1

Chapter 5The Time Value of Money

Some Important Concepts

2

Topics Covered

Future Values and Compound Interest Present ValuesMultiple Cash FlowsPerpetuities and AnnuitiesEffective Annual Interest Rates.Inflation

3

Future Values

Future Value - Amount to which an investment will grow after earning interest.

Compound Interest - Interest earned on interest.

Simple Interest - Interest earned only on the original investment.

4

Future Values

Example - Simple InterestInterest earned at a rate of 6% for five years on a principal balance of $100.

Today Future Years 0 1 2 3 4 5

Interest EarnedValue 100

6106

6112

6118

6124

6130

Value at the end of Year 5 = $130

5

Future Values

Example - Compound InterestInterest earned at a rate of 6% for five years on the previous year’s balance.

Today Future Years 0 1 2 3 4 5

Interest EarnedValue 100

6106

6.36112.36

6.74119.10

7.15126.25

7.57133.82

Value at the end of Year 5 = $133.82

6

Future Values

Future Value of $100 = FV

FV r t $100 ( )1

7

Future Values

0

1000

2000

3000

4000

5000

6000

7000

Number of Years

FV

of

$100

0%

5%

10%

15%

8

Future Values

Example – Manhattan Island Sale

Peter Minuit bought Manhattan Island for $24 in 1626. Was this a good deal?

trillion

FV

57.120$

)08.1(24$ 380

To answer, determine $24 is worth in the year 2006, compounded at 8%.

Financial calculator: n=380, i=8, PV=-24, PMT=0 FV= $120.57 trillion

Set up the Texas instrument

2nd, “FORMAT”, set “DEC=9”, ENTER2nd, “FORMAT”, move “↓” several times,

make sure you see “AOS”, not “Chn”.2nd, “P/Y”, set to “P/Y=1”2nd, “BGN”, set to “END” P/Y=periods per year, END=cashflow happens end of periods

10

Present Values

If the interest rate is at 6 percent per year, how much do we need to invest now in order to produce $106 at the end of the year?

How much would we need to invest now to produce $112.36 after two years?

100$06.1

106$

1.06

valuefuture(PV) luePresent va

100$06.1

36.112$

1.06

valuefuture(PV) luePresent va

22

11

Present Values

tr)+(1

periodsafter t Value Future=PV

Discount Factor (DF) = PV of $1

Discount Factors can be used to compute the present value of any cash flow.

PV: Discounted Cash-Flow

(DCF)

r: Discount Rate

DFr t

1

1( )

12

Present Values

0

20

40

60

80

100

120

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Number of Years

PV

of

$100

5%

10%

15%

Interest Rates

13

Present Values

Example

In 2007, Puerto Rico needed to borrow about $2.6 billion for up to 47 years. It did so by selling IOUs, each of which simply promised to pay the holder $1,000 at the end of that time. The market interest rate at the time was 5.15%. How much would you have been prepared to pay for one of these IOUs?

4.94$0944.000,1$)0515.1(

1000,1$

47PV

14

Present Values

Example

Kangaroo Autos is offering free credit on a $20,000 car. You pay $8,000 down and then the balance at the end of 2 years. Turtle Motors next door does not offer free credit but will give you $1,000 off the list price. If the interest rate is 10%, which company is offering the better deal?

15

Present Values

$8,000

$12,000

0 1 2

$8,000

$9,917.36

$17,917.36

PV at Time=0

2)10.1(

1000,12$

Total

16

Present Values

Example – Finding the Interest Rate

Each Puerto Rico IOU promised to pay holder $1,000 at the end of 47 years. It is sold for $94.4. What is the interest rate?

5.15%or 0.0515,r

0515.110.593 r)(1

593.10$94.4

$1,000r)(1

$1,000r)(1$94.4

1/47

47

47

Financial calculator: n=47, FV=1,000 PV=-94.4, PMT=0 i=5.15

17

Present Values

Example – Double your money

Suppose your investment advisor promises to double your money in 8 years. What interest rate is implicitly being promised?

9.05%or 0.0905,r

r)(1$1$2

r)(1PVFV8

t

Financial calculator: n=8, FV=2 PV=-1, PMT=0 i=9.05

18

Present Value of Multiple Cash Flows

Example

Your auto dealer gives you the choice to pay $15,500 cash now, or make three payments: $8,000 now and $4,000 at the end of the following two years. If your cost of money is 8%, which do you prefer?

$15,133.06 PVTotal

36.429,3

70.703,3

8,000.00

2

1

)08.1(

000,42

)08.1(

000,41

payment Immediate

PV

PV

What is the PV of this uneven cash flow stream?

0

100

1

300

2

300

310%

-50

4

90.91247.93225.39 -34.15530.08 = PV

Detailed steps (Texas Instrument calculator)

To clear historical data: CF, 2nd ,CE/CTo get PV:CF , ↓,100 , Enter , ↓,↓ ,300 , Enter, ↓,2,

Enter, ↓, 50, +/-,Enter, ↓,NPV,10,Enter, ↓,CPT

“NPV=530.0867427”

21

Future Value of Multiple Cash Flows

Example

You plan to save some amount of money each year. You put $1,200 in the bank now, another $1,400 at the end of the first year, and a third deposit of $1,000 at the end of the second year. If the interest rate is 8% per year, how much will be available to spend at the end of the 3 year?

22

Future Value of Multiple Cash Flows

$1,200

$1,400

$1,000

Year

0 1 2 3

$1,080.00

$1,632.96

$1,511.65

$4,224.61

FV in year 3

08.1000,1$ 2)08.1(400,1$ 3)08.1(200,1$

23

Perpetuities and Annuities

AnnuityEqually spaced level stream of cash flows

for a limited period of time.

PerpetuityA stream of level cash payments that never

ends.

24

Perpetuities and Annuities

rateinterest

paymentcash rCyPerpertuit of PV

25

Perpetuities and Annuities

Example - Perpetuity

In order to create an endowment, which pays $100,000 per year, forever, how much money must be set aside today in the rate of interest is 10%?

000,000,1$10.0000,100 PV

26

Perpetuities and Annuities

Example - continued

If the first perpetuity payment will not be received until three years from today, how much money needs to be set aside today?

446,826$2)10.01(

000,000,1

PV

27

Perpetuities and Annuities

Cash FlowYear 0 1 2 3 4 5 6 … Present

Value

1. Perpetuity A $1 $1 $1 $1 $1 $1 …

2. Perpetuity B $1 $1 $1 …

3. Three-year annuity $1 $1 $1

r

1

3)1(

1

rr

3)1(

11

rrr

28

Perpetuities and Annuities

C = cash payment r = interest rate t = Number of years cash payment is received

trrrCPV

)1(

11annuityyear - tof

29

Perpetuities and Annuities

PV Annuity Factor (PVAF) - The present value of $1 a year for each of t years.

trrrPVAF

)1(

11

30

Perpetuities and Annuities

Example – LotterySuppose you bought lottery tickets and won $295.7 million. This sum was scheduled to be paid in 25 equal annual installments of $11.828 million each. Assuming that the first payment occurred at the end of 1 year, what was the present value of the prize? The interest rate was 5.9 percent.

6.152$

])059.1(059.0

1

059.0

1[828.11$

25

PV

PV

Financial calculator: n=25, i=5.9, PMT=11.828, FV=0, PV=-152.6

31

Perpetuities and Annuities

Annuity Due Level stream of cash flows starting

immediately.

annuityan of PVr)(1 dueannuity an of PV

32

Perpetuities and Annuities

Example – Lottery (Continued)What is the value of the prize if you receive the first installment of $11.828 million up front and the remaining payments over the following 24 years?

7.161$059.16.152$

059.1])059.1(059.0

1

059.0

1[828.11$

25

PV

PV

33

Perpetuities and Annuities

Future Value of annual payments

r

rC

rtrrr

CFV

t

t

1)1(

)1()1(

11

34

Perpetuities and Annuities

Example - Future Value of annual payments

You plan to save $4,000 every year for 20 years and then retire. Given a 10% rate of interest, what will be the FV of your retirement account?

100,229$

20)10.01(20)10.01(10.0

110.01000,4

FV

FV

Financial calculator: n=20, i=10, PMT=-4,000, PV=0, FV=229,100

35

Effective Annual Interest Rate

Effective Annual Interest Rate - Interest rate that is annualized using compound interest.

Annual Percentage Rate (APR) - Interest rate that is annualized using simple interest.

36

Effective Annual Interest Rate

Example

Given a monthly rate of 1%, what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)?

12.00%or 0.12=12 x 0.01=APR

12.68%or 0.1268=1-0.01)+(1=EAR 12

Classifications of interest rates

1. Nominal rate (iNOM) – also called the APR, quoted rate, or stated rate. An annual rate that ignores compounding effects. Periods must also be given, e.g. 8% Quarterly.

2. Periodic rate (iPER) – amount of interest charged each period, e.g. monthly or quarterly. iPER = iNOM / m, where m is the number of

compounding periods per year. e.g., m = 12 for monthly compounding.

Classifications of interest rates

3. Effective (or equivalent) annual rate (EAR, also called EFF, APY) : the annual rate of interest actually being earned, taking into account compounding.

If the interest rate is compounded m times in a year, the effective annual interest rate is

1 1m

nomi

m

Example, EAR for 10% semiannual investment

EAR= ( 1 + 0.10 / 2 )2 – 1 = 10.25%

An investor would be indifferent between an investment offering a 10.25% annual return, and one offering a 10% return compounded semiannually.

EAR on a Financial Calculator

keys: description:

[2nd] [ICONV] Opens interest rate conversion menu

[↓] [EFF=] [CPT] 10.25

Texas Instruments BAII Plus

[↓][NOM=] 10 [ENTER] Sets 10 APR.

[↑] [C/Y=] 2 [ENTER] Sets 2 payments per year

Why is it important to consider effective rates of return?

An investment with monthly payments is different from one with quarterly payments.

Must use EAR for comparisons. If iNOM=10%, then EAR for different

compounding frequency:Annual 10.00%Quarterly 10.38%Monthly 10.47%Daily 10.52%

What’s the FV of a 3-year $100 annuity, if the quoted interest rate is 10%, compounded semiannually?

Payments occur annually, but compounding occurs every 6 months.

Cannot use normal annuity valuation techniques.

0 1

100

2 35%

4 5

100 100

61 2 3

Method 1:Compound each cash flow

110.25121.55331.80

FV3 = $100(1.05)4 + $100(1.05)2 + $100

FV3 = $331.80

0 1

100

2 35%

4 5

100

61 2 3

100

Method 2:Financial calculator

Find the EAR and treat as an annuity.EAR = ( 1 + 0.10 / 2 )2 – 1 = 10.25%.

INPUTS

OUTPUT

N I/YR PMTPV FV

3 10.25 -100

331.80

0

When is periodic rate used?

iPER is often useful if cash flows occur several times in a year.

48

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.

49

Inflation

1 real interest rate = 1+nominal interest rate1+inflation rate

approximation formula

Real int. rate nominal int. rate - inflation rate

50

Inflation

Example

If the interest rate on one year govt. bonds is 5.0% and the inflation rate is 2.2%, what is the real interest rate?

2.8%or .028=.022-.050=ionApproximat

2.7%or .027 = rateinterest real

1.027 =rateinterest real1.022+1.050+1=rateinterest real1

51

Inflation

Be consistent in how you handle inflation!!

Use nominal interest rates to discount nominal cash flows.

Use real interest rates to discount real cash flows.

You will get the same results, whether you use nominal or real figures

52

Inflation

ExampleYou own a lease that will cost you $8,000 next year, increasing at 3% a year (the forecasted inflation rate) for 3 additional years (4 years total). If discount rates are 10% what is the present value cost of the lease?

53

Inflation

Answer – Nominal figures

99.429,26$

78.59708741.82=8000x1.034

56.63768487.20=8000x1.033

92.68098240=8000x1.032

73.727280001

10% @ PVFlowCash Year

4

3

2

10.182.87413

10.120.84872

10.18240

1.108000

54

Inflation

Answer – Real Figures

Year Cash Flow PV@6.7961%

1 = 7766.99

2 = 7766.99

= 7766.99

= 7766.99

80001.03

7766.991.068

82401.03

8487.201.03

8741.821.03

2

3

4

7272 73

6809 92

3 6376 56

4 5970 78

26 429 99

7766 991 068

7766 991 068

7766 991 068

2

3

4

.

.

.

.

..

..

..

= $ , .