Chapter 2 – Time value of money
–
–
–
–
–
–
Interest: the cost of
money
Economic equivalence
Interest formulas –
single cash flows
Equal-payment series
Dealing with gradient
series
Composite cash flows.
Power-Ball Lottery
Decision dilemma – take a lump sum or
annual installments
A suburban Chicago couple
won the Power-ball.
They had to choose
between a single lump
sum $104 million, or $198
million paid out over 25
years (or $7.92 million
per year).
The winning couple opted
for the lump sum.
Did they make the right
choice? What basis do
we make such an
economic comparison?
Option A
(lump sum)
0
1
2
3
25
Option B
(installment plan)
$104 M
$7.92 M
$7.92 M
$7.92 M
$7.92 M
What do we need to know?
To make such comparisons (the lottery decision
problem), we must be able to compare the value
of money at different point in time.
To do this, we need to develop a method for
reducing a sequence of benefits and costs to a
single point in time. Then, we will make our
comparisons on that basis.
End-of-period convention
0
Beginning of
Interest period
0
1
End of interest
period
1
Methods of calculating interest
– Simple interest: rarely used; but the point is that
there are different types of interest
– Compound interest: what people generally mean
when they say “interest”
Investment variables
P Æ present value
F Æ future value
I Æ total interest
i Æ interest rate
N Æ number of interest periods
n Æ identifier of an interest period
Bn Æ balance at the end of interest period n
Compounding process
$1,080
$1,166.40
0
$1,259.71
1
$1,000
2
3
$1,080
$1,166.40
$1,259.71
0
1
2
3
$1,000
F = $1, 000(1 + 0.08)3
= $1, 259.71
Compound interest formula
n = 0:P
n = 1: F1 = P (1 + i )
n = 2 : F2 = F1 (1 + i ) = P (1 + i )
2
M
n = N : F = P (1 + i )
N
The fundamental law of engineering economy
(1 + i)N Æ compound amount factor
Practice problem: Warren Buffett’s
Berkshire Hathaway
– Went public in 1965: $18
per share
– Worth today (August 22,
2003): $76,200
– Annual compound growth:
24.58%
– Current market value:
$100.36 Billion
– If he lives till 100 (current
age: 73 years as of 2003),
his company’s total
market value will be ?
Market value
Assume that the company’s stock will continue to
appreciate at an annual rate of 24.58% for the next
27 years.
F = P(1 + i)N = $100.36(1 + 0.2458)27 = $37,902
$37.902 trillion
Practice problem
If you deposit $100 now (n = 0) and $200 two
years from now (n = 2) in an account that pays
10% interest, how much would you have at the
end of year 10?
Solution
F
0
1
2
3
4
5
6
7
8
9
10
$100(1+0.10)10 = $100(2.59) = $259
$100
$200
$200(1+0.10)8 = $200(2.149) = $429
F = $259 + $429 = $688
Practice problem
Consider the following sequence of deposits &
withdrawals over a period of 4 years. If you earn
10% interest, what would be the balance at the
end of 4 years?
$1,210
0
1
4
2
$1,000 $1,000
3
$1,500
?
?
$1,210
0
1
3
2
$1,000
$1,000
4
$1,500
$1,100
$1,000
$2,100
$2,310
$1,210
-$1,210
+ $1,500
$1,100
$2,710
$2,981
Solution
End of
Period
Beginning
balance
Deposit
made
Withdraw
Ending
balance
n=0
0
$1,000
0
$1,000
n=1
$1,000(1 + 0.10)
=$1,100
$1,000
0
$2,100
n=2
$2,100(1 + 0.10)
=$2,310
0
$1,210
$1,100
n=3
$1,100(1 + 0.10)
=$1,210
$1,500
0
$2,710
n=4
$2,710(1 + 0.10)
=$2,981
0
0
$2,981
Economic equivalence
What do we mean by “economic equivalence?”
Why do we need to establish an economic
equivalence?
How do we establish an economic equivalence?
Economic equivalence
– Economic equivalence exists between cash
flows that have the same economic effect and
could therefore be traded for one another.
– Even though the amounts and timing of the cash
flows may differ, the appropriate interest rate
makes them equal.
Equivalence in personal finance
F
If you deposit P dollars
today for N periods at i,
you will have F dollars at
the end of period N.
F = P(1+i)N
0
N
P ≡ F
P
Alternate way of defining equivalence
P
F dollars at the end of
period N is equal to a
single sum P dollars
now, if your earning
power is measured in
terms of interest rate i.
0
N
F
P = F (1+ i)− N
(1 +
i)-N
Æ present worth factor
0
N
Practice problem
At 8% interest, what is the equivalent worth
of $2,042 now, 5 years from now?
$2,042
0
If you deposit $2,042 today in an account
that pays 8% interest annually how much
would you have at the end of 5 years?
1
2
3
4
5
F
0
1
2
3
4
5
Solution
F = $2,042(1 + 0.08)5 = $3,000
Using interest tables: (F/P,8%,5) = 1.4693
$2,042 X 1.4693 = $3,000
At what interest rate
would these two amounts be equivalent?
$2,042
0
i=?
$3,000
5
Equivalence between two cash flows
Step 1: Determine the base
period, say, year 5.
Step 2: Identify the interest
rate to use.
Step 3: Calculate
equivalence value.
$2,042
$3,000
0
5
i = 6% , F = $2, 042 (1 + 0 .06 ) 5 = $2, 733
i = 8% , F = $2, 042 (1 + 0 .08 ) 5 = $3, 000
i = 10% , F = $2, 042 (1 + 0 .10 ) 5 = $3,289
Example - equivalence
Various dollar amounts that will be economically
equivalent to $3,000 in 5 years, given an interest rate
of 8%.
P=
$3,000
= $2,042
5
(1 + 0.08)
P
F
$2,042 $2,205
0
1
$2,382
$2,572
2
3
$2,778
4
$3,000
5
Example
V
$200
$150
$120
$100
$100
$80
0
1
2
3
4
5
0
1
2
3
4
Compute the equivalent lump-sum amount at n = 3 at 10% annual interest.
5
Approach
V
$200
$150
$120
$100
$100
$80
0
1
2
3
4
5
V3 = $511.90 + 264.46 = $776.36
V
$200(1 + 0.10)-1 + $100(1 + 0.10)-2
= $264.46
$200
$150
$120
$100
$100
$80
0
1
2
3
4
5
$100(1 + 0.10)3 + $80(1 + 0.10)2 + $120(1 + 0.10) + $150
= $511.90
Practice problem
How many years would it
take an investment to
double at 10% annual
interest?
2P
0
F = 2 P = P(1 + 0.10) N
2 = 1.1
log 2 = N log1.1
log 2
N=
log1.1
= 7.27 years
N
N=?
P
Rule of 72
Approximately how
long it will take for
a sum of money to
double
72
N≅
interest rate (%)
72
=
10
= 7.2 years
Practice problem
You just purchased 100 shares of stock at $60
per share. You will sell when the market price
has doubled. If you expect the stock price to
increase 20% per year, how long do you expect
to wait until selling?
Practice problem
$1,000
$500
Given: i = 10%,
A
0
Find: C that makes the
two cash flow streams
to be indifferent
1
2
C
C
3
B
0
1
2
3
Approach
Step 1: Select the base
period to use, say n = 2.
Step 2: Find the equivalent
lump sum value at n = 2
for both A and B.
Step 3: Equate both
equivalent values and
solve for unknown C.
$1,000
$500
A
0
1
2
C
C
3
B
0
1
2
3
Solution
$1,000
A
V2 = $500(1 + 0.10)2 + $1,000(+0.0)-1 =
$1,514.09
$500
A
0
B
1
2
C
C
3
V2 = C(1 + 0.10) + C = 2.1C
To find C:
2.1C = $1,514.09
C = $721
B
0
1
2
3
Practice problem
$1,000
At what interest rate
would you be
indifferent between the
two cash flows?
$500
A
0
1
2
3
$502 $502 $502
B
0
1
2
3
Approach
Step 1: Select the base period
to compute the equivalent
value (say, n = 3)
Step 2: Find the net worth of
each at n = 3.
$1,000
$500
A
0
1
2
$502
$502
3
$502
B
0
1
2
3
Establish equivalence at n = 3
Option A : F3 = $500(1 + i ) + $1, 000
3
Option B : F3 = $502(1 + i ) 2 + $502(1 + i ) + $502
– Find the solution by trial and error, say i = 8%
O ption A : F3 = $500(1.08) 3 + $1, 000
= $1, 630
O ption B : F3 = $502(1.08) 2 + $502(1.08) + $502
= $1, 630