The Time Value of Money

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1 The Time Value of Money 1

2 Learning Objectives The time value of money and its importance to business. The future value and present value of a single amount. The future value and present value of an annuity. The present value of a series of uneven cash flows. 2

3 The Time Value of Money Money grows over time when it earns interest. Therefore, money that is to be received at some time in the future is worth less than the same dollar amount to be received today. Similarly, a debt of a given amount to be paid in the future are less burdensome than that debt to be paid now. 3

4 The Future Value of a Single Amount Suppose that you have $100 today and plan to put it in a bank account that earns 8% per year. How much will you have after 1 year? 5 years? 15 years? After one year: $100 + (.08 x $100) = $100 + $8 = $108 OR: $100 x (1.08) 1 = $108 4

5 The Future Value of a Single Amount Suppose that you have $100 today and plan to put it in a bank account that earns 8% per year. How much will you have after 1 year? 5? 15? After one year: $100 x (1.08) 1 = $108 After five years: $100 x 1.08 x 1.08 x 1.08 x 1.08 x 1.08 = $100 x (1.08) 5 = $ After fifteen years: Equation: $100 x (1.08) 15 = $ FV = PV (1 + k) n 5

6 The Future Value of a Single Amount Graphical Presentation Different Interest Rates $ k = 8% k = 4% k = 0% Year 6

7 Present Value of a Single Amount Value today of an amount to be received or paid in the future. PV = FV n x 1 (1 + k) n Example: Expect to receive $100 in one year. If can invest at 10%, what is it worth today? PV = (1.10) 1= $100 7

8 Present Value of a Single Amount Value today of an amount to be received or paid in the future. PV = FV n x 1 (1 + k) n Example: Expect to receive $100 in EIGHT years. If can invest at 10%, what is it worth today? PV = 100 (1+.10) 8 = $100 8

9 Present Value of a Single Amount Graphical Presentation $ k = 0% k = 5% k = 10% Year 9

10 Financial Calculator Solution - PV Previous Example: Expect to receive $100 in EIGHT Using Formula: years. If can invest at 10%, what is it worth PV = 100 today? (1+.10) 8 = Calculator Enter: N = 8 I/YR = 10 FV = 100 CPT PV =? N I/YR PV PMT FV 8 10?

11 Financial Calculator Solution - FV Previous Example: You invest $200 at 10%. How much is it worth after 5 years? Using Formula: FV = $200 (1.10) 5 = $

12 Financial Calculator Solution - FV Previous Example: Previous Example: You invest $200 at 10%. How much is it worth after 5 years? Using Formula: FV = $200 (1.10) 5 = $ Calculator Enter: N = 5 I/YR = 10 PV = -200 CPT FV =? N I/YR PV PMT FV ?

13 Annuities An annuity is a series of equal cash flows spaced evenly over time. For example, you pay your landlord an annuity since your rent is the same amount, paid on the same day of the month for the entire year. Jan Feb Mar Dec $500 $500 $500 $500 $500 13

14 Future Value of an Annuity $0 $100 $100 $100 You deposit $100 each year (end of year) into a savings account. How much would this account have in it at the end of 3 years if interest were earned at a rate of 8% annually? 14

15 Future Value of an Annuity $0 $100 $100 $100 $100(1.08) 2 $100(1.08) 1 $100(1.08) 0 $ $ $ $ You deposit $100 each year (end of year) into a savings account. How much would this account have in it at the end of 3 years if interest were earned at a rate of 8% annually? 15

16 Future Value of an Annuity $0 $100 $100 $100 $100(1.08) 2 $100(1.08) 1 $100(1.08) 0 $ $ $ $ How much would this account have in it at the end of 3 years if interest were earned at a rate of 8% annually? n = 100 ( (1+.08) 3-1) FVA = PMTx( (1+k) - 1 ).08 k = 100(3.2464) =

17 Future Value of an Annuity Calculator Solution $0 $100 $100 $100 Enter: N = 3 I/YR = 8 PMT = -100 CPT FV =? N I/YR PV PMT FV ? 17

18 Present Value of an Annuity How much would the following cash flows be worth to you today if you could earn 8% on your deposits? $0 $100 $100 $100 18

19 Present Value of an Annuity How much would the following cash flows be worth to you today if you could earn 8% on your deposits? $100/(1.08) 1 $100 / (1.08) 2 $100 / (1.08) $0 $100 $100 $100 $92.60 $85.73 $79.38 $

20 Present Value of an Annuity How much would the following cash flows be worth to you today if you could earn 8% on your deposits? $0 $100 $100 $100 $92.60 $85.73 $79.38 $ PVA = PMTx( ) $100/(1.08) 1 $100 / (1.08) 2 $100 / (1.08) (1+k) n k = (1.08) 3.08 ( ) = 100(2.5771) =

21 Present Value of an Annuity Calculator Solution $0 $100 $100 $100 PV=? Enter: N = 3 I/YR = 8 PMT = 100 CPT PV =? N I/YR PV PMT FV 3 8?

22 Annuities An annuity is a series of equal cash payments spaced evenly over time. Ordinary Annuity: The cash payments occur at the END of each time period. Annuity Due: The cash payments occur at the BEGINNING of each time period. 22

23 Future Value of an Annuity Due $100 $100 $100 FVA=? You deposit $100 each year (beginning of year) into a savings account. How much would this account have in it at the end of 3 years if interest were earned at a rate of 8% annually? 23

24 Future Value of an Annuity Due $100 $100 $100 $100(1.08) 2 $100(1.08) 1 $100(1.08) 3 $108 $ $ $ You deposit $100 each year (beginning of year) into a savings account. How much would this account have in it at the end of 3 years if interest were earned at a rate of 8% annually? 24

25 Future Value of an Annuity Due $100 $100 $100 $100(1.08) 2 $100(1.08) 1 $100(1.08) 3 $108 $ $ $ How much would this account have in it at the end of 3 years if interest were earned at a rate of 8% annually? FVA = PMTx( (1+k) n -1 )(1+k) k (1+.08) 3-1 = 100 ( ) (1.08).08 =100(3.2464)(1.08)=

26 Present Value of an Annuity Due How much would the following cash flows be worth to you today if you could earn 8% on your deposits? $100 $100 $100 PV=? 26

27 Present Value of an Annuity Due How much would the following cash flows be worth to you today if you could earn 8% on your deposits? $100 $100 $100 $100/(1.08) 0 $100/(1.08) 1 $100 / (1.08) 2 $ $92.60 $85.73 $

28 Present Value of an Annuity Due How much would the following cash flows be worth to you today if you could earn 8% on your deposits? $100 $100 $100 $100/(1.08) 0 $100/(1.08) 1 $100 / (1.08) 2 $ $92.60 $85.73 $ (1+k) PVA = PMTx( n )(1+k) k = 100( (1.08) 3 )(1.08).08 = 100(2.5771)(1.08) =

29 Amortized Loans A loan that is paid off in equal amounts that include principal as well as interest. Solving for loan payments. 29

30 Amortized Loans You borrow $5,000 from your parents to purchase a used car. You agree to make payments at the end of each year for the next 5 years. If the interest rate on this loan is 6%, how much is your annual payment? $5,000 $? $? $? $? $? ENTER: N = 5 I/YR = 6 PV = 5,000 CPT PMT =? 1, N I/YR PV PMT FV 5 6 5,000? 30

31 Amortized Loans You borrow $20,000 from the bank to purchase a used car. You agree to make payments at the end of each month for the next 4 years. If the annual interest rate on this loan is 9%, how much is your monthly payment? 1 - (1+k) PVA = PMTx( n ) k (1.0075) 48 $20,000 = PMT ( ).0075 $20,000 = PMT( ) PMT =

32 Amortized Loans You borrow $20,000 from the bank to purchase a used car. You agree to make payments at the end of each month for the next 4 years. If the annual interest rate on this loan is 9%, how much is your monthly payment? ENTER: N = 48 I/YR =.75 PV = 20,000 CPT PMT =? Note: N = 4 * 12 = 48 I/YR = 9/12 = N I/YR PV PMT FV ,000? 32

33 Perpetuities A perpetuity is a series of equal payments at equal time intervals (an annuity) that will be received into infinity. PVP = PMT k 33

34 Perpetuities A perpetuity is a series of equal payments at equal time intervals (an annuity) that will be received into infinity. Example: A share of preferred stock pays a constant dividend of $5 per year. What is the present value if k =8%? PVP = PMT k 34

35 Perpetuities A perpetuity is a series of equal payments at equal time intervals (an annuity) that will be received into infinity. Example: A share of preferred stock pays a constant dividend of $5 per year. What is the present value if k =8%? PVP = PMT k If k = 8%: PVP = $5/.08 = $

36 Solving for k Example: A $200 investment has grown to $230 over two years. What is the ANNUAL return on this investment? $200 $230 FV = PV(1+ k) n 230 = 200(1+ k) = (1+ k) = (1+ k) = 1+ k k =.0724 = 7.24% 36

37 Solving for k - Calculator Solution Example: A $200 investment has grown to $230 over two years. What is the ANNUAL return on this investment? Enter known values: N = 2 I/YR =? PV = -200 FV = 230 Solve for: PMT. =? N I/YR PV PMT FV ?

38 Compounding more than Once per Year $500 invested at 9% annual interest for 2 years. Compute FV. $500(1.09) 2 = $ Annual $500(1.045) 4 = $ Semi-annual $500(1.0225) 8 = $ Quarterly $500(1.0075) 24 = $ Monthly $500( ) 730 = $ Daily Compounding Frequency 38

39 Continuous Compounding Compounding frequency is infinitely large. Compounding period is infinitely small. Example: $500 invested at 9% annual interest for 2 years with continuous compounding. FV = PV x e kn FV = $500 x e.09 x 2 = $

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