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Chapter 8: Problem 2

Portfolio beta An individual has \(\$ 35,000\) invested in a stock with a beta of 0.8 and another \(\$ 40,000\) invested in a stock with a beta of \(1.4 .\) If these are the only two investments in her portfolio, what is her portfolio's beta?

Short Answer

Expert verified
The portfolio's beta is approximately 1.12.

Step by step solution

01

Understand Portfolio Beta

The portfolio beta is a measure of the sensitivity of the portfolio's returns to the returns of the market index. It's calculated as the weighted average of the betas of the individual assets in the portfolio.
02

Determine the Weights

First, calculate the total investment in the portfolio: \( \(35,000 + \)40,000 = $75,000 \). Then, determine the weight of each investment. For the first stock: \( \frac{35,000}{75,000} = 0.4667 \). For the second stock: \( \frac{40,000}{75,000} = 0.5333 \).
03

Calculate Portfolio Beta

Use the formula for portfolio beta: \( \beta_{portfolio} = w_1 \beta_1 + w_2 \beta_2 \), where \( w_1 \) and \( w_2 \) are the weights, and \( \beta_1 \) and \( \beta_2 \) are the betas for the respective stocks. Substitute the values: \( \beta_{portfolio} = 0.4667 \times 0.8 + 0.5333 \times 1.4 \).
04

Complete the Calculation

Perform the multiplications: \( 0.4667 \times 0.8 = 0.3733 \) and \( 0.5333 \times 1.4 = 0.7466 \). Add the results: \( 0.3733 + 0.7466 = 1.1199 \). Therefore, the portfolio's beta is approximately 1.12.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Investment Portfolio
An investment portfolio represents the entire collection of financial investments an individual or entity holds. This can include stocks, bonds, mutual funds, and other assets. The primary goal of an investment portfolio is to diversify and maximize returns while managing risk.
When constructing a portfolio, investors consider their financial goals, risk tolerance, and investment timeline. By diversifying the portfolio, one can spread risk across different assets or asset classes, reducing the impact of any single investment's poor performance on the overall portfolio.
Key points to remember about an investment portfolio:
  • Diversification helps mitigate risk.
  • It can be tailored to specific investment goals and risk tolerance.
  • A well-balanced portfolio combines different asset types to optimize performance.
  • Regular review and adjustment of the portfolio are crucial to maintaining its alignment with the investor's goals.
Weighted Average
The weighted average is a method of calculating the average of a set of numbers where each number makes a different contribution to the final average. This is particularly useful in contexts like calculating portfolio beta where different components have varying levels of importance.
In the context of investment portfolios, each asset's weight is determined by its proportion relative to the total value of the portfolio. The weighted average formula helps in assessing parameters like expected returns and risks, taking into consideration these proportions.
How to calculate a weighted average:
  • Multiply each item by its respective weight.
  • Sum all these weighted values.
  • Divide the total by the sum of the weights.
The weighted average provides a comprehensive measure that reflects the significance of each individual component relative to the whole set.
Stock Beta
Stock beta is a measure that indicates the risk of a stock relative to the market. It reflects how much a stock's price moves in relation to market movements. If the beta is greater than 1, the stock is considered more volatile than the market. If it is less than 1, the stock is less volatile.
This metric is fundamental in assessing a stock's risk and potential returns, especially when used in combination with other analytical tools.
Understanding stock beta:
  • A beta of 1 implies that the stock's price moves with the market.
  • A beta of greater than 1 indicates higher risk but potentially higher returns.
  • A beta of less than 1 suggests lower risk and potentially lower returns.
Stock beta is crucial in the construction of investment portfolios as it informs decisions on how adding or reducing particular stocks can affect the overall risk and return profile of the portfolio. It's a vital part of risk management in financial investing.

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Most popular questions from this chapter

Stock \(X\) has a 10 percent expected return, a beta coefficient of \(0.9,\) and a 35 percent standard deviation of expected returns. Stock \(Y\) has a 12.5 percent expected return, a beta coefficient of \(1.2,\) and a 25 percent standard deviation. The riskfree rate is 6 percent, and the market risk premium is 5 percent. a. Calculate each stock's coefficient of variation. b. Which stock is riskier for a diversified investor? c. Calculate each stock's required rate of return. d. On the basis of the two stocks' expected and required returns, which stock would be more attractive to a diversified investor? e. Calculate the required return of a portfolio that has \(\$ 7,500\) invested in Stock \(X\) and \(\$ 2,500\) invested in Stock \(Y\) f. If the market risk premium increased to 6 percent, which of the two stocks would have the larger increase in its required return?

Stock \(\mathrm{R}\) has a beta of \(1.5,\) Stock \(\mathrm{S}\) has a beta of \(0.75,\) the expected rate of return on an average stock is 13 percent, and the risk-free rate of return is 7 percent. By how much does the required return on the riskier stock exceed the required return on the less risky stock?

Bradford Manufacturing Company has a beta of \(1.45,\) while Farley Industries has a beta of \(0.85 .\) The required return on an index fund that holds the entire stock market is 12.0 percent. The risk-free rate of interest is 5 percent. By how much does Bradford's required return exceed Farley's required return?

Given the following information, determine the beta coefficient for Stock J that is consistent with equilibrium: \(\hat{r}_{\mathrm{J}}=12.5 \% ; \mathrm{r}_{\mathrm{RF}}=4.5 \% ; \mathrm{r}_{\mathrm{M}}=10.5 \%\)

Suppose \(r_{\mathrm{RF}}=9 \%, \mathrm{r}_{\mathrm{M}}=14 \%,\) and \(\mathrm{b}_{\mathrm{i}}=1.3\) a. What is \(r_{i}\), the required rate of return on Stock i? b. Now suppose \(r_{\mathrm{RF}}\) (1) increases to 10 percent or (2) decreases to 8 percent. The slope of the SML remains constant. How would this affect \(r_{M}\) and \(r_{i} ?\) c. \(\quad\) Now assume \(r_{R F}\) remains at 9 percent but \(r_{M}(1)\) increases to 16 percent or (2) falls to 13 percent. The slope of the SML does not remain constant. How would these changes affect \(r_{i} ?\)
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