2 - 1CHAPTER 2

Discounted Cash Flow AnalysisTime Value of Money

Financial Mathematics

Future valuePresent valueRates of returnAmortizationAnnuities, ANDMany Examples

2 - 2

MINICASE 2SIMPLE?

p. 88

Also note financial mathematics problems at end of TAB &

Notes on Excel and LOTUS.

2 - 3

MINICASE 2

Why is financial mathematics (time value of money) so important in financial analysis?

2 - 4

a.Time lines show timing of cash flows.

ALWAYS A GOOD IDEA TO DRAW A TIME LINE.

CF0 CF1 CF3CF2

0 1 2 3i%

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

2 - 5

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

100

0 1 2 Years

i%

2 - 6

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

100 100100

0 1 2 3i%

2 - 7

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

2 - 8

b(1) What’s the FV of an initial$100 after 3 years if i = 10%?

FV = ?

0 1 2 310%

100

Finding FVs is compounding.

2 - 9

b(1) What’s the FV of an initial$100 after 3 years if i = 10%?

FV = ?

0 1 2 310%

100

Finding FVs is compounding.

110 ?

2 - 10

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)2

= $100(1.10)2

= $121.00.

2 - 11

FV3 = PV(1 + i)3

= 100(1.10)3

= $133.10.

In general,

FVn = PV(1 + i)n.

After 3 years

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Solve the equation with a regular calculator

Use tables

Use a financial calculator

Use a spreadsheet

Four Ways to Find FVs

2 - 13

USING TABLES: See handout

3 PERIODS10 %= 1.3310times 100 = $133.10SAY GOOD-BYE TO USING TABLES!

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Financial Calculator Solution

Financial calculators solve this equation:

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

FV PV inn 1 .

2 - 15

3 10 -100 0N I/YR PV PMT FV

Here’s the setup to find FV:

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

Set: P/YR = 1, END

133.10

INPUTS

OUTPUT

2 - 16

b(2) What’s the PV of $100 due in 3 years if i = 10%?

Finding PVs is discounting, and it’s the reverse of compounding.

100

0 1 2 310%

PV = ?

2 - 17

PV =

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

PV = $100/( ) =

= $100(0.7513) = $75.13.

FVn

(1 + i)n

n

1.103

2 - 18

Financial Calculator Solution

INPUTS

OUTPUT

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, take out $100 after 3 years.

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EXCEL SOLUTION

LOOK AT FUNCTION’S PAGE FOR EXCEL/LOTUS.

2 - 20

Spreadsheet Solution

Use the FV function: see spreadsheet in Ch 02 Mini Case.xls. = FV(Rate, Nper, Pmt, PV) = FV(0.10, 3, 0, -100) = 133.10

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Spreadsheet Solution

Use the PV function: see spreadsheet. = PV(Rate, Nper, Pmt, FV) = PV(0.10, 3, 0, 100) = -75.13

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

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

2 = 1(1.20)n

(1.20)n = 2

n ln (1.20) = ln 2n(0.1823) = 0.6931 n = 0.6931/0.1823 = 3.8 years.

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INPUTS

OUTPUT

Graphical Illustration:

01 2 3 4

1

2

FV

3.8

Years

20 -1 0 2N I/YR PV PMT FV

3.8 Beware:Some Calculators round up.

2 - 24

Spreadsheet Solution

Use the NPER function: see spreadsheet. = NPER(Rate, Pmt, PV, FV) = NPER(0.20, 0, -1, 2) = 3.8

Correction

2 - 25

ADDITIONAL QUESTION

A FARMER CAN SPEND $60/ACRE TO PLANT PINE TREES ON SOME MARGINAL LAND. THE EXPECTED REAL RATE OF RETURN IS 4%, AND THE EXPECTED INFLATION RATE IS 6%. WHAT IS THE EXPECTED VALUE OF THE TIMBER AFTER 20 YEARS?

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ADDITIONAL QUESTION

Bill Veeck once bought the Chicago White Sox for $10 million and then sold it five years later for $20 million. In short, he doubled his money in five years. What compound rate of return did Veeck earn on his investment?

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RULE OF 72

A good approximation of the interest rate--or number of years--required to double your money.

n * krequired to double = 72

In this case,5 * krequired to double = 72

• k = 14.4 Correct answer was 14.87, so for ball-park

approximation, use Rule of 72.

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ADDITIONAL QUESTION

John Jacob Astor bought an acre of land in Eastside Manhattan in 1790 for $58. If average interest rate is 5%, did he make a good deal?

2 - 29

d. What’s the difference between an ordinary annuity and an annuity due?

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d. What’s the difference between an ordinary annuity and an annuity due?

PMT PMTPMT

0 1 2 3i%

PMT PMT

0 1 2 3

i%

PMT

Annuity Due

Ordinary Annuity

36

2 - 31

HINT

ANNUITY DUE OF n PERIODS IS EQUAL TO A REGULAR ANNUITY OF (n-1) PERIODS PLUS THE PMT.

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e(1). What’s the FV of a 3-year ordinary annuity of $100 at 10%?

100 100100

0 1 2 310%

FV =

2 - 33

e(1). What’s the FV of a 3-year ordinary annuity of $100 at 10%?

100 100100

0 1 2 310%

110

121

FV = 331

2 - 34

FV Annuity Formula

The future value of an annuity with n periods and an interest rate of i can be found with the following formula:

.33110.

100

0.10

1)0(1

i

1i)(1PMT

3

n

2 - 35

Financial calculators solve this equation:

There are 5 variables. If 4 are known, the calculator will solve for the 5th.

.0i

1ni)(1PMTn

i1PVnFV

Financial Calculator Formula for Annuities

Correct but confusing!

2 - 36

3 10 0 -100

331.00N I/YR PV PMT FV

Financial Calculator Solution

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

INPUTS

OUTPUT

2 - 37

Spreadsheet Solution

Use the FV function: see spreadsheet. = FV(Rate, Nper, Pmt, Pv) = FV(0.10, 3, -100, 0) = 331.00

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e(2). What’s the PV of this ordinary annuity?

100 100100

0 1 2 310%

_____ = PV

2 - 39

What’s the PV of this ordinary annuity?

100 100100

0 1 2 310%

90.91

82.64

75.13248.69 = PV

2 - 40

3 10 100 0

-248.69

INPUTS

OUTPUTN I/YR PV PMT FV

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

2 - 41

Spreadsheet Solution

Use the PV function: see spreadsheet. = PV(Rate, Nper, Pmt, Fv) = PV(0.10, 3, 100, 0) = -248.69

2 - 42

e(3). Find the FV and PV if theannuity were an annuity due.

100 100

0 1 2 310%

100

2 - 43

3 10 100 0

-273.55

INPUTS

OUTPUTN I/YR PV PMT FV

Could, on the 12C, switch from “End” to “Begin”; i.e. f Begin.

Then enter variables to find PVA3 = $273.55.

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

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Another HINT

FV OF AN ANNUITY DUE OF n PERIODS IS EQUAL TO THE FV OF A REGULAR ANNUITY OF n PERIODS TIMES (1+k) (slide 30)

PV OF AN ANNUITY DUE OF n PERIODS IS EQUAL TO THE PV OF A REGULAR ANNUITY OF n PERIODS TIMES (1+k)

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HINT, illlustrated

The PV of this regular annuity was 248.69.

Multiply this by (1 + .10), and you get: 273.55, the PV of the annuity due.

This avoids the necessity of having to switch from end to begin.

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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

2 - 47

3 10 100 0

-273.55 N I/YR PV PMT FV

Switch from “End” to “Begin”.Then enter variables to find PVA3 = $273.55.

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

INPUTS

OUTPUT

2 - 48

Excel Function for Annuities Due

Change the formula to:

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

The fourth term, 0, tells the function there are no other cash flows. The fifth term tells the function that it is an annuity due. A similar function gives the future value of an annuity due:

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

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EXCEL SOLUTION

2 - 50

300

3

(f) What is the PV of this uneven cashflow stream?

0

100

1

300

210%

-50

4

______ = PV

2 - 51

300

3

(f) What is the PV of this uneven cashflow stream?

0

100

1

300

210%

-50

4

90.91247.93225.39

-34.15530.08 = PV

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Input in “CFLO” register:

CF0 = 0

CF1 = 100

CF2 = 300

CF3 = 300

CF4 = -50Enter I = 10%, then press NPV button

to get NPV = 530.09.

2 - 53

CALCULATOR SOLUTION

0 g CF0

100 g CFj

300 g CFj

2 g Nj

50 CHS g CFj

10 if NPV

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EXCEL SOLUTION

REMEMBER EXCEL/LOTUS READS THE 1ST CASH FLOW AS OCCURING ONE PERIOD HENCE.

2 - 55

Spreadsheet Solution

Excel Formula in cell A3:

=NPV(10%,B2:E2)

A B C D E

1 0 1 2 3 4

2 100 300 300 -50

3 530.09

2 - 56

g. 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

8.00%

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

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EXCEL SOLUTION

2 - 58

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

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h.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 more frequent than once a year--forexample, semiannually, quarterly,or daily--interest is earned on interest more often.

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0 1 2 310%

0 1 2 3

5%

4 5 6

134.01

100 133.10

1 2 30

100

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

Semiannually: FV6 = 100(1.05)6 = 134.01.

2 - 61

Periodic rate = iPer = iNom/m, where m is number of compounding 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%.

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Effective Annual Rate (EAR = EFF%):The annual rate which causes PV to grow to the same FV as under multiperiod compounding.

2 - 63

An investment with monthly compounding is different from one with quarterly compounding. Must put on EAR% basis to compare rates of return.

Banks say “interest paid daily.” Same as compounded daily.

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h(3).How do we find EAR for a nominal rate of 10%, compounded

semiannually?

EFF% = 1 + i

m - 1Nom

m

= 1+ 0.10

2 - 1.0

= 1.05 - 1.0= 0.1025 = 10.25%.

2

2

Or use a financial calculator (not 12C)

2 - 65EAR = (1+knom/m)m - 1

.10 ENT2 divide1 +2 Yx

1 -= .1025

(1 + EAR) = (1+knom/m)m

2 - 66

CALCULATOR

WHAT IS EAR IF COMPOUNDING QUARTERLY?

COMPOUNDING DAILY?COMPOUNDING CONTINUOUSLY?

2 - 67

EAR = EFF% of 10%

EARAnnual = 10%.

EARQ = (1 + 0.10/4)4 - 1 = 10.38%.

EARM = (1 + 0.10/12)12 - 1 = 10.47%.

EARD(360) = (1 + 0.10/360)360 - 1 = 10.52%.

2 - 68

For multiple years, n

(1 + EAR) = (1 + Knom/m)m

(1 + EAR)n = (1 + Knom/m)mn

To multiply by a $ of dollars, PRIN

PRIN * (1 + EAR)n = PRIN * (1 + Knom/m)mn

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i. Can the effective rate ever beequal to the nominal rate?

Yes, but only if annual compounding is 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. May be used in calculations or shown on time lines when compounding is annual.

OR USE EAR!

2 - 71

Used in calculations,shown on time lines.

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

Can always use EAR!

iPer:

2 - 72

EAR = EFF%: Used to compare returns on investments with different compounding patterns.

Also used for calculations if dealing with annuities where paymentsdon’t match interest compounding periods.

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FV of $100 after 3 years under 10% semiannual compounding? Quarterly?

Daily?

= $100(1.05)6 = $134.01.FV3Q = $100(1.025)12 = $134.49.

FV = PV 1 .+ imnNom

mn

FV = $100 1 + 0.10

23S

2x3

2 - 74

ALTERNATE SOLUTION USING EAR

FOR SEMIANNUAL COMPOUNDING, EAR = 10.25%

FOR 3 YEARS: 100*(1.1025)3 = $134.01

FOR Quarterly COMPOUNDING and 3 years:

100*(1.1038)3 = $134.49

2 - 75

j(3). What’s the value at the end of Year 3 of the following CF stream if the

quoted interest rate is 10%, compounded semi-annually?

0 1

100

2 35%

4 5 6 6-mos. periods

100 100

2 - 76

Payments occur annually, but compounding occurs each 6 months.

So we can’t use normal annuity valuation techniques.

2 - 77

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.

2 - 78

Could you find FV with afinancial calculator?

Yes, by following these steps:

a. Find the EAR for the quoted rate:

2nd Method: Treat as an AnnuityI.E. USE EAR

EAR = (1 + ) - 1 = 10.25%. 0.10

22

2 - 79

Or, to find EAR with a 17 OR 19b Calculator:

NOM% = 10 P/YR = 2 EFF% = 10.25

2 - 80

3 10.25 0 -100

INPUTS

OUTPUT

N I/YR PV FVPMT

331.80

b. The cash flow stream is an annual annuity whose EFF% = 10.25%.

c.

2 - 81

j(2) What’s the PV of this stream?

0

100

15%

2 3

100 100

2 - 82

What’s the PV of this stream?

0

100

15%

2 3

100 100

90.7082.2774.62

247.59

2 - 83

Calculator solution

100 PMT3 n10.25 if NPV=247.59

2 - 84What’s the FV of this stream under

quarterly compouning?

0

100

1 2 3

100 100

2 - 85

EAR WORKSHEET

EAR worksheet

knom = 0.12 INPUTm = 8 INPUTEAR = 0.126493

EAR= 0.1255 INPUTm = 4 INPUTknom= 0.119992

2 - 86

k. Amortization

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

2 - 87

Step 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

2 - 88

ALGEBRA

PMT/(1+k) + PMT/(1+k)2 + PMT/(1+k)3 = 1000

PMT [1/(1+k) + 1/(1+k)2 + 1/(1+k)3] = 1000, or

PMT = 1000/ [1/(1+k) + 1/(1+k)2 + 1/(1+k)3]

2 - 89

Step 2: Find interest charge for Year 1.

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

Step 3: Find repayment of principal in Year 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.

2 - 91

Interest declines. Tax implications.

BEG PRIN ENDYR BAL PMT INT REDUCTION BAL

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

2 698 402 70 332 366

3 366 402 37 366 0

TOT 1,206.34 206.34 1,000

2 - 92$

0 1 2 3

402.11Interest

302.11

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

Principal Payments

2 - 93

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

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

2 - 94

EXCEL SOLUTION

2 - 95

NEW PROBLEM:

On January 1 you deposit $100 in an account that pays a nominal interest rate of 10%, with daily compounding (365 days).

How much will you have on October 1, or after 9 months (273 days)? (Days given.)

2 - 96

iPer = 10.0% / 365= 0.027397% per day.

FV=?

0 1 2 273

0.027397%

-100

Note: % in calculator, decimal in equation.

FV = $100 1.00027397 = $100 1.07765 = $107.77.

273273

2 - 97

273 -100 0

107.77

INPUTS

OUTPUT

N I/YR PV FVPMT

iPer = iNom/m= 10.0/365= 0.027397% per day.

Enter i in one step.Leave data in calculator.

2 - 98

Now suppose you leave your money in the bank for 21 months, which is 1.75 years or 273 + 365 = 638 days.

How much will be in your account at maturity?

Answer: Override N = 273 with N = 638. FV = $119.10.

2 - 99

iPer = 0.027397% per day.

FV = 119.10

0 365 638 days

-100

FV = $100(1 + 0.10/365)638

= $100(1.00027397)638

= $100(1.1910)= $119.10.

2 - 100

ALTERNATIVE SOLUTION USING EAR

FIND EAR.10 ENTER365 DIVIDE1 +365 Yx [= (1 + EAR)]

.75 Yx [= (1 + EAR).75]

100 MULTIPLY = $107.79

2 - 101

MORE PRECISELY, instead of .75 exponent:Calculate [1+EAR] as above, then273 ENTER

365 DIVIDE [=(1+EAR)]

Yx [=(1+EAR)(273/365)]

100 MULTIPLY = $107.77

2 - 102

PROBLEM

SUPPOSE THAT YOU WERE TOLD THAT THE EFFECTIVE ANNUAL RATE WERE 10%, WITH DAILY COMPOUNDING. WHAT THE STATED, OR NOMINAL RATE BE IN THIS CASE?

2 - 103

ALGEBRA

(1 + EAR) = (1 + knom /m)m

(1 + EAR)(1/m) = (1 + knom /m)

(1 + EAR)(1/m) - 1 = knom /m

m*[(1 + EAR)(1/m) - 1] = knom

2 - 104m*[(1 + EAR)(1/m) - 1] = knom

Using the calculator, EAR = 10%, dailycompounding.

1.1 ENTER365 1/X Yx

1 -365 MULTIPLY= 9.53%

2 - 105

n. You are offered a note which pays $1,000 in 15 months (or 456 days) for $850. You have $850 in a bank which pays a 7.0% nominal rate, with 365 daily compounding, which is a daily rate of 0.019178% and an EAR of 7.25%. You plan to leave the money in the bank if you don’t buy the note. The note is riskless.

Should you buy it?

2 - 106

3 Ways to Solve:

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

iPer = 0.019178% per day.

1,000

0 365 456 days

-850

2 - 107

1. Greatest Future Wealth

Find FV of $850 left in bank for15 months and compare withnote’s FV = $1000.

FVBank = $850(1.00019178)456

= $927.67 in bank.

Buy the note: $1000 > $927.67.

2 - 108

456 -850 0

927.67

INPUTS

OUTPUT

N I/YR PV FVPMT

Calculator Solution to FV:

iPer = iNom/m= 7.0/365= 0.019178% per day.

Enter iPer in one step.

2 - 109

2. Greatest Present Wealth

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

PV = $1000/[(1.00019178)456]= $916.27.

2 - 110

456 .019178 0 1000

-916.27

INPUTS

OUTPUT

N I/YR PV FVPMT

7/365 =

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

2 - 111

Find the EFF% on note and compare with 7.25% bank pays, which is your opportunity cost of capital:

FVn = PV(1 + i)n

1000 = $850(1 + i)456

Now we must solve for i.

3. Rate of Return

2 - 112

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%.

2 - 113

Using interest conversion:

P/YR = 365NOM% = 0.035646(365) = 13.01 EFF% = 13.89

Since 13.89% > 7.25% opportunity cost,buy the note.

2 - 114

ADDITIONAL PROBLEM #2

IT IS NOW JANUARY 1. YOU PLAN TO MAKE 5 DEPOSITS OF $100 EACH, ONE EVERY 6 MONTHS, WITH THE FIRST PAYMENT BEING MADE TODAY. IF THE BANK PAYS A NOMINAL INTEREST RATE OF 12 PERCENT, SEMIANNUAL COMPOUNDING, HOW MUCH WILL BE IN YOUR ACCOUNT AFTER 10 YEARS?

2 - 115

ADDITIONAL PROBLEM #3 IT IS NOW JANUARY 1, 1997. YOU ARE

OFFERED A NOTE UNDER WHICH SOMEONE PROMISES TO MAKE 5 PAYMENTS OF $100 EACH, WITH THE FIRST PAYMENT ON JULY 1, 1997 AND SUBSEQUENT PAYMENTS ON EACH JULY 1 THEREAFTER THROUGH JULY 1, 2001, PLUS A FINAL PAYMENT OF $1000 TO BE MADE ON JANUARY 1, 2002. WITH A NOMINAL DISCOUNT RATE OF 10 PERCENT, QUARTERLY COMPOUNDING, WHAT IS THE PV OF THE NOTE?

2 - 116

JOHNM PROBLEMS