1

Chapter 4: Time Value of Money

Copyright, 2000 Prentice Hall ©Author Nick Bagley, bdellaSoft, Inc.

ObjectiveExplain the concept of compounding

and discounting and to provide examples of real life

applications

5

Compounding: Future Value of a Lump Sum

niPVFV )1(* FV with growths from -6% to +6%

0

500

1,000

1,500

2,000

2,500

3,000

3,500

0 2 4 6 8 10 12 14 16 18 20

Years

Fu

ture

Va

lue

of

$1

00

0

6%

4%

2%

0%

-2%

-4%-6%

10

The Frequency of Compounding

Annual Percentage Rate (APR) Effective Annual Rate (EFF): The

equivalent interest rate, if compounding were only once a year.

m

mAPR

EFF )1(1

11

Effective Annual Rates of an APR of 18%

Annual Percentage Rate

Frequency of Compounding

Annual Effective Rate

18 1 18.00 18 2 18.81 18 4 19.25 18 12 19.56 18 52 19.68 18 365 19.72

12

The Frequency of Compounding

Note that as the frequency of compounding increases, so does the annual effective rate

What occurs as the frequency of compounding rises to infinity?

111

APRm

me

mAPR

LimEFF

13

The Frequency of Compounding

11*

11

11

1

1

m

m

EFFmAPR

EFFm

APR

m

APREFF

m

14

The Frequency of Compounding

Annual Effective Rate

Compounding Frequency

Annual Percentage Rate

12 1 12.00 12 2 11.66 12 4 11.49 12 12 11.39 12 52 11.35 12 365 11.33 12 Infinity 11.33

15

Annuities

A level stream of Cash Flows or Payments

Immediate Annuity: The Cash Flows start immediately.

Ordinary Annuity: The Cash Flows start at the end of the current period.

16

Derivation of PV of Ordinary Annuity Formula

nn i

pmt

i

pmt

i

pmt

i

pmt

i

pmtPV

111

11

13

21

17

Derivation of PV of Ordinary Annuity Formula

}

1

1

1

1

1

1

1

1

1

1{*

13

21

nn iii

iipmtPV

18

PV of Ordinary Annuity Formula

n

n

iipmt

ii

pmt

PV

1

11*

}1

11{*

19

Annuity Formula: PV of Immediate Annuity

}1)1{(*

)1(*}11{*

)1(*

1 n

n

ordimm

iiipmt

iiipmt

iPVPV

20

Derivation of FV of Annuity Formula

11*

1*1

11*FV

sum) (lump 1*FV

annuity) (ord. 1

11*

n

n

n

n

n

iipmt

iii

pmt

iPV

iipmt

PV

21

Perpetual Annuities / Perpetuities

Recall the annuity formula:

niipmt

PV1

11*

• Let n -> infinity with i > 0:

ipmt

PV

22

Alternative Discounted Cash Flow Decision Rules

1. NPV rule: the NPV is the difference between the present value of all future cash inflows minus the present value of all current and future cash outflows. Accept a project if its NPV is positive.

23

DCF rules

Example: You have the opportunity to buy a piece of land for $10,000. You are sure that 5 years from now it will be worth $20,000. If you can earn 8% per year by investing your money in bank, is this investment in the land worthwhile?

24

NPV rule solution

03612$

612,13$000,10$08.1

000,20$000,10$

5

NPV

25

Alternative Discounted Cash Flow Decision Rules

2. FV rule: Invest if the future value of the investment is larger than the future value that can be obtained from the next best alternative.

26

FV rule solution

000,20$693,14$

08.1000,10$ 5

FV

27

Alternative Discounted Cash Flow Decision Rules

3. IRR rule: The IRR is the discount rate at which the NPV is zero. Invest if the IRR is greater than the opportunity cost of capital.

28

IRR rule solution

%8%87.14

)1(

000,20$000,10$

5

i

i

29

Alternative Discounted Cash Flow Decision Rules

4. Choose the investment alternative with fastest payback.

30

Payback rule solution

5908.1

000,20$000,10$

n

n

31

Loan Amortization

The process of paying a loan principal gradually over its term

Example: $100,000 mortgage loan, APR: 9%,

repaid in 3 annual installments pmt=? pmt=$39504.48

nii

pmtPV

1

11*

309.1

11*

09.0100000

pmt

32

Loan Amortization

First Year: Interest: (0.09)(100000)=9000 pmt: 39504.48 principal: 30504.48 Outstanding Balance: 69494.52

33

Loan Amortization

Second Year: Interest: (0.09)(69494.52)=6254.51 pmt: 39504.48 principal: 33250.97 Outstanding Balance: 36243.54

34

Loan Amortization

Third Year: Interest:(0.09)(36243.54)=3262 pmt: 39504.48 principal: 36244 Outstanding Balance: 0

35

Amortization of Principal

0.00

50000.00

100000.00

150000.00

200000.00

250000.00

300000.00

350000.00

400000.00

450000.00

0 24 48 72 96 120 144 168 192 216 240 264 288 312 336 360

Months

Ou

tsta

nd

ing

Bal

ance

36

Percent of Interest and Principal

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 24 48 72 96 120 144 168 192 216 240 264 288 312 336 360

Months

Per

cen

t

% Interest

% Principal

41

Computing NPV in Different Currencies

In any TVM calculation, the cash flows and the interest rate must be

denominated in the same currency.

42

Inflation and Future Values

Example: At age 20 you save $100 and invest it at a dollar interest rate of 8% per year, and you do not take it until age 65. If the inflation is estimated 5% per year, how much will you have accumulated in the account at that time in terms of real purchasing power?

43

Solution 1:

355$

02857.1100$

)1(

02857.11

)(1

)(.1)(1

45

FVreal

RPVFVreal

R

iInflation

rratenomRratereal

n

44

Solution 2:

355$985.8

192,3$.

985.8

05.145

192,3$

08.1100$45.

45

45

FPL

FVnomFVreal

yearsinlevelprice

yearsinFVnom

45

Inflation and Present Values

Example: Your daughter is 10 years old, and you are planning to open an account to provide for her college education. Tuition for a year of college is now $15,000. How much must you invest now in order to have enough to pay for her first year’s tuition 8 years from now, if you think you can earn a rate of interest that is 3% more than the inflation rate of 5%?

46

Solution:

Wrong:

841.11$03.1

000,15$8

PV

104.8$08.1

000,15$8

PV

47

Inflation and Present Values

Never use a nominal interest rate when discounting real cash flows or a real interest rate when discounting nominal cash flows.