The Time Value of Money

The Time Value of MoneyThe Time Value of Money

Chapter 9


Which would you rather have ? $100 today - or $100 one year from today Sooner is better !


How about $100 today or $105 one year from today?

We revalue current dollars and future dollars using the time value of money

Cash flow time line graphically shows the timing of cash flows

Time 0 is today; Time 1 is one period from today

Time 0 1 2 3 4 5

Cash Flows-100

5%

Outflow

Interest rate

105Inflow

Cash Flow Time LinesCash Flow Time Lines

Future ValueFuture Value

Compounding the process of determining the value of a

cash flow or series of cash flows some time in the future when compound interest is applied

Future ValueFuture Value PV = present value or starting amount,

say, $100 i = interest rate, say, 5% per year would

be shown as 0.05 INT = dollars of interest you earn during

the year $100 0.05 = $5 FVn = future value after n periods or

$100 + $5 = $105 after one year = $100 (1 + 0.05) = $100(1.05) = $105


i)PV(1 PV(i)PV

INTPVFV

1


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

Compounded InterestCompounded Interest

Interest earned on interest

nn i)PV(1FV


Time 0 1 2 3 4 5

Interest

-100

5%

5.00

Total Value 105.00

5.25

110.25

5.51

115.76

5.79

121.55

6.08

127.63

Future Value Interest Factor Future Value Interest Factor for i and n (FVIFfor i and n (FVIFi,ni,n))

The future value of $1 left on deposit for n periods at a rate of i percent per period

The multiple by which an initial investment grows because of the interest earned


FVn = PV(1 + i)n = PV(FVIFi,n)Period (n) 4% 5% 6%

1 1.0400 1.0500 1.0600 2 1.0816 1.1025 1.1236 3 1.1249 1.1576 1.1910 4 1.1699 1.2155 1.2625 5 1.2167 1.2763 1.3382 6 1.2653 1.3401 1.4185

For $100 at i = 5% and n = 5 periods

FVn = PV(1 + i)n = PV(FVIFi,n)Period (n) 4% 5% 6%

1 1.0400 1.0500 1.0600 2 1.0816 1.1025 1.1236 3 1.1249 1.1576 1.1910 4 1.1699 1.2155 1.2625 5 1.2167 1.2763 1.3382 6 1.2653 1.3401 1.4185

For $100 at i = 5% and n = 5 periods

$100 (1.2763) = $127.63


Financial Calculator Financial Calculator SolutionSolution

Five keys for variable input N = the number of periods I = interest rate per period

may be I, INT, or I/Y PV = present value PMT = annuity payment FV = future value

Find the future value of $100 at 5% interest per year for five years

1. Numerical Solution:

Two SolutionsTwo Solutions

Time 0 1 2 3 4 5

Cash Flows

-100

5%

5.00

Total Value 105.00

5.25

110.25

5.51

115.76

5.79

121.55

6.08

127.63

FV5 = $100(1.05)5 = $100(1.2763) = $127.63


2. Financial Calculator Solution:

Inputs: N = 5 I = 5 PV = -100 PMT = 0 FV = ?

Output: = 127.63

Graphic View of the Compounding Graphic View of the Compounding Process: GrowthProcess: Growth

Relationship among Future Value, Growth or Interest Rates, and Time

0

1

2

3

4

5

1

i= 15%

i= 10%

i= 5%

i= 0%

2 4 6 8 10Periods

Future Value of $1

0

Present ValuePresent Value

Opportunity cost the rate of return on the best available

alternative investment of equal risk

If you can have $100 today or $127.63 at the end of five years, your choice will depend on your opportunity cost


The present value is the value today of a future cash flow or series of cash flows

The process of finding the present value is discounting, and is the reverse of compounding

Opportunity cost becomes a factor in discounting

0 1 2 3 4 5

PV = ?

5%

127.63



Start with future value: FVn = PV(1 + i)n

nnnn

i1

1FV

i)(1

FVPV

Find the present value of $127.63 in five years when the opportunity cost rate is 5%

1. Numerical Solution:

110.25 115.76 121.55

5%0 1 2 3 4 5

PV = ? 127.63

105.00

÷ 1.05÷ 1.05÷ 1.05÷ 1.05

-100.00


100$)7835.0(63.127$

2763.1

$127.63

05.1

$127.63PV 5

Find the present value of $127.63 in five years when the opportunity cost rate is 5%

2. Financial Calculator Solution:

Inputs: N = 5 I = 5 PMT = 0 FV = 127.63 PV = ?

Output: = -100


Relationship among Present Value, Interest Rates, and Time

Graphic View of the Graphic View of the Discounting ProcessDiscounting Process

0

0.2

0.4

0.6

0.8

1

i= 15%

i= 10%

i= 5%

i= 0%

2 4 6 8 10Periods

Present Value of $1

12 14 16 18 20

Solving for Time and Solving for Time and Interest RatesInterest Rates

Compounding and discounting are reciprocals

FVn = PV(1 + i)n

nnnn

i1

1FV

i)(1

FVPV

Four variables: PV, FV, i and n

If you know any three, you can solve for the fourth

For $78.35 you can buy a security that will pay you $100 after five years

We know PV, FV, and n, but we do not know i

FVn = PV(1 + i)n

$100 = $78.35(1 + i)5 Solve for i

Solving for iSolving for i

0 1 2 3 4 5

-78.35

i = ?

100

FVn = PV(1 + i)n

$100 = $78.35(1 + i)5

0.051-1.05i

..i1

..$

$i1

05127631

276313578

100

51

5

Numerical SolutionNumerical Solution

Inputs: N = 5 PV = -78.35 PMT = 0 FV = 100 I = ?

Output: = 5 This procedure can be used for any rate

or value of n, including fractions


Solving for nSolving for n

Suppose you know that the security will provide a return of 10 percent per year, that it will cost $68.30, and that you will receive $100 at maturity, but you do not know when the security matures. You know PV, FV, and i, but you do not know n - the number of periods.

Solving for nSolving for n

FVn = PV(1 + i)n $100 = $68.30(1.10)n By trial and error you could substitute

for n and find that n = 4

0 1 2 n-1 n=?

-68.30

10%

100


Inputs: I = 10 PV = -68.30 PMT = 0 FV = 100 N = ?

Output: = 4.0

AnnuityAnnuity

An annuity is a series of payments of an equal amount at fixed intervals for a specified number of periods

Ordinary (deferred) annuity has payments at the end of each period

Annuity due has payments at the beginning of each period

FVAn is the future value of an annuity over n periods

Future Value of an AnnuityFuture Value of an Annuity

The future value of an annuity is the amount received over time plus the interest earned on the payments from the time received until the future date being valued

The future value of each payment can be calculated separately and then the total summed

If you deposit $100 at the end of each year for three years in a savings account that pays 5% interest per year, how much will you have at the end of three years?

100 100.00 = 100 (1.05)0

105.00 = 100 (1.05)1

110.25 = 100 (1.05)2

315.25


0 1 2 35%

100

1-n

0t

t1-n10n i1 PMTi)PMT(1i)PMT(1i)PMT(1FVA

i

i1PMTi1PMT

nn

t

tn 11

2531515253100

1100 .$).($

0.05

1.05$FVA

3

3



Financial calculator solution: Inputs: N = 3 I = 5 PV = 0 PMT = -

100 FV = ? Output: = 315.25 To solve the same problem, but for the

present value instead of the future value, change the final input from FV to PV

Annuities DueAnnuities Due

If the three $100 payments had been made at the beginning of each year, the annuity would have been an annuity due.

Each payment would shift to the left one year and each payment would earn interest for an additional year (period).

$100 at the end of each year

0 1 2 35%

100 100 100.00 = 100 (1.05)0

105.00 = 100 (1.05)1

110.25 = 100 (1.05)2

315.25


$100 at the start of each year

0 1 2 35%

100 100100105.00 = 100 (1.05)1

110.25 = 100 (1.05)2

331.0125115.7625 = 100 (1.05)3

Future Value of Future Value of an Annuity Duean Annuity Due

Numerical solution:

i1i

1i1PMT

i1i1PMT

i1PMTFVA(DUE)

n

n

1t

t-n

n

1t

tn


Numerical solution:

0125331

05115253100

0511051

100

.$

..$

.0.05

.$FVA(DUE)

3

n



Financial calculator solution: Inputs: N = 3 I = 5 PV = 0 PMT = -

100 FV = ? Output: = 331.0125

Present Value of an AnnuityPresent Value of an Annuity

If you were offered a three-year annuity with payments of $100 at the end of each year

Or a lump sum payment today that you could put in a savings account paying 5% interest per year

How large must the lump sum payment be to make it equivalent to the annuity?

0 1 2 35%

100 100 100

384.86

05.1

100

703.9005.1

100

238.9505.1

100

3

2

1

272.325



Numerical solution:

n

1t t

n21n

iPMT

iPMT

iPMT

iPMTPVA

1

1

1

1

1

1

1

1

3227272322100

1

11

1

11

1

1

1

1

1

1

1

1

.$).($0.05

.05 $100

ii

PMTi

PMT

iPMT

iPMT

iPMTPVA

3

nn

1t t

n21n



Financial calculator solution: Inputs: N = 3 I = 5 PMT = -100 FV

= 0 PV = ? Output: = 272.325

Present Value of Present Value of an Annuity Duean Annuity Due

Payments at the beginning of each year Payments all come one year sooner Each payment would be discounted for

one less year Present value of annuity due will exceed

the value of the ordinary annuity by one year’s interest on the present value of the ordinary annuity

0 1 2 35%

100 100

70390

051

100051

051

100

23895051

100051

051

100

000100051

100051

051

100

22

11

01

. .

..

. .

..

..

..

285.941


Numerical solution:

ii

iPMT

ii

PMTi

PMTPVA(DUE)

n

n

1t t

1-n

0t 1n

11

11

11

1

1

1


$285.941

(2.85941) $100

(1.05)][(2.72325) $100

...

$PV(DUE)3

051

050051

11

1003



Financial calculator solution: Switch to the beginning-of-period mode,

then enter Inputs: N = 3 I = 5 PMT = -100 FV

= 0 PV = ? Output: = 285.94 Then switch back to the END mode

-846.80

i = ?0 1 2 3 4

250 250 250 250

Solving for Interest Rates Solving for Interest Rates with Annuitieswith Annuities

Suppose you pay $846.80 for an investment that promises to pay you $250 per year for the next four years, with payments made at the end of each year

ii

$250 $846.8041

11


Numerical solution: Trial and error using different values for

i using until you find i where the present value of the four-year, $250 annuity equals $846.80. The solution is 7%.


Financial calculator solution: Inputs: N = 4 PV = -846.8 PMT = 250

FV = 0 I = ? Output: = 7.0

PerpetuitiesPerpetuities

Perpetuity - a stream of equal payments expected to continue forever

Consol - a perpetual bond issued by the British government to consolidate past debts; in general, and perpetual bond

i

PMT

RateInterest

PaymentPVP

Uneven Cash Flow StreamsUneven Cash Flow Streams

Uneven cash flow stream is 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

Present Value of Present Value of Uneven Cash Flow StreamsUneven Cash Flow Streams

PV of uneven cash flow stream is the sum of the PVs of the individual cash flows of the stream

n

1t tt

nn21

iCF

iCF

iCF

iCFPV

1

1

1

1

1

1

1

121

Future Value of Future Value of Uneven Cash Flow StreamsUneven Cash Flow Streams

Terminal value is the future value of an uneven cash flow stream

n

1t

t-nt

0n

2-n2

1-n1n

iCF

iCFiCFiCFFV

1

111

Solving for i with Solving for i with Uneven Cash Flow StreamsUneven Cash Flow Streams

Using a financial calculator, input the CF values into the cash flow register and then press the IRR key for the Internal Rate of Return, which is the return on the investment.

Compounding PeriodsCompounding Periods

Annual compounding interest is added once a year

Semiannual compounding interest is added twice a year 10% annual interest compounded semiannually

would pay 5% every six months adjust the periodic rate and number of periods

before calculating

Interest RatesInterest Rates

Simple (Quoted) Interest Rate rate used to compute the interest payment paid

per period

Effective Annual Rate (EAR) annual rate of interest actually being earned,

considering the compounding of interest

01.m

i1 EAR

msimple

Interest RatesInterest Rates

Annual Percentage Rate (APR) the periodic rate multiplied by the number of

periods per year this is not adjusted for compounding

More frequent compounding:

nmsimple

n m

i1PV FV

Amortized LoansAmortized Loans

Loans that are repaid in equal payments over its life

Borrow $15,000 to repay in three equal payments at the end of the next three years, with 8% interest due on the outstanding loan balance at the beginning of each year

0 1 2 38%

PMT PMT PMT15,000

3

3

000151t t

1t t

3213

1.08

PMT,$

i1

PMTi1

PMT

i1

PMT

i1

PMTPVA


Numerical Solution:

$5,820.502.5771

$15,000PMT

2.5771PMT$15,000

0.081.08

1-1

PMT1.08

1PMT

1.08

PMT$15,000

33

1tt

3

1tt



Financial calculator solution: Inputs: N = 3 I = 8 PV = 15000 FV =

0 PMT = ? Output: = -5820.5


Amortization Schedule shows how a loan will be repaid with a breakdown of interest and principle on each payment date

YearBeginning Amount (1)

Payment (2) Interesta (3)Repayment of Principalb (2)-

(3)=(4)

Remaining Balance (1)-

(4)=(5)1 15,000.00$ 5,820.50$ 1,200.00$ 4,620.50$ 10,379.50$ 2 10,379.50 5,820.50 830.36 4,990.14 5,389.36 3 5,389.36 5,820.50 431.15 5,389.35 0.01 c

aInterest is calculated by multiplying the loan balance at the beginning of the year by the interest rate. Therefor, interest in Year 1 is $15,000(0.08) = $1,200; in Year 2, it is $10,379.50(0.08)=$830.36; and in Year 3, it is $5,389.36(0.08) = $431.15 (rounded).

bRepayment of principal is equal to the payment of $5,820.50 minus the interest charge for each year.

cThe $0.01 remaining balance at the end of Year 3 results from rounding differences.

Comparing Interest RatesComparing Interest Rates

1. Simple, or quoted, rate, (isimple) rates compare only if instruments have the

same number of compounding periods per year

2. Periodic rate (iPER) APR represents the periodic rate on an

annual basis without considering interest compounding

APR is never used in actual calculations

Comparing Interest RatesComparing Interest Rates

3. Effective annual rate, EAR the rate that with annual compounding (m=1)

would obtain the same results as if we had used the periodic rate with m compounding periods per year

1

mPER

mSIMPLE

i1

1.0m

i1EAR

End of Chapter 9End of Chapter 9

The Time Value of Money

The Time Value of Money

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