6-1

Chapter 6The Time Value

of Money

Future ValuePresent Value

Rates of ReturnAmortization

6-2

Cash Flow Time Lines

CF0 CF1 CF3CF2

0 1 2 3i%

Tick marksTick marks at ends of periods, so t=0 is today;

t=1 is the end of Period 1; or the beginning of Period 2.

Graphical representations used to show the timing of cash flows.

6-3

Time line for a $100 lump sum due at the end of Year 2.

100

0 1 2 Year

i%

6-4

Time line for an ordinary annuity of $100 for 3 years.

100 100100

0 1 2 3i%

6-5

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

6-6

Future ValueThe amount to which a cash flow or series of cash flows will grow over a period of time when compounded at a given interest rate.

How much would you have at the end of one year if you deposited $100 in a bank account that pays 5 percent interest each year?

FVn = FV1 = PV + INT

= PV + (PV x i)

= PV (1 + i)

= $100(1+0.05) = $100(1.05) = $105

6-7

Finding FVs is Compounding.

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

FV = ?

0 1 2 3i=10%

100

6-8

In general, FVn = PV (1 + i)n

After 3 years: FV3 = PV(1 + i)3

= 100 (1.10)3

= $133.10.

After 2 years:FV2 = PV(1 + i)2

= $100 (1.10)2

= $121.00.

After 1 year:FV1 = PV + i = PV + PV (i)

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

Future Value

6-9

Solve the Equation with aRegular Calculator

Use Tables

Use a Financial Calculator

Three ways to find Future Values:

6-10

Financial Calculator Solution:

Financial calculators solve this equation:

There are 4 variables. If 3 are known, the calculator will solve for the 4th.

.i1PVFV n

n

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.

Set: P/YR = 1

Financial Calculator Solution:

6-12

Present Value

Present valuePresent value is the value today of a future cash flow or series of cash flows.

DiscountingDiscounting is the process of finding the present value of a cash flow or series of cash flows, the reverse of compounding.

6-13

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

100

0 1 2 310%

PV = ?

6-14

Solve FVn = PV (1 + i )n for PV:

n

i+11

nFV = ni+1

nFV =PV

PV = $1001

1.10

3 = $100PVIFi,n

= $1000.7513 = $75.13.

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

6-15

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

3 10 0 100N I/YR PV PMT FV

-75.13

INPUTS

OUTPUT

Financial Calculator Solution:

6-16

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

Solve for n:

FVn = 1(1 + i)n;

2 = 1(1.20)n .

6-17

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

INPUTS

OUTPUT

1 2 3 4Year0

1

2

FV

3.8

GraphicalIllustration:

Financial Calculator Solution:

6-18

Future Value of an Annuity

Annuity:Annuity: A series of payments of an equal amount at fixed intervals for a specified number of periods.

Ordinary (deferred) Annuity:Ordinary (deferred) Annuity: An annuity whose payments occur at the end of each period.

Annuity Due:Annuity Due: An annuity whose payments occur at the beginning of each period.

6-19

Ordinary Annuity Versus Annuity Due

PMT PMTPMT

0 1 2 3i%

PMT PMT

0 1 2 3i%

PMT

6-20

What’s the FV of a 3-yearOrdinary Annuity of $100 at 10%?

100 100100

0 1 2 310%

110

121

FV = 331

6-21

3 10 0 -100

331.00

INPUTS

OUTPUT

N I/YR PV PMT FV

Financial Calculator Solution:

6-22

Present Value of an Annuity

PVAn = the present value of an annuity with n payments

Each payment is discounted, and the sum of the discounted payments is the present value of the annuity

6-23

What is the PV of this Ordinary Annuity?

248.69 = PV

100 100100

0 1 2 310%

90.91

82.64

75.13

6-24

3 10 100 0

-248.69

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

Financial Calculator Solution:

N I/YR PV PMT FV

INPUTS

OUTPUT

6-25

Find the FV and PV if theAnnuity were an Annuity Due.

100 100

0 1 2 310%

100

6-26

3 10 100 0

-273.55

INPUTS

OUTPUT

N I/YR PV PMT FV

Switch from “End” to “Begin”.Switch from “End” to “Begin”.

Then enter variables to find PVA3 = $273.55.

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

Financial Calculator Solution:

6-27

Solving for Interest Rates with Annuities

250 250

0 1 2 3i = ?

- 864.80

4

250 250

You pay $864.80 for an investment that promises to pay you $250 per year for the next four years, with payments made at the end of the year. What interest rate will you earn on this investment?

6-28

Financial Calculator Solution:

PVAn = PMT(PVIFAi,n)

$846.80 = $250(PVIFA i = ?,4)

4 ? - 846.80 250 0

=7.0

INPUTS

OUTPUTN I/YR PV PMT FV

INPUTS

OUTPUTN I/YR PV PMT FV

6-29

Uneven Cash Flow Streams

A series of cash flows in which the amount varies from one period to the next. Payment (PMT) designates constant

cash flows Cash flow (CF) designates cash flows

in general, including uneven cash flows

6-30

What is the PV of this Uneven Cash Flow Stream?

0

100

1

300

2

300

310%

-50

4

90.91

247.93

225.39

-34.15

530.08 = PV

6-31

Input in “Cash Flow” register:

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.)

Financial Calculator Solution:

6-32

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

What interest rate would cause $100 to grow to $125.97 in 3 years?

3 -100 0 125.97

INPUTS

OUTPUT

N I/YR PV FVPMT

8%

6-33

Semiannual and Other Compounding Periods

Annual compoundingAnnual compounding is the arithmetic process of determining the final value of a cash flow or series of cash flows when interest is added once a year.

Semiannual compoundingSemiannual compounding is the arithmetic process of determining the final value of a cash flow or series of cash flows when interest is added twice a year.

6-34

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

Will the FV of a lump sum be larger or smaller if we compound more often, holding the state i% constant? Why?

6-35

0 1 2 310%

100133.10

0 1 2 35%

4 5 6

134.01

1 2 30

100

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

Semi-annually: FV6/2 = 100(1.05)6 = 134.01.

6-36

Distinguishing Between Different Interest Rates

iSIMPLE = Simple (Quoted) RateSimple (Quoted) Rate used to compute the interest paid per period

EAR = Effective Annual RateEffective Annual Ratethe annual rate of interest actually being earned

APR = Annual Percentage RateAnnual Percentage Rateperiodic rate X the number of periods per year

6-37

How do we find EAR for a simple rate of 10%, compounded semi-annually?

mEAR = 1 +

iSIMPLE

m

- 1

= 1+

0.102

- 1.0

= 1.05 - 1.0

= 0.1025 = 10.25%.

2

2

6-38

FV of $100 after 3 yearsunder 10% semi-annualcompounding? Quarterly?

FV = $100 1 + 0.10

23s

2x3

n x mSIMPLE

n mi + 1PV = FV

= $100(1.05)6 = $134.01

FV3Q = $100(1.025)12 = $134.49

6-39

0.75 10 - 100 0 ?

=107.41

INPUTS

OUTPUTN I/YR PV PMT FV

Fractional Time Periods

0 0.25 0.50 0.758%

- 100

1.00

FV = ?

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

6-40

Amortized Loans

Amortized Loan:Amortized Loan: A loan that is repaid in equal payments over its life.

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

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

6-41

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

Step 1: Find the required paymentsStep 1: Find the required payments

PMT PMTPMT

0 1 2 310%

-1000

3 10 -1000 0

INPUTS

OUTPUT

N I/YR PV FVPMT

402.11

6-42

Step 2: Find interest charge for Year 1Step 2: Find interest charge for Year 1INTt = Beg balt (i)INT1 = 1000(0.10) = $100.

Step 3: Find repayment of principal in Year 1Step 3: Find repayment of principal in Year 1Repmt. = PMT - INT

= 402.11 - 100= $302.11.

Step 4: Find ending balance after Year 1Step 4: Find ending balance after Year 1 End bal = Beg bal - Repmt

= 1000 - 302.11 = $697.89.

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

6-43

YR Beg Bal PMT INT Prin PMT End Bal

1 $1000 $402 $100 $302 $698

2 698 402 70 332 366

3 366 402 37 366 0

Total 1,206.34 206.34 1,000

Interest declines. Tax Implications.

Loan Amortization Table10 Percent Interest Rate

6-44$

0 1 2 3

402.11InterestInterest

Principal PaymentsPrincipal Payments

302.11

Level payments. Interest declines becauseoutstanding balance declines. Lender earns10% on loan outstanding, which is falling.

6-45

iSIMPLE: Written into contracts, quoted by banks

and brokers. Not used in calculations or shown on time lines.

iper: Used in calculations, shown on time lines.

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

EAR : Used to compare returns on investments with different payments per year.(Used for calculations if and only if dealing with annuities where payments don’t match interest compounding periods.)

Comparison of Different Types of Interest Rates

6-46

iSIMPLE is stated in contracts. Periods per year (m) must also be given.

Examples:

8%; Quarterly

8%, Daily interest (365 days)

Simple (Quoted) Rate

6-47

Periodic rate = iper = iSIMPLE/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%

Periodic Rate

6-48

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

Example: EAR for 10%, semiannual:

FV = (1 + iSIMPLE/m)m

= (1.05)2 = 1.1025.

EAR = 10.25% because

(1.1025)1 = 1.1025.Any PV would grow to same FV at 10.25% annually or 10% semiannually.

Effective Annual Rate