1. Time Value of Money Chapter 4

2. 2Barton College TVM on YouTube

3. 3Barton College Don’t forget to look at the Notes Pages (See Below how to do this..)

4. 4Barton College Chapter Topics TVM Introduction Single Period Investments Multiple Period Investments Compound Interest Future Value (FV) and Compounding Present Value (PV) & Discounting Determining the Discount Rate given the PV or FV Determining the Number of Periods given the PV or FV A few Problems for practice…..

5. 5Barton College Introduction to the Time Value of Money (TVM) Someone wins the PowerBall Lottery and the payout is $110M. Does the recipient really receive this amount or something less? One person invests $1M in savings bonds and another person invest $1M in a franchise restaurant chain. Are these investments similar, different, or contain different amounts of risk…..? –Understanding how to interpret the TVM will help the investor/manager choose a decision that best fits the particular purpose. –It also helps the investor understand why a dollar today is more than a dollar tomorrow (if held in one’s pocket and not invested)

6. 6Barton College Basic Definitions Future Value – later money on a time line Present Value – earlier money on a time line Interest rate – “exchange rate” between earlier money and later money –Discount rate –Cost of capital (wacc) –Opportunity cost of capital –Required return –Historical return –Expected return

7. 7Barton College Future Value Future Value is the amount an investment is worth after one or more periods. Board Example Single Period and Simple Interest –Suppose you invest $1000 for one year at 5% per year. What is the future value in one year? Interest = 1000(.05) = $50 Value in one year = principal + interest = 1000 + 50 = $1050 Future Value (FV) = 1000(1 +.05) = $1050 Multiple Periods: Suppose you leave the money in for another year. How much will you have two years from now? –FV = 1000(1.05)(1.05) = 1000(1.05) 2 = $1102.50

8. 8Barton College Compounding Effects Simple Interest vs. Compound Interest Consider the earlier example –FV with simple interest = 1000 + 50 + 50 = $1100 –FV with compound interest = 1102.50 FV = 1000(1.05)(1.05) = 1000(1.05) 2 = $1102.50 –The extra $2.50 comes from the interest of.05(50) = 2.50 earned on the first interest payment

9. 9Barton College 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 Future and Present Value Interest Factors can be found at the back of your book.

10. 10Barton College A few basic Definitions Compounding – The process of accumulation interest on an investment over time to earn more interest. Interest on Interest – Interest earned on the reinvestment of previous interest. Compound Interest – Interest earned on both the initial principal and the interest reinvested from prior periods. Simple interest – just the interest earned on the original principle.

11. 11Barton College Future Values – Example 2 Suppose you invest the $1000 from the previous example for 5 years. How much would you have? –i = 5%, t = 5, PV = $1000 –FV = PV(1+r) t –FV = 1000(1+.05) 5 –FV = 1000(1.05) 5 = 1276.28 The effect of compounding is small for a small number of periods, but increases as the number of periods increases. (Simple interest would have a future value of $1250, for a difference of $26.28.) Example of (100 @ 10% rate) = $10.00 Simple interest provides $10 x 5 = $150 Compound Interest provides $161.05 –Period 1: 100(1.10) = $110 –Period 2: 110(1.10) = $121 –Period 3: 121(1.10) = $133.10 –Period 4: 133.10(1.10) = $146.41 –Period 5: 146.41(1.10) = $161.05 FV = 100(1.10) 5 = $161.05 161.05 – 150 = $11.05

12. 12Barton College Future Values – Example 3 Suppose you had a deposit $10 at 5.5% interest 200 years ago. How much would the investment be worth today? –FV = PV(1+r) n = 10(1.055) 200 = 447,189.84 What is the effect of compounding? –Simple interest = 10 + 200(10)(5.5%) = 210.55 Compounding added $446,979.29 to the value of the investment Calculator: N = 200; I/Y = 5.5; PV = 10; CPT FV = -447,198.84

13. 13Barton College FV 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 Use Appendix D to help set up your calculator Calculator Page

14. 14Barton College Finding the Present Value or (Discounting a Future Value back to the Present)

15. 15Barton College 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.

16. 16Barton College 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? –PV = FV / (1+r) n = 10,000 / (1+.07) 1 –PV = 10,000 / (1.07) 1 = 9345.79 –Calculator 1 = N 7 = I/Y 10,000 = FV CPT PV = -9345.79 Calculator Example of Present Value (PV) http://movies.atomiclearning.com/movie/k12/21142/play

17. 17Barton College Present Values – Example 2 You want to begin saving for you 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? PV = FV / (1+r) n PV = 150,000 / (1.08) 17 = $40,540.34 Key strokes: 1.08 y x 17 = 3.70002 and $150,000/3.70002 = $40, 540.34 Calculator: N = 17; I/Y = 8; FV = 150,000; CPT PV = -40,540.34

18. 18Barton College 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? PV = FV / (1+r) n –PV = 19,671.51 / (1.07) 10 = 10,000 Calculator: N = 10; I/Y = 7; FV = 19,671.51; CPT PV = -10,000

19. 19Barton College 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 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 A dollar decreases in value more and more over time….. Or… the discount factor increases over time….

20. 20Barton College The Basic PV Equation - Refresher PV = FV / (1 + r) t There are four parts to this equation –PV, FV, r and t ( I will sometimes mix n and t for the same thing…) –If we know any three, we can solve for the fourth If you are using a financial calculator, be sure and remember the sign convention or you will receive an error when solving for r or t

21. 21Barton College 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 –FV/PV = (1+r) t –(FV/PV) 1/t = ((1+r) t ) 1/t –(FV/PV) 1/t = 1+r –r = (FV / PV) 1/t – 1 If you are using formulas, you will want to make use of both the y x and the 1/x keys

22. 22Barton College 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? –FV = 1200 –N = 5 –PV = 1000 r = ______ r = (FV / PV) 1/t – 1 –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%

23. 23Barton College 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? –r = (FV / PV) 1/t – 1 –r = (20,000 / 10,000) 1/6 – 1 =.122462 = 12.25% Calculator: N = 6; FV = 20,000; PV = 10,000; CPT I/Y = 12.25%

24. 24Barton College 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 $5000 to invest. What interest rate must you earn to have the $75,000 when you need it? –r = (FV / PV) 1/t – 1 –r = (75,000 / 5,000) 1/17 – 1 =.172688 = 17.27% Calculator: N = 17; FV = 75,000; PV = 5,000; CPT I/Y = 17.27%

25. 25Barton College Solving for the 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? Given: FV = PV (1+r) t Solve for t –FV/PV = (1+r) t –Using the ln and “The Power Rule for natural logs” –ln (FV/PV) = t ln(1+r) –t = ln(FV/PV) / ln(1+r) –t = ln(20,000 / 15,000) / ln(1.1) = 3.02 years –ln x t = t ln x (The Power Rule)

26. 26Barton College Number of Periods – Example 2 Continued To buy a house you have $15,000 but you need $21,750 that includes all your closing costs. Rates are currently 7.5%. How long will it take to reach the desired amount? –Down payment =.1(150,000) = 15,000 –Closing costs =.05(150,000 – 15,000) = 6,750 –Total needed = 15,000 + 6,750 = 21,750 Compute the number of periods –PV = -15,000 –FV = 21,750 –I/Y = 7.5 –CPT N = 5.14 years Using the formula –t = ln(21,750 / 15,000) / ln(1.075) = 5.14 years t = ln (FV/PV) / ln (1+r) To buy a House???

27. 27Barton College Spreadsheet Example Use the following formulas for TVM calculations –FV(rate,nper,pmt,pv) –PV(rate,nper,pmt,fv) –RATE(nper,pmt,pv,fv) –NPER(rate,pmt,pv,fv) The formula icon is very useful when you can’t remember the exact formula Click on the Excel icon to open a spreadsheet containing four different examples.

28. 28Barton College Practice Problems Complete the following problems in your book. Questions and Problems in your Book: –1-5, 12, 13, 14 & 15 –Try the problems several different ways, e.g., using the basic formulas, your calculator and Excel. –If you need help with using your calculator, please see the FNC 330 website. I have several user guide links to the HP 10-B and TI BA-II.

29. 29Barton College Future Value and Present Value Formulas