Introduction to Valuation: The Time Value of Money

1. Introduction to Valuation: The Time Value of Money Chapter 4

2. Prepare for Capital Budgeting Part 2: Understand financial statement and cash flow C2-Identify cash flow from financial statement C3-Financial statement and comparison Part 3: Valuation of future cash flow C4-Basic concepts C5-More exercise Part 4: Valuing stocks and bonds C6-Bond C7-Stock Part 5: Capital budgeting

3. Chapter Outline • Future Values: Definitions and Formula • Present Values • PV – Important Relationship • Calculate Rates and Number of Periods • Calculator Keys

4. 1. Example • Suppose you put $5000 in bank for one year at 3% interest rate per year. What is the value of your money in one year? • Interest = 5000(.03) = 150 • Value in one year = principal + interest = 5000+5000(.03) = 5000 + 150 = 5150 = 5000(1 + .03) = 5150 • Suppose you leave the money in for another year. How much will you have two years from now? • FV = [5000(1.03)](1.03) = 5000(1.03)2 = 5304.5

5. Example (cont..) • 1 year: Value = 5000(1+.03) • 2 years: Value = [5000(1+.03)](1+.03) • 3 years: Value = {[5000(1+.03)](1+.03)} (1+.03) • 4 years: Value = {{[5000(1+.03)](1+.03)} (1+.03)} (1+.03)

6. Example (cont..) • 1 year: Value = 5000(1+.03) = 5000(1+.03)1 • 2 years: Value = [5000(1+.03)](1+.03) = 5000(1+.03)2 • 3 years: Value = {[5000(1+.03)](1+.03)} (1+.03) = 5000(1+.03)3 • 4 years: Value = {{[5000(1+.03)](1+.03)} (1+.03)} (1+.03) = 5000(1+.03)4

7. Basic Definitions FV = PV(1 + r)t • 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 • Opportunity cost of capital • Required return • Time value of money: A dollar in hand today is worth more than a dollar promised at some time in the future.

8. Future Values: General Formula • FV = PV(1 + r)t • FV = future value • PV = present value • r = period interest rate • T = number of periods • Future value factor = (1 + r)t

9. Effects of Compounding • Compounding: the process of accumulating interest over time to earn more interest. • Compound interest: interest earned on both the initial principal and the interest earned from prior periods. • Compound interest (total interest) =Simple interest+Interest on interest • Simple interest: interest on principal

10. Illustration on Compounding 2004 2005 2006 2007 $5000 $5150 $5304.5 Interest earned (2004-2005)=$150 $150 Interest earned (2004-2006)=150+(150+4.5)=$304.5 $150$4.5 $150

11. Illustration on Compounding (cont..) 2004 2005 2006 2007 $5000 $5150 $5304.5 $5463.6 Interest earned (04-06)=150+(150+4.5)=$304.5 Interest earned(04-07)=150+(150+4.5)+(150+9.1)=$463.6 $150$4.5$9.1 $150 $150 Compounding effect=Total interest earned - simple interest =463.6 – 150 – 150 – 150 = 463.6 - 3(150) = 13.6

12. Figure 4.1

13. Future Values – Example Suppose you had a relative deposit $10 at 5.5% interest 200 years ago. How much would the investment be worth today? • FV = 10(1.055)200 = 447,189.84 What is the effect of compounding? • Total interest = FV- PV = 447, 179.84 • Simple interest = 200[(10)(.055)] = 110 • Compounding effect = Total interest – simple interest = 447,179.84 -110= 447,069.84 Compounding has added $447,069.84 to the value of the investment.

14. 2. 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 • Present Value factor (Discount factor)= 1 / (1 + r)t • When we talk about discounting, we mean finding the present value of some future amount. • When we just say the “value” of something, we are talking about the present value unless we specifically indicate that we want the future value.

15. Example If you want to have $5000 in your account this year, how much you should put in the bank last year given 3% interest rate per year? • (last year) vs. (this year) PV= 5000/(1+3%) = $4854

16. PV –Example • Suppose your grandmother know you need $2500 in one year for car down payment. If you can earn 3% quarterly interest when put the money in the bank, how much does she need to give you today? • PV = 2500 / (1.03)4 = 2221.2

17. 3. PV – Important Relationship I • For a given interest rate and future value – the longer the time period, the lower the present value • What is the present value of $500 to be received in 5 years? 10 years? The discount rate is 10% • 5 years: PV = 500 / (1.1)5 = 310.46 • 10 years: PV = 500 / (1.1)10 = 192.77 Future Value=Present Value+Interest Earned PV = FV / (1 + r)t

18. PV – Important Relationship II • For a given time period and future value – 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%? • Rate = 10%: PV = 500 / (1.1)5 = 310.46 • Rate = 15%; PV = 500 / (1.15)5 = 248.58 Future Value=Present Value+Interest Earned PV = FV / (1 + r)t

19. 4. 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

20. Discount Rate • Often we will want to know what the implied interest rate is in an investment • Rearrange the basic PV equation and solve for r • FV = PV(1 + r)t • r = (FV / PV)1/t – 1 • If you are using formulas, you will want to make use of both the yx and the 1/x keys

21. Discount Rate – Example • You are looking at an investment that will pay $1200 in 5 years if you invest $1000 today. What is the implied rate of interest? • r = (1200 / 1000)1/5 – 1 = .03714 = 3.714%

22. 6. Calculator Keys • Texas Instruments BA-II Plus • FV = future value • PV = present value • I/Y = period interest 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

23. Calculator Settings (Appendix D) • Compounding frequency • Set P/Y=1: Press [2 nd] [I/Y] (P/Y), show {P/Y}, [1] [ENTER] • Set C/Y=1: [ ] [1] [ENTER] [2 nd] [CPT] (QUIT) • End mode and annuities due • Start with [2 nd] [PMT] (BGN) • Switch between END and BGN using [2 nd] [ENTER] (SET) • End with [2 nd] [CPT] (QUIT)

24. Future Values – Example 1 • Suppose you invest $1000 for 5 years with 5% interest rate. How much would you have at the end of 5th year? • FV = 1000(1.05)5 = 1276.28 • Calculator: N = 5; I/Y = 5; PV = 1000; CPT FV = -1276.28

25. Present Value – Example 1 • Suppose your grandmother know you need $2500 in one year for car down payment. If you can earn 3% quarterly when put the money in the bank, how much does she need to give you today? • PV = 2500 / (1.03)4 = 2221.2 • Calculator • N=4 • I/Y=3 • FV=2500 • CPT PV = -2221.2

26. Using Financial Calculator • When using a financial calculator, be sure and remember the sign convention or you will receive an error when solving for r or t

27. Discount Rate – Example 1 • You are looking at an investment that will pay $1200 in 5 years if you invest $1000 today. What is the implied rate of interest? • r = (1200 / 1000)1/5 – 1 = .03714 = 3.714% • Calculator – the sign convention matters!!! • N = 5 • PV = -1000 (you pay 1000 today) • FV = 1200 (you receive 1200 in 5 years) • CPT I/Y = 3.714 (%)

28. 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? • Calculator: I/Y = 10; FV = 20,000; PV = -15,000; CPT N = 3.02 years

29. Review Questions 1. • Know how to calculate the future value, present value, and rate of return of an investment. • What is the difference between simple interest and compound interest? How to calculate compounding effect? • What is a compounding process and what is a discounting process?

30. Review Questions (cont..) 3. • How will discount factor and future value factor change with the interest rate and length of time? • As you increase the length of time involved, what happens to FV for a given PV and rate? What happens to PV for a given FV and rate? If you increase the rate, what happens to FV for a given PV and time length? What happens to PV for a given FV and time length?