Dr. Yi
Ch 5: Advanced Topics
on
Time Value of Money
Financial Calculator

A financial calculator will
– Make your life easier,
– But also it will …..
– Make you dumber.

Therefore, I will call it
– “MED”
– The Machine that make your life Easier but also make you
Dumber.
2
Outlines of Ch 4 and 5: Time Value of Money (TVM)

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3
Basics
Simple present / future value problem
Simple vs. Compound Interest (Power of
Compound)
Compounding Frequency
Annuity / Annuity Due / Perpetuity
Uneven cash flow
Amortized loan
Effective annual rate
Multiple CFs:
Annuities and Perpetuities Defined

Annuity – finite series of equal payments that
occur at regular intervals
– Two conditions
• Equal cash flows
• spaced evenly apart
– Ordinary annuity: “at the end of each period”
– Annuity due: “at the beginning of each period”

4
Perpetuity – infinite series of equal payments
Perpetuity

Definition
– infinite series of equal payments

Perpetuity formula
– PV = C / i

Example
– Valuing a share of stock with no definite maturity
like preferred stock
5
Does PV = C / i work?
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6
For example, you could invest $100 in a bank account paying
5% interest per year forever.
Suppose you withdraw $5 (=$100*5%) per year and leave $100
intact.
This means that you receive $5 perpetuity.
That is, PV = $100, C = $5, i = 5%.
PV * i = 100 * 5% = $5 = C
PV = C / i
Example 1: Endowing a Perpetuity

7
You want to endow an annual MBA graduation party
at your name recognition. You want the event to be a
memorable one, so you budget $30,000 per year
forever for the party. If the university earns 8% per
year on its investments, and if the first payment is in
one year from now, how much will you need to
donate to endow the party?
Example 2: Money Machine

Your buddy in mechanical engineering has invented a money
machine. The main drawback of the machine is that it is slow. It
takes one year to manufacture $100. However, once built, the
machine will last forever and will require no maintenance. The
machine can be built immediately, but it will cost $1000 to build.
Your buddy wants to know if he should invest the money to
construct it. If the interest rate is 9.5% per year, what should
your buddy do?
– Hint: Find the PV of $100 perpetuity starting at year 1 at 9.5% and
compare it to $1000.
8
Growing Perpetuities

Assume you expect the amount of your
perpetual payment to increase at a constant rate,
.
C
PV (growing perpetuity) 
i  g
9
Example 3: Endowing a Growing Perpetuity

In the earlier example, you planned to donate $30,000 per year to
fund an annual graduation party. Given an interest rate of 8% per
year, the required donation was the present value of $375,000.
Before accepting the money, however, the MBA student
association asked that you increase the donation to account for the
effect of inflation on the cost of the party in future years. Although
$30,000 is adequate for next year’s party, the students estimate
that the party’s cost will rise by 4% per year thereafter. To satisfy
their request, how much do you need to donate now?
10
Does PV = C / (i – g) work fine?
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11
Suppose you want to create a perpetuity growing at 2%,
so you invest $100 in a bank account that pays 5%
interest.
At the end of year, you will have $105 in the bank. If
you withdraw only $3, you will have $102 to reinvest –
2% more than the amount you had initially.
This amount will then grow to $102 * 1.05 = $107.10 in
the following year, and you can withdraw $3*1.02 =
$3.06.
At the end of first year, $105=PV * (1 + i)
At the end of first year, $102=PV * (1 + g)
At the end of first year, $105 – $102 = $3 = C
Thus, C = PV * (1+ i) – PV * (1+g) = PV (i – g)
Thus, PV = C / (i – g) Yes!
Future Value of an Annuity
Future value calculated by compounding forward one period at a time
0
2
1
3
4
5
Time
(years)
$0
$
0
$0
0
$2,200
2,000
x 1.1
$2,000
2,000
x 1.1
x 1.1
$4,200
$4,620
$7,282
2,000
2,000
$6,620 x 1.1
$9,282
$10,210.20
2,000
x 1.1
$12,210.20
Future value calculated by compounding each cash flow separately
0
2
1
3
4
5
Time
(years)
$2,000
$2,000
$2,000
$2,000
x 1.1
x
x
x
1.12
2,200.00
2,420.00
2,662.00
1.13
2.928.20
1.14
Total future value
14
$2,000.00
$12,210.20
Mathematically,

FV
= C * (1+i%)0 + C * (1+i%)1 +…….+ C * (1+i%)n-1
= 100*(1+10%)0 + 100*(1+10%)1 + ………… + 100 * (1+10%)n-1
= 100*{(1+10%)0 + (1+10%)1 + ………… + (1+10%)n-1}
= 100[{(1+i%)n-1}/i%]
= 100*FVIFA10%, 3
= 100*3.3100
= $331.00

PV
= C*{1/ (1+i%)1}+ C * {1/(1+i%)2} +……….+ C * {1/(1+i%)n }
= 100*(1/(1+10%)1) + 100*(1/(1+10%)2) + ……………. +100*(1/(1+10%)n)
= 100*{1/(1+10%)1 + 1/(1+10%)2 + ………… + 1/(1+10%)n}
= 100*[{1-1/(1+i%)n}/i%]
= 100*PVIFA10%, 3
= 100* 2.4869
= $248.69
Note: FVIFAi%, n = Future Value Interest Factor for an Annuity for i% for n periods
PVIFAi%, n = Present Value Interest Factor for an Annuity for i% for n periods
15
Deriving the Annuity Formula,
The Alternative Approach

Consider two different series of cash flows.
– Annuity (3-year) : 0, 100, 100, 100
– Perpetuity : 0, 100, 100, 100, ………

PV of Perpetuity = PV of 3-year annuity + PV of perpetuity starting
in year 4 and forever

C / i = PV of 3-year annuity + [C / i] / [(1 + i)3]

PV of 3-year annuity = C / i – C / [i * (1 + i)3]
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16
In general,
PV of N-year annuity = C * [1/i- 1/{i*(1+i)N}]
PV of N-year annuity = C * [1 – (1+i)-N] / i
What’s the FV of a 3-year ordinary annuity of
$100 at 10%?
0
1
2
100
100
3
10%
FV  100(1.1)2  100(1.1)1  100(1.1)0
100
110
121
FV = 331
(1  i ) N 1
FV  C
i
17
1.13  1
1.3310  1
 100
 100
 100(3.3100)  331
.1
.1
Financial Calculator Solution
INPUTS
OUTPUT
3
10
0
-100
N
I/YR
PV
PMT
FV
331.00
Have payments but no lump sum PV,
so enter 0 for present value.
18
What’s the PV of this ordinary annuity?
0
1
2
3
100
100
100
10%
90.91
82.64
75.13
248.69 = PV
Alternatively,
PV = FV/(1+i)N
= 331/1.13 = 248.69
19
PV 
100 100 100


1.11 1.12 1.13
1  (1  i)  N
PV  C
i
1  1.13
1  .7513
 100
 100
 100(2.4869)  248.69
.1
.1
Example 6: Annuity – Sweepstakes

Suppose you win the Publishers Clearinghouse
$10 million sweepstakes. The money is paid in
equal annual installments of $333,333.33 over 30
years. If the appropriate discount rate is 5%, how
much is the sweepstakes actually worth today?
Or, how much does this sweepstakes cost the
Publishers Clearinghouse?
– Similarly, invest $5,124,150.29 today at 5% per year
for 30 years, but withdraw $333,333.33 each year
starting one year from today. How much will the
ending balance be at the end of 30 years?
20
What’s the difference between an ordinary
annuity and an annuity due?
Ordinary Annuity
0
i%
1
2
3
PMT
PMT
PMT
1
2
3
PMT
PMT
Annuity Due
0
i%
PMT
PV
21
FV
How can I tell whether given annuity CFs are ordinary
annuity or annuity due?

Is there any CFs at year 0 or not?
– If no, ordinary annuity. If yes, annuity due.

Ask a question yourself: Are given CFs similar to
“installment loan” example, or “rent” example?
– Ordinary Annuity: Installment loan usually requires us to
pay interest payment at the end of every period until the loan
is paid off fully.
– Annuity Due: Rent is an example of annuity due. You are
usually required to pay rent when you first move in at the
beginning of the month, and then on the first of each month
thereafter.
22
Find the FV and PV if the annuity were an
annuity due.
0
1
2
100
100
10%
100
23
3
Switch from “End” to “Begin”.
Then enter variables to find PV = $273.55.
INPUTS
OUTPUT
3
10
N
I/YR
PV
100
0
PMT
FV
-273.55
Then enter FV = 0 and press PV to find
PV = $273.55.
24
Example 7: Annuity Due
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25
You are saving for a new house and you put
$10,000 per year in an account paying 8%. The
first payment is made today. How much will
you have at the end of 3 years?
Annuity Due Timeline
0
10000
1
10000
2
3
10000
32,464
35,061.12
26
Present Value of a Lottery Prize Annuity Due
Problem:
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27
You are the lucky winner of the $30 million state lottery.
You can take your prize money either as (a) 30 payments of $1
million per year (starting today), or (b) $15 million paid
today.
If the interest rate is 8%, which option should you take?
Present Value of an Annuity Due
Problem:

Your parents have made you an offer you can’t refuse.

They’re planning to give you part of your inheritance early.

They’ve given you a choice.
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28
They’ll pay you $10,000 per year for each of the next seven years
(beginning today) or they’ll give you their 2007 BMW M6
Convertible, which you can sell for $61,000 (guaranteed) today.
If you can earn 7% annually on your investments, which should you
choose?
Solving for Variables Other Than Present Value or
Future Value
In some situations, we use the present and/or
future values as inputs, and solve for the variable
we are interested in.
Solving for Cash Flows,
given Present Value
C
29
PV
 1  (1  i)  N

i




Solving for Cash Flows,
given Future Value
C
FV
 (1  i) N  1 


i


Computing a Loan Payment
Problem:
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30
Suppose you found a dream house.
Price tag = $250,000
Down payment = 20% or $50,000
Total borrowing = $200,000 for 30 years at APR of 6%.
Find the monthly payment to pay back $200k mortgage loan.
Computing a Loan Payment (cont’d)
Solution:

Note, we need to use the monthly interest rate. Since the
quoted rate is an APR, we can just divide the annual rate by 12:
i = .06/12 = .005
PV
200, 000
200, 000
C


 1,199.10
N
360
 1  (1  i )   1  1.005  166.7916

 

i
.005

 

Given:
360
0.5
200,000
Solve for:
31
Excel Formula: =PMT(RATE,NPER, PV, FV) =
PMT(0.005,360,200000,0)
0
-1199.10
Solving for Variables Other Than Present Value or
Future Value
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Suppose you have an investment opportunity that requires five
payments of $100 investment per year and will pay $1,000 in
five years.
What interest rate, i, would you need so that the future value of
what you get is exactly equal to the future value of what you
give up?
Future value calculated by compounding each cash flow separately
0
2
1
3
4
5
Time
(years)
$100
$100
$100
$100
x (1+i)
x (1+i)2
x
(1+i)3
?
?
?
x (1+i)4
?
Total future value
32
$100
$1,000
Solving for Variables Other Than Present Value or
Future Value
1,000=100(1+i) 4  100(1+i)3 +100(1+i) 2  100(1+i)1  100(1+i)0
(1  i )5  1
1, 000  100
i




The solution for i is the interest rate (investment return) that guarantees
$1,000 future payment in exchange for five payments of $100 per year.
There is no simple way to solve for the interest rate.
The only way to solve this equation is to guess at values for i until you
find the right one.
An easier solution is to use a financial calculator or a spreadsheet.
Given:
Solve for:
33
5
0
-100
35.24
Excel Formula: =RATE(NPER,PMT,PV,FV)=Rate(5,-100,0,1000)
1,000
Solving for the Number of Periods in a Savings Plan
Problem:
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34
You are dreaming about becoming a millionaire.
Your current income allows a saving of $10,000 per year (at
year-end).
Your financial advisor estimates a reasonable investment return
to be 6% per year.
How long will it take you to get to your goal of $1,000,000?
Solving for the Number of Periods in a Savings Plan
Solution:

The timeline for this problem is
0
1
2 ………
?
6%
10k
35
10k ……… 10k
…..
?
?
FV = $1M
Solving for the Number of Periods in a Savings Plan
Solution (cont’d):

We need to find N so that the future value of our planned
savings (which is an annuity) equals our desired amount.
(1.06) N  1
1,000,000  10,000
.06
Given:
Solve for:
6
0
-10,000
1,000,000
33.3953
Excel Formula: =NPER(RATE,PMT, PV, FV) = NPER(.06,-10000, 0,1000000)
36
Example 8: Investment Advice Example

An investor wishes to leave $5,000,000 to
charity at death. An investor has a life
expectancy of 20 years.
1. How much most the investor invest today in a
lump-sum? i= 10%
2. How much per year? i = 10%
3. An investor has $43,335 per year to invest. At
what rate?
4. An investor has $1,072,731 today. At what rate?
37
Ellen’s Retirement Savings Plan Annuity
Problem:



38
Ellen is 35 years old, and she has decided it is time to plan
seriously for her retirement.
At the end of each year until she is 65, she will save $10,000 in
a retirement account.
If the account earns 10% per year, how much will Ellen have
saved at age 65?
Adam’s Retirement Savings Plan Annuity
Problem:

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39
Adam is 25 years old, and he has decided it is time to plan
seriously for his retirement.
He will save $10,000 in a retirement account at the end of each
year until he is 45.
At that time, he will stop paying into the account, though he
does not plan to retire until he is 65.
If the account earns 10% per year, how much will Adam have
saved at age 65?
Solving for the Number of Periods in a Savings Plan
Problem:



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40
Let’s return to Ellen and Adam.
Suppose Ellen decides she will continue working until she has
as much at retirement as her brother, Adam, will have when he
retires.
She will continue to contribute $10,000 each year to her
retirement account.
How much longer will she need to work to tie the competition
with her brother?
Solving for the Number of Periods in a Savings Plan
Solution:



We need to find N so that the FV of the $1,645,000 she’ll have at age
65 plus the $10,000 she’ll contribute each year is equal to $3,850,000.
Ellen will have to work until she’s 73 ½ years old.
(Here’s hoping she really loves her job!)
N
1
.
1
1
N
3,850,000  1,645,000(1.1)  10,000
.1
Given:
Solve for:
10
-1645000
8.57
Excel Formula: =NPER(RATE,PMT, PV, FV) =
NPER(0.10,-10000,-1645000,3850000)
41
-10,000
3850000
Uneven Cash Flow:
What is the PV of this uneven cash flow stream?
0
1
2
3
4
100
300
300
-50
10%
90.91
247.93
225.39
-34.15
530.08 = PV
42

Input in “CFj” register:
CF0
CF1
CF2
CF3
CF4

43
= 0
= 100
= 300
= 300
= -50
Enter I = 10%, then press NPV button to get
NPV = 530.09. (Here NPV = PV.)
What is the FV of this uneven cash flow
stream?
Will this cash flow earn interest?
0
1
2
3
4
100
300
300
-50
10%
330
363
133.10
Financial Calculator Solution? FV =776.10
44
Mike Piazza’s Two Contracts

Piazza got two contracts:
–
–
–
–
$91 million from Yankee (but spread out for next several years)
$80 million from Mets on the table
Which one is better?
Assume the interest rate is 3% per year.
Mike Piazza's Contract (million)
Contract Sum
Signing Bonus
$
7.5
Salary
$
83.5
Total
$
91.0
Today
1998
1999
2000
2001
$ 4.0
$
$ 6.0 $ 11.0 $ 12.5 $
$ 10.0 $ 11.0 $ 12.5 $
2002
2003
2004
2005
3.5
9.5 $ 14.5 $ 15.0 $ 15.0
13.0 $ 14.5 $ 15.0 $ 15.0
1
2
3
9.7 $ 10.4 $ 11.4 $
4
5
6
7
11.6 $ 12.5 $ 12.6 $ 12.2
0
Discount Rate
3%
45
Value in 1998 Dollars
3 $
80.3
$
Three Kinds of Loans


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46
Pure discount loan
Interest only loan
Amortized Loan
Pure Discount Loans
Treasury bills are excellent examples of pure
discount loans. The principal amount is repaid
at some future date, without any periodic
interest payments.
 If a T-bill promises to repay $10,000 in 2 years
and the market interest rate is 4 percent, how
much does the bill sell for in the market today?

– PV = 10,000 / 1.042 = 9246.6
47
Interest Only Loan - Example

Consider a 5-year, interest only loan with a 7%
interest rate. The principal amount is $10,000.
Interest is paid annually.
– What would the stream of cash flows be?
• Years 1 – 4: Interest payments of .07(10,000) = 700
• Year 5: Interest + principal = 10,700

48
This cash flow stream is similar to the cash
flows on corporate bonds and we will talk about
them in greater detail later.
Amortized Loan with Fixed Payment
Each payment covers the interest expense plus
reduces principal
 Example

– Mortgage payment
– Automobile loan
49
Amortized Loan


50
Amortization tables are widely used - for home
mortgages, auto loans, business loans,
retirement plans, and so on. They are very
important!
Financial calculators (and spreadsheets) are
great for setting up amortization tables.
51
Example 10: Amortized Loan Example
Construct an amortization schedule for
a $1,000, 10% annual rate loan with 3
equal payments.
52
Find the required payments.
0
1
2
3
PMT
PMT
PMT
10%
-1,000
INPUTS
OUTPUT
53
3
10
-1000
N
I/YR
PV
0
PMT
402.11
FV
Amortization Table
Beg. Bal.
54
Payment
Interest
Paid
Principal
Paid
End. Bal.
1
1000.00
$402.11
100.00
$302.11
697.89
2
697.89
$402.11
69.79
$332.33
365.56
3
365.56
$402.11
36.56
$365.56
0.00
Total
$1,206.34
$206.34
$1,000.00
$
402.11
Interest
302.11
Principal Payments
0
1
2
3
Level payments. Interest declines because
outstanding balance declines. Lender earns
10% on loan outstanding, which is falling.
55
Interest declines. Tax implications
More Challenging One - Automobile Example


Suppose you want to buy a Ford Mustang at
$20,000 and finance it with 5-year, 10% stated
APR loan.
Since you pay automobile monthly, we assume
a monthly compounding.
– Find a monthly payment.
– Find total dollar payment over 5-year period.
– Find total interest payment over 5-year period.

56
Do the same analysis with 5-year but 5% stated
APR loan.
Automobile Loan:
$20,000 with 5-year, 10% stated APR
Interest Principal
Beg Bal
Payment
paid
paid
End Bal
1 20,000.00 $424.94
166.67 $258.27 19,741.73
2 19,741.73 $424.94
164.51 $260.43 19,481.30
59
60
839.38
421.43
$424.94
$424.94
Total
Payment
Total
Interest Paid
57
6.99
3.51
$417.95
$421.43
$25,496.45
$5,496.45
421.43
0.00
What if we could lower APR to 5%?
Interest
Principal
Beg Bal
Payment
paid
paid
End Bal
1 20,000.00
$377.42
83.33 $294.09 19,705.91
2 19,705.91
$377.42
82.11 $295.32 19,410.59
58
59
750.16
$377.42
3.13
$374.30
375.86
60
375.86
$377.42
1.57
$375.86
0.00
Total
Payment
Total
Interest Paid
$22,645.48
$2,645.48
How much could we save?
$5,496 - 2,645
=$2,851 !!!
59
Credit Quality and Mortgage Rate
60
Example 11: Another Example


Ten years ago your firm borrowed $3 million to
purchase an office building using a loan with
7.80% APR and monthly payments for 30
years.
What is the monthly payment?
• Monthly pmt = 21,596.12

How much do you owe on the loan today?
• Find it without using spreadsheet.
61
Step-by-Step Calculator Instructions:
Home Mortgage
30-year, $100K loan @ 7%  Compute PMT
12 p/yr
30 yrs * 12 months = 360 payments  360 N
7 I/YR
$100,000 PV
(0 FV)
Hit PMT key to compute
PMT = $665.30
62
Home Mortgage
Each payment = $665.30 = principal + interest
1st payment:
1 INPUT 1 <GOLD> AMORT = = =
(“beginning with payment 1, ending with payment 1”)
$665.30 = $81.97 principal + $583.33 interest
New loan balance = $100K - $81.97 = $99,918
63
Home Mortgage
Each payment = $665.30 = principal + interest
2nd payment:
2 INPUT 2 <GOLD> AMORT = = =
(“beginning with payment 2, ending with payment 2”)
$665.30 = $82.45 principal + $582.85 interest
New loan balance = $99,918 - $82.45 = $99,836
64
Home Mortgage
1st year  12 payments = $665.30*12 = $7,984
12 payments:
1 INPUT 12 <GOLD> AMORT = = =
(“beginning with payment 1, ending with payment 12”)
$7,984 = $1,016 principal + $6,968 interest
New loan balance = $100,000 - $1016 = $98,984
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Home Mortgage
2nd year  12 payments = $665.30*12 = $7,984
12 payments:
13 INPUT 24 <GOLD> AMORT = = =
(“beginning with pmt 13, ending with pmt 24”)
$7,984 = $1,089 principal + $6,895 interest
New loan balance = $ 98,984 - $1089 = $97,895
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Nominal, Periodic, and Effective
Interest Rates
We will deal with 3 different
rates:
iNom = nominal, or stated, or
quoted, rate per year.
iPer = periodic rate.
effective annual
EAR = EFF% =
.
rate
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A Simple but Important Question,

If a bank quotes a car loan at 12 percent
APR, is the consumer actually paying 12
percent?
– Surprisingly, the answer is NO!
– Since the car loan is paid monthly, the periodic rate
per month is 12%/12 = 1% per month.
– Actually, the consumer pays (1+12%/12)12 – 1 =
1.0112 – 1 = 12.68%
– Therefore, EAR = 12.68% ( > 12%)
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What does this mean to the borrower?



Borrowing at 12% APR (or, nominal rate)
compounded monthly, and
Borrowing at 12.68% EAR
Are the same thing !!!
– 12% APR compounded monthly = 12.68% EAR
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iNom = nominal, or stated, or
quoted, rate per year.
i
is stated in contracts, quoted by banks.
Periods per year (m) must also be given.
Examples:
 Nom


8%; Quarterly
 8%, Daily interest (365 days)


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The other name for Nominal Rate =
Annual Percentage Rate (APR)
iPer = periodic rate.
i = iNom/m = Periodic rate, where m is
number of compounding periods per year.
m = 4 for quarterly, 12 for monthly, and 365
for daily compounding.
Examples:
 Per


8% quarterly: iPer = 8%/4 = 2%.
8% daily (365): iPer = 8%/365 = 0.021918%.
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Effective Annual Rate

Effective annual rate (EAR=EFF%)
– The rate that would produce the same ending
(future) value if annual compounding had been used.

EAR = (1 + iNom / m)m – 1

Example: The EAR for 10%, semiannual is
FV = (1 + iNom/m)m = (1 + 10% / 2)2 = (1.05) 2 = 1.1025.
EAR or EFF% = 1.1025 – 1 = .1025 = 10.25%
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Nominal Rate = 12%, EARs of 12% are….
EARAnnual
= 12%.
EARSemi
= (1 + 0.12/2)2 - 1
= 12.36%.
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%.
If m > 1, EFF% will always be greater than the nominal rate.
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EAR, Effective Annual Rate

Used to compare returns on investments
with different payments per year.
– An investment with monthly payments is
different from one with quarterly payments.
Must put on EFF% basis to compare rates of
return.


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Use EFF% only for comparisons, but use
APR as a stated interest rate.
Banks say “interest paid daily.” Same as
compounded daily.
EAR Example


Example 12: Universal Bank pays 7% interest,
compounded annually on time deposits.
Regional Bank pays 6 percent interest,
compounded quarterly. Based on effective
interest rate, in which bank would you prefer to
deposit your money?
Example 13: Bobcat television ads say you can
get a fitness machine that sells for $999 for $33
a month for 36 months. What rate of interest
are you paying on this loan? APR = ? EAR = ?
– 11.62%, 12.26%
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EAR: What does it really mean?

Suppose an investment opportunities stipulates that
(1) You pay $999 today, then
(2) They pay you $33 per month (ordinary annuity) for next 36 months.


Putting numbers into a TED, you will get 11.62%, which is the
APR.
Let’s say now that you are smart enough to re-invest $33 annuity
payments at APR until the end of 36th month.
–Then, our TED says that your future value would be $1,413.29.

Now, the question becomes simpler to understand. The deal is that
(1) You pay $999 today, then
(2) You will get paid $1,413.29 in THREE years.
(3) Using a TED, the average annual compounding return for this three-year
investment should be 12.26%.
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Example 14: Another Example

Suppose every two years, you donate $1,000 to
your church because your company pays
biannual bonus. You keep doing this for next
10 years (i.e., 5 payments). The first payment
will be made in two years from now. How
much will this cost you today if you finance
donation by putting lump sum dollars into an
account earning 4% per year?

Hint: Use the 2-year rate, instead of 1-year rate.
$3,975.93

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Video Clip: EAR
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Computing APRs from EARs

If you have an effective rate, how can you
compute the APR? Rearrange the EAR
equation and you get:

APR  m (1  EAR)

79
1
m

-1

Example 15: Finding the APR

Suppose you want to earn an effective rate of
12% and you are looking at an account that
compounds on a monthly basis. What APR must
they pay?


APR  12 (1  .12)  1  .1138655152
or 11.39%
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12
Example 16: Various Ways to Compound
Interest

Suppose you want to buy a new computer system and the
store is willing to sell it to allow you to make monthly
payments. The entire computer system costs $3500. The
loan period is for 2 years and the interest rate is 16.9%
with monthly compounding. What is your monthly
payment?
– PMT = 172.88

Suppose you deposit $50 a quarter into an account that has
an APR of 9%, based on quarterly compounding. How
much will you have in the account in 35 years?
– FV = $47,856.34

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You need $15,000 for a new car. If you can deposit
$10,000 today into an account that pays an APR of 5.5%
based on daily compounding, how long will it takes for
you to be able to buy the new car?
– N = 2,691 days or 7.37 years
Practice Problems

P 1: Tonya just won the contest held in one of national TV
stations. The contest rules states that she can choose from two
payment options. She plans to finance her new BMW with contest
money in three years from now.
– Option A: pay $10,000 one year from today, pay $20,000 two
years from now, and pay $30,000 in three years from now.
– Option B: pay three constant, $20,000 per year. Which option
would she choose? Assume interest rate is 10% per year.
– Key: Option B

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P 2: Want to buy a business in 7 years from now for $50,000.
– Parents give you $10,000 now
– Sisters give you $5,000 2 years from now
– Brothers give you $11,000 4 years from now
– Given R=10%, is that enough? If not, how much would you
have to invest today to have enough funds?
– Key: $4,013
Practice Problems
P 3: The present value of the following cash flow stream
is $5979 when discounted at 10% annually. What
is the value of the missing (t=2) cash flow?
• Cash Flow in Year 1= $1,000
• Cash Flow in Year 2= ?
• Cash Flow in Year 3= $2,000
• Cash Flow in Year 4 =$2,000
– Solution: Find the present value of all the cash flows
assuming that CF2=0. You will get $3,777.75 and then take
the difference of the two PVs: 5979-3777.75=2201.25 It
became PV=2201, n=2, i=10, and then hit FV.
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Practice Problems
84

P 4: What is the present value of $920 per year, at a discount
rate of 10 percent, if the first payment is received 5 years from
now and the last payment is received 20 years from now?
– Key: PV= $4,916.20

P 5: If you ran a bank, which rate would you rather advertise on
monthly-compounded loans, the EAR or the APR? Which rate
would you rather advertise on quarterly compounded savings
accounts, the EAR or the APR? Explain. As a consumer, which
would you prefer to see and why?
– Answer: A bank would rather advertise the APR on loans
since this rate appears to be lower and the EAR on savings
accounts since this appears to be higher. As a consumer, the
EAR is the more important rate since it represents the rate
actually paid or earned.
Practice Problems

85
P 6: Some financial advisors recommend you increase the amount
of federal income taxes withheld from your paycheck each month
so that you will get a larger refund come April 15th. That is, you
take home less today but get a bigger lump sum when you get
your refund. Based on your knowledge of the time value of
money, what do you think of this idea? Explain.
– Answer: Some students may slip in a discussion about the
benefits of forced savings, etc., but these issues are based on
preferences, not the time value of money. Based on the time
value of money, the students should recommend the opposite
strategy, that is, withhold as little as possible and pay the tax
bill when it comes the following year. This is the usual dollar
today versus a dollar tomorrow argument. Of course, the
astute student will note the potential tax complications of this
strategy, namely the IRS penalty for insufficient withholding,
but the basic argument still applies.
Practice Problems

P7: You work for a furniture store. You
normally sell a living room set for $4,000 and
finance the full purchase price for 24 monthly
payments at 24% APR. You are planning to run
a zero-interest financing sale during which you
will finance the set over 24 months at 0%
interest. How much do you need to charge for
the bedroom set during the sale in order to earn
your usual combined return on the sale and the
financing?
– $5,076
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Practice Problems

P8: You have just turned 30 years old, and have
accepted the first job after earning the MBA.
You want to contribute to the retirement
account earning 7% per year, and you cannot
withdraw until you retire on your 65th birthday.
After that point, you will need $100,000 per
year starting at the end of the first year of
retirement and ending on your 100th birthday.
What is your annuity payment starting at the
end of every year that you work?
– $9366.29
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At-the-end-of-chapter problems
See class syllabus
Practice! Practice! Practice!
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