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1. Introduction to Valuation: The Time Value of Money Chapter 5 Copyright © 2013 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin

2. 5- Chapter Outline Time and Money Future Value and Compounding Present Value and Discounting More about Present and Future Values

3. 5- Chapter Outline Time and Money Future Value and Compounding Present Value and Discounting More about Present and Future Values

4. 5- Time and Money The single most important skill for a student to learn in this course is the manipulation of money through time.

5. The idea that money available at the present time is worth more than the same amount in the future due to its potential earning capacity . it is better to have money now rather than later ? You can do much more with the money if you have it now because over time you can earn more interest on your money. Time value of money

6. The Timeline A friend owes you money. He has agreed to repay the loan by making two payments of $10,000 at the end of each of the next two years

7. Example of timeline You’re still feeling generous and have agreed to lend your brother $10,000 today. Your brother has agreed to repay this loan in two installments of $6000 at the end of each of the next two years Suppose you must pay tuition of $10,000 per year for the next two years. Your tuition payments must be made in equal installments at the start of each semester. What is the timeline of your tuition payments? Assuming that each year, we have two semesters.

8. INTEREST and INTEREST RATES Interest  in general is the cost of borrowing money. An  interest rate  is the cost stated as a percent of the amount borrowed (principal) per period of time, usually one year. Interest Compound interest Simple interest

9. INTEREST RATES: Simple and Compound interest Simple Interest Simple interest is calculated on the  original principal   only . Accumulated interest from prior periods is NOT used in calculations for the following periods. Simple interest is normally used for a single period of less than a year, such as 30 or 60 days. Simple Interest = PV* i * n where :     PV = principal (original amount borrowed or loaned)     i = interest rate for one period     n = number of periods

10. INTEREST RATES: Simple and Compound interest Simple Interest Example 1 : You borrow $10,000 for 3 years at 5% simple annual interest. Example 2 : You borrow $10,000 for 60 days at 5% simple interest per year (assume a 365 day year).

11. INTEREST RATES: Simple and Compound interest Compound Interest Compound interest is calculated each period on the  original principal and all interest accumulated during past periods . The compounding periods can be annually, semiannually or quarterly…

12. Future Value is the value of an asset or cash at a specified date in the future that is equivalent in value to a specified sum today. FUTURE VALUE

13. Suppose you invest $100 in a bank account that pays interest of r = 7% a year. In the first year you will earn interest of .07 x $100 = $7 and the value of your investment will grow to $107: Value of investment after 1 year = $100 x (1 + 7%) = $107  By investing, you give up the opportunity to spend $100 today and you gain the chance to spend $107 next year. FUTURE VALUE (cont)

14. Year Beginning balance Interest Ending balance 1 C C *r C *(1+r) 2 C *(1+r) C *(1+r)*r C *(1+r) 2 ……. ………… ………. ……………. t C *(1+r) t-1 C *(1+r) t-1 * r C *(1+r) t Initial investment is C . The interest rate is r. FV: The future value.

15. FV = C *(1+r) t Compounding ( Tính lãi gộp – Lãi tính chồng lên lãi ) For example: The interest of the second year = C *(1+r)*r = C *r + (C *r)*r In which: C *r is the interest on principal ; (C *r)*r is the interest on interest. INTEREST RATES: Simple and Compound interest

16. INTEREST RATES: Simple and Compound interest Compound Interest For example, you borrow $10,000 for three years at 5% annual interest compounded annually. What is your wealth at the end of the investment period?

17. 5- Effects of Compounding Simple interest Compound interest Consider the previous example: FV with simple interest = 1,000 + 50 + 50 = $1,100 FV with compound interest = $1,102.50 The extra $2.50 comes from the interest of .05(50) = $2.50 earned on the first interest payment or “interest on interest”

18. Example If you have $1 today, calculate your wealth at the end of the following years using simple and compound interest? (R=9%) Year 1 Year 2 Year 3

19. INTEREST RATES: Simple and compound interest rate

20. The government of France and Germany pays interest on its bonds annually . In the United States and Britain government bonds pay interest semiannually . So if the interest rate on a U.S. government bond is quoted as 10%, the investor receives interest of 5% every six months. 1.05 = 1+10%/2 I f you invest $100 in a bond that pays interest of 10% compounded semiannually, your wealth will grow to 1.05 x $100 = $105 by the end of six months and to 1.05 x $105 = $110.25 by the end of the year. An interest rate of 10% compounded semiannually is equivalent to 10.25% compounded annually. 10% is called the quoted annual interest rate 10.25% is called the effective annual interest rate INTEREST RATES: Quoted and Effective annual interest rate

21. EAR: Effective annual rate 6-

22. Quoted annual interest rate Effective annual interest rate Interest paid once a year = Interest paid more frequently (semiannually, quarterly, …) < INTEREST RATES: Quoted and Effective annual interest rate Effective annual interest rate = Where: r: quoted annual interest rate m: number of compounding periods

23. Benjamin Franklin’s statement, “Money makes money and the money that makes money makes more money,”

24. Effective annual rate Example: 1) The bank A is lending money at the rate of 12%, compounded semi-annually. The bank intends to move to a new plan where the interest is compounded quarterly. Find the new quoted interest rate (APR: annual percentage rate) that the bank will announce?. 2) Youbank offers personal loans at 10%, compounded quarterly. SaveBank , on the other hand, offers similar loans at 10.5%, compounded annually. Which bank should you borrow from?

25. Wheat and the chessboard

26. How an investment of $100 grows with compound interest at different interest rates. FUTURE VALUE $672.75 $265.33 $1,636.65 $100

27. Example CF = $10,000 CF t is the future value of CF r = 9% t = 10 years If interest is compounded annually, the future value is: If interest is compounded monthly, the future value is: Impact of frequency?

28. Example CF = $10,000 CF t is the future value of CF r = 9% t = 10 years If interest is compounded annually, the future value is: CF t = $10,000(1 + .09)10 = $23,674 If interest is compounded monthly, the future value is: CF t = $10,000(1 + .09/12)120 = $24,514 --> when you increase the frequency of compounding, you also increase the future value of your investment

29. PRESENT VALUE How much you need to invest today to produce $114.49 at the end of the second year? What is the present value (PV) of the $114.49 payoff? OR  Run the future value calculation in reverse :

30. PRESENT VALUE How much you need to invest today to produce $114.49 at the end of the second year? What is the present value (PV) of the $114.49 payoff? OR  Run the future value calculation in reverse :

31. PRESENT VALUE: formula CF 1 is cash flow at date 1 CF t is cash flow at date t r is the appropriate interest rate t is the number of compounding periods = discount factor

32. Example

33. 5- Chapter Outline Time and Money Future Value and Compounding Present Value and Discounting More about Present and Future Values

34. 5- 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 Opportunity cost of capital Required return or required rate of return

35. 5- Future Value as a General Growth Formula The formula for growth works for money , but it also works for numerous other variables: Bacteria Housing Epidemics Production

36. 5- Future Value as a General Growth Formula Suppose your company expects to increase unit sales of widgets by 15% per year for the next 5 years. If you sell 3 million widgets in the current year, how many widgets do you expect to sell in the fifth year? 5 N;15 I/Y; 3,000,000 PV CPT FV = -6,034,072 units (remember the sign convention)

37. 5- Quick Quiz What is the difference between simple interest and compound interest? Suppose you have $500 to invest and you believe that you can earn 8% per year over the next 15 years. How much would you have at the end of 15 years using compound interest? How much would you have using simple interest?

38. 5- Chapter Outline Time and Money Future Value and Compounding Present Value and Discounting More about Present and Future Values

39. 5- PV and FV Finance uses “ compounding ” as the verb for going into the future and “ discounting” as the verb to bring funds into the present. Today 1 2 3 4 5 FV PV Today 1 2 3 4 5 FV PV Compounding Discounting

40. 5- Quick Quiz II What is the relationship between present value and future value? Suppose you need $15,000 in 3 years. If you can earn 6% annually, how much do you need to invest today? If you could invest the money at 8%, would you have to invest more or less than at 6%? How much?

41. 5- Chapter Outline Time and Money Future Value and Compounding Present Value and Discounting More about Present and Future Values

42. 5- 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

43. 5- 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 = (1,200 / 1,000) 1/5 – 1 = .03714 = 3.714% Calculator note – the sign convention matters (for the PV)! N = 5 PV = -1,000 (you pay 1,000 today) FV = 1,200 (you receive 1,200 in 5 years) CPT I/Y = 3.714%

44. 5- 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? N = 6 PV = -10,000 FV = 20,000 CPT I/Y = 12.25%

45. 5- 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? N = 17; PV = -5,000; FV = 75,000 CPT I/Y = 17.27%

46. 5- Quick Quiz III What are some situations in which you might want to know the implied interest rate? You are offered the following investments: You can invest $500 today and receive $600 in 5 years. The investment is low risk. You can invest the $500 in a bank account paying 4%. What is the implied interest rate for the first choice, and which investment should you choose?

47. 5- Finding the Number of Periods Start with the 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.

48. 5- 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? I/Y = 10; PV = -15,000; FV = 20,000 CPT N = 3.02 years

49. 5- Number of Periods: Example 2 Suppose you want to buy a new house. You currently have $15,000, and you figure you need to have a 10% down payment plus an additional 5% of the loan amount for closing costs. Assume the type of house you want will cost about $150,000 and you can earn 7.5% per year. How long will it be before you have enough money for the down payment and closing costs?

50. 5- Number of Periods: Example 2 (Continued) How much do you need to have in the future? 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

51. 5- Quick Quiz IV When might you want to compute the number of periods? Suppose you want to buy some new furniture for your family room. You currently have $500, and the furniture you want costs $600. If you can earn 6%, how long will you have to wait if you don’t add any additional money?

52. 5- 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.

53. 5- Finance Formulas

54. 5- Work the Web Many financial calculators are available online. Click on the web surfer to go to Investopedia’s web site and work the following example: You need $50,000 in 10 years. If you can earn 6% interest, how much do you need to invest today? You should get $27,919.74

55. 5- Comprehensive Problem You have $10,000 to invest for five years. How much additional interest will you earn if the investment provides a 5% annual return, when compared to a 4.5% annual return? How long will it take your $10,000 to double in value if it earns 5% annually? What annual rate has been earned if $1,000 grows into $4,000 in 20 years?

56. 5- Terminology Future Value Present Value Compounding Discounting Simple Interest Compound Interest Discount Rate Required Rate of Return

57. 5- Formulas FV = PV(1 + r) t PV = FV / (1 + r) t r = ( FV / PV ) 1/t – 1 t = ln( FV / PV ) / ln(1 + r)

58. 5- Key Concepts and Skills Compute the future value of an investment made today Compute the present value of an investment made in the future Compute the return on an investment and the number of time periods associated with an investment

59. 5- Time changes the value of money as money can be invested. 2. Money in the future is worth more than money received today. 3. Money received in the future is worth less today. What are the most important topics of this chapter?

60. 5- The interest rate (or discount rate) and time determine the change in value of an investment. 5. The longer money is invested, the more compounding will increase the future value. What are the most important topics of this chapter?

61. 5- Questions?