Time Value of Money1

7/28/2019 Time Value of Money1

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Chapter 3

The Time Valueof Money

2005 Thomson/South-Western


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2 2005 Thomson/South-Western

Time Value of Money

The most important concept in financeUsed in nearly every financial decision

Business decisions

Personal finance decisions


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Cash Flow Time Lines

CF0 CF1 CF3CF2

0 1 2 3k%

Time 0 is todayTime 1 is the end of Period 1 or the beginningof Period 2.

Graphical representations used toshow timing of cash flows


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100

0 1 2 Year

k%

Time line for a $100 lump sumdue at the end of Year 2


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Time line for an ordinary annuity

of $100 for 3 years

100 100100

0 1 2 3k%


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Time line for uneven CFs

- $50 at t = 0 and $100, $75, and $50at the end of Years 1 through 3

100 5075

0 1 2 3

k%

-50


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The amount to which a cash flow orseries of cash flows will grow over aperiod of time when compounded at

a given interest rate.

Future Value


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FVn = FV1 = PV + INT

= PV + (PV x k)

= PV (1 + k)

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

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

Future Value


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FV = ?

0 1 2 310%

100

Finding FV is Compounding.

Whats the FV of an initial $100after 3 years if k = 10%?


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After 1 year:FV1 = PV + Interest1 = PV + PV (k)

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

After 2 years:

FV2 = PV(1 + k)2

= $100 (1.10)2= $121.00.

After 3 years:

FV3 = PV(1 + k)3

= 100 (1.10)3= $133.10.

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

Future Value


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Three Ways to Solve TimeValue of Money Problems

Use EquationsUse Financial Calculator

Use Electronic Spreadsheet


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Solve this equation by plugging in theappropriate values:

Numerical (Equation) Solution

nn k)PV(1FV

PV = $100, k = 10%, and n =3

$133.10)$100(1.331

$100(1.10)FV 3n


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Present Value

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

Discountingis the process of finding the

present value of a future cash flow or seriesof future cash flows; it is the reverse ofcompounding.


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100

0 1 2 310%

PV = ?

What is the PV of $100 due in3 years if k = 10%?


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

n

nn

n

k+1

1FV=

k+1

FV=PV

$75.13=0.7513$100=1.10

1$100=PV

3


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Future Value of an Annuity

Annuity: A series of payments of equal

amounts at fixed intervals for a specifiednumber of periods.

Ordinary (deferred) Annuity: An annuitywhose payments occur at the end of each

period.Annuity Due: An annuity whose payments

occur at the beginning of each period.


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PMT PMTPMT

0 1 2 3

k%

PMT PMT

0 1 2 3k%

PMT

Ordinary Annuity VersusAnnuity Due

Ordinary Annuity

Annuity Due


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100 100100

0 1 2 310%

110

121

FV = 331

Whats the FV of a 3-yearOrdinary Annuity of $100 at 10%?


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Numerical Solution (usingtable):

$331.0000)$100(3.310

0.10

1(1.10)

$100FVA

3

3


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Present Value of an Annuity

PVAn = the present value of an annuitywith n payments.

Each payment is discounted, and thesum of the discounted payments is thepresent value of the annuity.


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248.69 = PV

100 100100

0 1 2 310%

90.91

82.64

75.13

What is the PV of thisOrdinary Annuity?


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2005 Thomson/South-Western 22 2005 Thomson/South-Western

Using table:

$248.685)$100(2.486

PVA3


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100 100

0 1 2 310%

100

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


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Numerical Solution

$273.5553)$100(2.735 1.10(2.48685)$100PVA3


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250 250

0 1 2 3k = ?

- 864.80

4

250 250

You pay $864.80 for an investment that promises to

pay you $250 per year for the next 4 years, withpayments made at the end of each year. Whatinterest rate will you earn on this investment?

Solving for Interest Rateswith annuities


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Uneven Cash Flow Streams

A series of cash flows in which the amountvaries from one period to the next:

Payment (PMT) designates constant cashflowsthat is, an annuity stream.

Cash flow (CF) designates cash flows ingeneral, both constant cash flows anduneven cash flows.


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0

100

1

300

2

300

3

10%-50

4

90.91

247.93

225.39

-34.15

530.08 = PV

What is the PV of thisUneven Cash Flow Stream?


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Numerical Solution

nn2211 k)(1

1CF...

k)(1

1CF

k)(1

1CFPV

4321 (1.10)

150)(

(1.10)

1300

(1.10)

1300

(1.10)

1100PV

$530.09

01)$50)(0.683(31)$300(0.75145)$300(0.82609)$100(0.909


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Semiannual and OtherCompounding Periods

Annual compounding is the process ofdetermining the future value of a cash flowor series of cash flows when interest isadded once a year.

Semiannual compounding is the processof determining the future value of a cashflow or series of cash flows when interest isadded twice a year.


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30

0 1 2 310%

100 133.10

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.

Compounding

Annually vs. Semi-Annually


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kSIMPLE = Simple (Quoted) Rate

used to compute the interest paid per period

EAR = Effective Annual Ratethe annual rate of interest actually beingearned

APR =Annual Percentage Rate = kSIMPLEperiodic rate X the number of periods per year

Distinguishing BetweenDifferent Interest Rates


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1-m

k+1=EAR

m

SIMPLE

10.25%=0.1025=1.0-1.05=

1.0-20.10+1=

2

2

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


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nmSIMPLE

nm

k+1PV=FV

$134.0110)$100(1.3402

0.10

+1$100=FV

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23

FV of $100 after 3 years ifinterest is 10% compoundedsemi-annual? Quarterly?

$134.4989)$100(1.3444

0.10+1$100=FV

34

43


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Amortized LoansAmortized Loan: A loan that is repaid in equal

payments over its life.

Amortization tables are widely used for homemortgages, auto loans, business loans,retirement plans, and so forth to determine howmuch of each payment represents principal

repayment and how much represents interest.They are very important, especially to homeowners


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Step 1: Construct an amortizationschedule for a $1,000, 10% loan thatrequires 3 equal annual payments.

PMT PMTPMT

0 1 2 3

10%

-1,000


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Step 2: Find interest chargefor Year 1

INTt = Beginning balancet x (k)

INT1 = 1,000 x 0.10 = $100.00

Repayment = PMT - INT= $402.11 - $100.00= $302.11.

Step 3: Find repayment ofprincipal in Year 1


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2005 Thomson/South-Western

End bal = Beginning bal. - Repayment

= $1,000 - $302.11 = $697.89.

Repeat these steps for the remainder of thepayments (Years 2 and 3 in this case)to complete the amortization table.

Step 4: Find ending balanceafter Year 1


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Interest declines, which has tax implications.

Loan Amortization Table

10% Interest RateYR Beg Bal PMT INT Prin PMT End Bal

1 $1000.00 $402.11 $100.00 $302.11 $697.89

2 697.89 402.11 69.79 332.32 365.57

3 365.57 402.11 36.56 365.55 0.02

Total 1,206.33 206.35 999.98 *

* Rounding difference

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Time Value of Money1