Time Value of Money (TVM) Professor: Burcu Esmer

Transcription

1 Time Value of Money (TVM) Professor: Burcu Esmer

2 Time Value of Money You ve just won the Itsy Bitsy Lottery, with a prize of $000. You can choose between receiving your prize now, or in 50 years. What do you choose? Why? 2

3 Basic Finance Principles A dollar today is worth more than a dollar tomorrow Time Value of Money A safe dollar is worth more than a risky one Risk and Return 3

4 Time Value of Money People prefer to have a dollar today rather than a dollar tomorrow They must be compensated for waiting to consume! So, markets form that allow some people to invest and earn interest for delaying consumption, and other people to borrow money and pay interest to be able to consume now Given some cash flows in the future, how do we find the present value, or how much people would be willing to pay for those future cash flows? 4

5 Future Value: FV If I deposit $00 in a bank account that pays 0%, how much will I have after year? $00 FV=? r =0% 0 $00+(.0)(00) = $0 or $00(+.0) = $0 5

6 FV (cont d) After 2 years? $0+(.0)(0) = $2 or $00(+.0)(.0) = $00(.0) 2 = $2 After 30 years? $00(.0) 30 = $, After t years? $00(.0) t = FV 6

7 Present Value: PV (**i.e. value of any asset**) Now let s reverse the problem. How much will I pay today to receive $0 one year from today if I require a 0% return on my money? P 0 = PV=? 0 r =0% 0 P 0 PV $0 ( 0. ) $00 7

8 PV (Cont d) What about PV of $2 in 2 years? P 0 = PV=? 2 r =0% 0 2 PV $2 ( 0. ) 2 $00 $,000 in t years? P 0 = PV=? 000 r =0% 0 N PV $,000 t (.0) 8

9 Time Value of Money FV PV ( r) t or rearranging terms, we get PV FV FV t ( r) ( r) t 9

10 Example Calculate Present Value Your job promises a one-time retirement benefit of $00,000 on the day of your retirement. You plan to work for 5 more years, and require a return of 8% on investments. What is this benefit worth? 0

11 Present Value: Another Example Always ahead of the game, Tommy, at 8 years old, believes he will need $00,000 to pay for college. He goes to college at 8. If he can invest at a rate of 7% per year, how much money should he ask his rich Uncle GQ to give him? FV $00, 000 t 0 yrs r 7% PV FV $00,000 0 $50,835 (.07) ( r) t Note: Ignore inflation/taxes

12 Example Calculate r What rate of return will double my money if I invest for 9 years? (i.e., PV=$ FV=$2 r =? t=9) 2

13 Example Calculate time (t) How long will it take to double my money if I invest it at 8%? (i.e. PV=$ FV=$2 r =8% t=?) 3

14 Relationship between Present Value and Interest Rate Your aunt promises to give you $00 in 0 years. If the interest rate you could earn is 5% per year, what is the promise worth in today s terms? If the interest rate you could earn is 8% per year, what is the promise worth in today s terms? 4

15 Present Values: Changing Discount Rates The present value of $00 to be received in to 20 years at varying discount rates: PV of $ Discount Rates 0% 5% 0% 5% Number of Years 5

16 Multiple Cash Flows 6

17 PV of Multiple Cash Flows The present value of multiple cash flows can be calculated: Denote : C C C 2 t The cash flow in year The cash flow in year 2 The cash flow in year t (with any number of cash flows in between) PV C C... t 2 2 ( r) ( r) ( r) C t 7 Recall: r = the discount rate

18 Multiple Cash Flows: Example Your auto dealer gives you the choice to pay $5,500 cash now or make three payments: $8,000 now (that is at t=0) and $4,000 at the end of the following two years ($4,000 payment at t= and $4,000 payment at t=2). If your cost of money (discount rate) is 8%, which do you prefer? Initial Payment* 8, PV of C PV of C 4,000 (.08) 4,000 2 (.08) 2 3, , Total PV $5, * The initial payment occurs immediately and therefore would not be discounted.

19 Example Multiple Cash Flows What is the present value of the following cashflow stream if the interest rate is 8%? What is the value of the same cash-flow stream at time t=, t=2, t=4, CF=300 9

20 Multiple Cash Flows Perpetuities 20

21 Perpetuities (lasting for eternity) A perpetuity is an infinite sequence of equal cash flows i.e. $ per year forever Present value of a perpetuity: PV t PMT r t 2

22 Perpetuities (cont.) Examples of Perpetuities: preferred stock dividends: amount of dividend is fixed and occurs on a regular basis, presumably continuing forever perpetual bond (consol): bond issued (by British government) that pays interest forever 22

23 Perpetuity Formula Derivation r x x xr x x r x x for solve and from Subtract r r r x r r r x xr ; ) ( : 2) )... ) ( 2)... )

24 Preferred Stock Example: CHS Inc s preferred stock promises a $2 dividend that does not change year to year. Estimate its price if investors require a 9% return. P $ $22.22 If it currently trades at $25.24, what is the implied required rate of return? $2 $2 $ r or r $ % 24

25 Perpetuities: Another Example In order to create an endowment, which pays $85,000 per year forever, how much money must be set aside today if the rate of interest is 8%? PV.08 $2,32,500 85,000 What if the first payment won t be received until 3 years from today? PV 2,32,500 2 $,982, (.08)

26 Multiple Cash Flows Annuity 26

27 Annuity annuity : Finite level stream of cash flows ordinary annuity vs. annuity due 27

28 PV of an annuity How much would you pay to receive $00 every year for the next 4 years if you require a 0% return on your investment? P 0 =? $00 $00 $00 $00 00 (. 0) (. 0) (. 0) (. 0) Total = PV = P 0

29 The present value of the annuity is PV ( 0. ) ( 0. ) ( 0. ) ( 0. ) ( 0. ) ( 0. ) ( 0. ) ( 0. ) $

30 Present Value of an Annuity Let: C = yearly cash payment r = interest rate n = number of years cash payment is received PV C r r( r) t The terms within the brackets are collectively called the annuity factor. 30

31 Back to example, Recall, CF=00, r =0%, and t=4 PVA CF r r ( r) t ( 0.0) 4 00[3.699]

32 Annuities: Example You are purchasing a home and are scheduled to make 30 annual installments of $0,000 per year. Given an interest rate of 5%, what is the price you are paying for the house (i.e. what is the present value)? PV PV $0, 000 $53, (.05) 30 32

33 FV of an annuity How much would be in your bank account if you deposit $00 at the end of every year for the next 4 years, and your account earns 0% a year? FV=? $00 $00 $00 $00 0 r=0% ( 0. ) 00( 0. ) 00( 0. ) 00( 0. ) ( 0. ) ( 0. ) ( 0. ) ( 0. ) $

34 FV of an Annuity FV PV t ( PV 0 )( r) t CF r r ( r) t ( r) t CF ( r) r t r CF ( r) r t 34

35 Future Value of Annuities: Example You plan to save $4,000 every year for 20 years and then retire. Given a 0% rate of interest, how much will you have saved by the time you retire? FV FV $4, (.0) 20.0(.0) $229,

36 Annuity Formula Notes The PV r,t formula is constructed to discount all the annuity payments back to period before the st cash flow! PV CF CF CF CF N The FV r,t formula is constructed to compound all the annuity payments forward to the day of the last cash flow! CF CF CF CF FV N Remember these rules! 36

37 Ordinary Annuity Payments or receipts occur at the end of each period. End of Period End of Period 2 End of Period $00 $00 $00 Today Equal Cash Flows Each Period Apart 37

38 Annuity Due Payments or receipts occur at the beginning of each period. Beginning of Period Beginning of Period 2 Beginning of Period $00 $00 $00 Today Equal Cash Flows Each Period Apart 38

39 For example:- You are receiving $,000 a year for 3 years and you deposit each annual receipt into a savings account earning 7% interest. How much money will you have at the end of three years? 39

40 Example of an Ordinary Annuity - FVA Cash flows occur at the end of the period %,000,000,000,070,45 3,25 = FVA 3 40

41 Example of an Annuity Due - FVAD Cash flows occur at the beginning of the period %,000,000,000,070,45,225 3,440 = FVAD 3 4

42 Let s assume the cash flows of $,000 a year for three years represent withdrawals from a savings account earning 7% compound annual interest. How much money would you have to place in the account right now (time period 0) such that you would end up with a zero balance after the last $,000 withdrawal? 42

43 Example of Ordinary Annuity - PVA Cash flows occur at the end of the period % 2, = PVA 3,000,000,000 43

44 Example of an Annuity Due - PVAD Cash flows occur at the beginning of the period %,000.00,000, , = PVAD n 44

45 Annuity Due What is it? How does it differ from an ordinary annuity? PV PV ( r) Annuity Due Annuity How does the future value differ from an ordinary annuity? FV FV ( r) Annuity Due Annuity 45 Recall: r = the discount rate

46 Annuity Due: Example Assuming a 0% discount rate, how much would you pay for a 5-year annuity that begins by making the first payment today? - $00 $00 $00 $00 $00 0 r=0% Due means payments arrive at the beginning of the period! If we treat it as a regular annuity, each cash flow is discounted too many times. So, we need a second step that brings the cash flows forward period. PV(Annuity Due) = (PV (of regular annuity) 0%,5 )(.0) 46

47 ) Discount each cash flow individually: PV( AnnuityDue) 00 ( 0. ) ( 0. ) ( 0. ) ( 0. ) ) Now factor out (+r) 00 ( 0. ) ( 0. ) ( 0. ) ( 0. ) ( 0. ) ( 0. ) =(.0)(PV (of regular annuity) 0%,5 ) = $46.99 In general PV(Annuity Due)= (+r)(pv(of regular annuity) r,t ) 47

48 Annuities Due: Example FV AD FV Annuity ( r) Example: Suppose you invest $ annually at the beginning of each year at 0% interest. After 50 years, how much would your investment be worth? FV AD FV Annuity ( r) FV FV AD AD ($500,000) $550,000 (.0) 48

49 Formulas For single cash flows (lump sums) For Streams PV r, t ( r) t PV ( Perpetuity) t CF r t FV r) r, t ( t PVA CF t [ r r ( r) r, t t ] FVA r, t CF ( r) r t Note: r = discount rate t= # periods Note: r = discount rate t= # periods = # payments 49

50 Example How much do you need to put into your retirement account each year if you plan to retire in 30 years, and if you will need $2 million? (assume will earn a 2% rate of return) Organize information CF CF CF CF r = 2% FV = $2million 50

51 $2million (.2) CF.2 30 CF(24.33) CF $2million $8, PRACTICE: What if the rate were 8%? 6%? Answers: At 8%, A=$7, At 6%, A=$3,77.37 What if we wait 5-years to start saving (i.e. N=25, r =2%)? Answer: At 2% with 25 years, A=$4,

52 How much would you pay to receive $00 every year for 4 years but the payments don t start for 3 years? (assume you require a 0% return on your investment? P 0 =? $00 $00 $00 $ Step : Lump the cash flows PV 2 CF r r ( r) t (.0) 4 00(3.699)

53 Step 2: Move the lump to t=0 PV PV 0 2 (.0) (0.8264) Alternatively, we could lump the cash flows at t=6 and move them from t=6 to t=0! FV 6 ( 00 r) r (.0) (4.640) t 4 PV FV (.0) 464.0( )

54 What would you pay for a 0-year $500 annuity if you require a 0% return and the annuity makes it s first payment 3 years from today? Step : Organize information r = 0% Decompose into two parts: ) Treat like a normal annuity that starts paying in -year 2) Discount the PV of that annuity 2-years Part2 Annuity Part 54

55 Step 2: write out equation and solve X CF r r ( r) t X (.0) 0 500(6.446) X=$3, PV $3, (.0) 2 $2,

56 Example: Loan Amortization Consider a 4-year amortizing loan. You borrow $,000 initially, and repay it in four equal annual year-end payments. A. If the interest rate is 8%, find the annual payment. B. Fill in the table: Time Loan Balance Year-End Year-End Amortization Interest Due Payment of Loan 0 $, $80.00 $30.92 $22.92 $ $62.25 $30.92 $ $538.4 $43.07 $30.92 $ $ $22.36 $30.92 $ $0.00 $0.00 C. Show that the loan balance after year is equal to the yearend payment of $ times 3 year annuity factor. PV= $30.92 $ (.08) 56

57 Loan Amortization 57

58 Interest rate 58

59 What s in an interest rate? Forgone consumption All else equal, I would prefer to have it now rather than wait for a year. Inflation Things will cost more in the future Risk How bad is the downside? 59

60 Difference bw simple and compound interest pay attention to the difference between simple interest and compound interest!!! Future Value: Amount to which an investment will grow after earning interest. Let r = annual interest rate Let t = # of years Simple Interest FV = Initial investment ( rt) Simple Compound Interest FV = Initial investment ( r) t Compound 60

61 Simple Interest: Example Interest earned at a rate of 7% for five years on a principal balance of $00. Example - Simple Interest Today Future Years Interest Earned Value Value at the end of Year 5: $35 6

62 Compound Interest: Example Interest earned at a rate of 7% for five years on the previous year s balance. Example - Compound Interest Interest Earned Value 00 Today Future Years Value at the end of Year 5 = $

63 Future Value The Power of Compounding Interest earned at a rate of 7% for the first forty years on the $00 invested using simple and compound interest. $,600 $,400 $,200 $,000 Simple Interest Compound Interest $800 $600 $400 $200 $ Year 63

64 Effective Interest Rates Effective Annual Interest Rate - Interest rate that is annualized using compound interest. Annual Percentage Rate - Interest rate that is annualized using simple interest. 64

65 Compounding Most rates are Stated Annual Interest Rates, a.k.a APR - Annual Percentage Rate Credit cards (2% compounded monthly) Certificate of deposit rates (5% with daily compounding) Coupon rates on bonds (0% with semiannual compounding) Stated rates ignore compounding! CANNOT BE USED FOR PV or FV! 65

66 Translating Stated Rates into Effective Rates A stated rate of 2% with monthly compounding means you pay % a month for twelve months! months % -year From the stated rate (APR), the only thing useful is the periodic rate: r Periodic r. 2 Stated. 0 N 2 66 # of compounding periods

67 Does 2% really mean 2%? If we deposit $ today in a bank account that has a stated rate of 2% with monthly compounding, what will we have at the end of the year? FV=(.0) 2 =$.268 So what is the actual return? 67

68 Effective Rates (cont d) $. 268 $ $ % Effectively, we ve earned 2.68% in a year So what s the rule? APR EAR( EffectiveAnnualRate) m m: number of compounding periods in a year m 68

69 Credit card statement rate Credit card statement rate Summary: monthly rate =.65% nominal annual rate = 9.8% (=.65% x 2) What is EAR? (assume interest is compounded monthly) (+0.065) 2 - = 2.7% 69

70 0% Stated rate: Compounding N Formula Effective Annual rate Annual ( ) 0% Semiannual 2 Monthly 2 Daily 365 Continuous ( ( ( ) 2. 0 ) 2. 0 ) 365 e % 0.47% 0.52% 0.52% 70

71 Real vs. Nominal Interest Rate 7

72 Inflation Inflation - Rate at which prices as a whole are increasing. Nominal Interest Rate - Rate at which money invested grows. Real Interest Rate - Rate at which the purchasing power of an investment increases. 72

73 Nominal vs. Real Returns nominal return: raw % return, not adjusted for inflation real return: nominal return less inflation e.g. You invested in a CD for one year that earned you a total of 5.5% on your investment. If prices of goods and services went up by 3% over the same period, your real return is about 2.5%. 73

74 Inflation rate - Turkey 74

75 Returns on a $ investment in 900 (nominal returns) $ $0.000 Common Stock Bonds Bills 5,578 Dollars $.000 $ $0 $ Source: Brealey, Myers, and Allen Start of Year

76 Returns on a $ investment in 900 (real returns) Dollars $.000 $00 $0 Common Stock Bonds Bills $ Start of Year Source: Brealey, Myers, and Allen

77 Nominal vs. Real Returns (cont.) real int erest rate ( no min al ( inf int erest lation) rate) Approximately ; Real interest rate nominal interest rate inflation rate e.g. Suppose that you invest your funds at an interest rate of 8%. What will be your real interets if the inflation is zero? What if it is 5%? Answer: +real interest rate= (+8%) / (+0) => real interest rate : 8 % +real interest rate= (+8%) / (+5%) => real interest rate : 2.857% 77

78 Inflation: Example If the nominal interest rate on your interest-bearing savings account is 2.0% and the inflation rate is 3.0%, what is the real interest rate? real interest rate = real interest rate = real interest rate = or -.97% Approximation = =.0 % 78

79 Example 2 How much would you invest today to earn $00 in a year if the discount rate is 0%? Answer= 00 /. = $90.9 How would your answer change if the inflation rate is 7%? The real value of $00 will be 00/.07= $93.46 The real interest rate = 2.8% Answer = 93.46/.028 = $90.9 Nominal cash flows must be discounted by the nominal interest rate and real cash flows must discounted by the real interest rate! 79