Download Chapter 4

Finance
School of Management
Chapter 4: Time Value of Money
Objective
Explain the concept of compounding
and discounting and to provide
examples of real life
applications
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Chapter 4 Contents






Compounding
Frequency of Compounding
Present Value and
Discounting
Alternative Discounted
Cash Flow Decision Rules
Multiple Cash Flows
Annuities





Perpetual Annuities
Loan Amortization
Exchange Rates and Time
Value of Money
Inflation and Discounted
Cash Flow Analysis
Taxes and Investment
Decisions
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Financial Decisions
– Costs and benefits being spread out over time
– The values of sums of money at different dates
– The same amounts of money at different dates
have different values.
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Time Value of Money
– Interest
– Purchasing power
– Uncertainty
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Compounding
– Present value (PV)
– Future value (FV)
– Simple interest: the interest on the original
principal
– Compound interest: the interest on the interest
– Future value factor
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Value of Investing $1 at an Interest
Rate of 10%
– Continuing in this manner you will find that the
following amounts will be earned:
1 Year
$1.1
Simple interest: 0.1
2 Years
$1.21
Simple interest:
0.1+0.1=0.2
Compound interest: 0.01
3 Years
$1.331
4 Years
$1.4641
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Value of $5 Invested
– More generally, with an investment of $5 at
10% we obtain
1 Year
$5*(1+0.10)
$5.5
2 years
$5.5*(1+0.10)
$6.05
3 years
$6.05*(1+0.10)
$6.655
4 Years
$6.655*(1+0.10)
$7.3205
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Value of $5 Invested
– If we can earn 10% interest on the principal $5,
then after 4 years
FV  5  (1  0.1)  7.2305
4
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Future Value of a Lump Sum
FV  PV * (1  i ) n
F V w ith gro w th s fro m -6 % to +6 %
F utu re V a lue o f $1 0 0 0
3 ,5 0 0
6%
3 ,0 0 0
2 ,5 0 0
4%
2 ,0 0 0
1 ,5 0 0
2%
1 ,0 0 0
0%
-2 %
-4 %
-6 %
500
0
0
2
4
6
8
10
12
14
16
18
Y ea rs
9
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School of Management
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Example: Future Value of a Lump Sum


Your bank offers a CD
(Certificate of Deposit)
with an interest rate of
3% for a 5 year
investments.
You wish to invest
$1,500 for 5 years, how
much will your
investment be worth?
FV  PV * (1  i ) n
 $ 1500 * (1  0 .03 ) 5
 $ 1738 .1111145
n
i
PV
FV
Result
5
3%
1,500
?
1738.911111
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Finance
Example: Reinvesting at a Different Rate





You have $10,000 to invest
for two years.
Two years CDs and one
year CDs are paying 7%
and 6% per year
respectively.
What should you do?
Reinvestment rate?
You are sure the interest
rate on one-year CDs will
be 8% next year.

With the two-year CD
FV  $10,000  (1.07 ) 2
 $11,449

With the sequence of
two one-year CDs
FV  $10,000 1.06 1.08
 $11, 448
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Frequency of Compounding
– Annual percentage rate (APR)
– Effective annual rate (EFF)

Suppose you invest $1 in a CD, earning interest at a
stated APR of 6% per year compounded monthly.
FV  $1 (1  0.06 / 12)12  1.06168

General formula
APR 

EFF  1 

m 

m
1
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Effective Annual Rates of an APR of 18%
Annual Percentage Frequency of
rate
Compounding
Annual Effective
Rate
18
18
18
18
18
18
18.00
18.81
19.25
19.56
19.68
19.72
1
2
4
12
52
365
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The Frequency of Compounding
Note that as the frequency of compounding
increases, so does the annual effective rate.
 What occurs as the frequency of compounding
rises to infinity?

m
 APR 
APR
EFF  Lim 1 

e
1

m  
m 
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Present Value
– In order to reach a target amount of money at
a future date, how much should we invest
today?
– Discounting
– Discounted-cash-flow (DCF)
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Present Value of a Lump Sum
FV  PV * (1  i ) n
Divide both sides by (1  i ) to obtain :
n
FV
n
PV 
 FV * (1  i )
n
(1  i )
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Finance
School of Management
Example: Present Value of a Lump Sum
You have been offered
$40,000 for your
printing business,
payable in 2 years.
 Given the risk, you
require a return of 8%.
 What is the present
value of the offer?

FV
PV 
(1  i ) n
40,000

(1  0.08) 2
 34293.55281
 $34,293.55 today
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Solving Lump Sum Cash Flow for
Interest Rate
FV  PV * (1  i ) n
FV
 (1  i ) n
PV
FV
n
(1  i ) 
PV
FV
n
i
1
PV
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Example: Interest Rate on a Lump
Sum Investment
If you invest $15,000
for ten years, you
receive $30,000.
 What is your annual
return?

FV
i
1
PV
1
30000
10
10
10

1  2 1  2 1
15000
 0.071773463
n
 7.18% (to the nearest basis point)
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Solving Lump Sum Cash Flow for
Number of Periods
FV  PV * (1  i ) n

ln 

FV
 (1  i ) n
PV
FV 
n

ln
(
1

i
)
 n * ln 1  i 

PV 


 FV 
ln 

PV  ln  FV   ln  PV

n

ln 1  i 
ln 1  i 
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
School of Management
Finance
NPV (Net Present Value) Rule
– NPV
– NPV rule: Accept a project if its NPV is positive.
– Opportunity cost of capital: The rate (of return) we
could earn somewhere else if we did not invest in the
project under evaluation.
– Yield to maturity or Internal Rate of Return (IRR)
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Finance
Example: Evaluate a Project
A five-year savings
bond with face value
$100 is selling for a
price of $75.
 Your next-best
alternative for
investing is an 8%
bank account.
 Is the savings bond a
good project?

NPV 
$100
 $75
1.08 5
 $68 .06  $75
 $6.94
FV  $75  1.08 5
 $110 .20
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Example: Evaluate a Project
IRR  (100 / 75)
1/ 5
 1  5.92%
 100 
n  ln 
 ln 1.08  3.74
 75 
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Example: Borrowing




You need to borrow
$5,000 to buy a car.
A bank can offer you a
loan at an interest rate of
12%.
A friend says he will
lend the $5,000 if you
pay him $9,000 in four
years.
Should you borrow from
the bank or the friend?
$9,000
NPV  $5,000 
1.124
 $719.66
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Finance
PV of Annuity Formula
PV 
PMT
(1  i )1


PMT
(1  i ) 2
PMT 1  (1  i )  n

i


PMT
(1  i ) n
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Finance
School of Management
Example: Buying an Annuity




You are 65 years old and
NPV  $10,000
expect to live until age 80.
$1,000  1  (1  8%) 15

For a cost of $10,000, an
8%
insurance company will pay
 $1,440 .52
you $1,000 per year for the
rest of your life.
i  5.56%
You can earn 8% per year on
your money in a bank
account.
n  21
Does it pay to buy the
insurance policy?

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
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Perpetual Annuities / Perpetuities

Recall the annuity formula:
pmt 
1 

PV 
* 1 
n 
i
1  i  


Let n -> ∞ with i > 0:
pmt
PV 
i
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Loan Amortization





Home mortgage loans or car loans are repaid in
equal periodic installments.
Part of each payment is interest on the outstanding
balance of the loan.
Part is repayment of principal.
The portion of the payment that goes toward the
payment of interest is lower than the previous
period’s interest payment.
The portion that goes toward repayment of principal
is greater than the previous period’s.
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Finance
Calculator Solution
n
i
PV
3
9%
100,000
FV PMT
0
?
Result
-39,505.48
This is
the yearly
repayment
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Amortization Schedule for 3-Year
Loan at 9%
Beginning
Year
Banlance
1
2
3
100,000
69,495
36,244
Total
Payment
Interest
Paid
Principal
Paid
Remaining
Balance
39,505
39,505
39,505
9,000
6,255
3,262
30,505
33,252
36,244
69,495
36,244
0
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Example: Exchange Rates
Investing $10,000 in dollar-denominated bonds
offering an interest rate of 10% per year
 Investing in yen-denominated bonds offering an
interest rate of 3% per year
 The exchange rate for the yen is now $0.01 per
yen.
 Which is the better investment for the next year?

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Time
U.S.A.
$10,000
Japan
0.01 $/¥
10% $/$ (direct)
$11,000 ¥
1,000,000¥
3% ¥ / ¥
? $/¥
1,030,000¥
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School of Management
Finance
Time
U.S.A.
$10,000
Japan
0.01 $/¥
10% $/$ (direct)
$11,124
$11,000 ¥
1,000,000¥
3% ¥/¥
0.0108 $/¥
1,030,000¥
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School of Management
Finance
Time
U.S.A.
$10,000
Japan
0.01 $/¥
10% $/$ (direct)
$10,918 ¥
$11,000 ¥
1,000,000¥
3% ¥ / ¥
0.0106 $/¥
1,030,000¥
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School of Management
Finance
Time
U.S.A.
$10,000
Japan
0.01 $/¥
10% $/$ (direct)
$11,000 ¥
$11,000 ¥
1,000,000¥
3% ¥ / ¥
0.01068 $/¥
1,030,000¥
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Finance
School of Management
The Real Rate of Interest
1  Nominal interest rate
1  Real interest rate 
1  Rate of inflation
Nominal interest rate - Rate of inflation
Real interest rate 
1  Rate of inflation
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Finance
School of Management
Switch to a Gas Heat ?






You currently heat your house with oil and your
annual heating bill is $2,000.
By converting to gas heat, you estimate that this year
you could cut your heating bill by $500.
You think the cost differential between gas and oil is
likely to remain the same for many years.
The cost of installing a gas heating system is $10,000.
Your alternative use of the money is to leave it in a
bank account earning an interest rate of 8% per year.
Is the conversion worthwhile?
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Switch to a Gas Heat ?




Assume that the $500 cost differential will remain forever.
The investment in switch of heating is a perpetuity, i.e. paying
$10,000 now for getting $500 per year forever.
i  $500 / $10,000  5%
If the $500 cost differential will increase over time with the
general rate of inflation, then the 5% rate of return is a real
rate of return.
The conversion is not worthwhile unless the rate of inflation
is greater than 2.875% per year.
(0.08  0.05) /1.05  2.875%
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Taxes
After tax interest rate 
(1  Tax rate)  Before tax interest rate
39
Finance
School of Management
Taking Advantage of a Tax Loophole




You are in a 40% tax bracket
and currently have $100,000
invested in municipal bonds
earning a tax-exempt rate of
interest of 6% per year.
Now you buy a house at a
cost of $100,000.
A bank offers a loan for you
at an interest rate of 8% per
year.
Does it pays for you to
borrow?
After tax interest rate 
(1  0.4)  8%  4.8%
40