1
Finance School of Management
Objective
Explain the concept of compounding
and discounting and to provide
examples of real life
applications
Chapter 4,Time Value of Money
2
Finance School of Management
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
3
Finance School of Management
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,
4
Finance School of Management
Time Value of Money
– Interest
– Purchasing power
– Uncertainty
5
Finance School of Management
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
6
Finance School of Management
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 Sim ple i n terest, 0.1
2 Years $1.21 Sim ple i n terest,
0.1+0.1=0.2
Compound inte rest,0.01
3 Years $1.331
4 Years $1.4641
7
Finance School of Management
Value of $5 Invested
– More generally,with an investment of $5 at
10% we obtain
1 Year $5 *(1+0.1 0) $5,5
2 y ea rs $5,5* (1+0.10 ) $6,05
3 y ea rs $6,05 *(1+0.1 0) $6,65 5
4 Years $6,65 5*(1+0.10 ) $7,32 05
8
Finance School of Management
Value of $5 Invested
– If we can earn 10% interest on the principal $5,
then after 4 years
2 3 0 5.7)1.01(5 4 ????FV
9
Finance School of Management
Future Value of a Lump Sum
niPVFV )1(* ??
F V w i t h g r o w t h s f r o m - 6 % t o + 6 %
0
500
1,0 0 0
1,5 0 0
2,0 0 0
2,5 0 0
3,0 0 0
3,5 0 0
0 2 4 6 8 10 12 14 16 18 20
Y e a r s
F
u
t
u
r
e
V
a
l
u
e
o
f
$
1
0
0
0
6%
4%
2%
0%
- 2 %
- 4 %
- 6 %
10
Finance School of Management
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
$1,500 for 5 years,how
much will your
investment be worth?
1 1 1 1 1 4 5.1 7 3 8$
)03.01(*1 5 0 0$
)1(*
5
?
??
??
n
iPVFV
n 5
i 3%
PV 1,50 0
FV?
Re sul t 17 38,9 11 111
11
Finance School of Management
Example,Reinvesting at a Different Rate
? You have $10,000 to invest
for two years,
? Two years CDs and one
year CDs are paying 7%
and 6% per year
respectively,
? What should you do?
? Reinvestment rate?
? You are sure the interest
rate on one-year CDs will
be 8% next year,
? With the two-year CD
4 4 9,11$
)07.1(0 0 0,10$ 2
?
??FV
? With the sequence of
two one-year CDs
448,11$
08.107.1000,10$
?
???FV
12
Finance School of Management
Frequency of Compounding
– Annual percentage rate (APR)
– Effective annual rate (EFF)
? Suppose you invest $1 in a CD,earning interest at a
stated APR of 6% per year compounded monthly,
0 6 1 6 8.1)12/06.01(1$ 12 ????FV
? General formula
11 ??
?
?
?
?
? ??
m
m
A P R
E F F
13
Finance School of Management
Effective Annual Rates of an APR of 18%
An nual Percentage
rat e
Frequency o f
Com poundi ng
An nual Ef f e cti ve
Rate
18 1 18.00
18 2 18.81
18 4 19.25
18 12 19.56
18 52 19.68
18 365 19.72
14
Finance School of Management
The Frequency of Compounding
11 ???
?
?
?
?
?
??
??
A P R
m
m
e
m
A P R
L imE F F
? Note that as the frequency of compounding
increases,so does the annual effective rate,
? What occurs as the frequency of compounding
rises to infinity?
15
Finance School of Management
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)
16
Finance School of Management
Present Value of a Lump Sum
n
n
n
n
iFV
i
FV
PV
i
iPVFV
?
??
?
?
?
??
)1(*
)1(
:o b t a i n t o)1(b y s i de sb o t h Di v i de
)1(*
17
Finance School of Management
Example,Present Value of a Lump Sum
? You have been offered
$40,000 for your
printing business,
payable in 2 years,
? Given the risk,you
require a return of 8%,
? What is the present
value of the offer?
t o d a y55.293,34$
5 5 2 8 1.3 4 2 9 3
)08.01(
000,40
)1(
2
?
?
?
?
?
?
n
i
FV
PV
18
Finance School of Management
Solving Lump Sum Cash Flow for
Interest Rate
1
)1(
)1(
)1(*
??
??
??
??
n
n
n
n
PV
FV
i
PV
FV
i
i
PV
FV
iPVFV
19
Finance School of Management
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? p o i n t ) b a si sn e a r e st t h e( t o %18.7
0 7 1 7 7 3 4 6 3.0
12121
1 5 0 0 0
3 0 0 0 0
1
10
1
10
10
?
?
??????
?? n
PV
FV
i
20
Finance School of Management
Solving Lump Sum Cash Flow for
Number of Periods
? ? ? ?
? ?
? ? ? ?
? ?i
PVFV
i
PV
FV
n
ini
PV
FV
i
PV
FV
iPVFV
n
n
n
?
?
?
?
?
?
?
?
?
?
?
?????
?
?
?
?
?
??
??
1ln
lnln
1ln
ln
1ln*)1(lnln
)1(
)1(*
21
Finance School of Management
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)
22
Finance School of Management
Example,Evaluate a Project
? A five-year savings
bond with face value
$100 is selling for a
price of $75,
? Your next-best
alternative for
investing is an 8%
bank account,
? Is the savings bond a
good project?
94.6$
75$06.68$
75$
08.1
100$
5
??
??
??NP V
20.1 1 0$
08.175$ 5
?
??FV
23
Finance School of Management
Example,Evaluate a Project
%92.51)75/100( 5/1 ???IR R
74.308.1ln
75
1 0 0
ln ??
?
?
?
?
?
?n
24
Finance School of Management
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 $9,000 in
four years,
? Should you borrow
from the bank or the
friend?
66.719$
12.1
000,9$
000,5$
5
??
?
??N P V
25
Finance School of Management
PV of Annuity Formula
? ?
i
iP M T
i
P M T
i
P M T
i
P M T
PV
n
n
?
??
?
?
??
?
?
?
?
)1(1
)1()1()1(
21
?
26
Finance School of Management
Example,Buying an Annuity
? You are 65 years old and
expect to live until age 80,
? For a cost of $10,000,an
insurance company will
pay you $1,000 per year for
the rest of your life,
? You can earn 8% per year
on your money in a bank
account,
? Does it pay to buy the
insurance policy?
? ?
52.440,1$
%8
%)81(1000,1$
000,10$
15
??
???
?
??
?
NP V
%56.5?i
21?n
27
Finance School of Management
Perpetual Annuities / Perpetuities
? Recall the annuity formula,
? ? ?
?
?
?
??
?
?
?
?? n
ii
p m t
PV
1
1
1*
? Let n -> ∞ with i > 0,
i
p m t
PV ?
28
Finance School of Management
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,
29
Finance School of Management
Calculator Solution
n i PV FV PMT Re sul t
3 9% 10 0,000 0? - 39,505.48
This is
the yearly
repayment
30
Finance School of Management
Amortization Schedule for 3-Year
Loan at 9%
Year
Be gin ni ng
Ba nlance
Total
Paym ent
In te rest
Paid
Pr incipal
Paid
Rem ai nin g
Ba la nce
1
2
3
100,0 00
69,4 95
36,2 44
39,50 5
39,50 5
39,50 5
9,000
6,255
3,262
30,50 5
33,25 2
36,24 4
69,49 5
36,24 4
0
31
Finance School of Management
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?
32
Finance School of Management
$10,000
$11,000 ￥
1,000,000￥
1,030,000￥
Time
10% $/$ (direct)
0.01 $/￥
3% ￥ / ￥
$/￥
U.S.A,Japan
33
Finance School of Management
$10,000
$11,124
$11,000 ￥
1,000,000￥
1,030,000￥
Time
10% $/$ (direct)
0.01 $/￥
3% ￥/￥
0.0108 $/￥
U.S.A,Japan
34
Finance School of Management
$10,000
$10,918 ￥
$11,000 ￥
1,000,000￥
1,030,000￥
Time
10% $/$ (direct)
0.01 $/￥
3% ￥ / ￥
0.0106 $/￥
U.S.A,Japan
35
Finance School of Management
$10,000
$11,000 ￥
$11,000 ￥
1,000,000￥
1,030,000￥
Time
10% $/$ (direct)
0.01 $/￥
3% ￥ / ￥
0.01068 $/￥
U.S.A,Japan
36
Finance School of Management
The Real Rate of Interest
i n f l a t i o n of R a t e
r a t e i n t e r e s t No m i n a lr a t e i n t e r e s t R e a l
?
???
1
11
i n f l a t i o n of R a t e
i n f l a t i o n of R a t e-r a t e i n t e r e s t No m i n a lr a t e i n t e r e s t R e a l
?
?
1
37
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?
38
Finance School of Management
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,
? If the $500 cost differential will increase over time with the
general rate of inflation,then the 5% rate of return is a real
rate of return,
? The conversion is not worthwhile unless the rate of inflation
is greater than 2.875% per year,
%875.205.1/)05.008.0( ??
%50 0 0,10$/5 0 0$ ??i
39
Finance School of Management
Taxes
r a t e i n t e r e s t t a x B e f o r er a t e T a x
r a t e i n t e r e s t t a x A f t e r
??
?
)1(
40
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?
%8.48)4.01( ???
?
%
r a t e i n t e r e s t t a x A f t e r