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Future value

Present value

Rates of return

Amortization

Chapter 2Time Value of Money

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Time lines show timing of cash flows.

CF0 CF1 CF3CF2

0 1 2 3i%

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

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Time line for a $100 lump sum due at

the end of Year 2.

100

0 1 2 Yeari%

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Time line for an ordinary annuity of

$100 for 3 years.

100 100100

0 1 2 3i%

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Time line for uneven CFs: -$50 at t = 0

and $100, $75, and $50 at the end ofYears 1 through 3.

100 5075

0 1 2 3i%

-50

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Whats the FV of an initial $100 after 3

years if i = 10%?

FV = ?

0 1 2 3

10%

Finding FVs (moving to the righton a time line) is called compounding.

100

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After 1 year:

FV1 = PV + INT1 = PV + PV (i)= PV(1 + i)= $100(1.10)

= $110.00.After 2 years:

FV2 = FV1(1+i) = PV(1 + i)(1+i)

= PV(1+i)2

= $100(1.10)2= $121.00.

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After 3 years:

FV3 = FV2(1+i)=PV(1 + i)2(1+i)

= PV(1+i)3

= $100(1.10)3

= $133.10.

In general,

FVn = PV(1 + i)n.

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Three Ways to Find FVs

Solve the equation with a regularcalculator.

Use a financial calculator.

Use a spreadsheet.

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Financial calculator: HP10BII

Adjust display brightness: hold downON and push + or -.

Set number of decimal places todisplay: Orange Shift key, then DISPkey (in orange), then desired decimalplaces (e.g., 3).

To temporarily show all digits, hitOrange Shift key, then DISP, then =

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HP10BII (Continued)

To permantly show all digits, hitORANGE shift, then DISP, then .(period key)

Set decimal mode: Hit ORANGE shift,then ./, key. Note: many non-UScountries reverse the US use of

decimals and commas when writing anumber.

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HP10BII: Set Time Value Parameters

To set END (for cash flows occuringat the end of the year), hit ORANGEshift key, then BEG/END.

To set 1 payment per period, hit 1,then ORANGE shift key, then P/YR

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Financial calculators solve thisequation:

There are 4 variables. If 3 areknown, the calculator will solvefor the 4th.

.0ni1PVnFV

Financial Calculator Solution

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3 10 -100 0

N I/YR PV PMT FV133.10

Heres the setup to find FV:

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

Set: P/YR = 1, END.

INPUTS

OUTPUT

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Spreadsheet Solution

Use the FV function: see spreadsheetin Ch 02 Mini Case.xls.

= FV(Rate, Nper, Pmt, PV)

= FV(0.10, 3, 0, -100) = 133.10

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10%

Whats the PV of $100 due in 3 years if

i = 10%?

Finding PVs is discounting, and its

the reverse of compounding.

100

0 1 2 3

PV = ?

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Solve FVn

= PV(1 + i )n for PV:

PV = FV1+ i = FV 11+ in n nn

PV = $100 11.10= $100 0.7513 = $75.13. 3

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3 10 0 100

N I/YR PV PMT FV-75.13

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

INPUTS

OUTPUT

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Use the PV function: see spreadsheet.

= PV(Rate, Nper, Pmt, FV)

= PV(0.10, 3, 0, 100) = -75.13

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Finding the Time to Double

20%

2

0 1 2 ?

-1 FV = PV(1 + i)n$2 = $1(1 + 0.20)n

(1.2)n = $2/$1 = 2nLN(1.2) = LN(2)

n = LN(2)/LN(1.2)n = 0.693/0.182 = 3.8.

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20 -1 0 2

N I/YR PV PMT FV3.8

INPUTS

OUTPUT

Financial Calculator

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Use the NPER function: seespreadsheet.

= NPER(Rate, Pmt, PV, FV)

= NPER(0.10, 0, -1, 2) = 3.8

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Finding the Interest Rate

?%

2

0 1 2 3

-1 FV = PV(1 + i)n$2 = $1(1 + i)3

(2)(1/3) = (1 + i)1.2599 = (1 + i)

i = 0.2599 = 25.99%.

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3 -1 0 2

N I/YR PV PMT FV25.99

INPUTS

OUTPUT

Financial Calculator

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Use the RATE function:

= RATE(Nper, Pmt, PV, FV)

= RATE(3, 0, -1, 2) = 0.2599

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Ordinary Annuity

PMT PMTPMT

0 1 2 3

i%

PMT PMT

0 1 2 3i%

PMT

Annuity Due

Whats the difference between an

ordinary annuity and an annuity due?

PV FV

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Whats the FV of a 3-year ordinary

annuity of $100 at 10%?

100 100100

0 1 2 310%

110

121FV = 331

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FV Annuity Formula

The future value of an annuity with nperiods and an interest rate of i canbe found with the following formula:

.33110.

100

0.10

1)0(1

i

1i)(1PMT

3

n

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Financial calculators solve thisequation:

There are 5 variables. If 4 areknown, the calculator will solvefor the 5th.

.0i

1ni)(1PMTni1PVnFV

Financial Calculator Formula

for Annuities

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3 10 0 -100

331.00

N I/YR PV PMT FV


Have payments but no lump sum PV,so enter 0 for present value.

INPUTS

OUTPUT

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Use the FV function: see spreadsheet.

= FV(Rate, Nper, Pmt, Pv)

= FV(0.10, 3, -100, 0) = 331.00

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Whats the PV of this ordinary annuity?

100 100100

0 1 2 310%

90.91

82.6475.13

248.69 = PV

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PV Annuity Formula

The present value of an annuity with nperiods and an interest rate of i canbe found with the following formula:

69.24810.

100

0.10

)0(1

11-

i

i)(1

11-

PMT

3

n

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

3 10 100 0N I/YR PV PMT FV

-248.69

INPUTS

OUTPUT


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Use the PV function: see spreadsheet.

= PV(Rate, Nper, Pmt, Fv)

= PV(0.10, 3, 100, 0) = -248.69

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Find the FV and PV if the

annuity were an annuity due.

100 100

0 1 2 310%

100

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PV and FV of Annuity Due

vs. Ordinary Annuity

PV of annuity due:

= (PV of ordinary annuity) (1+i)

= (248.69) (1+ 0.10) = 273.56

FV of annuity due:

= (FV of ordinary annuity) (1+i)

= (331.00) (1+ 0.10) = 364.1

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3 10 100 0

-273.55

N I/YR PV PMT FV

Switch from End to Begin.Then enter variables to find PVA

3=

$273.55.

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

INPUTS

OUTPUT

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Excel Function for Annuities Due

Change the formula to:

=PV(10%,3,-100,0,1)

The fourth term, 0, tells the functionthere are no other cash flows. Thefifth term tells the function that it is an

annuity due. A similar function givesthe future value of an annuity due:

=FV(10%,3,-100,0,1)

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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

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Financial calculator: HP10BII

Clear all: Orange Shift key, then C Allkey (in orange).

Enter number, then hit the CFj key.

Repeat for all cash flows, in order.

To find NPV: Enter interest rate (I/YR).

Then Orange Shift key, then NPV key(in orange).

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Financial calculator: HP10BII (more)

To see current cash flow in list, hitRCL CFj CFj

To see previous CF, hit RCL CFj

To see subseqent CF, hit RCL CFj +

To see CF 0-9, hit RCL CFj 1 (to see

CF 1). To see CF 10-14, hit RCL CFj .(period) 1 (to see CF 11).

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Input in CFLO register:

CF0 = 0

CF1 = 100

CF2 = 300

CF3 = 300

CF4

= -50

Enter I = 10%, then press NPV buttonto get NPV = 530.09. (Here NPV = PV.)

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Excel Formula in cell A3:

=NPV(10%,B2:E2)

A B C D E

1 0 1 2 3 42 100 300 300 -50

3 530.09

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Nominal rate (iNom)

Stated in contracts, and quoted by banksand brokers.

Not used in calculations or shown on timelines

Periods per year (m) must be given.

Examples:

8%; Quarterly

8%, Daily interest (365 days)

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Periodic rate (iPer )

iPer = iNom/m, where m is number ofcompounding periods per year. m = 4 forquarterly, 12 for monthly, and 360 or 365

for daily compounding. Used in calculations, shown on time lines.

Examples:

8% quarterly: iPer = 8%/4 = 2%.8% daily (365): iPer = 8%/365 =

0.021918%.

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Will the FV of a lump sum be larger or

smaller if we compound more often,holding the stated I% constant? Why?

LARGER! If compounding is morefrequent than once a year--forexample, semiannually, quarterly,

or daily--interest is earned on interestmore often.

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FV Formula with Different CompoundingPeriods (e.g., $100 at a 12% nominal rate with

semiannual compounding for 5 years)

= $100(1.06)10 = $179.08.

FV = PV 1 .+i

m

nNom

mn

FV = $100 1 +0.12

25S

2x5

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FV of $100 at a 12% nominal rate for 5years with different compounding

FV(Annual)= $100(1.12)5 = $176.23.

FV(Semiannual)= $100(1.06)10=$179.08.

FV(Quarterly)= $100(1.03)20 = $180.61.

FV(Monthly)= $100(1.01)60 = $181.67.

FV(Daily) = $100(1+(0.12/365))(5x365)

= $182.19.

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Effective Annual Rate (EAR = EFF%)

The EAR is the annual rate which causes PVto grow to the same FV as under multi-periodcompounding Example: Invest $1 for oneyear at 12%, semiannual:

FV = PV(1 + iNom/m)m

FV = $1 (1.06)2 = 1.1236.

EFF% = 12.36%, because $1 invested for one

year at 12% semiannual compounding wouldgrow to the same value as $1 invested for oneyear at 12.36% annual compounding.

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An investment with monthlypayments is different from onewith quarterly payments. Must

put on EFF% basis to comparerates of return. Use EFF% onlyfor comparisons.

Banks say interest paid daily.Same as compounded daily.

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How do we find EFF% for a nominal

rate of 12%, compoundedsemiannually?

EFF% = - 1(1 + )iNommm

= - 1.0(1 + )0.122

2

= (1.06)2

- 1.0= 0.1236 = 12.36%.

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Finding EFF with HP10BII

Type in nominal rate, then Orange Shiftkey, then NOM% key (in orange).

Type in number of periods, then Orange

Shift key, then P/YR key (in orange).

To find effective rate, hit Orange Shiftkey, then EFF% key (in orange).

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EAR (or EFF%) for a Nominal Rate of

of 12%

EARAnnual = 12%.

EARQ = (1 + 0.12/4)4 - 1 = 12.55%.

EARM = (1 + 0.12/12)12 - 1 = 12.68%.

EARD(365) = (1 + 0.12/365)365 - 1 = 12.75%.

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Can the effective rate ever be equal to

the nominal rate?

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

If m > 1, EFF% will always be greater

than the nominal rate.

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When is each rate used?

iNom: Written into contracts, quoted

by banks and brokers. Notused in calculations or shownon time lines.

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iPer: Used in calculations, shown ontime lines.

If iNom has annual compounding,then iPer = iNom/1 = iNom.

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(Used for calculations if and only ifdealing with annuities wherepayments dont match interestcompounding periods.)

EAR = EFF%: Used to comparereturns on investmentswith different payments

per year.

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Amortization

Construct an amortization schedule

for a $1,000, 10% annual rate loanwith 3 equal payments.

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Step 1: Find the required payments.

PMT PMTPMT

0 1 2 310%

-1,000

3 10 -1000 0INPUTS

OUTPUTN I/YR PV FVPMT

402.11

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Step 2: Find interest charge for Year 1.

INTt = Beg balt (i)INT1 = $1,000(0.10) = $100.

Step 3: Find repayment of principal inYear 1.

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

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Step 4: Find ending balance after

Year 1.

End bal = Beg bal - Repmt

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

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

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Interest declines. Tax implications.

BEG PRIN ENDYR BAL PMT INT PMT BAL

1 $1,000 $402 $100 $302 $698

2 698 402 70 332 3663 366 402 37 366 0

TOT 1,206.34 206.34 1,000

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$

0 1 2 3

402.11Interest

302.11

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

Principal Payments

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Amortization tables are widelyused--for home mortgages, autoloans, business loans, retirement

plans, and so on. They are veryimportant!

Financial calculators (and

spreadsheets) are great forsetting up amortization tables.

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On January 1 you deposit $100 in anaccount that pays a nominal interestrate of 11.33463%, with daily

compounding (365 days).How much will you have on October1, or after 9 months (273 days)?(Days given.)

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iPer = 11.33463%/365

= 0.031054% per day.

FV=?

0 1 2 273

0.031054%

-100

Note: % in calculator, decimal in equation.

FV = $100 1.00031054= $100 1.08846 = $108.85.273273

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273 -100 0

108.85

INPUTS

OUTPUT

N I/YR PV FVPMT

iPer = iNom/m

= 11.33463/365= 0.031054% per day.

Enter i in one step.Leave data in calculator.

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Whats the value at the end of Year 3 of

the following CF stream if the quotedinterest rate is 10%, compoundedsemiannually?

0 1

100

2 35%

4 5 6 6-mos.periods

100 100

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Payments occur annually, butcompounding occurs each 6months.

So we cant use normal annuityvaluation techniques.

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1st Method: Compound Each CF

0 1

100

2 35%

4 5 6

100 100.00110.25121.55331.80

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

= $331.80.

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Could you find the FV with afinancial calculator?

Yes, by following these steps:

a. Find the EAR for the quoted rate:

2nd Method: Treat as an Annuity

EAR = (1 + ) - 1 = 10.25%.0.1022

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3 10.25 0 -100INPUTS

OUTPUT

N I/YR PV FVPMT

331.80

b. Use EAR = 10.25% as the annual ratein your calculator:

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Whats the PV of this stream?

0

100

15%

2 3

100 100

90.70

82.2774.62247.59

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You are offered a note which pays$1,000 in 15 months (or 456 days)for $850. You have $850 in a bankwhich pays a 6.76649% nominal rate,

with 365 daily compounding, whichis a daily rate of 0.018538% and anEAR of 7.0%. You plan to leave themoney in the bank if you dont buy

the note. The note is riskless.Should you buy it?

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3 Ways to Solve:

1. Greatest future wealth: FV2. Greatest wealth today: PV3. Highest rate of return: Highest EFF%

iPer = 0.018538% per day.

1,000

0 365 456 days

-850

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1. Greatest Future Wealth

Find FV of $850 left in bank for15 months and compare withnotes FV = $1,000.

FVBank = $850(1.00018538)456

= $924.97 in bank.

Buy the note: $1,000 > $924.97.

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456 -850 0

924.97

INPUTS

OUTPUT

N I/YR PV FVPMT

Calculator Solution to FV:

iPer = iNom/m= 6.76649%/365= 0.018538% per day.

Enter iPer in one step.

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2. Greatest Present Wealth

Find PV of note, and comparewith its $850 cost:

PV = $1,000/(1.00018538)456= $918.95.

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456 .018538 0 1000

-918.95

INPUTS

OUTPUT

N I/YR PV FVPMT

6.76649/365 =

PV of note is greater than its $850cost, so buy the note. Raises yourwealth.

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Find the EFF% on note andcompare with 7.0% bank pays,which is your opportunity cost of

capital:

FVn = PV(1 + i)n

$1,000 = $850(1 + i)456

Now we must solve for i.

3. Rate of Return

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456 -850 0 1000

0.035646%per day

INPUTS

OUTPUT

N I/YR PV FVPMT

Convert % to decimal:

Decimal = 0.035646/100 = 0.00035646.

EAR = EFF% = (1.00035646)365 - 1= 13.89%.

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Using interest conversion:

P/YR = 365NOM% = 0.035646(365) = 13.01

EFF% = 13.89

Since 13.89% > 7.0% opportunity cost,buy the note.

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