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Intermediate Financial Management

Chapter Two
THE TIME VALUE OF MONEY – Conventions & Definitions
Introduction
Now, we are going to learn one of the most important topics in finance, that is, the time
value of money
...
Hence, it is a good idea to spend a fair
amount of time in learning the concepts
...
The conventions used in the study of time value of money
2
...
You will
seldom find any transaction either in the real world or in the academics that is
based on the simple rate of interest
...

Otherwise the mathematical foundations and the resultant applications are almost
impossible to deal with mathematically
...

In the simple rate of interest we will learn
a
...
Present value of an amount
c
...
Present value of an annuity
3
...
The topic that will be covered can be broadly categorized as in two
main categories
a
...
The time value concepts under a series of payments case
In lump sum case we will learn
a
...
Future value of an amount
c
...
Finding the unknown time period
Under the series of payments topic we will learn present and future value of a
series of payments including future and present value of annuities and annuity

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Intermediate Financial Management

dues
...
All of the above concepts will be dealt with under annual
compounding, compounding ‘m’ (m>1) times per year and continuous
compounding
...
Special topics
Under this we will discuss the concept of time value of money for such topics but
not limited to cash flows growing at a constant rate (under the constant growth
rate) and/or under constant level increments, when compounding and deposit
intervals are different etc
...

For example,( READ CAREFULLY)
Whether the problem tells you or not, if it is not mentioned that whether the
deposits (withdrawals) are made at the end or the beginning of the period; it
(deposits or withdrawals) is always assumed to be done at the end of the period
...
Hence, it will always be assumed that the deposits are made at
the end of the period
...

Having said that we enumerate the conventions as follows:
1
...
Time Line
Time line is simply a straight line with numbers 0,1,2,3…n written under it
...
The beginning of period 3 will be at point 2,
which is the end of period 2
...
Similarly, the point n is the end of
period n
...

For example, in the above diagram the interval between one and two should be read as
during period 2
...
Note that in number 2 we did not specify any particular “time unit” for the word period
...
The general tendency is that when somebody mentions the
word “period”, the students usually assume the period as one year long
...

4
...

For example, if you call a mortgage broker to ask him/her the current rate of interest
on a 30 year mortgage
...
The very statement “rate of interest 4%” means that it is the
annual rate
...
For
example, if you borrow $100 from Mr
...
Then it is, of course, 9 percent per week
...
In the computational formula (will come later), the rate of interest always enters in
fractions
...
10
0
...
005
If you are confused as what fraction to use, just divide the annual percentage rate by
100
...
2% will be equal to 0
...
002 etc
...
However, unlike the above convention (number 5), in case of using calculator the rate
of interest is entered as percent
...
There are some specific
conventions related to a particular topic
...

Before we start discussing the actual computational procedure we introduce the following
two definitions
...


7
...

First, by convention, the future value of an annuity is computed just after the last
deposit unless stated otherwise; and second, the number of interest earning periods is
one less than the number of deposits
...

In case of the annuity-due the number of interest earning periods is the same as the
number of deposits
...
For example, suppose one is going to make n deposits over an n-year
period, and wants to compute the FV of the account at the end of year n
...
However, if all the
deposits are made in the beginning of the year, then the FVAD formula should be used
because the FV is computed one year after the last deposit
...
While this argument is not invalid, the
application of the formula should be based on the timing of the FV computation rather
than the timing of the deposits
...
The interest earned at the end of the period on the principal will not earn
interest in any of the subsequent periods
$100

$110

0

1

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3

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Intermediate Financial Management

Consider the above time line
...
At the end of period 1, you will have $110 in the account
...
10 x $100 = $ 10
Total amount = $100 + $10 = $110
Under the simple rate of interest in the successive period only the principal ($100) will
earn the interest
...
e
...
Therefore, the total amount in the account at the end of the period
2 will be $120
...
e
...
10 x $100 = $ 10
Interest earned in period 2 = 0
...

For example:
In the above diagram at the end of period 1, the account will have $110 under both
approaches (simple interest and compound interest)
...

Break down is as follows:
Principal $100
Interest earned in period 1 = 0
...
10 x $100 + 0
...
The difference of $1 between compound interest and simple
interest approach is due to interest of $1 earned in period 2 on $10 interest that was
earned in period 2
...


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Intermediate Financial Management

For example,
“Annual compounding” means that the interest earned during the year also starts earning
interest in the next period, that is, at the end of one year
...

Definition 4
Unless stated otherwise, a “calendar time period” will be defined as follows:
1 year = 12 months
1 year = 365 days
1 year = 52 weeks
1 year = 4 quarters
1 quarter = 3 months
1 month = 4 weeks
1 month = 30 days
Note that the above calendar definition of time period is a standard
...

For example,
If month = 4 weeks and year = 12 months, then ideally there should be 48 weeks in a
year
...

Definition 5
In the process “compounding” we move from present to future
...

Having equipped ourselves with these conventions and definitions we now proceed to the
topic of simple rate of interest
...
What will be the future value of these P
dollars if left in the account for n years? The mathematical basis can be developed with
the help of the following diagram:
Deposit $P
Year

0

1

2

3

4 …

n-1

n

Today

Remembering that with simple interest only the principal (P, in this case) earns interest,
we have the following formulas for interest earned at the end of
Year 1 =
Year 2 =
Year 3 =

...


kP
2kP
3kP

Year n = nkP
Therefore, the principal P will grow to the future value (FV) as follows:
FV = P + nkP = P(1+ nk)

(3
...


What if the interest rate does not stay the same over time? For example, the simple
interest rate is k1, for the first n1 years, k2 for the next n2 years, k3 for the next n3 years,
etc
...
This implies that the future value
under changing simple interest rates is
FV = P(1 + n1k1 + n2k2 + n3k3 +
...
1: Suppose an account earns 15 percent simple interest per year
...
15 left for 5 years?
(c) $1 left for 2,000 years?
Answers
(a)

P = 15, k =
...
15 ×10) = $37
...
15, k =
...
15(1 +
...
2625
(c) P = 1, k =
...
15 × 2,000) = $301

Example 3
...
What will
be the future value of the account if:
(a) the annual simple interest rate is 7% for the first 5 years, 10% for the next 10 years,
and 12% for the last 5 years?
(b) 5% for the first 10 years, 10% for the next 10 years, 15% for the last 10 years?
Answers:
(a)

P = $1,000, k1 = 7%, n1 = 5
k2 = 10%, n2 = 10
k3 = 12%, n3 = 5
FV = $1,000(1 +
...
10 × 10 +
...
95) = $2,950
...
05 × 10 +
...
15 × 10)
= $1,000(4) = $4,000
...
1)
...
Thus, solving for P
and calling it instead the present value, PV, we have
PV =

F
1 + nk

(3
...


The present value (PV) of an amount F received at the end of n years
with a simple rate of interest k is given by
PV =

F
1 + nk

The formula in the box assumes that rate of interest k will remain constant in each of
the periods
...
etc
...


Example 3
...
What
will the present value be of:
(a) $37
...
2625 to be received at the end of 5 years?
(c) $301 to be received at the end of 2,000 years?
Answers:
(a) F = 37
...
15, n = 10

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Intermediate Financial Management

PV =

37
...
15 ×10

(b) F = 0
...
15, n = 5
PV =

0
...
15
1 +
...
15, n = 2000
PV =

301
= $1
...
15 × 2000

One easily can see that the answers correspond to the original principal balances in the
previous example on future value
...
Below we
discuss the procedure to obtain the present and future value of series of payments
...

Consider the following time line
...
As per the
convention stated above, the above statement “Deposits of $10 are made each year for six
years”
...
The deposits are of the same amount, known as level deposits in each period
2
...

For example,
Every year, every month, every ten minutes, etc
3
...

Note that in the simple rate of interest the question of the above statement (number 3)
doesn’t arise
...
The mathematical
formula can be obtained with the help of the following diagram
...
n - 1

A
n

Applying formula (3
...
3)

Intermediate Financial Management

The future value of an annuity (FVA) of n deposits made at the beginning or the end
of each year at simple rate of interest k is given by
⎡ (n − 1)k ⎤
FVA = nA⎢1 +
2 ⎥⎦
⎣

Another commonly used terminology, the future value of an annuity due, is computed
one period after the last deposit is made
...
4)

⎡ (n + 1)k ⎤
= nA⎢1 +
2 ⎥⎦
⎣

Example 3
...

What will the future value be of an annual deposit of:
(a) $15 at the end of each year for four years?
(b) $15 at the beginning of each year for four years?
Since the number of deposits is not too large, we can obtain the answer in two
ways-either directly or by using the expression (3
...


Answer: Direct method:
15

15

15

15

Amount deposited
End of year

0
1
2
3
4
-------->------ >------- >------- >

FVA = 15(1 +
...
15 × 2) + 15(1 +
...
75 + 19
...
25 + 15 = $73
...
3),
Number of deposits = n = 4
Number of interest earning periods = (n – 1) = 3
A = 15; k =
...
15 ⎤
= (4)(15) ⎢1 +
2 ⎥⎦
⎣
= $73
...
15 × 3) + 15(1 +
...
15) + 15
Using expression (3
...
15 ⎤
FVA = nA⎢1 +
= (4)(15) ⎢1 +
= $73
...

⎥
2 ⎦
2 ⎥⎦
⎣
⎣

Notice that in computing the future value of an annuity it does not matter whether the
deposits are made at the end or the beginning of the period as long as the number of
deposits remains the same
...
Let an amount A be deposited at the end of each year in an account that earns
simple interest k annually
...
n -1

n

Applying formula (3
...
5)

No further simplification is possible
...
If the deposits are made at the beginning of each year, the
formula for a total of n deposits would be
PVAe = A +

A
A
A
+
+ ⋅⋅⋅ +
(1 + k ) (1 + 2k )
[1 + (n − 1)k ]

(3
...

As stated earlier, the only thing simple about the simple rate of interest is its name
...

For example,
The mortgage payments on your house or lease payment on your car is computed on the
basis of compound rate of interest
As you see the formula 2
...
5 are not capable of any further simplification
...

This statement will be clear to you when we deal with compound rate of interest problem
in the next section
...


It should be noted that PVAe is simply called the present value of an annuity,
whereas PVAb is sometimes referred to as the present value of an annuity due
...
6% per year at the end
of the 5th year
...
076 )(5)] = $13
...
If the simple rate of interest is 8
...
083)]

PV = $57
...
5% per year?
⎛ 260 ⎞
⎜
⎟ −1
100 ⎠
⎝
n=
= 320 years

...
03448 = 3
...
If k = 8% per year
...
08) ⎤
FVA = (10)(20)⎢1 +
⎥⎦
2
⎣
FVA = $352 + [(10)(20)(1 + (0
...
If the rate of interest is 6%,
obtain the present value of this annuity as well as annuity due:
PVA =

8
8
8
8
+
+
+
...
06 1 + 2(0
...
06)
1 + 20(0
...
+
1 + 0
...
06) 1 + 3(0
...
06)

7) In how many years will $100 become $320 if k = 11%?
⎛ 320 ⎞
− 1⎟
⎜
100 ⎠
⎝
n=
= 20 years
0
...
If however, if you can do the following problems without looking at the solutions
consider yourself well versed in the time value of money under the simple rate of interest
...
Assuming that you make the first deposit today what will be
the future value of the deposits in 100 years
...
Furthermore, you are making 50 deposits in
an account beginning today, i
...
, at point 0
...
)

© Arun J Prakash

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Intermediate Financial Management

As noted earlier, the topic of simple rate of interest is of academic interest only
...


© Arun J Prakash

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Intermediate Financial Management

Chapter Four
THE TIME VALUE OF MONEY WITH ANNUAL COMPOUNDING
By convention, the interest rate is always quoted in annual terms, but the
compounding interval can be of any length of time
...
The mathematical formulas that follow determine future and present values with
compounded rates that assume ANNUAL COMPOUNDING
...
What
will its future value be at the end of n years? The mathematical formula can be developed
as follows:
P
Deposit
Year

0

1

2

3

4

……

n

Since the interest earned on principal P together with the principal starts earning interest
k at the end of each year, we have the following situation:
Principal at the end of the year:
0 (today)
1
2
3

...


...

Therefore, the future value of P is
FV = P(1 + k)n
...
1)

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Intermediate Financial Management

The future value (FV) at the end of n years of an amount P deposited today at an
annual rate of interest k compounded annually is
FV = P(1 + k)n
...
1: Suppose at the time of your birth, 25 years ago, your father deposited
$1,200 in an account at an annual interest rate of 15 percent
...
15, n = 25
FV = 1,200(l +
...
74
...

Example 4
...

Today is your thirty-fifth birthday
...
15)18 = $14,850
...

Since half of that must be withdrawn:
Amount withdrawn = 14,850
...
27
...
35

Intermediate Financial Management

The future value of the remaining balance of $7,425
...
27(l +
...
29
...
The last 17 years accumulate nearly
5 times as much as the first 18 years
...
3: Suppose Mr
...
, to take delivery on a
car in four years for $100,000
...
Mr
...
He made a promise to himself that he
would withdraw half the accumulated amount from this account each year
...
How much
money must Mr
...
)
Answer: The amount remaining at the end of each year withdrawal is shown below:
Year

0

1

2

3

4

Amount
deposited A

A(1 + k ) 2 1
×
2
2

A(1 + k )
2

A(1 + k ) 3 1
×
4
2

A(1 + k ) 4 1
× = $66,000
8
2

Therefore,
A(1 + k ) 4
= 66,000
8

or
A=

66,000 × 8
= $301,885
...
15)4

Thus Mr
...

Present Value of an Amount

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Intermediate Financial Management

In expression (4
...
If we solve
equation (4
...
We can write this as
PV =

F
(1 + k )n

(4
...

Example 4
...
, will need $15,000 to enter college on his eighteenth birthday
...
15
...
08
(1 +
...
5: The Roles Rice, Inc
...
Smith, who wanted to
buy a customized car
...

(b) An initial down payment of $34,000 will be required
...
71 in a joint account (Roles Rice and Mr
...

(d) Mr
...

In the fourth year, the company will withdraw the remaining amount
...
Smith accepted the offer
...
Thus the amount of
money each year before the withdrawal by Mr
...
71

=

1
2 × 4F
(1 + k ) 3

8F
(1 + k ) 4

2

3

4

2 × 2F
(1 + k ) 2

2F
(1 + k )

F

(Read from right to left)
Therefore,
F=

$301,885
...
15) 4
= $66,000
8

Again, this problem just reverses the question asked of the future value problem
...
Sometimes, we are interested in obtaining the number of
years needed to reach a given target with an initial deposit or the annual interest rate that
must be received for the initial deposit to grow to a given target
...

Finding the Unknown Period
Suppose a given amount P is deposited today in an account with an annual rate of
interest k
...
Then, using expression (4
...
3)

At an annual interest rate k, the number of years required for an amount P to grow
to a target level, T, is given by
n=

ln( F / P)

...
In order to solve let us again consider
equation 4
...

P=

F
(1 + k )n

or

(1 + k )n = F

P

or
1n

(1 + k ) = ⎛⎜ F ⎞⎟
⎝P⎠
or
1n

⎛F⎞
k =⎜ ⎟
⎝P⎠

© Arun J Prakash

−1

(4
...
We developed equation (4
...

What did we do?
We knew the value of P, k and n and obtained F
...
Therefore, if 3 of them are
known one can solve for the unknown as follows
...

b) Known F, k, n, what is P?
P=

F
(1 + k )n

c) Known F, P, k, what is n?
n=

ln(F P )
ln(1 + k )

d) Known F, P, n, what is k?
1n

⎛F⎞
k =⎜ ⎟
⎝P⎠

−1

The above derivation tells you how simple it is to deal with time value of money
problems if you understand what you are doing
...

An interesting problem that is often found in basic finance textbooks is that of
finding how many years it will take for an initial deposit to double at an annual interest
rate k
...

Applying formula (4
...

ln (1 + k ) = k −

k2 k3
+ + ⋅⋅⋅
2
3

= k (taking the final approximation)

Therefore,
ln 2
k

...
31472
=
100k
69
n=
+ a correction factor
100k
n=

Since we have approximated ln(1 + k) with a Taylor series expansion
...
35
...
35
100k

(2
...
This is
known as the rule of 69
...
35
100k

years where100 k is the rate of interest in percentage
...

k=

100(n −
...


It should be noted that the value of n, obtained using the rule of 69, is
underestimated for low interest rates, almost the same for intermediate values of the
interest rate, and overestimated for higher values of the interest rate
...

Example 4
...
005,
...
05,
...
15,
...
25,
...
40, and
...
12) and the rule of 69, the exact and approximate values of n
for various values of k are obtained as follows:

k

Exact n using
equation (2
...
005

...
05

...
15

...
25

...
40

...
98
69
...
21
7
...
95
3
...
11
2
...
06
1
...
12)
overestimated (+)
138
...
35
14
...
25
4
...
80
3
...
65
2
...
73

27/29

-0
...
31
-0
...
02
0
0
0
+
...
015
+
...


Rule of 72: The time required for an initial amount to double at annual interest
rate k is approximately
n=

72
100k

years
...
03874 In (I + k) Overestimate
k
> 1
...
The following example illustrates this
...
7: Using the rule of 72, obtain n for the various values of k given in the
previous example and compare the results
...
11)

Approximate n
using
rule of 72

138
...
66
14
...
27
4
...
80
3
...
64
2
...
71

144
...
00
14
...
20
4
...
60
2
...
40
1
...
44


...
01

...
10

...
20

...
30

...
50

Underestimated or
overestimated
+5
...
19
-
...
15
-
...
23
-
...
26
-
...
Another
variation of this type of problem is to ask how many years it will take for an amount to

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Intermediate Financial Management

triple at an annual interest rate k
...
11); however, a quick answer can be obtained
by using the rule of 110
...
52
100k

years
...
01 is almost negligible, as can be
seen from the following example
...
8: In how many years will $100 grow to $300 at annual rates of interest of

...
01,
...
05,
...
15,
...
25,
...
35,
...
50,
...
90,
...
99, and 1
...
11)

Approximate n
using
rule of 110


...
01

...
05

...
15

...
25

...
35

...
50

...
90

...
99
1
...
27
110
...
48
22
...
52
7
...
03
4
...
18
3
...
27
2
...
96
1
...
64
1
...
58

220
...
52
55
...
52
11
...
85
6
...
92
4
...
66
3
...
72
1
...
74
1
...
63
1
...
25
+
...
04
0
0
-
...
01
-
...
01
0
0
-
...
01
-
...
02
-
...
04



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