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 find out how much you should invest in the risk-free security, we can use the concept of the weighted average of expected returns for the portfolio.
First, let's calculate the expected return of the overall portfolio. We can do this by finding the weighted average of the expected returns of the two stocks:
Expected Return of Stock A = 13.6%
Expected Return of Stock B = 11.4%
Since you want the portfolio to have an expected return equal to that of the market, which is 10%, the weighted average of the two stocks should also be 10%.
Let's assume the amount invested in Stock A is X. Since you want to own $30,000 of Stock B, the amount invested in Stock B would be $30,000.
Now we can set up the equation:
(X * 0.136) + ($30,000 * 0.114) + (Amount invested in the risk-free security * 0.04) = $75,000 * 0.10
Simplifying the equation:
0.136X + 0.114 * $30,000 + 0.04 * Amount invested in the risk-free security = $7,500
Now you can solve for the amount invested in the risk-free security:
0.136X + 0.114 * $30,000 + 0.04 * Amount invested in the risk-free security = $7,500
0.04 * Amount invested in the risk-free security = $7,500 - 0.114 * $30,000 - 0.136X
Amount invested in the risk-free security = ($7,500 - 0.114 * $30,000 - 0.136X) / 0.04
Now you can plug in the values to find the amount invested in the risk-free security.