Ch 08 Show

8/14/2019 Ch 08 Show

1/71

8/14/2019 Ch 08 Show

2/71

8 - 2

Copyright 2002 by Harcourt, Inc. All rights reserved.

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.

8/14/2019 Ch 08 Show

3/71

8 - 3


Time line for a $100 lump sum due at

the end of Year 2.

100

0 1 2 Year i%

8/14/2019 Ch 08 Show

4/71

8 - 4


Time line for an ordinary annuity of

$100 for 3 years.

100 100100

0 1 2 3i%

8/14/2019 Ch 08 Show

5/71

8 - 5


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

8/14/2019 Ch 08 Show

6/71

8 - 6


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

8/14/2019 Ch 08 Show

7/71

8 - 7


After 1 year:

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

= $110.00.After 2 years:

FV2 = PV(1 + i)2

= $100(1.10)2

= $121.00.

8/14/2019 Ch 08 Show

8/71

8 - 8


After 3 years:

FV3 = PV(1 + i)3

= $100(1.10)3

= $133.10.

In general,

FVn = PV(1 + i)n.

8/14/2019 Ch 08 Show

9/71

8 - 9


Three Ways to Find FVs

Solve the equation with a regularcalculator.

Use a financial calculator.

Use a spreadsheet.

8/14/2019 Ch 08 Show

10/71

8 - 10


Financial calculators solve thisequation:

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

( )FV PV in n= +1 .

Financial Calculator Solution

8/14/2019 Ch 08 Show

11/71

8 - 11


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

8/14/2019 Ch 08 Show

12/71

8 - 12


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 = ?

8/14/2019 Ch 08 Show

13/71

8 - 13


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

8/14/2019 Ch 08 Show

14/71

8 - 14



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

8/14/2019 Ch 08 Show

15/71

8 - 15


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.

8/14/2019 Ch 08 Show

16/71

8 - 16


20 -1 0 2

N I/YR PV PMT FV3.8

INPUTS

OUTPUT

Financial Calculator

8/14/2019 Ch 08 Show

17/71

8 - 17


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

8/14/2019 Ch 08 Show

18/71

8 - 18


Whats the FV of a 3-year ordinary

annuity of $100 at 10%?

100 100100

0 1 2 310%

110

121FV = 331

8/14/2019 Ch 08 Show

19/71

8 - 19


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

8/14/2019 Ch 08 Show

20/71

8 - 20


Whats the PV of this ordinaryannuity?

100 100100

0 1 2 310%

90.91

82.6475.13

248.69 = PV

8/14/2019 Ch 08 Show

21/71

8 - 21


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

8/14/2019 Ch 08 Show

22/71

8 - 22


A B C D

1 0 1 2 32 100 100 100

3 248.69

Spreadsheet Solution

Excel Formula in cell A3:

=NPV(10%,B2:D2)

8/14/2019 Ch 08 Show

23/71

8 - 23


Special Function for Annuities

For ordinary annuities, this formula incell A3 gives 248.96:

=PV(10%,3,-100)

A similar function gives the futurevalue of 331.00:

=FV(10%,3,-100)

8/14/2019 Ch 08 Show

24/71

8 - 24


Find the FV and PV if the

annuity were an annuity due.

100

100

0 1 2 310%

100

8/14/2019 Ch 08 Show

25/71

8 - 25


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

8/14/2019 Ch 08 Show

26/71

8 - 26


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)

8/14/2019 Ch 08 Show

27/71

8 - 27


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

8/14/2019 Ch 08 Show

28/71

8 - 28


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.)

8/14/2019 Ch 08 Show

29/71

8 - 29


Spreadsheet Solution

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

8/14/2019 Ch 08 Show

30/71

8 - 30


What interest rate would cause $100 to

grow to $125.97 in 3 years?

3 -100 0 125.97

N I/YR PV FVPMT

8%

$100(1 + i )3 = $125.97.(1 + i)3 = $125.97/$100 = 1.2597

1 + i = (1.2597)1/3 = 1.08i = 8%.

INPUTS

OUTPUT

8/14/2019 Ch 08 Show

31/71

8 - 31


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.

8/14/2019 Ch 08 Show

32/71

8 - 32


0 1 2 3

10%

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.

8/14/2019 Ch 08 Show

33/71

8 - 33


We will deal with 3 differentrates:

iNom = nominal, or stated, or

quoted, rate per year.iPer = periodic rate.

EAR= EFF% = .

effective annual

rate

8/14/2019 Ch 08 Show

34/71

8 - 34


iNom is stated in contracts. Periods

per year (m) must also be given.

Examples:s8%; Quarterly

s8%, Daily interest (365 days)

8/14/2019 Ch 08 Show

35/71

8 - 35


Periodic rate = iPer = iNom/m, where m isnumber of compounding periods peryear. 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%.

8/14/2019 Ch 08 Show

36/71

8/14/2019 Ch 08 Show

37/71

8 - 37


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.

8/14/2019 Ch 08 Show

38/71

8 - 38


How do we find EFF% for a nominal

rate of 10%, compoundedsemiannually?

Or use a financial calculator.

EFF% = - 1(1 + )iNomm

m

= - 1.0(1 + )0.102

2

= (1.05)2

- 1.0= 0.1025 = 10.25%.

8/14/2019 Ch 08 Show

39/71

8 - 39


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%.

8 0

8/14/2019 Ch 08 Show

40/71

8 - 40


FV of $100 after 3 years under 10%

semiannual compounding? Quarterly?

= $100(1.05)6 = $134.01.

FV3Q = $100(1.025)12 = $134.49.

FV = PV 1 .+i

m

nNom

mn

FV = $100 1 +0.10

23S

2x3

8 41

8/14/2019 Ch 08 Show

41/71

8 - 41


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.

8/14/2019 Ch 08 Show

42/71

8 43

8/14/2019 Ch 08 Show

43/71

8 - 43


iPer: Used in calculations, shown ontime lines.

If iNom has annual compounding,

then iPer = iNom/1 = iNom.

8/14/2019 Ch 08 Show

44/71

8 45

8/14/2019 Ch 08 Show

45/71

8 - 45


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

8 46

8/14/2019 Ch 08 Show

46/71

8 - 46


Payments occur annually, butcompounding occurs each 6months.

So we cant use normal annuityvaluation techniques.

8 47

8/14/2019 Ch 08 Show

47/71

8 - 47


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.

8 48

8/14/2019 Ch 08 Show

48/71

8 - 48


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

8 49

8/14/2019 Ch 08 Show

49/71

8 - 49


3 10.25 0 -100

INPUTS

OUTPUT

N I/YR PV FVPMT

331.80

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

8 50

8/14/2019 Ch 08 Show

50/71

8 - 50


Whats the PV of this stream?

0

100

15%

2 3

100 100

90.70

82.2774.62247.59

8 51

8/14/2019 Ch 08 Show

51/71

8 - 51


Amortization

Construct an amortization schedule

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

8 52

8/14/2019 Ch 08 Show

52/71

8 - 52


Step 1: Find the required payments.

PMT PMTPMT

0 1 2 310%

-1,000

3 10 -1000 0

INPUTS

OUTPUT

N I/YR PV FVPMT

402.11

8 53

8/14/2019 Ch 08 Show

53/71

8 - 53


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.

8 54

8/14/2019 Ch 08 Show

54/71

8 - 54


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.

8 - 55

8/14/2019 Ch 08 Show

55/71

8 - 55


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

8 - 56

8/14/2019 Ch 08 Show

56/71

8 - 56


$

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

8 - 57

8/14/2019 Ch 08 Show

57/71

8 - 57


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.

8/14/2019 Ch 08 Show

58/71

8 - 59

8/14/2019 Ch 08 Show

59/71

8 59


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

8 - 60

8/14/2019 Ch 08 Show

60/71

8 60


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.

8 - 61

8/14/2019 Ch 08 Show

61/71

8 61


Now suppose you leave your moneyin the bank for 21 months, which is1.75 years or 273 + 365 = 638 days.

How much will be in your account atmaturity?

Answer: Override N = 273 with N =

638. FV = $121.91.

8 - 62

8/14/2019 Ch 08 Show

62/71

8 62


iPer= 0.031054% per day.

FV = 121.91

0 365 638 days

-100

FV = $100(1 + 0.1133463/365)638

= $100(1.00031054)638= $100(1.2191)= $121.91.

8 - 63

8/14/2019 Ch 08 Show

63/71

8 63


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,which is a daily rate of 0.018538%and an EAR of 7.0%. You plan toleave the money in the bank if you

dont buy the note. The note isriskless.

Should you buy it?

8 - 64

8/14/2019 Ch 08 Show

64/71

8 64


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

8/14/2019 Ch 08 Show

65/71

8 - 66

8/14/2019 Ch 08 Show

66/71


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.

8 - 67

8/14/2019 Ch 08 Show

67/71


2. Greatest Present Wealth

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

PV = $1,000/(1.00018538)456

= $918.95.

8 - 68

8/14/2019 Ch 08 Show

68/71


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.

8 - 69

8/14/2019 Ch 08 Show

69/71


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

capital:

FVn = PV(1 + i)n

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

Now we must solve fori.

3. Rate of Return

8 - 70

8/14/2019 Ch 08 Show

70/71


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%.

8 - 71

8/14/2019 Ch 08 Show

71/71

Using interest conversion:

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

EFF% = 13.89

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

Documents

Ch 08 Show