Rate of Return Rule Accept investments that offer rates of return in excess of their opportunity cost of capital Example In the project listed below, the foregone investment opportunity is 12%. Should we do the project? Calculating Present Values When There Are Multiple Cash Flows For multiple periods we have the discounted cash flow (DCF) formula Figure 2.5 NPV Calculation How to Value Perpetuities Sometimes there are shortcuts that make it very easy to calculate the present value of an asset that pays off in different periods. These tools allow us to cut through the calculations quickly. Shortcuts Perpetuity: Financial concept in which a cash flow is theoretically received forever. Shortcuts Continued Perpetuity: Financial concept in which a cash flow is theoretically received forever. Present Values Example What is the present value of $1 billion every year, for all eternity, if you estimate the perpetual discount rate to be 10%?

Present Values Continued Example continued What if the investment does not start making money for 3 years? How to Value Annuities Annuity: An asset that pays a fixed sum each year for a specified number of years Perpetuities & Annuities PV Annuity Factor (PVAF): The present value of $1 a year for each of t years Figure 2.7 Annuity Figure 2.8 Costing an Installment Plan Valuing Annuities Due Annuity due: Level stream of cash flows starting immediately How does it differ from an ordinary annuity? How does the future value differ from an ordinary annuity? Example 2.3 Paying off a Bank Loan Table 2.1 Amortizing Loan Example Future Value of an Annuity Example: Suppose you invest $429.59 annually at the beginning of each year at 10% interest. After 50 years, how much would your investment be worth? Future Value of an Annuity Continued Future Value of an Annuity: The future value of an asset that pays a fixed sum each year for a specified number of years.

Future Value of an Annuity Concluded Example What is the future value of $20,000 paid at the end of each of the following 5 years, assuming your investment returns 8% per year? Growing Perpetuities Present value of growing perpetuity g = the annual growth rate of the cash flow Growth Perpetuity Example Example What is the present value of $1 billion paid at the end of every year in perpetuity, assuming a rate of return of 10% and a constant growth rate of 4%? How Interest is Paid and Quoted Annual Percentage Rate: Interest rate that is annualized using simple interest Effective Annual Interest Rate: Interest rate that is annualized using compound interest EAR & APR Formulas Annual Percentage Rate (APR): Effective Annual Interest Rate (EAR): *where MR = monthly interest rate Effective Interest Rates Example: Given a monthly rate of 1%, what is the effective annual rate (EAR)? What is the annual percentage rate (APR)?