The Time Value of MoneyChapter 8

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.

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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 is less burdensome than that debt to be paid now.

4

Future Value of a Lump Sum

• 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 = $146.93$100 x (1.08)5 = $146.93

– After fifteen years:$100 x (1.08)15 = $317.22

– Equation:FV = PV (1 + k)n

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Future Value of a Lump SumGraph of the Effect of Compounding

$1000

900

800

700

600

400

500

300

200

0

Year0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

k = 8%

k = 4%

k = 0%

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Present Value of a Lump Sum

• Value today of an amount to be received or paid in the future.

– Example: Expect to receive $100 in one year. If we can invest at 10%, what is it worth today?

PV = FVn x1

(1 + k)n

0 1 2

$100PV = 100 (1.10)1

= 90.90

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Present Value of a Lump Sum

• Value today of an amount to be received or paid in the future.

– Example: Expect to receive $100 in EIGHT years. If we can invest at 10%, what is it worth today?

PV = FVn x1

(1 + k)n

0 1 2 3 4 5 6 7 8

$100PV = 100 (1.10)8

= 46.65

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Present Value of a Lump SumGraph of the Effect of Discounting

$100

90

80

70

60

40

50

30

20

0

Year0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

k = 10%

k = 5%

k = 0%

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Financial Calculator Solution - PV

• Previous Example: Expect to receive $100 in EIGHT years. If we can invest at 10%, what is it worth today? PV = 100

(1+.10)8 = 46.65Using Formula:

Calculator Enter:N = 8I/YR = 10FV = 100CPT PV = ?

N I/YR PV PMT FV

- 46.65

8 10 ?

100

10

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 = $322.10

Calculator Enter:N = 5I/YR = 10PV = -200CPT FV = ?

10

N I/YR PV PMT FV

322.10

5 10 -200 ?

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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.

$500 $500 $500 $500 $500

Jan Feb Mar Dec

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Future Value of an Annuity

• 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?

0 1 2 3

$0 $100 $100 $100

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Future Value of an Annuity• 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?

0 1 2 3

$0 $100 $100 $100$100(1.08)2 $100(1.08)1

$108.00$116.64$324.64

$100.00

$100(1.08)0

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Future Value of an Annuity

0 1 2 3

$0 $100 $100 $100$100(1.08)2 $100(1.08)1

$108.00$116.64$324.64

$100.00

$100(1.08)0

= 100(3.2464) = 324.64FVA = PMTx( ) (1+k) - 1

k

n = 100

(1+.08)3 - 1 .08( )

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Future Value of an AnnuityCalculator Solution

0 1 2 3

$0 $100 $100 $100

Enter:N = 3I/YR = 8PMT = -100CPT FV = ?

N I/YR PV PMT FV

3 8 -100 ?

324.64

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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 1 2 3

$0 $100 $100 $100

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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 1 2 3

$0 $100 $100 $100

$100 / (1.08)2

$92.60$85.73$79.38

$100/(1.08)1 $100 / (1.08)3

$257.71

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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 1 2 3

$0 $100 $100 $100

$100 / (1.08)2

$92.60$85.73$79.38

$100/(1.08)1 $100 / (1.08)3

$257.71 PVA = PMTx( )

1(1+k)n1 -

k= 100(2.5771) = 257.71

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Present Value of an AnnuityCalculator Solution

0 1 2 3

$0 $100 $100 $100

Enter:N = 3I/YR = 8PMT = 100CPT PV = ?

N I/YR PV PMT FV

3 8 ? 100

-257.71

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Annuities and Annuities Due

• 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.

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Future Value of an Annuity Due

• You deposit $100 each year at the beginning of the 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?

0 1 2 3

$100 $100 $100

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Future Value of an Annuity Due• You deposit $100 each year (begin. of year) into 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?

0 1 2 3

$100 $100 $100$100(1.08)2 $100(1.08)1$100(1.08)3

$108.00

$116.64$125.97$350.61

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Future Value of an Annuity Due• You deposit $100 each year (begin. of year) into 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?

0 1 2 3

$100 $100 $100$100(1.08)2 $100(1.08)1$100(1.08)3

$108.00

$116.64$125.97$350.61

FVADUE = PMTx( ) (1+k) (1+k)n - 1 k =100(3.2464)(1.08)= 350.61

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Future Value of an Annuity DueCalculator Solution

0 1 2 3

$100 $100 $100

Enter:N = 3I/YR = 8PMT = -100CPT FV = ?

N I/YR PV PMT FV

3 8 -100 ?

350.61

Set Calculator to“Begin Mode”

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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?

0 1 2 3

$100 $100 $100$100 / (1.08)2

$92.60$85.73

$100/(1.08)1

$100.00

$100/(1.08)0

$278.33

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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?

0 1 2 3

$100 $100 $100$100 / (1.08)2

$92.60$85.73

$100/(1.08)1

$100.00

$100/(1.08)0

$278.33PVADUE= PMTx( )

1(1+k)n1 -

k(1+k)

= 100(2.5771)(1.08)

= 278.33

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Amortized Loans

• A loan that is paid off in equal amounts that include principal as well as interest.

• Solving for loan payments.• Solving for interest and principal paid.

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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?

0 1 2 3 4 5

$5,000 $? $? $? $? $?

ENTER:N = 5I/YR = 6PV = 5,000CPT PMT = ?

N I/YR PV PMT FV

–1,186.98

5 6 5,000 ?

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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?

$20,000 = PMT(40.184782)PVA = PMTx( )

1(1+k)n1 -

kPMT = 497.70

= PMT .0075

1 - 1 (1.0075)48

$20,000 ( )

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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 = 48I/YR = 9PV = 20,000CPT PMT = ?

N I/YR PV PMT FV

– 497.70

48 9 20,000 ?Note: N = 4 * 12 = 48

And set your P/Y to 12!!

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Perpetuities

• A perpetuity is a series of equal payments at equal time intervals (an annuity) that will be received into infinity.

PMT k

PVP =

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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%?

PMT k

PVP =

If k = 8%: PVP = $5/.08 = $62.50

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PV of Uneven Cash Flows

• How much would the following cash flows be worth to you today if you could earn 8% on your deposits?

$6mil /(1.08)3

$3,429,355$4,762,993$4,410,179

$4mil/(1.08)2 $6mil /(1.08)4

$21,454,380

0 1 2 3 4

$7,000,000 $2,000,000 $4,000,000 $6,000,000 $6,000,000

$1,851,852$2mil/(1.08)1$7,000,000

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Solving for k• Example: A $200 investment has grown to $230 over two years.

What is the ANNUAL return on this investment? 0 1 2

$230$200

FV = PV(1+ k)n

230 = 200(1+ k)2

1.15 = (1+ k)2

1.0724 = 1+ k

1.15 = (1+ k)2

k = .0724 = 7.24%

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Solving for k• Example: A $200 investment has grown to $230 over two years.

What is the ANNUAL return on this investment?

Enter known values: N = 2PMT = 0PV = -200FV = 230

Solve for: I/YR = ?

N I/YR PV PMT FV

2 -200 230?

7.24

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Compounding more than Once per Year

• $500 invested at 9% annual interest for 2 years. Compute FV.

$500(1.09)2 = $594.05 Annual

$500(1.045)4 = $596.26 Semi-annual

$500(1.0225)8 = $597.42 Quarterly

$500(1.0075)24 = $598.21 Monthly

$500(1.000246575)730 = $598.60 Daily

Compounding Frequency

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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 ekn

FV = $500 x e.09 x 2 = $598.61