Mastering Time Value of Money: Future and Present Value Concepts

1. CHAPTER 5 TIME VALUE OF MONEY

2. Chapter Outline • Introduction • Future value • Present value • Multiple cash flow • Annuities • Perpetuities • Amortization

3. Introduction • Is that the value of one Ringgit today is the same as the value of one Ringgit ten years ago or ten year to come? • What will happen to the money if we invest or we save it in the bank or financial institutions?

4. Future Value • Refer to the amount of money an investment will grow to over some period of time at some given interest rate. • Cash value of an investment at some time in future.

5. Future Values • Suppose you invest $1,000 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?

6. 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

7. Example: • What will be the future value of RM 10, 000 in 5 years, given the rate of return of 15%? • You plan to save in an account that pays 10% return per year. You put in RM 20,000 in the first year, RM 30,000 in the second year and another RM 50,000 in the third year . How much will you have by the end of 10 years?

8. Present Value • Value today of a sum of money to be received in future. • Value that we should invest or deposit, if we want some specific value in the future. • How much do I have to invest today to have some amount in the future?

9. Present Values • 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.

10. Present Value – One Period Example • Suppose you need $10,000 in one year for the down payment on a new car. If you can earn 7% annually, how much do you need to invest today?

11. Present Values – Example 2 • You want to begin saving for your daughter’s college education and you estimate that she will need $150,000 in 17 years. If you feel confident that you can earn 8% per year, how much do you need to invest today?

12. Present Values – Example 3 • 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?

13. Discount Rate • Often we will want to know what the implied interest rate is on an investment • Rearrange the basic PV equation and solve for r • FV = PV(1 + r)t • r = (FV / PV)1/t – 1

14. Discount Rate – Example 1 • 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? • r = (FV / PV )1/t – 1

15. Discount Rate – Example 2 • Suppose you are offered an investment that will allow you to double your money in 6 years. You have $10,000 to invest. What is the implied rate of interest?

16. Discount Rate – Example 3 • Suppose you have a 1-year old son and you want to provide $75,000 in 17 years towards his college education. You currently have $5,000 to invest. What interest rate must you earn to have the $75,000 when you need it?

17. Finding the Number of Periods • Start with basic equation and solve for t (remember your logs) • FV = PV(1 + r)t • t = ln(FV / PV) / ln(1 + r) • You can use the financial keys on the calculator as well; just remember the sign convention.

18. Number of Periods – Example 1 • 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?

19. Multiple Cash Flows – Present Value Example 6.3 • Find the PV of each cash flows and add them • Year 1 CF: N = 1; I/Y = 12; FV = 200; CPT PV = -178.57 • Year 2 CF: N = 2; I/Y = 12; FV = 400; CPT PV = -318.88 • Year 3 CF: N = 3; I/Y = 12; FV = 600; CPT PV = -427.07 • Year 4 CF: N = 4; I/Y = 12; FV = 800; CPT PV = - 508.41 • Total PV = 178.57 + 318.88 + 427.07 + 508.41 = 1,432.93

20. 0 1 2 3 4 200 400 600 800 178.57 318.88 427.07 508.41 1,432.93 Example 6.3 Timeline

21. Multiple Cash Flows – PV Another Example • 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? • N = 1; I/Y = 10; FV = 1,000; CPT PV = -909.09 • N = 2; I/Y = 10; FV = 2,000; CPT PV = -1,652.89 • N = 3; I/Y = 10; FV = 3,000; CPT PV = -2,253.94 • PV = 909.09 + 1,652.89 + 2,253.94 = 4,815.93

22. 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

23. Annuities and Perpetuities – Basic Formulas • Perpetuity: PV = C / r • Annuities:

24. Annuity – Sweepstakes Example • Suppose you win the Publishers Clearinghouse $10 million sweepstakes. The money is paid in equal annual end-of-year installments of $333,333.33 over 30 years. If the appropriate discount rate is 5%, how much is the sweepstakes actually worth today? • 30 N; 5 I/Y; 333,333.33 PMT; CPT PV = 5,124,150.29

25. Amortized Loan with Fixed Principal Payment - Example • Consider a $50,000, 10 year loan at 8% interest. The loan agreement requires the firm to pay $5,000 in principal each year plus interest for that year. • Click on the Excel icon to see the amortization table

27. Amortized Loan with Fixed Payment - Example • Each payment covers the interest expense plus reduces principal • Consider a 4 year loan with annual payments. The interest rate is 8% and the principal amount is $5,000. • What is the annual payment? • 4 N • 8 I/Y • 5,000 PV • CPT PMT = -1,509.60 • Click on the Excel icon to see the amortization table