Math Problem Statement

You own a portfolio equally invested in a risk-free asset and two stocks. One of the stocks has a beta of 1.16 and the total portfolio is equally as risky as the market. What must the beta be for the other stock in your portfolio?

Note: Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.

Solution

To determine the beta for the other stock, we use the fact that the total portfolio beta is equally as risky as the market (beta = 1). Here's how to solve:

Key Points:

  1. Portfolio is equally invested in:
    • Risk-free asset (beta = 0)
    • Stock A (beta = 1.16)
    • Stock B (unknown beta = βB\beta_B)
  2. The weights of the three components are equal, i.e., w=13w = \frac{1}{3} for each.
  3. Total portfolio beta is the weighted average of the betas of its components.

Portfolio Beta Formula:

βportfolio=w1β1+w2β2+w3β3\beta_{\text{portfolio}} = w_1 \beta_1 + w_2 \beta_2 + w_3 \beta_3

Substitute the values: 1=13(0)+13(1.16)+13(βB)1 = \frac{1}{3}(0) + \frac{1}{3}(1.16) + \frac{1}{3}(\beta_B)

Simplify: 1=0+0.3867+13βB1 = 0 + 0.3867 + \frac{1}{3} \beta_B

10.3867=13βB1 - 0.3867 = \frac{1}{3} \beta_B

0.6133=13βB0.6133 = \frac{1}{3} \beta_B

βB=0.6133×3\beta_B = 0.6133 \times 3

βB=1.84\beta_B = 1.84

Final Answer:

The beta for the other stock in your portfolio is 1.84.


Do you want further details or have any questions? Here are 5 related questions for further exploration:

  1. How does the beta of a portfolio change if weights shift between the assets?
  2. What is the significance of a beta being greater than 1 or less than 1?
  3. How does adding more risk-free assets affect the portfolio's beta?
  4. Can a portfolio have a negative beta? Under what conditions?
  5. How would diversification impact the overall risk of this portfolio?

Tip: Always verify weights and beta contributions to ensure proper calculation of portfolio risk.

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Math Problem Analysis

Mathematical Concepts

Portfolio Theory
Weighted Averages
Risk Metrics

Formulas

Portfolio Beta Formula: β_portfolio = w1 * β1 + w2 * β2 + w3 * β3

Theorems

Portfolio Theory: Weighted average of betas

Suitable Grade Level

Undergraduate (Finance/Investment Basics)