You want to create a $75,000 portfolio comprised of two stocks plus a risk-free security. Stock A has an expected return of 13.6 percent and stock B has an expected return of 11.4 percent. You want to own $30,000 of stock B. The risk-free rate is 4 percent and the expected return on the market is 10 percent. If you want the portfolio to have an expected return equal to that of the market, how much should you invest in the risk-free security?
To determine how much to invest in the risk-free security, we can use the formula for expected return of a portfolio:
Expected Return = Weight of Stock A * Expected Return of Stock A + Weight of Stock B * Expected Return of Stock B + Weight of Risk-free Security * Expected Return of Risk-free Security
Given the information provided, let's assign variables:
- Expected Return of Stock A = 13.6%
- Expected Return of Stock B = 11.4%
- Weight of Stock B = $30,000 / Total Portfolio Value
- Expected Return of Risk-free Security = 4%
- Expected Return of Market = 10%
The goal is to make the portfolio's expected return equal to that of the market (10%). We have two unknowns: the weight of Stock A and the weight of the Risk-free Security. Let's solve for those unknowns.
Step 1: Determine the weight of Stock A.
Since the portfolio comprises only two stocks and a risk-free security, the sum of the weights should equal 1.
Weight of Stock A + Weight of Stock B + Weight of Risk-free Security = 1
We know the weight of Stock B is $30,000 / Total Portfolio Value.
So, Weight of Stock A + $30,000 / Total Portfolio Value + Weight of Risk-free Security = 1
Step 2: Set up the equation for expected return and solve for the weight of the risk-free security.
To set up the equation, substitute the variables in the formula for expected return:
10% = Weight of Stock A * 13.6% + Weight of Stock B * 11.4% + Weight of Risk-free Security * 4%
Now substitute the value of the weight of Stock B:
10% = Weight of Stock A * 13.6% + ($30,000 / Total Portfolio Value) * 11.4% + Weight of Risk-free Security * 4%
Rearrange the equation:
Weight of Risk-free Security * 4% = 10% - Weight of Stock A * 13.6% - ($30,000 / Total Portfolio Value) * 11.4%
Simplify the right side of the equation:
Weight of Risk-free Security * 4% = 10% - (Weight of Stock A * 0.136) - (0.114 * $30,000 / Total Portfolio Value)
Step 3: Calculate the weight of the risk-free security.
We can now calculate the weight of the risk-free security by dividing both sides of the equation by 4%:
Weight of Risk-free Security = [10% - (Weight of Stock A * 0.136) - (0.114 * $30,000 / Total Portfolio Value)] / 4%
Step 4: Calculate the weight of Stock A.
Substitute the calculated weight of the risk-free security into the previous equation from Step 1 to solve for the weight of Stock A:
Weight of Stock A + $30,000 / Total Portfolio Value + [10% - (Weight of Stock A * 0.136) - (0.114 * $30,000 / Total Portfolio Value)] / 4% = 1
Rearrange and simplify the equation:
Weight of Stock A - (Weight of Stock A * 0.136) - (0.114 * $30,000 / Total Portfolio Value) + 0.25 * [10% - (Weight of Stock A * 0.136) - (0.114 * $30,000 / Total Portfolio Value)] = 1 - $30,000 / Total Portfolio Value
Combine like terms:
Weight of Stock A - 0.136 * Weight of Stock A - 0.114 * $30,000 / Total Portfolio Value + 0.25 * 10% - 0.25 * Weight of Stock A * 0.136 - 0.25 * 0.114 * $30,000 / Total Portfolio Value = 1 - $30,000 / Total Portfolio Value
Simplify further:
Weight of Stock A * (1 - 0.136 - 0.25 * 0.136) - 0.114 * $30,000 / Total Portfolio Value * (1 - 0.25 * 0.114) = 1 - $30,000 / Total Portfolio Value
Combine like terms:
Weight of Stock A * 0.6344 - 0.108516 * $30,000 / Total Portfolio Value = 1 - $30,000 / Total Portfolio Value
Now isolate Weight of Stock A:
Weight of Stock A * 0.6344 = 1 - $30,000 / Total Portfolio Value + 0.108516 * $30,000 / Total Portfolio Value
Combine like terms:
Weight of Stock A * 0.6344 = (1 - 0.891484 * $30,000 / Total Portfolio Value)
Now divide both sides by 0.6344 to get the weight of Stock A:
Weight of Stock A = [1 - 0.891484 * $30,000 / Total Portfolio Value] / 0.6344
Step 5: Calculate the value of the risk-free security.
We can now substitute the calculated weight of Stock A and Stock B into the equation for expected return:
Weight of Risk-free Security = [10% - (Weight of Stock A * 0.136) - (Weight of Stock B * 0.114)] / 4%
Finally, calculate the value of the risk-free security by multiplying its weight by the total portfolio value:
Value of Risk-free Security = Weight of Risk-free Security * Total Portfolio Value
With these calculations, you should be able to determine how much to invest in the risk-free security to achieve an expected return equal to that of the market.