1. Chapter 5 Time Value of Money

2. Basic Definitions Present Value – earlier money on a time line Future Value – later money on a time line Interest rate – “exchange rate” between earlier money and later money Discount rate Cost of capital Required return 5C-2

3. Future Values Suppose you invest $100 for one year at 5% per year. What is the future value in one year? Suppose you leave the money in for another year. How much will you have two years from now? 5C-3

4. Future Values: General Formula FV = PV(1 + r) t FV = future value PV = present value r = period interest rate, expressed as a decimal t = number of periods Future value interest factor = (1 + r) t 5C-4

5. Calculator Keys Texas Instruments BA-II Plus FV = future value PV = present value I/Y = period interest rate P/Y must equal 1 for the I/Y to be the period rate Interest is entered as a percent, not a decimal N = number of periods Remember to clear the registers (CLR TVM) after each problem Other calculators are similar in format 5C-5

6. Effects of Compounding Suppose you invest $100 for ten year at 5% per year. What is the future value in ten years if you earn simple interest ? Suppose you invest $100 for ten year at 5% per year. What is the future value in ten years if you earn compound interest ? 5C-6

7. Present Values How much do I have to invest today to have some amount in the future? FV = PV(1 + r) t Rearrange to solve for PV = FV / (1 + r) t When we talk about discounting, we mean finding the present value of some future amount. When we talk about the “value” of something, we are talking about the present value unless we specifically indicate that we want the future value. 5C-7

8. Present Values Your parents set up a trust fund for you 10 years ago that is now worth $19,671.51. If the fund earned 7% per year, how much did your parents invest? 5C-8

9. Present Value – Important Relationship I For a given interest rate – the longer the time period, the lower the present value What is the present value of $500 received in 5 years ? 10 years ? The discount rate is 10% 5C-9

10. Present Value – Important Relationship II For a given time period – the higher the interest rate, the smaller the present value What is the present value of $500 received in 5 years if the interest rate is 10%? 15%? 5C-10

11. The Basic PV Equation - Refresher PV = FV / (1 + r) t There are four parts to this equation PV, FV, r and t If we know any three, we can solve for the fourth If you are using a financial calculator, be sure to remember the sign convention or you will receive an error (or a nonsense answer) when solving for r or t 5C-11

12. Solving for Discount Rate You are looking at an investment that will pay $1,200 in 5 years if you invest $1,000 today. What is the implied rate of interest? 5C-12

13. Discount Rate What are some situations in which you might want to know the discount rate? You are offered the following investments with similar risk: You can invest $500 today and receive $600 in 5 years. You can invest the $500 in a bank account paying 4%. What is the discount rate for the first choice, and which investment should you choose? 5C-13

14. Solving for Number of Periods You want to purchase a new car, and you are willing to pay $20,000. If you can invest at 10% per year and you currently have $15,000, how long will it be before you have enough money to pay cash for the car? 5C-14

15. FV of Multiple Cash Flows Suppose you plan to deposit $100 into an account in one year and $300 into the account in three years. How much will be in the account in five years if the interest rate is 8%?

16. PV of Multiple Cash Flows You are considering an investment that will pay you $1,000 in one year, $2,000 in two years and $3,000 in three years. If you want to earn 10% on your money, how much would you be willing to pay?

17. Annuities and Perpetuities Defined Annuity – finite series of equal payments that occur at regular intervals If the first payment occurs at the end of the period, it is called an ordinary annuity If the first payment occurs at the beginning of the period, it is called an annuity due Perpetuity – infinite series of equal payments

18. Present Value of Annuities 0 1 2 3 4 5 6 $100 $100 $100 $100 $100 Example 1: Find the PV of the following stream of cash flows. The discount rate is 5%.

19. Present Value of Annuities 0 1 2 3 4 5 6 $100 $100 $100 $100 $100 Example 2: Find the PV of the following stream of cash flows. The discount rate is 5%.

20. Future Value of Annuities 0 1 2 3 4 5 $100 $100 $100 $100 $100 Example 1: Find the FV of the following stream of cash flows at the end of 5 years. The discount rate is 5%.

21. Future Value of Annuities 0 1 2 3 4 5 $100 $100 $100 $100 $100 Example 2: Find the FV of the following stream of cash flows at the end of 5 years. The discount rate is 5%.

22. Future Value of Annuities Retirement Example Suppose you begin saving for your retirement by depositing $2,000 per year (at the end-of- year) in an IRA. If the interest rate is 7.5%, how much will you have in 40 years?

23. Perpetuity Perpetuity formula: PV = C / r You are considering preferred stock that pays an annual dividend of $1.50. If your desired return is 3% per year, how much would you be willing to pay?

24. Finding the Payment Suppose you want to borrow $20,000 for a new car. You can borrow at 8% per year. If you take a 4-year loan, what is your annual payment?

25. Finding the Number of Payments Suppose you borrow $2,000 at 5%, and you are going to make annual payments of $734.42. How long before you pay off the loan?

26. Finding the Rate Suppose you borrow $10,000 from your parents to buy a car. You agree to pay $1000 per year for 15 years. What is the annual interest rate?

27. Annual Percentage Rate This is the annual rate that is quoted by law By definition APR = period rate times the number of periods per year Consequently, to get the period rate we rearrange the APR equation: Period rate = APR / number of periods per year You should NEVER divide the effective rate by the number of periods per year – it will NOT give you the period rate

28. Computing APRs What is the APR if the monthly rate is.5%? What is the APR if the semiannual rate is.5%? What is the monthly rate if the APR is 12% with monthly compounding?

29. Effective Annual Rate (EAR) This is the actual rate paid (or received) after accounting for compounding that occurs during the year If you want to compare two alternative investments with different compounding periods, you need to compute the EAR and use that for comparison.

30. EAR - Formula Remember that the APR is the quoted rate, and m is the number of compounding periods per year 6C-30

31. Decisions, Decisions You are looking at two savings accounts. One pays 5.25%, with daily compounding. The other pays 5.3% with semiannual compounding. Which account should you use? First account: Second account: Which account should you choose and why?

32. Continuous Compounding Sometimes investments or loans are figured based on continuous compounding EAR = e q – 1 The e is a special function on the calculator normally denoted by e x Example: What is the effective annual rate of 7% compounded continuously?

33. Future Values with Monthly Compounding Suppose you deposit $50 a month into an account that has an APR of 9%, based on monthly compounding. How much will you have in the account in 35 years?

34. Present Value with Daily Compounding You need $15,000 in 3 years for a new car. If you can deposit money into an account that pays an APR of 5.5% based on daily compounding, how much would you need to deposit?

35. Computing Payments with APRs Suppose you want to buy a new computer system and the store is willing to allow you to make monthly payments. The entire computer system costs $3,500. The loan period is for 2 years, and the interest rate is 16.9% APR with monthly compounding. What is your monthly payment?

36. Amortized Loan with Fixed Payment - Example Each payment covers the interest expense plus reduces principal Construct an amortization schedule for a 3 year car loan with annual payments. The interest rate is 8%, and the principal amount is $5,000. Construct an amortization schedule for a 30 year home loan with monthly payments. The APR is 7% with monthly compounding, and the principal amount is $200,000.