6-1

CHAPTER 5Time Value of Money

Read Chapter 6 (Ch. 5 in the 4th edition)

Future value Present value Rates of return Amortization

6-2

Time Value of Money Problems

Use a financial calculator Bring your calculator to class Will need on exams We will not use the tables

6-3

Time lines show timing of cash flows.

CF0 CF1 CF3CF2

0 1 2 3i%

Tick marks at ends of periods, so Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period 2.

6-4

A. (1) a. Time line for a $100 lump sum due at the

end of Year 2.

100

0 1 2 Year

i %

6-5

A. (1) b. Time line for anordinary annuity of $100 for

3 years.

100 100100

0 1 2 3i%

6-6

A. (1) c. Time line for uneven CFs -$50 at t=0 and$100, $75, and $50 at the end of Years 1 through 3.

100 50 75

0 1 2 3i%

-50

6-7

What’s the FV of an initial$100 after 3 years if i = 10%?

FV = ?

0 1 2 310%

100

Finding FVs is Compounding.

6-8

After 1 year:

FV1 = PV + I1 = PV + PV (i)= PV(1 + i)= $100 (1.10)= $110.00.

After 2 years:

FV2 = PV(1 + i)2

= $100 (1.10)2

= $121.00.

6-9

After 3 years:

FV3 = PV(1 + i)3

= 100 (1.10)3

= $133.10.

In general,

FVn = PV (1 + i)n

6-10

Three ways to find FVs:

1. ‘Solve’ the Equation with aScientific Calculator

2. Use Tables (the book describes this but not for use in this class)

3. Use a Financial Calculator4. Spreadsheet (has built-in formulas) -- won’t work on exams

6-11

3 10 -100 0N I/YR PV PMT FV

133.10

INPUTS

OUTPUT

Here’s the setup to find FV:

Clearing automatically sets everything to0, but for safety enter PMT = 0.

Check your calculator. Set: P/YR = 1 and END (“BEGIN” should not show on the display)

6-12

What’s the PV of $100 duein 3 years if i = 10%?

Finding PVs is discounting,and it’s the reverse of compounding.

100

0 1 2 310%

PV = ?

6-13

Financial Calculator Solution:

3 10 0 100N I/YR PV PMT FV

-75.13

INPUTS

OUTPUT

Either PV or FV must be negative. HerePV = -75.13. Put in $75.13 today, take out $100 after 3 years.

6-14

If sales grow at 20% per year,how long before sales double?

Solve for n:

FVn = 1(1 + i)n; In our case 2 = (1.20)n .Take the log of both sides:ln(2) = n ln(1.2)n = ln(2)/ln(1.2)=.693…/0.1823.. =3.8017

6-15

20 -1 0 2N I/YR PV PMT FV3.8

INPUTS

OUTPUT

Graphical Illustration:

01 2 3 4

1

2

FV

3.8

Year

Financial calculator solution

6-16

What’s the differencebetween an ordinary

annuity and an annuitydue?

6-17

Ordinary vs. Annuity Due

PMT PMTPMT

0 1 2 3i%

PMT PMT

0 1 2 3i%

PMT

6-18

What’s the FV of a 3-yearordinary annuity of $100 at

10%?

100 100100

0 1 2 310%

110

121

FV = 331

6-19

3 10 0 -100

331.00

INPUTS

OUTPUT

N I/YR PV PMT FV

Financial Calculator Solution:

If you enter PMT of 100, you get FV of

-331.

Get used to the fact that you have to figure out the sign.

6-20

What’s the PV of this ordinaryannuity?

100 100100

0 1 2 310%

90.91

82.64

75.13

248.69 = PV

6-21

3 10 100 0

-248.69

INPUTS

OUTPUT

N I/YR PV PMT FV

Have payments but no lump sum FV,so enter 0 for future value.

Financial Calculator Solution:

6-22

Technical Aside:

Your calculator really is assuming a NPV equation, with PV as a time zero cash flow as follows:

nn

)i1(FVi

)i1(1PMTPVNPV

When you use the top row of calculator keys, the calculator assumes NPV=0 and solves for one variable.

6-23

Find the FV and PV if theannuity were an annuity due.

100 100

0 1 2 310%

100

6-24

3 10 100 0

-273.55

INPUTS

OUTPUT

N I/YR PV PMT FV

Switch from “End” to “Begin”.

Then enter variables to find PVA3 = $273.55.

Then enter PV = 0 and press FV to findFV = $364.10.

6-25Alternative:

The first payment is in the present and thus has a PV of 100.

The next two payments comprise a two period ordinary annuity -- use the formula with n=2, PMT=100, and i=.10.

Sum the above two for the present value. If you already have the PV, multiply by

To get FV

3)i1(

6-26

Perpetuities A perpetuity is a stream of regular payments that goes on forever

An infinite annuity Future value of a perpetuity

Makes no sense because there is no end point Present value of a perpetuity

A diminishing series of numbers

• Each payment’s present value is smaller than the one before

p

PMTPV

k

6-27

Perpetuities—Example E

xam

ple

p

PMT $5PV $250

k 0.02 You may also work this by inputting a

large n into your calculator (to simulate infinity), as shown below.

PV

N

PMT

I/Y

250

999

5

2

0FV

Answer

Q: The Longhorn Corporation issues a security that promises to pay its holder $5 per quarter indefinitely. Money markets are such that investors can earn about 8% compounded quarterly on their money. How much can Longhorn sell this special security for?

A: Convert the k to a quarterly k and plug the values into the equation.

6-28

What is the PV of this uneven cashflow stream?

0

100

1

300

2

300

310%

-50

4

90.91

247.93

225.39

-34.15

530.08 = PV

6-29

Input in “CFLO” register ( CFj ):

CF0 = 0

CF1 = 100

CF2 = 300

CF3 = 300

CF4 = -50 Enter I = 10%, then press NPV button to

get NPV = 530.09. (Here NPV = PV.)

6-30

What’s Project L’s NPV?

10 8060

0 1 2 310%

Project L:

-100.00

9.09

49.59

60.11

18.79 = NPVL

11.1

21.1

31.1

6-31

Calculator Solution:

Enter in CFLO for L:

-100

10

60

80

10

CF0

CF1

NPV

CF2

CF3

i = 18.78 = NPVL

6-32TI Calculators

•BA-35 doesn’t appear to do uneven cash flows (NPV and IRR)

BA II PLUSCF

CF0= -100 Enter

C01= 10 Enter F01= 1.00

C02= 60 Enter F02= 1.00

C03= 80 Enter F03= 1.00 NPV I=10 Enter CPT NPV= 18.78

IRR CPT IRR= 18.13

6-33

The Sinking Fund Problem

Companies borrow money by issuing bonds for lengthy time periods

No repayment of principal is made during the bonds’ lives

• Principal is repaid at maturity in a lump sum– A sinking fund provides cash to pay off a

bond’s principal at maturity• Problem is to determine the periodic

deposit to have the needed amount at the bond’s maturity—a future value of an annuity problem

6-34

The Sinking Fund Problem –Example

Q: The Greenville Company issued bonds totaling $15 million for 30 years. The bond agreement specifies that a sinking fund must be maintained after 10 years, which will retire the bonds at maturity. Although no one can accurately predict interest rates, Greenville’s bank has estimated that a yield of 6% on deposited funds is realistic for long-term planning. How much should Greenville plan to deposit each year to be able to retire the bonds with the money put aside?

A: The time period of the annuity is the last 20 years of the bond issue’s life. Input the following keystrokes into your calculator.

PMT

N

FV

I/Y

407,768.35

20

15,000,000

6

0PV

Answer

Exa

mpl

e

6-35

What interest rate wouldcause $100 to grow to

$125.97 in 3 years?

3 -100 0 125.97

INPUTS

OUTPUT

N I/YR PV FVPMT

8%

$100 (1 + i )3 = $125.97.

6-36Will the FV of a lump sum belarger or smaller if we

compound more often, holdingthe stated i% constant? Why?

LARGER! If compounding is morefrequent than once a year--forexample, semi-annually, quarterly,or daily--interest is earned on interestmore often.

6-37

0 1 2 310%

0 1 2 35%

4 5 6

134.01

100133.10

1 2 30

100

Semi-annually:

Annually: FV3 = 100(1.10)3 = 133.10.

FV6/2 = 100(1.05)6 = 134.01.

6-38

We will deal with 3different rates:

iNom = nominal, or stated, or quoted, rate per year.

iPer = periodic rate. The literal rate applied each period

EAR= EFF% = effective annual rate.

6-39

iNom is stated in contracts. Periods per year (m) must also be given. Sometimes (incorrectly) referred to as the “simple” interest rate.

Examples:• 8%, Daily interest (365 days)• 8%; Quarterly

6-40

Periodic rate = iPer = iNom/m, where m is periods per year. m = 4 for quarterly, 12 for monthly, and 360 or 365 for daily compounding.

Examples:8% quarterly: iper = 8/4 = 2%

8% daily (365): iper = 8/365 = 0.021918%

6-41 Effective Annual Rate (EAR = EFF%):

The annual rate which cause PV to grow to the same FV as under multiperiod compounding.

Example: EFF% for 10%, semiannual:

FV = (1 + inom/m)m

= (1.05)2 = 1.1025.

Any PV would grow to same FV at 10.25% annually or 10% semiannually:

(1.1025)1 = 1.1025

(1.05)2 = 1.1025

6-42

Comparing Financial Investments

An investment with monthly payments is different from one with quarterly payments. Must put on EFF% basis to compare rates of return. Use EFF% only for comparisons.

Banks say “interest paid daily.” Same as compounded daily.

6-43

How do we find EFF% for a nominal rate of 10%, compounded

semi-annually?

EFF% = 1 + im

- 1nomm

= 1+ 0.10

2 - 1.0

= 1.05 - 1.0= 0.1025 = 10.25%.

2

2

6-44

EAR = EFF% of 10%

105170918.1e:Continuous 10.

EARAnnual = 10%.

EARQ = (1 + 0.10/4)4 - 1 = 10.38%.

EARM = (1 + 0.10/12)12 - 1 = 10.47%.

EARD = (1 + 0.10/360)360 - 1= 10.5155572%.

6-45

Can the effective rate ever beequal to the nominal rate?

Yes, but only if annual compounding is used, i.e., if m = 1.

If m > 1, EFF% will always be greater than the nominal rate.

6-46

When is each rate used?

inom: Written into contracts,quoted by banks andbrokers. Not used incalculations or shownon time lines.

6-47

iper: Used in calculations,shown on time lines.

If inom has annual compounding,then iper = inom/1 = inom.

6-48

EAR = EFF%: Used to compare returnson investments with different paymentsper year and in advertising of deposit interestrates.

(Used for calculations if and only ifdealing with annuities where paymentsdon’t match interest compounding periods.)

6-49

FV of $100 after 3 yearsunder 10% semi-annual

compounding? Quarterly?

FV = PV 1 + imnnom

mn

FV = $100 1 + 0.10

23s

2x3

= $100(1.05)6 = $134.01

FV3Q = $100(1.025)12 = $134.49

6-50What’s the value at the endof Year 3 of the following CF stream if the quoted interest

rate is 10%, compoundedsemi-annually?

0 1

100

2 35%

4 5 6

100 100

6-month periods

6-51

Payments occur annually, but compounding occurs each 6 months.

So we can’t use normal annuity valuation techniques.

6-52

1st method: Compound each CF

0 1

100

2 35%

4 5 6

100 100.00

110.25

121.55

331.80

FVA3 = 100(1.05)4 + 100(1.05)2 + 100= 331.80

2)05.1(100

4)05.1(100

6-53

What’s the PV of this stream?

0

100

15%

2 3

100 100

90.70

82.27

74.62

247.59

Years

2)05.1(100

4)05.1(100

6)05.1(100

6-54

Second Method: use yourfinancial calculator!

Follow these two steps:

a. Find the EAR for the quoted rate:

EAR = 1 + 0.10

2 - 1 = 10.25%.

2

This is the iper for a period of one year. Use in formula (or calculator) with the period equal to a year.

6-55

0

100

110.25%

2 3

100 100

Time line

6-56

3 10.25 0 -100

INPUTS

OUTPUT

N I/YR PV FVPMT

331.80

b. Calculator inputs

6-57

N I PV PMT FV

10 10 100 0  

5 8   0 100

7  -500 100 0

15  -750 100 1000

240 8/12 -100,000   0

50 10 100 10  

Calculator Workout: fill in the blanks

6-58

0.75 10 - 100 0 ?

=107.41

INPUTS

OUTPUTN I/YR PV PMT FV

Fractional Time Periods

0 0.25 0.50 0.7510%

- 100

1.00

FV = ?

Example: $100 deposited in a bank at 10% interest for 0.75 of the year

6-59

AMORTIZATION

Construct an amortization schedulefor a $1,000, 10% annual rate loanwith 3 equal payments.

6-60

This is what an amortization schedule looks like.

Amortization Table

Beginning Ending

Principal Total Interest Principal Principal

Period Balance Payment Payment Payment Balance

1 $1,000.00 $402.11 $100.00 $302.11 $697.89

2 $697.89 $402.11 $69.79 $332.33 $365.56

3 $365.56 $402.11 $36.56 $365.56 $0.00

6-61

Step 1: Find the required payment.

PMT PMTPMT

0 1 2 310%

-1000

3 10 -1000 0

INPUTS

OUTPUT

N I/YR PV FVPMT

402.11

6-62

Step 2: Find interest chargefor Year 1.

INTt = Beg balt (i)INT1 = 1000(0.10) = $100.

Step 3: Find repayment of principal in Year 1.

Repmt. = PMT - INT= 402.11 - 100= $302.11.

6-63

Step 4: Find ending balanceafter Year 1.

End bal = Beg bal - Repmt = 1000 - 302.11 = $697.89.

Repeat these steps for Years 2 and 3to complete the amortization table.

6-64

Amortization Table

Beginning Ending

Principal Total Interest Principal Principal

Period Balance Payment Payment Payment Balance

1 $1,000.00 $402.11 $100.00 $302.11 $697.89

2 $697.89 $402.11 $69.79 $332.33 $365.56

3 $365.56 $402.11 $36.56 $365.56 $0.00

Interest declines. Tax Implications.

6-65

Amortization tables are widely used-- for home mortgages, auto loans, business loans, retirement plans, etc. They are very important!

Financial calculators (and spreadsheets) are great for setting up amortization tables.

6-66

Amortized Loans—Example E

xam

ple

PMT

N

PV

I/Y

293.75

48

10,000

1.5

0FV

Answer

This can also be calculated using the PVA formula of PVA = PMT[PVFAk, n] with an n of 48 and a k of 1.5%,

resulting in $10,000 = PMT[34.0426] = $293.75.

Q: Suppose you borrow $10,000 over four years at 18% compounded monthly repayable in monthly installments. How much is your loan payment?

A: Adjust your interest rate and number of periods for monthly compounding and input the following keystrokes into your calculator.

6-67

Amortized Loans—Example

PV

N

FV

I/Y

15,053.75

36

0

1

500PMT

Answer

Exa

mpl

e

This can also be calculated using the PVA formula of PVA = PMT[PVFAk, n] with an n of 36 and a k of 1%,

resulting in PVA = $500[30.1075] = $15,053.75.

Q: Suppose you want to buy a car and can afford to make payments of $500 a month. The bank makes three-year car loans at 12% compounded monthly. How much can you borrow toward a new car?

A: Adjust your k and n for monthly compounding and input the following calculator keystrokes.

6-68

Loan Amortization Schedules—Example E

xam

ple

Q: Develop an amortization schedule for the loan demonstrated in Example 5.12.

Note that the Interest portion of the payment is decreasing

while the Principal portion is increasing.