Chapter 4: Time Value of
Money
Objective
Explain the concept of compounding
and discounting and to provide
examples of real life
applications
Copyright, 2000 Prentice Hall ©Author Nick Bagley, bdellaSoft, Inc.
1
Compounding: 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
20
Y ea rs
5
The Frequency of Compounding


Annual Percentage Rate (APR)
Effective Annual Rate (EFF): The
equivalent interest rate, if
compounding were only once a
year.
APR m
1  EFF  (1 
)
m
10
Effective Annual Rates of an APR
of 18%
Annual
Percentage
Rate
18
18
18
18
18
18
Frequency of Annual
Compounding Effective
Rate
1
18.00
2
18.81
4
19.25
12
19.56
52
19.68
365
19.72
11
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?
  APR 
EFF  Lim 1 

m 
m 

m

  1  e APR  1


12
The Frequency of Compounding
 APR 
1  EFF  1 

m 

1
APR
m
 1  EFF 
1
m

m

APR  m * 1  EFF   1
1
m
13
The Frequency of Compounding
Annual
Effective
Rate
12
12
12
12
12
12
12
Compounding Annual
Frequency
Percentage
Rate
1
12.00
2
11.66
4
11.49
12
11.39
52
11.35
365
11.33
Infinity
11.33
14
Annuities



A level stream of Cash Flows or
Payments
Immediate Annuity: The Cash Flows
start immediately.
Ordinary Annuity: The Cash Flows
start at the end of the current
period.
15
Derivation of PV of Ordinary
Annuity Formula
pmt
pmt
PV 


1
2
1  i  1  i 
pmt
pmt
pmt


3
n 1
n
1  i 
1  i 
1  i 
16
Derivation of PV of Ordinary
Annuity Formula
1
1
PV  pmt *{


1
2
1  i  1  i 
1
1
1


}
3
n 1
n
1  i 
1  i 
1  i 
17
PV of Ordinary Annuity Formula
1
pmt *{1 
}
n

1 i
PV 
i
pmt 
1 


* 1 
n 
i  1  i  
18
Annuity Formula: PV of Immediate
Annuity
PVimm  PVord * (1  i )
pmt
n

*{1  1  i  } * (1  i )
i
pmt
1 n

*{(1  i )  1  i  }
i
19
Derivation of FV of Annuity
Formula
pmt 
1 
 (ord. annuity)
PV 
* 1 
n 
i  1  i  
FV  PV * 1  i  (lump sum)
n
pmt 
1 
n
 * 1  i 
FV 
* 1 
n 
i  1  i  


pmt
n

* 1  i   1
i
20
Perpetual Annuities / Perpetuities

Recall the annuity formula:
pmt 
1 

PV 
* 1 
n 
i  1  i  
• Let n -> infinity with i > 0:
pmt
PV 
i
21
Alternative Discounted Cash Flow
Decision Rules
1.
NPV rule: the NPV is the difference
between the present value of all
future cash inflows minus the
present value of all current and
future cash outflows. Accept a
project if its NPV is positive.
22
DCF rules
Example: You have the opportunity to
buy a piece of land for $10,000. You
are sure that 5 years from now it
will be worth $20,000. If you can
earn 8% per year by investing your
money in bank, is this investment in
the land worthwhile?
23
NPV rule solution
20,000
NPV  $10,000 $
5
1.08
 $10,000 $13,612
 $3612 0
24
Alternative Discounted Cash Flow
Decision Rules
2.
FV rule: Invest if the future value
of the investment is larger than
the future value that can be
obtained from the next best
alternative.
25
FV rule solution
FV  $10,0001.08
5
 $14,693 $20,000
26
Alternative Discounted Cash Flow
Decision Rules
3.
IRR rule: The IRR is the discount
rate at which the NPV is zero.
Invest if the IRR is greater than
the opportunity cost of capital.
27
IRR rule solution
$20,000
$10,000 
5
(1  i )
i  14.87%  8%
28
Alternative Discounted Cash Flow
Decision Rules
4.
Choose the investment alternative
with fastest payback.
29
Payback rule solution
$20,000
$10,000 
n
1.08
n95
30
Loan Amortization

The process of paying a loan principal
gradually over its term
Example:
$100,000 mortgage loan, APR: 9%,
repaid in 3 annual installments

pm t 
1
pmt=?

PV 
*
1
n 

i
1  i  

pmt=$39504.48 100000  pmt * 1  1 3 
0.09  1.09 


31
Loan Amortization

First Year:
Interest:
(0.09)(100000)=9000
pmt:
39504.48
principal:
30504.48
Outstanding Balance:
69494.52
32
Loan Amortization

Second Year:
Interest: (0.09)(69494.52)=6254.51
pmt:
39504.48
principal:
33250.97
Outstanding Balance:
36243.54
33
Loan Amortization

Third Year:
Interest:(0.09)(36243.54)=3262
pmt:
39504.48
principal:
36244
Outstanding Balance:
0
34
Amortization of Principal
450000.00
Outstanding Balance
400000.00
350000.00
300000.00
250000.00
200000.00
150000.00
100000.00
50000.00
0.00
0
24
48
72
96
120 144 168 192 216 240 264 288 312 336 360
Months
35
Percent of Interest and Principal
100%
90%
80%
Percent
70%
% Interest
60%
50%
40%
% Principal
30%
20%
10%
0%
0
24
48
72
96
120 144 168 192 216 240 264 288 312 336 360
Months
36
Computing NPV in Different Currencies
In any TVM calculation, the cash flows
and the interest rate must be
denominated in the same currency.
41
Inflation and Future Values
Example: At age 20 you save $100
and invest it at a dollar interest rate
of 8% per year, and you do not
take it until age 65. If the inflation
is estimated 5% per year, how
much will you have accumulated in
the account at that time in terms of
real purchasing power?
42
Solution 1:
1  nom. rate(r )
1  real rate( R ) 
1  Inflation(i )
1  R  1.02857
real FV  PV (1  R)
n
real FV  $1001.02857
 $355
45
43
Solution 2:
nom. FV in 45 years  $100 1.08
 $3,192
45
price level in 45 years  1.05
 8.985
nom.FV $3,192
real FV 

FPL
8.985
 $355
45
44
Inflation and Present Values
Example: Your daughter is 10 years
old, and you are planning to open
an account to provide for her
college education. Tuition for a year
of college is now $15,000. How
much must you invest now in order
to have enough to pay for her first
year’s tuition 8 years from now, if
you think you can earn a rate of
interest that is 3% more than the
inflation rate of 5%?
45
Solution:
$15,000
PV 

$
11
.
841
8
1.03
Wrong:
$15,000
PV 

$
8
.
104
8
1.08
46
Inflation and Present Values
Never use a nominal interest rate
when discounting real cash flows or
a real interest rate when
discounting nominal cash flows.
47