You would like to create a portfolio that is equally invested in a risk-free asset and two stocks. The one stock has a beta of .80. What does the beta of the second stock have to be if you want the portfolio risk to equal that of the overall market?
A risk-free asset has a beta of 0. The average market beta is 1.0
0.8 + x = 3.0
x = 3 - 0.8
x = 2.2
To determine the beta of the second stock required for the portfolio's risk to equal that of the overall market, we need to calculate the portfolio's overall beta.
Here's how you can calculate it step by step:
Step 1: Assume the proportion of investment in the risk-free asset, the first stock, and the second stock. Let's say you invest x proportion in the risk-free asset, and (1 - x) proportion is equally divided among the two stocks.
Step 2: The beta of the risk-free asset is 0 since it has no market risk.
Step 3: The beta of the first stock is given as 0.80.
Step 4: Let's assume the beta of the second stock as β. Since the (1 - x) proportion is divided equally, each stock's proportion will be (1 - x)/2.
Step 5: Calculate the portfolio's overall beta using the weighted average of the individual stocks' beta:
Portfolio Beta = (x * 0) + ((1 - x)/2 * 0.80) + ((1 - x)/2 * β)
To have the portfolio's risk equal that of the overall market, the portfolio beta should be 1.
Step 6: Set up the equation and solve for β:
1 = ((1 - x)/2 * 0.80) + ((1 - x)/2 * β)
Step 7: Simplify the equation:
1 = (1 - x)(0.80 + β)/2
Step 8: Multiply both sides by 2 to simplify further:
2 = (1 - x)(0.80 + β)
Step 9: Distribute the (0.80 + β) term:
2 = 0.80 - 0.80x + β - βx
Step 10: Combine similar terms:
0 = 0.80 - 0.80x + β - βx - 2
Step 11: Rearrange the equation:
1.20x - βx = -1.20
Step 12: Factor out the common x:
x(1.20 - β) = -1.20
Step 13: Divide both sides by (1.20 - β):
x = -1.20 / (1.20 - β)
Therefore, the proportion of investment in the risk-free asset is equal to -1.20 divided by (1.20 - β).
By determining the value of x, you can calculate the required beta of the second stock by substituting it back into the equation.
To determine the beta of the second stock that would make the portfolio's risk equal to that of the overall market, we need to understand the concept of beta and how it relates to portfolio risk.
Beta measures the volatility or systematic risk of a stock relative to the overall market. A beta of 1 indicates that the stock tends to move in line with the market, while a beta greater than 1 implies higher volatility compared to the market, and a beta less than 1 suggests lower volatility.
In this case, you want to create a portfolio that equals the risk of the overall market. The overall market is assumed to have a beta of 1. By investing in a risk-free asset and two stocks, you need to find the beta of the second stock that balances out the portfolio's risk.
The formula to calculate the portfolio's beta is as follows:
Portfolio Beta = (Weight of Stock1 * Beta of Stock1) + (Weight of Stock2 * Beta of Stock2) + (Weight of Risk-free asset * Beta of Risk-free asset)
Since the risk-free asset has a beta of 0, the formula simplifies to:
Portfolio Beta = (Weight of Stock1 * Beta of Stock1) + (Weight of Stock2 * Beta of Stock2)
For the portfolio risk to equal that of the overall market (beta of 1), we can set up the equation as follows:
1 = (Weight of Stock1 * 0.8) + (Weight of Stock2 * Beta of Stock2)
Assuming an equal weight for both stocks (i.e., 50% each), the equation becomes:
1 = (0.5 * 0.8) + (0.5 * Beta of Stock2)
Simplifying the equation:
1 = 0.4 + 0.5 * Beta of Stock2
Rearranging the equation to solve for Beta of Stock2:
0.5 * Beta of Stock2 = 1 - 0.4
0.5 * Beta of Stock2 = 0.6
Beta of Stock2 = 0.6 / 0.5
Beta of Stock2 = 1.2
Therefore, the beta of the second stock needs to be 1.2 for the portfolio's risk to equal that of the overall market.