Question

1. Calculate the present value of a growing perpetuity with the first cash flow (occurring in one year) being $10 million and every subsequent year’s cash flow growing at a constant 6% rate (i.e, the cash flow at the end of two years is $10M(1.06) = $10.6 million, the cash flow at the end of three years is 10M(1.06)^2 = $11.236 million, etc.). The cost of capital for this calculation is 12%.

2. The firm has to spend $50 million immediately and $25 million one year from now to start a new investment project. The firm’s cost of capital is 12%. What is the present value of the firm’s expenditures to start the new investment project?

3. The firm is considering the investment project in problem 2, where they invest $50 million immediately and $25 million additionally one year from now. There are no other cash inflows or outflows for the project until the end of five years, at which time the first cash inflow from the project is $10 million. After this first cash inflow, the cash flows grow each year at 6%, indefinitely. The cost of capital for this project is 12%. Calculate the Net Present Value of this investment.

4. The NPV of the investment in problem 3 depends on the growth rate in perpetuity of the cash inflows. For example, if the growth rate were 4% rather than six percent, the NPV of the project would be lower. For what value of the growth rate is the NPV of the project zero? (In other words, how high does the growth rate in perpetuity have to be for this project to be positive NPV?)

   The following questions use the Capital Asset Pricing Model. You have the following market- wide parameters: the risk-free rate of interest in the economy is 3% and the risk premium on the market portfolio (i.e., the market risk premium) is 6%.

5. What is the expected return on a stock with a beta of 1.5?

6. What is the expected return on a stock the return on which is uncorrelated with the return on the market portfolio?

7. What is the beta of a portfolio where half of the money in the portfolio is invested in the market index and half the money in the portfolio is invested in risk free securities? What is the expected return on such a portfolio?

Answer 1

1) Present Value of growing perpetuity = Cash flow/(cost of capital-growth rate)

=10,000,000/(0.12-0.06)

=$166,666,667

2) Present value of firm's expenditure =$50+$25*0.893

=$72.325 million

3) Net present value of investment = PV of inflows - PV of outflows

=(10*1/(1.12)⁵} *{1/ (0.12-0.06)} -$72.325

=10*0.567*16.6666667 -72.325

=94.50-72.325

=$22.175 million

4.In order to know NPV as zero cash inflow should be equal to cash outflow

72.325 =10*0.567*(1/0.12-g)

72.325(0.12-g) =5.67

8.679-72.325g =5.67

g=3.009/72.325

g= 4.16 %

The NPV is zero at 4.16 that means if the growth rate is above 4.16% NPV will be positive+

5. Expected return on stock = r_f + ( r_m - r_f ) Beta

= 3% + 6%*1.5

= 12%

6. If a stock is uncorrelated with the return on the market portfolio, the beta will be zero

Expected return on stock = r_f + ( r_m - r_f ) Beta

= 3%+ 6%*0

=3%

7. Portfolio beta is the weighted average of the beta of risk free asset and the risky asset The beta of risk free asset is zero. Therefore,

Beta of portfolio =0.5*1.5+0.5*0

=0.75

Expected return of portfolio is the weighted average of the return  of risk free asset and the risky asset

Expected return of portfolio =0.50*12+0.5*3

=7.5%