Chapter 3

The Time Value of Money

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Time Value of Money

The most important concept in finance

Used in nearly every financial decisionBusiness decisionsPersonal 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 beginning of Period 2.

Graphical representations used to show timing of cash flows

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100

0 1 2 Year

k%

Time line for a $100 lump sum due 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%

6

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 3k%

-50

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The 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.

Future Value

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FVFVnn = FV= FV1 1 = PV + INT= 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 if you 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.

What’s the FV of an initial $100 after 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 Time Value of Money Problems

Use EquationsUse Financial CalculatorUse Electronic Spreadsheet

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Solve this equation by plugging in the appropriate values:

Numerical (Equation) Solution

nn k)PV(1FV

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

$133.100)$100(1.331

$100(1.10)FV 3n

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Present Value

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

Discounting is the process of finding the present value of a future cash flow or series of future cash flows; it is the reverse of compounding.

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100

0 1 2 310%

PV = ?

What is the PV of $100 due in 3 years if k = 10%?

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

n

nnn

k+11

FV = 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 specified number of periods.

Ordinary (deferred) Annuity: An annuity whose 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 3k%

PMT PMT

0 1 2 3k%

PMT

Ordinary Annuity Versus Annuity Due

Ordinary Annuity

Annuity Due

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100 100100

0 1 2 310%

110

121

FV = 331

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

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Numerical Solution (using table):

$331.0000)$100(3.310

0.101(1.10)

$100FVA3

3

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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.

21248.69 = PV

100 100100

0 1 2 310%

90.91

82.64

75.13

What is the PV of this Ordinary Annuity?

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Using table:

$248.6985)$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, with payments made at the end of each year. What interest 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 amount varies from one period to the next:Payment (PMT) designates constant cash

flows—that is, an annuity stream.Cash flow (CF) designates cash flows in

general, both constant cash flows and uneven cash flows.

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0

100

1

300

2

300

310%

-50

4

90.91

247.93

225.39

-34.15530.08 = PV

What is the PV of this Uneven Cash Flow Stream?

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Numerical Solution

nn2211k)(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 Other Compounding Periods

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

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

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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.

Compounding Annually vs. Semi-Annually

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kSIMPLE = 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 Rate = kSIMPLE periodic rate X the number of periods per year

Distinguishing Between Different Interest Rates

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1 - m

k + 1 = EAR

mSIMPLE

10.25% = 0.1025 =1.0 - 1.05 =

1.0 - 2

0.10 +1 =

2

2

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

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nmSIMPLE

n mk

+ 1PV = FV

$134.0110)$100(1.3402

0.10 + 1$100 = FV

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FV of $100 after 3 years if interest is 10% compounded semi-annual? Quarterly?

$134.4989)$100(1.3444

0.10 + 1$100 = FV

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Amortized Loans

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, and so forth to determine how much 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 amortization schedule for a $1,000, 10% loan that requires 3 equal annual payments.

PMT PMTPMT

0 1 2 310%

-1,000

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Step 2: Find interest charge for 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 of principal in Year 1

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End bal = Beginning bal. - Repayment

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

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

Step 4: Find ending balance after Year 1

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Interest declines, which has tax implications.

Loan Amortization Table10% Interest Rate

YR 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