?

(c)    If a portfolio of two assets has an expected return of 10%, what is beta?

(d)   If a portfolio of two assets has a beta of 2.26, what are portfolio weights?

Answer 1

Explanation:

part a):

Weight of Stock = 0.50
Expected Return of Stock = 12.10%
Weight of Risk-free Asset = 0.50
Expected Return of Risk-free Asset = 3.60%

Expected Return of Portfolio = Weight of Stock * Expected Return of Stock + Weight of Risk-free Asset * Expected Return of Risk-free Asset
Expected Return of Portfolio = 0.50 * 12.10% + 0.50 * 3.60%
Expected Return of Portfolio = 7.85%

Part b:

Let Weight of Stock be x and Weight of Risk-free Asset be (1 - x)

Portfolio Beta = Weight of Stock * Beta of Stock + Weight of Risk-free Asset * Beta of Risk-free Asset
0.50 = x * 1.13 + (1 - x) * 0.00
0.50 = x * 1.13
x = 0.4425

Weight of Stock = 0.4425 or 44.25%

Weight of Risk-free Asset = 1 - Weight of Stock
Weight of Risk-free Asset = 1 - 0.4425
Weight of Risk-free Asset = 0.5575 or 55.75%

Part c:

Let Weight of Stock be x and Weight of Risk-free Asset be (1 - x)

Expected Return of Portfolio = Weight of Stock * Expected Return of Stock + Weight of Risk-free Asset * Expected Return of Risk-free Asset
0.10 = x * 0.1210 + (1 - x) * 0.0360
0.10 = x * 0.1210 + 0.0360 - x * 0.0360
0.0640 = x * 0.0850
x = 0.7529

Weight of Stock = 0.7529 or 75.29%

Weight of Risk-free Asset = 1 - Weight of Stock
Weight of Risk-free Asset = 1 - 0.7529
Weight of Risk-free Asset = 0.2471 or 24.71%

Portfolio Beta = Weight of Stock * Beta of Stock + Weight of Risk-free Asset * Beta of Risk-free Asset
Portfolio Beta = 0.7529 * 1.13 + 0.2471 * 0.00
Portfolio Beta = 0.85

part d:

Let Weight of Stock be x and Weight of Risk-free Asset be (1 - x)

Portfolio Beta = Weight of Stock * Beta of Stock + Weight of Risk-free Asset * Beta of Risk-free Asset
2.26 = x * 1.13 + (1 - x) * 0.00
2.26 = x * 1.13
x = 2.00

Weight of Stock = 2.00 or 200%

Weight of Risk-free Asset = 1 - Weight of Stock
Weight of Risk-free Asset = 1 - 2.00
Weight of Risk-free Asset = -1.00 or -100%

Negative weight of risk-free asset means that you have to borrow funds at risk-free rate and invest it in stock.

Answer to A.

Expected Return of Portfolio = 7.85%

Answer to B

Weight of Risk-free Asset = 0.5575 or 55.75%

Answer to C

Portfolio Beta = 0.85

Answer to D

Weight of Risk-free Asset = -1.00 or -100%