An investor who believes the economy is slowing down wishes to reduce the risk of her portfolio. She currently owns 12 securities, each with a market value of $3,000. The current beta of the portfolio is 1.21, and the beta of the riskiest security is 1.62. What will the portfolio beta be if the riskiest security is replaced with a security of equal market value but a beta of 0.80?
A) 1.14
B) 1.18
C) 1.05
D) 1.10

Question

Quizplus · Accepted Answer

First, calculate the current portfolio beta using the formula:

Portfolio beta = Sum of (security beta x security weight)

Assuming all securities have equal weights, the current portfolio beta can be calculated as:

1.21 = (12 x 1.62 x 0.0833) + (11 x 1.21 x 0.0833) + … + (1 x beta x 0.0833)

Simplifying the equation, we get:

1.21 = (12 x 1.62 x 0.0833) + (11 x 1.21 x 0.0833) + … + (1 x beta x 0.0833)
1.21 = 1.62 x 0.10 + 1.21 x 0.10 + … + beta x 0.0833
1.21 = 0.162 + 0.121 + … + 0.0833beta
1.21 = 1.95 + 0.0833beta
0.74 = 0.0833beta
beta = 8.88

This means the current portfolio is very risky, as the overall beta is much higher than the market average of 1.0.

To reduce the risk, we need to replace the riskiest security (with a beta of 1.62) with a less risky one (with a beta of 0.80).

Assuming equal weights, the new portfolio beta can be calculated as:

New portfolio beta = Sum of (security beta x security weight)

However, since we are only replacing one security, the new portfolio beta can be simplified as:

New portfolio beta = (Current portfolio beta x Total portfolio value – Value of riskiest security) / (Total portfolio value – Value of riskiest security + Value of new security)

Plugging in the values, we get:

New portfolio beta = (1.21 x 12 x 3000 – 1.62 x 3000) / (12 x 3000 – 3000 + 3000)
New portfolio beta = (41,220 – 4,860) / 33,000
New portfolio beta = 1.14

This means that replacing the riskiest security with a less risky one will lower the overall risk of the portfolio, as the new portfolio beta is closer to the market average.

An Investor Who Believes the Economy Is Slowing Down Wishes