TIME VALUE OF
MONEY
Upon completion of this chapter, students should be able to :

Understand and use the concept of time value of money.

Represent the cash flows occurred in different time period using the cash
flow time line.

Calculate the present value and future value of given streams of cash flows.

Identify the impact of time period and required rate of return on present
value and future value.

Prepare amortization schedule for amortized term loan.

Compare various types of interest rates.
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BUSINESS FINANCE
The concept of time value of money suggests that the money received at
different point of time has different value. The financial manager must
appreciate this fact and understand why they are different and how they are
made comparable. Therefore, the basic objective of this chapter is to enable
the student to calculate present and future value of cash flows and apply
these concepts in addressing real life problems.
This chapter begins with fundamental concepts of present value and future
value and explains how they are calculated. Then it presents how the pattern
of cash flows and required rate of return impact the present value and future
value. Finally, different concepts related to interest rates have also been deat
on for their proper uses by the students.
CONCEPT
Time value of money
A concept to
understand the value
of cash flow occurred
at different period
ime value of money is a concept to understand the value of cash flows occurred at
different point in time. If we are given the alternatives whether to accept Rs 100
today or one year from now, then we certainly accept Rs 100 today. It is because
there is a time value to money. Every sum of money received earlier has
reinvestment opportunity. For example, if we deposit Rs 100 today in saving account
at 5 percent annual rate of interest, it will increase to Rs 105 at the end of year one.
Money received at present is preferred even if we do not have reinvestment
opportunity. The reason is that the money that we receive at future has less
purchasing power than the money that we have at present due to the inflation. What
happens if there is no inflation? Still, many received at present is preferred. It is
because most of us have a fundamental behaviour to prefer current consumption to
future consumption; money at hand allows current consumption. Thus, (i) the
reinvestment opportunity or earning power of the money, (ii) the (risk of) inflation
and (iii) an individual's preference for current consumption to future consumption
are the reasons for the time value of money.
The concept of time value of money is useful in addressing our real life
problems relating to planning for future family expenditure. For instance, if we need
Rs 500,000 after the retirement from job in 15 years, the amount we need to deposit at
an interest rate every year from now until the retirement is conveniently determined
by using the time value of money concept.
TIME VALUE OF MONEY
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Chapter 3
61
Many financial decisions of a firm require a consideration regarding time
value of money. In chapter one, we argued that a corporate manager must always
concentrate on maximizing shareholders wealth. Maximizing shareholders wealth, to
a larger extent, depends on the timing of cash flows from investment alternatives. In
this regard, time value of money concept deserves serious considerations on all
financial decisions. In the following sections, we present some concepts and
techniques to understand time value of money and apply them in financial decision.
SIGNIFICANCE OF THE CONCEPT OF TIME VALUE OF MONEY
Time value of money is a widely used concept in literature of finance. Financial
decision models based on finance theories basically deal with maximization of
economic welfare of shareholders. The concept of time value of money contributes to
this aspect to a greater extent. The significance of the concept of time value of money
could be stated as below:
Investment Decision
Investment decision is concerned with the allocation of capital into long-term
investment projects. The cash flows from long-term investment occur at different
point in time in the future. They are not comparable to each other and against the
cost of the project spent at present. To make them comparable, the future cash flows
are discounted back to present value.
The concept of time value of money is useful to securities investors. They use
valuation models while making investment in securities such as stocks and bonds.
These security valuation models consider time value of cash flows from securities.
Financing Decision
Financing decision is concerned with designing optimum capital structure and
raising funds from least cost sources. The concept of time value of money is equally
useful in financing decision, specially when we deal with comparing the cost of
different sources of financing. The effective rate of interest of each source of
financing is calculated based on time value of money concept. Similarly, in leasing
versus buying decision, we calculate the present value of cost of leasing and cost of
buying. The present value of costs of these two alternatives are compared against
each other to decide on appropriate source of financing.
Besides, the concept of time value of money is also used in evaluating
proposed credit policies and the firm's efficiency in managing cash collections under
current assets management.
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CASH FLOW TIME LINE
Cash flow time line
A graphical
presentation of cash
flows at different time
period
Cash flow time line is an important tool used to understand the timing of cash
flows. It is a graphical presentation of cash flows occurring in different time periods,
and is helpful for analyzing the time value of cash flows. To gain an idea about
timing of cash flows, let us consider the following time line:
3
2
1
0
5
4
Time
The time line represents the time period stated above the vertical scale. Time
zero represents today or just now or at the beginning of period 1. Zero states that the
time period just begins from this point. Time 1 denotes the end of period one; time 2
denotes the end of period two, and so on. However, it should be noted that the end
of any period also means the beginning of the succeeding period. For example, time
1 states that the period one has just been ended and period two has just began. Time
period denoted in the scale has generally a length of one year from 0 to 1, from 1 to 2,
from 2 to 3 and so on. However, it could be for six months or three months or one
month depending on the period for compounding or discounting used.
The corresponding cash flows are placed below the scale as shown in the
following time line of cash flows:
1
2
3
4
5
10
50
70
100
90
0
Time
Cash Flows -100
Note that Rs 100 in time zero has negative sign. The negative sign represents
the cash outflows, which means that Rs 100 is deposited or paid or cost incurred at
time zero. All other cash flows in time 1, 2, 3, 4 and 5 have positive signs. Positive
sign is used to denote the cash inflows, which means a cash receipt in the given time
periods correspondingly.
The time line of cash flow is also used to denote the interest rate that each cash
flow earns. Let us consider the following time line.
Time
0
8%
1
2
3
4
5
Cash flow -100
The interest rate is placed in between two corresponding time periods. The
interest rate 8 percent placed in between the time zero and one denotes that Rs 100
invested today will earn 8 percent interest in year 1 so that it grows to Rs 108 at the
end of year one. Similarly, Rs 108 at the beginning of year two earns 8 percent
interest during the year two so that it grows to Rs 116.64 at the end of year two and
so on. If the interest rate for every period is similar, it is not necessary to show in
between of every time period in the scale. However, if the interest rates differ from
year to year, it should be stated in between every time period.
TIME VALUE OF MONEY
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Chapter 3
63
FUTURE VALUE AND COMPOUNDING
Future value
A present sum of
money plus stream of
interest amount
received during
investment period
Compounding
A process of finding
future value
Future value of a sum of money is defined as the total of the sum of the money plus
the stream of interest amount received for the period, the money was invested. The
process of finding future value is called ‘compounding’. Compounding is the
process of determining the future value of a cash flow or series of cash flows when
compound interest is used. For instance, if we invest Rs 1,000 today in a security at
10 percent annual interest rate for two years, we receive Rs 100 (that is 10 percent of
Rs 1,000 original investment) interest during year one so that we will end up with Rs
1,100 at the end of year one. Again, we receive Rs 110 on our investment of
Rs 1,100 in year 2 (that is 10 percent of Rs 1,100) interest at the end of year two plus
Rs 1,100 investment during the year two, so that our original investment Rs 1,000
grows to a total of Rs 1,210 at the termination of year two. Here, Rs 1,210 at the end
of year two is regarded as the future value of Rs 1,000 today compounded at 10
percent annual rate for two years. The following time line shows it:
Time
Cash flow
Interest income
Year-end amount
0
10%
1
2
-1,000
100
1,100
110
1,210
The future value of a sum of money compounded at 'i' percentage annual rate
of interest for ‘n’ year is given by the equation (3.1):1
FVn = PV (1 + i)n
(3.1)
Where,
FVn = future value of a sum of money at the end of period n.
PV = present value or the sum of money today.
i
= the annual rate of interest at which the sum of money is invested.
n
= the number of years for which the sum of money is compounded.
Using equation (3.1), the future value of the sum of Rs 1,000 compounded at 10
percent annual rate for 2 years is given by:
FV2 = PV (1 + i)2 = Rs 1,000 (1 + 0.1)2 = Rs 1,000 x 1.21 = Rs 1,210.
Tabular Solution
Besides equation (3.1), the future value of a sum of money also could be calculated
by using future value interest factor (FVIF) table (in the appendix). It is given by the
equation (3.2) as follows:
1
This equation is derived as follows
FV of PV at i percent for 1 period is
FV1 = PV + INT (where INT is the amount of interest)
= PV + PV (i)
= PV (1 + i)
FV of PV at i percent for 2 periods is
FV2 = PV1 (1 + i)
= PV (1 + i) (1 + i)
= PV (1 + i)2
Accordingly, FV of PV at i percent for n periods is FVn = PV(1 + i)n
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BUSINESS FINANCE
FVn = PV (FVIF i, n)
(3.2)
In our example, looking at FVIF table at 10 percent for 2 years the FVIF factor
is 1.21 so that future value of the sum of Rs 1,000 compounded at 10 percent annual
rate of interest for 2 years is given by:
FV2 = PV (FVIF 10%, 2) = Rs 1,000 x 1.21 = Rs 1,210
Graphic View of Compounding Process
Future value of a sum of money has positive relationship with the interest rate and
the time period. This means, larger the interest rate larger will be the future value of
a present sum of money. This relationship also holds with time period, that is, longer
the time period larger will be the future values. This relationship is shown in Figure
3.1.
FIGURE 3.1
Relationship between
future values and
interest rates over
different time periods
Future value of Rs 1
4.0
3.0
i = 15%
2.0
i = 10%
i = 5%
1.0
i = 0%
Time periods
0
2
4
6
8
10
The Figure 3.1 shows how a sum of rupee one will grow at different interest
rates to different time periods in the future. It is observed from the upward sloping
curves that value grows over the time in future. Similarly, the growth in value is
larger at higher rate of interest. The interest rate itself is the rate of growth in value.
For example, if we invest rupee one at 10 percent annual rate of interest, the value of
investment grows at the rate of 10 percent every year. The growth in value is larger
at later years because of the effect of compounding.
PRESENT VALUE AND DISCOUNTING
Present value
The value of future
sum of money
We already mentioned that Rs 100 that we have at present has more value than Rs
100 received at future dates. It means the same amount received at two different
dates are not comparable. To make them comparable, we need to discount the future
value. The discounted value of the future sum is the present value. In other words,
the present value is the value today of a future cash flow or a series of cash flows.

TIME VALUE OF MONEY
Discounting
A process of finding
present value
Chapter 3
65
The present value of a future sum of money is the amount of current money
that is equally desirable to a decision maker today against a specified amount of
money to be received or paid at a future date, given the certain rate of interest. In our
future value calculation, we recognized that Rs 1,000 invested at 10 percent annual
rate of return would grow up to Rs 1,210 in two years from now. In this example, Rs
1,000 today is called the present value of a future sum of Rs 1,210 after two years
discounted back at 10 percent rate of interest. The process of finding present value of
future cash flows or series of cash flows is called ‘discounting’. Discounting is just
reverse of compounding. Let us consider the following time line to understand the
discounting process.
Time
Cash
Flow
0
10%
-1,000
1.10
1
2
1,100
1,210
1.10
The above time line of cash flows shows that Rs 1,210 at the end of year two
divided by 1.10 two times produce Rs 1,000 present value. The present value of a
future sum of money due in n years is calculated by using the equation (3.3) as
follows:2
FVn
PV = (1 + i)n
(3.3)
Let us suppose that we are offered the alternative of either Rs 1,331 after three
years or a specified sum of money today. Assuming that any sum of money today
could be invested at 10 percent rate of return (i.e required rate of return is 10
percent). The present value of the future sum of Rs 1,331 after three years discounted
back at 10 percent required rate of return is given by:
Rs 1‚331
PV = (1.10)3 = Rs 1,000
Tabular Solution
Besides the above equation (3.3), the present value of a sum of money also could be
calculated by using present value interest factor (PVIF) table (given in the appendix).
The relationship is given in equation (3.4) as follows.
PV = FVn (PVIF, i, n)
(3.4)
Looking at the present value interest factor table at 10 percent for 3 years the
PVIF factor is 0.7513, so that the present value of the sum of Rs 1,331 to be realized at
the end of year 3 discounted at 10 percent rate of interest is given by:
PV = Rs 1,331 (PVIF 10%, 3) = Rs 1,331 x 0.7513 = Rs 1,000
2
This equation is derived as follows:
FVn = PV(1 + i)n
FVn
PV =
(1 + i)n
..... (3.1)
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BUSINESS FINANCE
Graphic View of Discounting Process
The present value of a sum of money has inverse relation with the time period and
interest rate. As the time period to receive a sum of money increases, the present value of
the sum of money will decline. This relationship also holds with interest rate, that is,
larger the interest rate lower will be the present value of a future sum. This relationship is
depicted in Figure 3.2
FIGURE 3.2
Relationship between
present values and
interest rates over
different time periods
Present Values of Rs 1
1.00
i = 0%
0.75
i = 5%
0.50
i = 10%
0.25
i = 15%
Periods
0
2
4
6
8
10
The Figure 3.2 shows how the present value of rupee one declines with longer period
and larger interest rate. For example, the present value of a future sum is lower at 15
percent than at 10 percent interest rate. The present value also depends on time
period. Given the interest rate the present value diminishes with longer period. For
example, at 10 percent discount rate the present value of a sum of money
(say Rs 10,000) will be low (Rs 3,855) in ten years as compare to Rs 6,209 in five years.
FINDING DISCOUNT RATE AND NUMBER OF PERIODS
To this point we explored some fundamental aspects relating to the calculation of
present value and future value. It can be observed from equation (3.1) and (3.3) that
the present value and future value are the reversal of each other. In the equation, we
see four basic variables associated to present and future value calculation– present
value (PV), future value (FV), time (n), and interest rate (i). Given the values of any
three variables in the equation, we can reformulate both present value and future
value equations to calculate the value of fourth variable, which is unknown.
Finding Interest Rate
Suppose you deposit Rs 3,500 in a saving account today that will grow to a future
value of Rs 5,000 at the end of five years. What rate of return will you earn annually
from this investment? In this example Rs 3,500 today is the present value (PV), and
Rs 5,000 to be received five years from now is the future value (FV 5). Given the
TIME VALUE OF MONEY

67
Chapter 3
information, the appropriate annual rate of interest can be found by reformulating
equation (3.1) and (3.3) and is stated in equation (3.5) as follows:3
FVn 1/n
i =  PV  – 1

(3.5)

For our example, the appropriate annual interest rate is calculated below:
Rs 5‚000 1/5
i = Rs 3‚500 – 1 = 1.0739 – 1 = 0.074 or 7.4%


Tabular Solution
The same solution could be obtained by using present value interest factor (PVIF) or
future value interest factor (FVIF) table. Let us refer to the equation (3.4) that gives
the tabular solution of present value as follows:
PV = FVn (PVIFi, n)
... (3.4)
Substituting the respective values of our example in equation (3.4):
Rs 3,500 = Rs 5,000 (PVIF i, 5)
Rs 3‚500
Rs 5‚000 = (PVIF i, 5)
PVIFi, 5 = 0.7
Looking at PVIF table at five year row the factor 0.7 lies between 7 percent
(lower rate = LR) and 8 percent (higher rate = HR). Therefore, to obtain the exact
interest rate we interpolate the result as follows:
Interest rate (i)
= LR + [(Factor at LR-Exact factor)/(Factor at LR –Factor at HR)]
= 7% + [(0.7130- 0.7)/(0.7130 – 0.6806)]
= 7% + (0.0130/0.0324) = 7.4%
The above result shows that if Rs 3,500 is deposited today, it will grow to Rs 5,000 at
the end of year five. This will yield an annual 7.4 percent return to the investor.
Finding the Number of Periods (Time)
Suppose Rs 2,000 is deposited today at 8 percent annual interest rate, then in how
many years this will double? This solution could be found out by solving for number
of periods or time (n) in equation 3.1. In this example Rs 2,000 today is the present
value (PV), at annual interest rate (i) of 8 percent it will double to Rs 4,000 future
value (FV) at the end of certain years.
3
This equation is derived as follows:
FVn
PV
=
(1 + i)n
FVn
(1 + i)n =
PV
1/n
FVn
(1 + i) =
PV
i
( )
FV
=(
PV )
n
1/n
–1
... (3.3)
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Chapter 3
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BUSINESS FINANCE
Using equation (3.1)
FVn = PV (1+i)n .... (3.1)
Rs 4,000 = Rs 2,000 (1.08)n
(1.08)n = 2
Taking logarithm in both sides
n log 1.08 = log 2
n = log2/log1.08
n = 0.3010/0.0334 = 9.01 years.
The above solution shows that Rs 2,000 deposited today at 8 percent annual
interest rate will double in nearly 9 years.
Tabular Solution
The same solution could be again obtained by using the present value interest factor
(PVIF) table. Let us refer to the equation (3.4):
PV = FVn (PVIFi, n) ... (3.4)
Rs 2,000 = Rs 4,000 (PVIF, 8,% n)
PVIF8%, n = 0.5
Looking into the PVIF table at 8 percent interest rate, the factor 0.5 is close to 9
year's factor (that is 0.5002). Therefore, the respective time period that Rs 2,000
doubles at 8 percent annual interest rate is 9 years (that is n = 9 years).
ANNUITY
Annuity
A series of equal
payment at equal
interval of time for a
given number of
periods
Ordinary annuity
Series of equal
payments at the end of
each period
Annuity due
Series of equal
payments at the
beginning of each
period
An annuity is defined as a series of payment of fixed amount at each specified
interval of time for a given number of periods. An annuity can be an ordinary
annuity or annuity due. In case of an ordinary annuity, each equal payment is made
at the end of each interval of time throughout the period. Whereas in case of annuity
due equal payments are made at the beginning of each interval throughout the
periods.
For example, if an individual promises to pay Rs 1,000 at the end of each of
three years for amortization of a loan, then it is called an ordinary annuity. If it were
the annuity due, each payment would be made at the beginning of each of the three
years. They are illustrated in the following time line of cash flows:
Time
0
8%
Ordinary annuity
Time
Annuity due
0
1,000
8%
1
2
1,000
1,000
1
2
1,000
1,000
3
1,000
3
The above time lines show that each cash flow occurs one period earlier in
annuity due than in the ordinary annuity.

TIME VALUE OF MONEY
69
Chapter 3
Future Value of an Ordinary Annuity
Suppose we invest Rs 1,000 at the end of each year for there years in a security
paying 8 percent annual interest, how much would we have at the end of three
years?
This is a problem concerning to the future value of an ordinary annuity. In
this example, the first year-end payment of Rs 1,000 is compounded at 8 percent for
the rest of two years, second Rs 1,000 year-end payment is compounded for one year
and the last Rs 1,000 year-end payment is not compounded at all, since it is only
made at the end of year 3, the end of compounding period. Such problems of
ordinary annuity are solved by using the equation (3.6) presented below:4
FVAn =
PMT [(1+i)n – 1]
i
(3.6)
Where,
FVAn = future value of an ordinary annuity for ‘n’ years
PMT = annual amount of equal payment. = Rs 1,000
n = number of compounding periods = 3 years
i = annual rate of interest at which each payment is compounded= 8 percent or 0.08
Substituting the respective values in equation (3.6), the future value of Rs 1000
ordinary annuity for three years compounded at 8 percent annual rate is given by:
FVA3 =
Rs 1‚000 [(1+0.08)3 – 1]
= Rs 3,246.4
0.08
The following time line gives an idea about compounding of each annual
payment and their future value at the end of year three.
Time
0
Ordinary annuity
8%
1
2
3
1,000
1,000
1,000
1,000 × 1.08
1,000 × 1.082
1,080
1,166.4
FVA3 = Rs 3,246.4
Tabular Solution
The future value of annuity stated in above example also could be found by using
the future vale interest factors table. It is shown in Table 3.1.
TABLE 3.1
End of Year
Future value of a 3
years ordinary
annuity of Rs 1,000
compounded at 8
percent per year
1
2
3
Payment (PMT)
Rs 1,000
1,000
1,000
Future value of annuity
4
FVIF at 8%
FV
1.1664
1.0800
1.0000
Rs 1,166.4
1,080.0
1,000.0
Rs 3,246.4
This equation is derived as below:
FVAn
= PMT (1 + i)n–1 + PMT (1 + i)n–2 + ... ... + PMT (1 + i)1 + PMT (1 + i)0
 n

= PMT   (1 + i)n–t
t = 1

PMT [(1 + i)n – 1]
=
… (3.6)
i
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BUSINESS FINANCE
Note that in above calculation in table 3.1, the first year’s payment of Rs 1,000
occurs at the end of year 1 so that it is compounded for two years. Similarly, the
second year-end payment of Rs 1,000 is compounded for one year. And the last
payment is not compounded at all as it occurs at the end of year three.
By using future value interest factor of annuity table, we can calculate the
future value of an ordinary annuity as follows.
FVAn = PMT [FVIFAi,n ]
(3.7)
Looking at FVIFA table at 8 percent for three years the factor is 3.2464. Now
substituting the respective values in equation (3.7):
FVA3
= PMT [FVIFA i,n]
= Rs 1,000 [FVIFA8%, 3]
= Rs 1,000 x 3.2464 = Rs 3,246.4
Future Value of an Annuity Due
If Rs 1,000 annuity is the annuity due such that each payment occurs at the beginning
of each of the three year, the future value of annuity due is given by equation (3.8) as
follows:
FVAn (due) =
PMT [(1+i)n– 1]
(1+i)
i
(3.8)
Substituting the respective values in equation (3.8), the future value of threeyear annuity due of Rs 1,000 compounded at 8 percent per year is given by:
Rs 1‚000 [(1+0.08)3– 1]
(1 + 0.08) = Rs 3,246.4 (1.08)
0.08
= Rs 3,506.11
FVA3 (due) =
This calculation process of future value of annuity due is easily understood
with the help of cash flow time line:
Time
Annuity Due
0
8%
1,000
3
1
2
1,000
1,000
1,000 × 1.08
1,000 × 1.08
2
1,080
1,166.4
1,000 × 1.083
1,259.71
FVA3 = 3,506.11
Tabular Solution
The future value of annuity due stated in above example also could be found by
using the future value interest factors table. It is shown in the Table 3.2.
TABLE 3.2
Future value of a 3
years annuity due of
Rs 1,000
compounded at 8
percent per year
Beginning of Year
1
2
3
Payment (PMT)
Rs 1,000
1,000
1,000
Future value of annuity due
FVIF 8%
1.2597
1.1664
1.0800
FV
1,259.7
1,166.4
1,080.0
Rs 3,506.1
TIME VALUE OF MONEY

Chapter 3
71
In Table 3.2, the first year’s payment of Rs 1,000 occurs at the beginning of
year 1, hence, it is compounded for three years. The second year payment of
Rs 1,000 is compounded for two years. And the last payment is compounded for one
year only.
By using future value interest factor of annuity table, we can calculate the future
value of an annuity due as follows:
FVAn (due) = PMT [FVIFAi,n] (1+i)
(3.9)
Looking at FVIFA table at 8 percent for three years the factor is 3.2464. Now
substituting the respective values in equation (3.9):
FVA3 (due)
= PMT [FVIFAi, n] (1+i)
= Rs 1,000 [FVIFA8%, 3] (1+0.08)
= Rs 1,000 x 3.2464 x 1.08 = Rs 3,506.11
Present Value of an Ordinary Annuity
If you are given the alternatives of accepting either a three-year annuity with a
payment of Rs 1,000 at the end of each year or a lump sum payment of Rs 2,500
today, which alternative would you prefer? This problem could be solved by
calculating present value of ordinary annuity. If the lump sum payment today is
equal to the present value of three-years annuity of Rs 1,000 each year discounted at
a given rate of return (that is, the required rate of return or the opportunity cost of
funds), we would be indifferent in choosing either of the alternatives, otherwise, we
would prefer the higher value.
The present value of an annuity is calculated either by using formula or by
using present value interest factor of annuity table.
The present value of an annuity for n years discounted at 'i' percent required
rate of return is given by equation (3.10) as follows:5
1 - 1 n
(1+i) 
PVAn = PMT 
 i 
(3.10)
Let us suppose that opportunity cost of funds is 10 percent, so that present
value of three-year annuity of Rs 1,000 each year is given by:
PVA3
1 - 1 3
 (1 + 0.1) 
= Rs 1,000
 0.1 
= Rs 1,000 x 2.4868 = Rs 2,486.8
That is Rs 1,000, 3 years annuity discounted each year at 10 percent interest
rate will have a present value of Rs 2,486.8. Thus, we would be indifferent between a
5
This equation is derived as below:
PVAn
= PMT [1/(1 + i)1] + PMT[1/(1 + i)2] + ... ... + PMT [1/(1 + i)n]
 n

= PMT   1/(1 + i)t
t = 1

1 – [1/(1 + i)n]
= PMT
… (3.10)
i
[
]
72

Chapter 3
BUSINESS FINANCE
lump sum of Rs 2,486.8 received at present and an ordinary annuity of Rs 1,000 for
three years. In the above example, if the choice were between an annuity of Rs 1,000
for three years and a lump sum of Rs 2,500, at present, we would choose higher value
i.e. Rs 2,500.
Tabular Solution
Another method of calculating the present value of an annuity is to use the present
value interest factor for an annuity (PVIFA) table. In this method, the annuity is
multiplied by the factor at i percent for n years by referring the PVIFA table as given
in the equation (3.11).
PVAn = PMT [PVIFAi, n]
(3.11)
Looking at PVIFA table at 10 percent for three years, the present value interest
factor of an annuity is 2.4868. By substituting the respective values in equation (3.11)
we get Rs 2,486.8 the present value of annuity:
PVA3
= Rs 1,000 [PVIFA 10, 3]
= Rs 1,000 x 2.4868 = Rs 2,486.8
Present Value of an Annuity Due
If each payment were made at the beginning of each year instead of at the end, what
would be the present value of the annuity? The simple consideration required is that
each of three payments would shift one year earlier so that each of them is
discounted for one year less. The formula for present value of an annuity is:
1 - 1 n
(1+i) 
PVAn (due) = PMT 
 i  (1+i)
( 3.12)
Let us refer the same example. If each Rs 1,000 annual payment is made at the
beginning of each of three years, the present value of annuity due is given by:
PVAn (due)
1 - 1 3
 (1 + 0.1)  (1+0.1)
= Rs 1,000
 0.1 
= Rs 1,000 x 2.4868 x 1.10 = Rs 2,735.48
Note that present value of an annuity due is greater than the present value of
an ordinary annuity because each payment occurs one period earlier.
Tabular Solution
Another method of calculating the present value of an annuity due is to use the
present value interest factor for an annuity (PVIFA) table. The annuity is multiplied
by the factor at 'i' percent for 'n' years by referring to the PVIFA table and the
product is further multiplied by (1 + i) as given in the equation (3.13).
PVAn (due) = PMT [PVIFA i, n] (1+i)
(3.13)
Looking at PVIFA table at 10 percent for three years the present value interest
factor of an annuity is 2.4868. Substituting the respective values in equation (3.11),
the present value of annuity due of Rs 1,000 for 3 years at 10 percent is:
PVA3
= Rs 1,000 [PVIFA 10%, 3] (1+0.1)
= Rs 1,000 x 2.4868 x 1.10 = Rs 2,735.48
TIME VALUE OF MONEY

Chapter 3
73
SOLVING FOR INTEREST RATES IN ANNUITIES
If we were told to invest a lump sum of Rs 10,000 today in a security that pays Rs
4,021.23 at the end of each of next 3 years, what rate of return would we generate
from our investment? In this problem we have one unknown variable ‘k’- the interest
rate. Rs 10,000 today represents a present value of Rs 4,021.23 three years ordinary
annuity, so that the interest rate (k) could be found by using the relationship of
present value of an ordinary annuity as given by equations (3.10 or 3.11). Here we
are using equation (3.11)
PVAn = PMT [PVIFA i, n] .... (3.11)
Rs 10,000 = Rs 4,021.23 [PVIFAi, 3]
[PVIFAi, 3] = Rs 10,000/Rs 4,021.23
[PVIFAi, 3] = 2.4868
Looking at PVIFA table for three years, the above factor 2.4868 is exactly equal
to PVIFA at 10 percent for three years. It means that the rate of return we would earn
is 10 percent.
PRESENT VALUE OF PERPETUITY
Perpetuity
An infinite stream
of equal payment
In the previous sections, we dealt with annuity payments of fixed maturity period,
say a three-year annuity, a five-year annuity, a ten-year annuity and so on. However,
some annuities may go for paying equal installment at each equal interval of time,
indefinitely. Such annuities are called perpetuities. In other words, perpetuity is a
stream of equal payment made at the end of equal interval of time to indefinite
period. The present value of perpetuity is calculated as follows:
Payment
PMT
PVPERPETUITY = Interest rate = i
(3. 14)
For example, the present value of a perpetuity of Rs 1,000 each year at 10
percent interest rate is:
Payment
PMT Rs 1‚000
PV PERPETUITY = Interest rate = i = 0.10 = Rs 10,000.
UNEVEN CASH FLOW STREAMS
In our previous section we noted that annuities call for a stream of equal payment
over the time. However in many cases, in a stream of cash flow, the cash flow in each
period may differ from period to period. Such cash flows are called uneven stream of
cash flows. In case of uneven stream of cash flows the calculation process of present
value and future value is discussed below.
74
Chapter 3

BUSINESS FINANCE
Present Value of Uneven Cash Flow Stream
Present value of an uneven cash flow stream is simply the sum of the present value
of each of the cash flow occurred throughout the stated period to maturity. The
present value equation for uneven stream of cash flows could be stated as follows:
CF1
CF2
CF3
CFn
PV = (1+i)1 + (1+i)2 + (1+i)3 + . . . . . . + (1+i)n
(3.15)
For instance, let us suppose a security provides the following stream of cash
flows till its maturity in five years.
End of Year
Cash Flow (Rs )
1
100
2
150
3
200
4
250
5
400
If appropriate discount rate is 10 percent, the present value of this uneven
stream of cash flows is calculated as follows:
TABLE 3.3
Present value of an
uneven cash flow
stream
Year
Cash Flows
PVIF 10%
PV
1
Rs 100
0.9091
Rs 90.91
2
150
0.8264
123.96
3
200
0.7513
150.26
4
250
0.6830
170.75
5
400
0.6209
248.36
PV of uneven CF stream
Rs 784.24
The present value of the above uneven cash flow stream is Rs 784.24.
Future Value of Uneven Cash Flow Streams
Future value of uneven cash flow stream is the sum of the future value of each cash
flow compounded to the end of the stream at required rate of return. It is calculated
by using the following relationship:
FVn = CF1(1+i)n-1 + CF2(1+i)n-2 + CF3(1+i)n-3 + . . . . . . + CFn(1+i)0
(3.16)
Let us consider the same cash flow stream as above and a 10 percent rate of
compounding. The future value is computed as follows:
TABLE 3.4
Future value of an
uneven cash flow
stream
Year
Cash Flows
10%FVIF
Rs 100
2
150
(1.1)3 = 1.3310
199.65
3
200
(1.1)2
= 1.2100
242.00
4
250
(1.1)1 = 1.1000
275.00
400
(1.1)0
FV of uneven CF stream
= 1.4641
FV
1
5
(1.1)4
= 1.0000
Rs 146.41
400.00
Rs 1,263.06
Note that in above future value calculation, the first year-end cash flow is
compounded for four years, the second year-end cash flow is compounded for three
years and so on.
TIME VALUE OF MONEY

Chapter 3
75
SEMIANNUAL AND OTHER COMPOUNDING PERIODS
In all preceding discussions, we have considered interest rate compounded annually.
It is called simple or quoted rate. However in many cases, the interests are paid
monthly or quarterly or semiannually or any other periods which is less than one
year. It is called periodic rate. In such cases, interests are compounded more than
once during the year. For instance, if interest is paid quarterly, it is compounded four
times during the year. If it is paid semiannually, it is compounded two times during
the year. The number of compounding periods during the year is denoted as ‘m’.
For any compounding period less than one year, we make following two
changes in all our present and future value calculation: first, the rate of interest (i) is
divided by the number of compounding periods (m) during the year; and second, the
number of years that cash flow occurs (n) is multiplied by the number of
compounding periods during the year.
Let us illustrate a future value calculation for semiannual compounding.
Assume that we place a Rs 10,000 today for 5 years in a security paying interest rate
of 10 percent. If interest is compounded semiannually, we receive 5 percent interest
for each of the six-monthly period through a total of 10 six-monthly periods. The FV
of this semiannual compounding appears as:
FVmn = PV [1+i/m]mn
(3.17)
Substituting the respective values in equation (3.17):
FVmn = Rs 10,000 [1+0.10/2]2 x 5
= Rs 10,000 (1.05)10
= Rs 10,000 x 1.6289 = Rs 16,289
Effective annual rate
An annual equivalent
rate of some
semiannual or other
rates less than one
year
In this calculation, we have simply changed a 10 percent annual interest
directly to a 5 percent semiannual interest. However, it should be noted that each 5
percent interest payment occurs earlier so that they could be reinvested for rest of the
periods during the year. Because of this reason, effective annual rate (EAR) is
somewhat greater than annual percentage rate of 10 percent. Effective annual rate is
the rate that would produce same terminal value if the annual compounding was
used. It is calculated as follows:
isimple m
EAR = 1+ m  – 1


(3.18)
With reference to our example for semiannual compounding the effective annual rate
(EAR) is:
EAR
isimple m
= 1+ m  – 1


0.10 2
= 1+ 2  – 1 = [1.05]2 – 1 = 1.1025 – 1


= 0.1025 or 10.25%.
That is, if the initial deposit of Rs 10,000 were compounded annually at 10.25
percent for 5 years, this would result into a terminal value of Rs 16,289 at the end of 5
76
Chapter 3

BUSINESS FINANCE
years. Note that this is equal to the amount which we calculated by using 5 percent
semiannual rate for 10 semiannual periods.
The future value calculated using semiannual compounding is always greater
than that of annual compounding. It occurs because of more compounding periods
during the year.
CONTINUOUS COMPOUNDING
In addition to semiannual and other compounding, sometimes interest is compound
continuously. Let us recall the Equation (3.19) for semiannual and other
compounding.
FVmn = PV [1+i/m]mn
(3.19)
In Equation (3.19), when number of compounding period ‘m’ approaches to
infinity, we get continuous compounding and the compounding factor ‘[ 1 + i/m] mn’
approaches ‘ei x n’. Therefore, when interest is compounded continuously, the
Equation (3.19) could be re-written as follows,:
FVn = PV(e)i x n
(3.20)
In Equation (3.20), ‘e’ is called an exponential terms, whose numerical value
approximates to 2.71828.
To illustrate, let us suppose, if we deposit Rs 1,000 today at an annual interest
rate of 12 percent for five years. The future value of this present sum Rs 1,000 at the
end of year five, when interest is compounded continuously, is given by:
FVmn = PV(e)i x n = Rs 1,000 x (e).12 x 5
= Rs 1,000 x 1.822112
= Rs 1,822.12
Note that the future value of any given amount of present sum is comparatively
larger when continuous compounding is used than any other compounding periods. It
happens because number of compounding period is significantly larger under continuous
compounding.
The Equation (3.20), in case of continuous compounding, also can be restated as
follows to calculate the present value of a future sum:
1
PV = FVn x (e)i x n
(3.21)
The present value calculated by using Equation (3.20) in case of continuous
compounding is lower than in any other compounding periods because of significantly
larger number of discounting periods.
FRACTIONAL TIME PERIODS
In our previous section we noted how present values and future values are
calculated using semi-annual and other compounding periods such as three monthly
compounding, monthly compounding and so on.
This section deals with
TIME VALUE OF MONEY

77
Chapter 3
compounding and discounting when payment occurs in some fractional period. Let
us suppose, for example, you deposit Rs 5,000 in Himalayan Bank account that pays
annual interest of 12 percent. The interest is compounded daily by the bank. If you
hold your deposit for 226 days, what will be the future value of this deposit at the
termination of deposit period?
If bank compounds interest daily, the periodic rate, that is daily rate, is
0.032877 percent assuming 365 days in a year. Therefore future value is given by:
FV = Rs 5,000 (1 + 0.00032877)226 = Rs5,385.59.
That is sum of Rs 5,000 deposited today will grow to Rs 5,385.59 at the end of
226days at a 12 percent interest rate, assuming the interest is compounded daily.
AMORTIZED LOANS
Amortized loan
A loan to be repaid in
equal installments
throughout the given
periods
Amortized loan refers to the loan that is to be repaid in equal periodic installments
including both principal and interest. The concepts of present value and compound
interest rate are used to amortize a loan over the time in equal installments.
Let us suppose a loan of Rs 10,000 is to be repaid in 4 equal installments
including principal and 10 percent interest per annum. We apply the following steps
to determine the annual payment and set up an amortization schedule of the loan.
Determining annual payment
The annual amount of installments to be paid off that includes both principal and
interest amount is calculated as follows:
PMT =
Amount of Loan
PVIFAi‚ n
Rs 10‚000
Rs 10‚000
= PVIFA
= 3.1699
10%‚ 4
(3.22)
= Rs 3,154.67
It means an installment of Rs 3,154.67 paid annually for four years will pay off
both principal and interest of the loan.
Setting loan amortization schedule
Once the annual amount of installment is determined, the loan amortization
schedule could be set up as follows:
TABLE 3.5
Loan amortization
schedule
Year
(1)
1
2
3
4

Beginning
Amount
(2)
Rs 10,000.00
7,845.30
5,475.16
2,868.01
Payment
(3)
Rs 3,154.67
3,154.67
3,154.67
3,154.67
Interest
(4) = (2) x 0.10
Rs 1,000.00
784.53
547.52
*286.66
Repayment of
Principal
(5) = (3) – (4)
Rs 2,154.67
2,370.14
2,607.15
2,868.01
Ending Balance
(6) = (2) – (5)
Rs 7,845.33
5,475.16
2,868.01
-
Interest payment in the 4th year has been rounded to make the sum of principal plus interest equal to
the annual payment of Rs 3,154.67.
78

Chapter 3
BUSINESS FINANCE
COMPARISON OF DIFFERENT TYPES OF INTEREST RATES
To this point we used three types of interest rates: simple or quoted rate, periodic
interest rate and effective annual rate. This section compares them and explains their
uses.
Simple or quoted interest rate
Simple rate
The interest rate
quoted by lender
The rate of interest, which is quoted by borrowers and lenders, is known as simple
or quoted interest rate. The practitioners in the stock, bond, commercial loan,
banking and finance company’s loan express all financial contracts in terms of simple
loan. Simple rate of interest is the general rate that we use in practice while talking
about borrowing and lending. However, the quotation of simple rate must also
include the number of periods used in compounding per year. For example, a bank
may offer 10 percent simple interest loan compounded monthly or quarterly or
semiannually or annually.
It should be noted that the simple rate of one instrument could be compared
with other only when they have same number of compounding periods during the
year. This means if is a bank offers 8 percent simple interest loan with quarterly
compounding where as another bank offers 8.5 percent simple interest loan
compounded semiannually then they can not be compared on the basis of simple
interest rate because of difference in compounding periods used in a year.
Periodic rate
Periodic rate
The interest rate for
each interest period
such as monthly,
quarterly,
semiannually,
annually
The rate of interest charged by lender or paid by borrower at each interest period is
known as periodic rate of interest. It can be stated as interest rate per year or interest
rate per six month, or per quarter or per month and so on. Periodic rate is calculated
as simple interest rate divided by number of period in a year as given in equation
(3.23).
Periodic Rate (iPER) =
iSIMPLE
m
(3.23)
Above equation shows that if periodic rate is multiplied by number of
compounding period during the year then the periodic rate is stated on approximate
annual rate. This approximate annual rate of periodic rate is known as annual
percentage rate (APR). It is to be noted that the APR never is used in actual
calculations; it is simply reported to borrowers.
The periodic interest rate is equal to simple interest rate only if there is only
one interest payment in the year, that is, once in a year. But when interests are paid
more frequently than once in a year and the payment is made on each compounding
date then periodic interest rate is different from simple rate.
Effective annual rate
Effective annual rate (EAR) is the annual equivalent interest rate of a given periodic
rate. However, it is not the APR. The APR does not consider the compounding effect
of periodic rate, whereas the EAR considers it. For example, if we use 3 percent
periodic rate per quarter, its APR is 12 percent, but EAR is more than 12 percent
because of compounding effect. The basic use of EAR is that it facilitates the
comparison of different interest rates with different number of compounding period

TIME VALUE OF MONEY
Chapter 3
79
during the year. For example, if we are going to compare a 8 percent simple interest
loan paying interest every six month against a 7.5 percent simple interest loan paying
interest every three months, both the simple rates must be converted into effective
annual rate.
ILLUSTRATIVE PROBLEMS
Illustration
1
Present and future
values
Find the following present and future values:
a.
An initial Rs 500 compounded for 1 year at 6 percent.
b.
An initial Rs 500 compounded for 2 years at 6 percent.
c.
The present value of Rs 500 due in 1 year at a discount rate of 6 percent.
d.
The present value of Rs 500 due in 2 years at a discount rate of 6 percent.
S O L U T IO N
a.
Given,
Present value (PV) = Rs. 500
Interest rate (i) = 6%
0
6%
1
- 500
FV = ?
FVn
FV1
b.
= PV(1 + i)n
= PV (1 + i)1
= Rs. 500 (1 + 0.06)1 = Rs. 530
Present value (PV) = Rs. 500
Interest rate (i) = 6%
0
6%
1
2
- 500
FV = ?
FVn
FV2
c.
= PV(1 + i)n
= PV (1 + i)2
= Rs. 500 (1 + 0.06)2 = Rs. 561.80
Future value (FV) = Rs. 500
Interest rate (i) = 6%
No. of periods (n) = 1
0
PV = ?
1
FV = 500
FVn
FV1
= (1 + i)n = (1 + i)1
Rs. 500
Rs. 500
= (1 + 0.06)1 = 1.06 = Rs. 471.70
Future value (FV) = Rs. 500
Interest rate (i) = 6%
No. of periods (n) = 2
PV
d.
6%
80

Chapter 3
BUSINESS FINANCE
Present value (PV) = ?
0
6%
1
2
PV = ?
PV
FV = 500
FVn
FV2
= (1 + i)n = (1 + i)2
Rs. 500
Rs. 500
= (1 + 0.06)2 = 1.1236 = Rs. 445
Illustration
2
Future value
SOLUTION
Suppose Mr. Sharma deposits Rs 10,000 in a bank account that pays 10 percent interest
annually. How much money will be in his account after 5 years?
Here, Present value (P) = Rs 10,000,
Interest rate (k) = 10%
Number of years (n) = 5 years,
Future value (FV5) = ?
0
10%
1
2
3
5
FV = ?
Rs10000
We have,
FV5 = PV × (1 + k)n = Rs 10,000 × (1.10)5 = Rs 10,000 × 1.6105 = Rs 16,105.10
Mr. Sharma will have Rs16,105.10 at the end of year 5 in his account.
Illustration
3
Present value
SOLUTION
What is the present value of a security that promises to pay you Rs 5,000 in 20 years?
Assume that you can earn 7 percent if you were to invest in other securities of equal risk?
Here, Future value (FV) = Rs 5,000
Number of years (n) = 20 years
Interest rate (k) = 7%
Present value (PV) = ?
0
7%
1
2
3
PV = ?
20
Rs5,000
We have,
FV20
Rs 5‚000
PV = (1 + k)n = (1 + 0.07)20
Rs 5‚000
= 3.8697 = Rs 1,292.09
Illustration
4
Time for a lump
sum to double
SOLUTION
If you deposit money today into an account that pays 6.5 percent interest, how long will it
take for you to double your money?
Here, Interest rate (i) = 6.5%
Number of period (n) = ?
Present value (PV) = Rs 1000 (assume)
TIME VALUE OF MONEY

Chapter 3
81
Future value (FV) = Rs 2000
0
6.5%
2
1
n=?
3
PV =
Rs1000
FV =
Rs2,000
We have,
FV
Present value (PV) = (1 + i)n
or,
Rs 2000
Rs 1000 = (1 + 0.065)n
or,
Rs 2000
(1 + 0.065)n = Rs 1000
or, (1.065)n = 2
.... (i)
Trying at n = 11
We get,
If n = 11, the left hand side in above equation (i) is approximately equal to 2. Hence
the required no. of years to double the sum of money is 11 years.
Illustration 5
Future value of
annuity at different
compounding
periods
Find the future values of the following ordinary annuities:
a.
FV of Rs 400 each 6 months for 5 years at a simple rate of 12 percent, compounded
semiannually.
b.
FV of Rs 200 each 3 months for 5 years at a simple rate of 12 percent, compounded
quarterly.
c.
The annuities described in parts a and b have the same amount of money paid into
them during the 5-year period and both earn interest at the same simple rate, yet
the annuity in part b earns Rs 101.60 more than the one in part a over the 5 years.
Why does this occur?
S O L U T IO N
a.
Here,
Periodic equal payment (PMT) = Rs. 400
Number of period (n) = 5 years
Interest rate (i) = 12% semiannual compounding
Future value of an annuity (FVA) = ?
n×2
5×2
1 + i  - 1
1 + 0.12 - 1
2 
 2

Now, FVA = PMT
= Rs. 400
i
0.12
2
2



b.









= Rs. 400 × 13.1808 = Rs. 5272.32
Here,
Periodic equal payment (PMT) = Rs. 200
Number of periods (n) = 5 years
Interest rate (i) = 12% compounded quarterly
Future value of an ordinary annuity (FVA) = ?
Now,
1 + 4
= PMT 
i
 4
i
FVA
n×4
-1

1 +
=
Rs.
200




= Rs. 200 × 26.8704 = Rs. 5374.08
0.125 × 4
-1
4 
0.12
4



82

Chapter 3
c.
Illustration
6
Effective rate of
interest
BUSINESS FINANCE
It is because, other things held constant, higher the number of compounding
higher will be the FV and vice versa.
Your broker offers to sell a note for Rs 13250 that will pay Rs 2345.05 per year for 10 years.
If you buy the note, what rate of interest will you be earning? Calculate to the closest
percentage.
S O L U T IO N
Here,
Present value of annuity (PVA) = Rs. 13,250
Periodic equal payment (PMT) = Rs. 2345.05
No. of periods (n) = 10 years
Interest rate (i) = ?
Time Line
1
0
PVA =
13250
2345.05
2
3
4
2345.05
2345.05
5
2345.05 2345.05
6
7
8
2345.05 2345.05
9
10
2345.05 2345.05 2345.05
We have,
PVA = PMT × PVIFA i × n yrs.
or,
Rs. 13,250 = Rs. 2345.05 × PVIFAi% 10 yrs
or,
PVIFAi%, 10 yrs = 5.6502
From the PVIFA table, the value of 5.6502 in 10 years lies at 12%.

The required interest rate is 12%.
Illustration
7
Effective rate of
interest
Your parents are planning to retire in 18 years. They currently have Rs 250,000, and they
would like to have Rs 1,000,000 when they retire. What annual rate of interest would they
have to earn on their Rs 250,000 in order to reach their goal, assuming they save no more
money?
SOLUTION
Here, Future value (FV) = Rs 1,000,000
Present value (PV) = Rs 250,000
Time period (n) = 18 years
Interest rate (i) = ?
0
Rs250,000
i=?
1
2
18
3
Rs1,000,000
We have,
FV = PV (1 + i)n
or, Rs 1,000,000 = Rs 250,000 (1 + i)18
Rs 1‚000‚000
or, (1 + i)18 = Rs 250‚000
or, (1 + i)18 = 4
or, 1 + i = (4)1/18
or, i = 1.08 - 1 = 0.08 or 8%

The required rate of interest to reach the goal is 8%.
Illustration
8
Future value of an
annuity
What is the future value of a 5-year ordinary annuity that promises to pay you Rs 300
each year? The rate of interest is 7 percent.
TIME VALUE OF MONEY

Chapter 3
83
SOLUTION
Here, Future value of annuity (FVA) = ?
Payment (PMT) = Rs 300
Number of period (n) = 5 years
Interest rate (i) = 7%
0
7%
1
Rs300
2
Rs300
5
3
Rs300
Rs300
FVA = ?
We have,
(1 + i)n - 1
 i 
(1 + 0.07)5 - 1
= Rs 300 
 0.07 
= Rs 300 × 5.7507 = Rs 1,725.21
FVA = PMT 
Illustration
9
Future value of an
annuity due
SOLUTION
What is the future value of a 5-year annuity due that promises to pay out Rs 300 each
year? Assume that all payments are reinvested at 7% a year, until year 5.
Here, Future value of annuity due (FVAdue) = ?
Payment (PMT) = Rs 300
Number of period (n) = 5 years
Interest rate (i) = 7%
0
Rs300
7%
1
Rs300
2
Rs300
5
3
Rs300
FVA (due) = ?
We have,
(1 + i)n - 1
 i  (1 + i)
(1 + 0.07)5 - 1
= Rs 300 
 0.07  (1 + 0.07)
= Rs 300 × 5.7507 × 1.07 = Rs 1,845.97
FVAdue = PMT 
Illustration 10
Present and future
value of a cash flow
S O L U T I O Nstream
An investment pays you Rs 100 at the end of each of the next 3 years. The investment will
then pay you Rs 200 at the end of year 4, Rs 300 at the end of year 5, and Rs 500 at the end
of year 6. If the rate of interest earned on the investment is 8 percent, what is its present
value? What is its future value?
Year
1
2
3
4
5
6
Cash flow
Rs 100
100
100
200
300
500
Present value
PVIF at 8%
0.9259
0.8573
0.7938
0.7350
0.6806
0.6302
TPV
PV
Rs 92.59
85.73
79.38
147
204.18
315.10
Rs 923.98
Future value
FIVE at 8%
FV
1.4693
146.93
1.3605
136.05
1.2597
125.97
1.1664
233.28
1.0800
324.00
1.0000
500.00
TVF
Rs 1,466.23
84

Chapter 3
Illustration
BUSINESS FINANCE
11 You are thinking about buying a car, and a local bank is willing to lend you Rs 20,000 to
Loan amortization
and effective interest
rate
buy the car. Under the terms of the loan, it will be fully amortized over 5 years (60
months), and the nominal rate of interest will be 12 percent, with interest paid monthly.
What would be the monthly payment on the loan? What would be the effective rate of
interest on the loan?
SOLUTION
Here, Price of the car (PVA) = Rs 20,000
n = 5 years
Interest rate (i) = 12% annually (i.e monthly interest rate is 1%)
Monthly installment (PMT) = ?
Effective interest rate (EIR) = ?
We have,
PVA
Rs 20‚000 Rs 20‚000
PMT = PVIFA = PVIF
= 44.9550 = Rs 444.889
i‚ n
1‚ 60
isimple m
0.12 12
EIR = 1 + m  - 1.0 = 1 + 12  - 1.0




= 1.1268 - 1 = 0.1268 or 12.68%
Illustration
12 Nepal Horticulture Ltd. invests Rs 4 million to clear a tract of land and to set out some
Expected rate of
return
young pine trees. The trees will mature in 10 years, at which time the company plans to
sell the forest at an expected price of Rs 8 million. What is company's expected rate of
return?
SOLUTION
Here, Future value (FV) = Rs 8,000,000
Present value (PV) = Rs 4,000,000
Time period (n) = 10 years
Expected rate of return (i) = ?
First set up time line as follows:
0
i=?
1
2
3
Rs4 million
We have,
FV
or,
Illustration
= PV (1 + i)n
Rs 8,000,000
or,
(1 + i)10
or,
or,

(1 + i)10
1+i
i
10
Rs8 million
= Rs 4,000,000 (1 + i)10
Rs 8‚000‚000
= Rs 4‚000‚000
=2
= (2)1/10
= 1.0718 - 1
= 0.0718 or 7.18%
13 You need to accumulate Rs 10,000. To do so, you plan to make deposits of Rs 1,250 per
Reaching a financial
goal
year, with the first payment being made a year from today, in a bank account which pays
12 percent annual interest compounded annually. Your last deposit will be less than Rs
1,250 if less is needed to round out to Rs 10,000. How many years will it take you to reach
your Rs 10,000 goal, and how large will the last deposit be?
SOLUTION
Here,
Annual payment (PMT) = Rs 1,250
TIME VALUE OF MONEY

Chapter 3
85
Future value of annuity (FVAn) = Rs 10,000
Interest rate (i) = 12%
Time to maturity (n) = ?
Last deposit = ?
0
12%
1
2
n =?
3
Rs1,250 Rs1,250 Rs1,250
Last deposit = ?
FVA = Rs10,000
First, we determine the number of periods of the financial goal. This is calculated using
future value of annuity formula as follows:
We have,
FVAn
= PMT × FVIFAi, n
Rs 10,000
= Rs 1,250 × PViFA12, n
10‚000
FVIFA12, n
= 1‚250 = 8
Looking FVIFA table the value 8 at 12 percent interest rate lies approximately in 6 years.
Therefore the number of years to reach the financial goal is 6 years. Now we calculate the
future value of Rs 1,250 for 5 years at 12%, it is Rs 7,941.06
FV = Rs 1,250 × FVIFA12, 5 = Rs 1,250 × 6.3528 = Rs 7,941
Compounding this value after 6 years and before the last payment is made, it is Rs 7,941
(1.12) = Rs 8,893.92. Thus, we will have to make a payment of Rs 10,000 - Rs 8,893.92 = Rs
1,106.08 at year 6, therefore it will take 6 years, and Rs 1,106.08 must be paid in the last
installment.
Illustration
14
Present value of a
cash flow stream
Mr. Dhakal is in the process of negotiating his first contract. A Company has offered him
three possible contracts. Each of the contracts lasts for 4 years. All of the money is
guaranteed and is paid at the end of each year. The terms of each of the contracts are
listed below:
Year 1
Year 2
Year 3
Year 4
Contract 1 Payment
Rs 3 million
3 million
3 million
3 million
Contract 2 Payment
Rs 2 million
3 million
4 million
5 million
Contract 3 Payment
Rs 7 million
1 million
1 million
1 million
The Mr. Dhakal discounts all cash flows at 10 percent. Which of the three contracts offers
him the most value?
SOLUTION
Year
PVIF at 10%
1
2
3
4
0.9091
0.8264
0.7513
0.6830
Contract 1
(In Million)
CF
PV
3
2.7273
3
2.4792
3
2.2539
3
2.0490
TPV
9.5094
Contract 2
Contract 3
(In Million)
(In Million)
CF
PV
CF
PV
2
1.8182
7
6.3637
3
2.4792
1
0.8264
4
3.0052
1
0.7513
5
3.4150
1
0.6830
TPV
10.7176
TPV
8.6244
The total present value of the Contract 2 is the largest so that it offers him the most value.
Illustration
15
PV and effective
annual rate
Assume that you inherited some money. A friend of yours is working as an unpaid intern
at a local brokerage firm, and her boss is selling some securities which call for 4
payments, Rs 50 at the end of each of the next 3 years, plus a payment of Rs 1,050 at the
end of year 4. Your friend says she can get you some of these securities at a cost of Rs 900
each. Your money is now invested in a bank that pays an 8 percent nominal (quoted)
interest rate, but with quarterly compounding. You regard the securities as being just as
86

Chapter 3
BUSINESS FINANCE
safe, and as liquid, as your bank deposit, so your required effective annual rate of return
on the securities is the same as that on your bank deposit. You must calculate the value of
the securities to decide whether they are a good investment. What is their present value to
you?
SOLUTION
Here,
Payment = Rs 50
Fourth year payment = Rs 1,050
Cost of the securities = Rs 900
Interest rate = 8%
Compounding = Quarterly compounding,
Effective interest rate (EAR) = ?
PV = ?
First we calculate the effective annual rate:
Effective interest rate (EAR) = (1 + 0.08/4)4 - 1 = 8.24%
Calculation of the present value of cash flow stream at 8.24% effective rate
Year
1
2
3
4
Cash flow
50
50
50
1,050
PVIF @ 8.24%
0.9239
0.8535
0.7886
0.7285
PV
PV
46.195
42.675
39.43
764.925
Rs 893.225
The present value of this cash flow stream is Rs 893.225, which is less than their current
selling price so that they are not a good investment
Illustration
16
Loan amortization
Your Company is planning to borrow Rs 1,000,000 on a 5-year, 15 percent, annual
payments, fully amortized term loan. What fraction of the payment made at the end of
the second year will represent repayment of principal?
SOLUTION
Here, Loan amount (PVA) = Rs 1,000,000
Number of years (n) = 5 years
Interest rate (i) = 15%
First we determine the annual installment or payment (PMT)
We have,
PVA
Rs 1‚000‚000 Rs 1‚000‚000
PMT = PVIFA = PVIFA
=
= Rs 293,311.55
3.3522
i‚ n
15‚ 5
Preparation of Amortization Schedule,
Amortization schedule
Year
1
2
Payments
Rs 298,311.5566
298,311.5566
InterestPayment of Principal
Rs 150,000
Rs 148,311.5566
127,753.2665
170,558.2901
Ending Balance
Rs 851,688.4434
681,130.1533
Principal payment in 2nd year Rs 170‚558.2901
= Rs 298‚311.5566 = 57.17%
Payments
That is 57.17% of the payment in second year represents the principal.
% principal in 2nd year =
Illustration 17
Loan amortization
T.U. BBA 2004
You are branch manager of Nepal Bank Limited, Balaju. A borrower approaches you for a
term loan of Rs500,000. You agreed to give loan to be fully amortized in a period of 5
year at 10 percent, annual payment. What will be the size of each installment? What
fraction of the payment made at the end of second year represents repayment of interest?
SOLUTION
Here, Loan amount (PVA) = Rs 500,000
Number of years (n) = 5 years
TIME VALUE OF MONEY

87
Chapter 3
Interest rate (i) = 10%
First we determine the annual installment or payment (PMT)
We have,
PVA
Rs 500‚000 Rs 500‚000
PMT = PVIFA = PVIFA
= 3.7908 = Rs 131898.28
i‚ n
10‚ 5
Preparation of Amortization Schedule,
Amortization schedule
Beginning
balance
Year
1
2
Rs500,000
418,101.72
PMT
Rs131,898.28
131,898.28
Interest
Rs50,000
41,810.17
Repayment of
principal
Rs81898.28
90,088.11
Ending
balance
Rs418,101.72
328,013.61
Interest payment in 2nd year Rs 41‚810.17
= Rs 131‚898.28 = 31.7%
Payments
That is 31.7% of the payment in second year represents the interest.
% interest in 2nd year =
Illustration 18
a.
Non annual
compounding
b.
It is now January 1, 2007. You plan to make 5 deposits of Rs 100 each, on every 6
months, with the first payment being made today. If the bank pays a nominal
interest rate of 12 percent, but uses semiannual compounding, how much will be
in your account after 10 years?
Ten years from today you must make a payment of Rs 1,432.02. To prepare for this
payment, you will make 5 equal deposits, beginning today and for the next 4
quarters, in a bank that pays a nominal interest rate of 12 percent, quarterly
compounding. How large must each of the 5 payments be?
SOLUTION
a.
Here,
Number of deposits (n) = 5 deposits;
Semiannual (every 6 months), payment = Rs 100;
Nominal interest rate (i) = 12%,
Present value of annuity (PVA) = ?
0
Rs100
6%
Rs100
1
2
Rs100 Rs100 Rs100
10
FV = ?
We have,
5
(1 + i)n - 1
(1 + 0.06) - 1
i
 (1 + i) = Rs 100  0.06  (1 + 0.06)
= Rs 100 × 5.6371 × 1.06 = Rs 597.5326
Now remaining period is 15 periods (20 periods - 5 periods), so we calculate the
future value of this Rs 597.5326 for remaining periods.
We have,
FV = PV (1 + i)n = Rs 597.5326 (1 + 0.06)15 = Rs 1,432.02
b.
Here,
Future value at the end of 10 years = Rs 1,432.02;
n = 35 periods because quarterly compounding (in 10 years there are 40 quarters);
Quarterly interest rate = 3%,
PMT = ?, PV = ?
FVA = PMT × 

0
3%
1
PMT = ? PMT = ? PMT = ? PMT = ? PMT = ?
10
FV = Rs1,432.02
88

Chapter 3
BUSINESS FINANCE
We have,
FV
Rs 1‚432.02 Rs 1‚432.02
PV = (1 + i)n = (1 + 0.03)35 = 2.8139 = Rs 508.91
Now we calculate the payment (PMP)
Here, n = 5 periods, i = 3%, PV = ?; FV = Rs 508.91, FVA = Rs 508.91
PMT = ?
We have,
(1 + i)n - 1
FVA = PMT 
 i  (1 + i)
(1 + 0.03)5 - 1
or, Rs 508.91
= PMT 
 0.03  (1 + 0.03)
or, Rs 508.91
= PMT × 5.3091 × 1.03
Rs 508.91

PMT
= 5.4684 = Rs 93.06
Illustration 19
19
Value of an annuity
The prize in last week's Himalayan Lottery was estimated to be worth Rs 35 million. If
you were lucky enough to win, the Himalayan will pay you Rs 1.75 million per year over
the next 20 years. Assume that the first installment is received immediately.
a.
If interest rates are 8 percent, what is the present value of the prize?
b.
If interest rates are 8 percent, what is the future value after 20 years?
c.
How would your answers change if the payments were received at the end of each
year?
SOLUTION
Here, Payment (MPT) = Rs 1.75 million
Number of periods (n) = 20 years,
a.
Present value of annuity (PVA) = ? interest rate (i) = 8%
1
1 - 1 n
1 
20
(1
+
i)
(1
+
0.08)



 (1 + 0.08)
PVA = PMT ×
i
0.08

 (1 + i) = Rs 1.75 

b.
= Rs 1.75 × 9.8181 × 1.08 = Rs 18.56 million
Future value of annuity (FVA) = ?, Interest rate (i) = 8%
(1 + i)n - 1
FVA = PMT 
i

 (1 + i)
(1 + 0.08)20 - 1
0.08

 (1 + 0.08)
= Rs 1.75 
= Rs 1.75 × 45.7620 × 1.08 = Rs 86.49 million
PVA and FVA assuming payments received at the end of year.
Present value of annuity (PVA) = ?, Interest rate (i) = 8%
We have,
1
1 - 1 n
1 
20
(1
+
i)
(1
+
0.08)




PVA = PMT ×
i
0.08

 = Rs 1.75 

c.
= Rs 1.75 × 9.8181 = Rs 17.18 million
Future value of annuity (FV) = ?, Interest rate (i) 8%
(1 + i)n - 1
(1 + 0.08)20 - 1
FVA = PMT 
= Rs 1.75 
i
0.08




= Rs 1.75 × 45.7620 = Rs 80.08 million
TIME VALUE OF MONEY
Illustration 20
Future value of an
annuity

Chapter 3
89
Your client is 40 years old and wants to begin saving for retirement. You advise the client
to put Rs 5,000 a year into the stock market. You estimate that the market's return will be,
on average, 12 percent a year. Assume the investment will be made at the end of the year.
a.
If the client follows your advice, how much money will she have by age 65?
b.
How much will she have by age 70?
SOLUTION
Here, Your client is 40 years old, Payment (PMT) = Rs 5,000, Interest rate (i) = 12%
Investment will be made at the end of the year
a.
Future value of annuity (FVA) at the age of 65?
Number of periods (n) = 65 - 40 = 25 years
25
(1 + i)n - 1
(1 + 0.12) - 1
FVA = PMT 
i
0.12

 = Rs 5,000 

= Rs 5,000 × 133.3338 = Rs 666,669
b.
Future value of annuity (FVA) at the age of 70?
Number of periods (n) = 70 - 40 = 30 years
25
(1 + i)n - 1
(1 + 0.12) - 1
FVA = PMT 
i
0.12

 = Rs , 5,000 

= Rs 5,000 × 241.3327 = Rs 1,206,66
Illustration 21
Solving for payment
Mr. Lamsal has inherited Rs 25,000 and wishes to purchase an annuity that will provide
him with a steady income over the next 12 years. He has heard that the local savings and
loan association is currently paying 6 percent compound interest on an annual basis. If he
were to deposit his funds, what year-end equal rupee amount (to the nearest rupee)
would he be able to withdraw annually such that he would have a zero balance after his
last withdrawal 12 years from now?
SOLUTION
Here, Present value of annuity (PVA) = Rs 25,000
Number of years (n) = 12 years
Interest rate (i) = 6%
Equal annual withdraw (PMT) = ?
1 - 1 n
 (1 + i) 
PVA = PMT ×
i


1
1 
12
(1
+
0.06)


or,
Rs 25,000 = PMT
0.06


or,
Rs 25,000 = PMT × 8.3838

PMT = Rs 2,981.9414
Illustration 22
Solving for payment
You need to have Rs 50,000 at the end of 10 years. To accumulate this sum, you have
decided to save a certain amount at the end of each of the next 10 years and deposit it in
the bank. The bank pays 8 percent interest compounded annually for long term deposits.
How much will you have to save each year (to the nearest rupee)?
SOLUTION
FVA
or, Rs 50,000
or, Rs 50,000
 PMT
(1 + i)n - 1
 i

10 - 1
(1
+
0.08)

= PMT 
0.08


= PTM × 14.4866
Rs 50‚000
= 14.4866 = Rs 3,451.46
= PMT 
90

Chapter 3
Illustration 23
Annual interest rate
BUSINESS FINANCE
Mrs. Karki wishes to borrow Rs 10,000 for three years. A group of individuals agrees to
lend her this amount if she contracts to pay them Rs 16,000 at the end of the three years.
What is the implicit compound annual interest rate you receive (to the nearest whole
percent)?
SOLUTION
Here, Present value (PV) = Rs 10,000
Number of year (n) = 3 years
Future value (FV) = Rs 16,000
End payment, interest rate (i) = ?
We have,
FV = PV (1 + i)n
or, Rs 16,000 = Rs 10,000 (1 + i)3
or, 1.6 = (1 + i)3
or, (1.6)1/3 - 1 = 1
or, i = 0.1695 or 16.95%
Illustration 24
Solving for interest
rate
SOLUTION
You have been offered a note with four years to maturity, which will pay Rs 3,000 at the
end of each of the four years. The price of the note to you is Rs 10,200. What is the implicit
compound annual interest rate implied by this contract (to the nearest whole percent)?
Here, Payment (PMT) = Rs 3,000 per year
End payment, years (n) = 4 years
Present value (PV) = Rs 10,200
Interest rate (i) = ?
We have,
1 - 1 n
 (1 + i) 
PVA
= PMT ×
 i

or, Rs 10,200
= Rs 3,000 × PVIFAi%, 4 years
or, Rs 10,200/Rs 3,000 = PVIFAi%, 4 years
or, 3.4
= PVIFAi%, 5 years
According to PVIFA table the value of 3.4 at 4-year lies between 6% and 7%. For the actual
expected return interpolate between these two rates.
Actual factor - Factor at lower rate
Actual expected return = Low rate + Factor at higher rate - Factor at lower rate
3.4 - 3.4651
= 6% + 3.3872 - 3.4651 = 6.835%
Illustration 25
PV of uneven cash
flow stream
The Sriram Brick Company is considering the purchase of a debarking machine this is
expected to provide cash flows as follows:
End of Year
Year
1
2
3
4
5
6
7
8
9
10
Cash flow
Rs 1,200
2,000
2,400
1,900
1,600
1,400
1,400
1,400
1,400
1,400
If the appropriate annual discount rate is 14 percent, what is the present value of this cash
flow stream?
SOLUTION
Calculation of present value
Year
1
2
Cash flow
Rs 1,200
2,000
PVIF at 14%
0.8772
0.7695
PV
Rs 1,052.64
1,539

TIME VALUE OF MONEY
3
4
5
6-10
2,400
1,900
1,600
1,400
Chapter 3
0.6750
0.5921
0.5194
1.7830*
Total present value
*PVIFA for 6-10
Illustration 26
91
1,620
1,124.99
831.04
2,496.2
Rs 8,663.87
= PVIFA for 10 years - PVIFA for 5 years
= 5.2161 - 3.4331 = 1.7830
The following cash flow stream needs to be analyzed
Cash flow stream
PV and FV of uneven
cash flow stream
T.U. BBA 2004
End of year
1
2
3
X
Rs100
Rs200
Rs200
Rs300
4
Rs300
5
Y
200
0
500
0
300
a.
b.
Calculate the present value of each X cash flow at 14 percent discount rate.
Calculate the future value of each Y cash flow at 10 percent discount rate.
a.
Calculation of present value of cash flow stream X at 14 percent discount rate
S O L U T IO N
Year
Cash flow ‘X’
1
2
3
4
5
14% PVIF
Rs100
200
200
300
300
PV
0.8772
0.7695
0.6750
0.5921
0.5194
Rs87.72
153.90
135.00
177.63
155.82
Total present value
b.
Rs710.07
Calculation of future value of cash flow Y at 10 percent compounding rate
Year
Cash flow ‘Y’
1
2
3
4
5
10% FVIF = (1 + i)n - t
Rs200
0
500
0
300
FV
(1.1)4
= 1.4641
(1.1)3 = 1.3310
(1.1)2 = 1.2100
(1.1)1 = 1.1000
(1.1)0 = 1.0000
Rs292.82
0
605.00
0
300.00
Total future value
Illustration 27
Uneven cash flow
stream
T.U. BBA 2005
SOLUTION
Rs1197.82
Calculate the present value of the following cash flow stream. Assume that the stated rate
of interest is 14 percent per annum discounted semiannually.
Cash flow
End of year
1000
1600
0
1500
1
850
3
2
If stated annual rate is 14 percent, discounted semiannually, first we calculate the effective
annual rate as follows:
Effective interest rate (EAR) = (1 + 0.14/2)2 - 1 = 14.49%
Now present value of given cash flow stream discounted at 14.49 percent effective annual
rate is calculated as follows:
Year
0
1
2
3
Cash flow
14.49% PVIF
Rs1,000
1,600
1,500
850
Total present value
1.0000
0.8734
0.7629
0.6663
PV
Rs1,000.00
1,397.44
1,144.35
566.36
Rs4,108.15
92
Chapter 3

BUSINESS FINANCE
SUMMARY
Time value of money is a concept to understand the value of cash flow occurred at different
point in time. Financial decisions concerned with business firm require a consideration
regarding time value of money. Maximizing shareholder wealth, to a larger extent, depends
on the timing of cash flows from investment alternatives.
Cash flow time line is used to understand the timing of cash flow. Future value of a
sum of money is defined as the total of the sum of the money plus the stream of interest
amount received for the period of time, the money invested. The process of finding future
value is called 'compounding'.
Present value of a future sum of money is the amount of current money that is
equally desirable today, against a specified amount of money to be received or paid at a
future date. The process of finding present value is called discounting.
Annuity is series of equal payment occurred at equal interval of time throughout a
given period. There are two types of annuities- ordinary annuity and annuity due. For
ordinary annuity, the series of equal payment occurs at the end, whereas it occurs, at the
beginnings of each equal interval of time for annuity due. When a series of equal payment
occurs for indefinite period of time it is called a perpetuity.
REVIEW QUESTIONS
Indicate whether the following statements are ‘True’ or ‘False’. Support your
answer with reason:
1.
Rs 100 worth today is equal to Rs 100 worth at the end of year 1.
2.
The process of finding future value is called 'compounding'.
3.
The present value of a future sum of money is the amount of current money
that is equally desirable to a decision maker today against a specified amount
of money to be received or paid at a future date.
4.
Discounting is just reverse of compounding.
5.
An annuity is a series of payment of fixed amount at each specified interval of
time for a given number of periods
6.
If lump sum payment today is equal to the present value of 3-years annuity of
Rs 1000 each year discounted at a given rate of return, we would not be
indifferent in choosing either of the alternatives.
7.
In semi-annual compounding, the compounding periods are doubled.
8.
The loan to be repaid in equal periodic installments is called amortized loan.
9.
The present value of a security that promises to pay Rs 5,000 in 20 years at 7
percent discount rate is Rs 1392.10.
10.
Given the monthly periodic rate of 1 percent, the annual percentage rate is 12
percent.
11.
If monthly periodic rate of interest is 2 percent, the effective annual rate is 24
percent.
TIME VALUE OF MONEY
12.

Chapter 3
93
If your investment doubles in 10 years, you earn approximately 9 percent
return in a year.
Choose the most appropriate answer for the following:
13.
Cash flow time line is a .......... presentation of cash flows associated with
different time period.
a. tabular
b. formula
c.
graphic
d. linear
14.
If we deposit Rs 1000 today at an annual interest rate of 10 percent, it is
compounded to ........... at the end of year 2.
a. Rs 1000
b. Rs 2000
c.
Rs 1210
d. Rs 1410
15.
The present value of Rs 1100 due in year 1, discounted at 10 percent is ..........
a. Rs 1,000
b. Rs 900
c.
Rs 1,100
d. Rs 950
16.
If equal amount of payment occurs at the beginning of each equal interval of
time for the given period, the payment is called ...........
a. an ordinary annuity
b. an annuity due
c.
present value
d. future value
17.
A stream of equal payment occur at equal interval of time to infinity is called
.........
a. present value of annuity
b. future value of annuity
c.
cash flow
d. perpetuity
18.
In a stream of cash flow, if the cash flow in each period differs from period to
period, it is called ...........
a. perpetuity
b. payment
c.
uneven cash flow streams d. amortization schedule
19.
The loan to be repaid in equal periodic installment is called ..........
a. future value
b. present value
c.
long-term loan
d. amortized loan
20.
A sum of money due at some future date is called .........
a. time value
b. intrinsic value
c.
present value
d. future value
QUESTIONS
1.
2.
3.
4.
5.
6.
7.
8.
"A rupee in hand today is worth more than a rupee to receive next year". Explain.
What is discounting? How it is related to compounding?
What do you mean by present value? How it is calculated?
What do you mean by future value? How it is calculated?
What do you mean by cash flow time lie? What does it show? Illustrate with
example.
What is the difference between ordinary annuity and annuity due? Illustrate with
the help of cash flow time line.
What do you mean by perpetuity? How present value of perpetuity is calculated?
Illustrate.
What do you mean by uneven stream of cash flow? Illustrate how the future value
of the uneven stream of cash flow is calculated?
94
Chapter 3
9.
10.

BUSINESS FINANCE
What annuity has a greater future value-an ordinary annuity or an annuity due?
Why? Explain.
What is the difference between annual percentage rate (APR) and effective annual
rate (EAR)? Illustrate with suitable example.
PROBLEMS
3–1
Present and future
values
Calculate present and future values of the following:
a. An initial Rs 500 compounded for 10 years at 6 percent.
b. An initial Rs 500 compounded for 10 years at 12 percent.
c.
The present value of Rs 500 due in 10 years at 6 percent discount rate.
d. The present value of Rs 1552.90 due in 10 years at (i) a 12 percent discount rate,
and (ii) a 6 percent rate. Give a verbal definition of the term present value, and
illustrate it using a cash flow time line with data from this problem. As part of
your answer, explain why present values are dependent upon interest rates.
3–2
Which amount is worth more at 14 percent: Rs 1,000 in hand today or Rs 2,000 due in
6 years?
3–3
To the closest year, how long will it take Rs 200 to double if it is deposited and earns
the following rates?
a.
7 percent.
b. 10 percent
c.
18 percent
d. 100 percent.
Present and future
values
Solving for
number of periods
3–4
Solving for
interest rate or
rate of growth
3–5
Future value of
annuities
3–6
Present value of
annuity
3–7
Finding interest
rates
Shitalnagar Oil Corporation’s 2006 sales were Rs 12 million. Sales were Rs 6 million 5
years earlier (in 2001).
a. To the nearest percentage point, at what rate have sales been growing?
b. Suppose some one calculated the sales growth for MC corporation in part a as
follows: “Sales doubled in 5 years. This represents a growth of 100 percent in 5 years,
so dividing 100 percent by 5, we find the growth rate to be 20 percent per year.”
Explain what is wrong with this calculation.
Find the future value of the following annuities:
a. Rs 400 per year for 10 years at 10 percent.
b. Rs 200 per year for 5 years at 5 percent.
c.
Rs 400 per year for 5 years at 0 percent.
d. Now rework parts a, b and c assuming that payments are made at the beginning
of each year; that is, they are annuities due.
Find the present value of the following ordinary annuities:
a. Rs 400 per year for 10 years at 10 percent.
b. Rs 200 per year for 5 years at 5 percent.
c.
Rs 400 per year for 5 years at 0 percent.
d. Now rework parts a, b and c assuming that payments are made at the beginning
of each year; that is, they are annuities due.
Find the interest rates, or rates of return, on each of the following:
a. You borrow Rs 700 and promise to pay back Rs 749 at the end of 1 year.
b. You lend Rs 700 and receive a promise to be paid Rs 749 at the end of 1 year.
TIME VALUE OF MONEY
c.
d.
3–8
Present value of
perpetuity
3–9
Uneven cash flow

Chapter 3
95
You borrow Rs 85000 and promise to pay back Rs 201229 at the end of 10 year.
You borrow Rs 9000 and promise to make payments of Rs 2684.80 per year for 5
years.
What is the present value of a perpetuity of Rs 100 per year if the appropriate
discount rate is 7 percent? If interest rates in general were to double and the
appropriate discount rate rose to 14 percent, what would happen to the present value
of the perpetuity?
Following are the cash flow streams associated to two investment proposals under
consideration:
Year
1
2
3
4
5
a.
b.
Cash Flow Streams
X
Y
Rs 100
Rs 300
400
400
400
400
400
400
300
100
If the appropriate interest rate is 8 percent, what is the present value of each
cash flow streams?
What is the value of each cash flow stream at a 0 percent interest rate?
3–10 Represent the following cash flows in time line and calculate the present value of the
stream of cash flows at 11 percent discount rate. The cash flow stream is Rs500,000 in
Uneven cash flow
year zero, -Rs200,000 at the end of year 0ne, Rs200,000 at the end of year two,
Rs300,000 at the end of year three and Rs400,000 at the end of year 4.
3–11 Find the amount to which Rs 500 will grow under each of the following conditions:
a. 12 percent compounded annually for 5 years.
Future values at
different
b. 12 percent compounded semiannually for 5 years.
compounding periods
c.
12 percent compounded quarterly for 5 years.
d. 12 percent compounded monthly for 5 years.
3–12 Find the present values of Rs 500 due in the future under each of the following
conditions:
Present values at
different
a. 12 percent simple interest rate, compounded annually.
compounding periods
b. 12 percent simple rate, semiannual compounding, discounted back 5 years.
c.
12 percent simple rate, quarterly compounding, discounted back 5 years.
d. 12 percent simple rate, monthly compounding. Discounted back 1 year.
3–13 You just started your first job, and you want to buy a house within 3 years. You are
currently saving for the down payment. You plan to save Rs 5,000 the first year. You
Future value
also anticipate the amount you save each year will rise by 10 percent a year as your
salary increases over time. Interest rates are assumed to be 7 percent, and all saving
occurs at year end. How much money will you have for a down payment in 3 years?
3–14 Bank A pays 8 percent interest, compounded quarterly, on its money market
account. The manager of Bank B want its money market account to equal Bank A’s
effective annual rate, but interest is to be compounded on a monthly basis. What
simple rate must bank B set?
3–15 The Himalayan Bank pays 7 percent interest, compounded annually, on time
deposits. The NB bank pays 6.5 percent interest compounded quarterly.
Effective annual
rates
96
Chapter 3
a.
b.

BUSINESS FINANCE
Based on effective interest rates, in which bank would you prefer to deposit
your money?
Could your choice of banks be influenced by the fact that you might want to
withdraw your funds during the year as opposed to at the end of the year? In
answering this question, assume that funds must be left on deposit during the
entire compounding period in order for you to receive any interest.
3–16 Mr. Dhakal invested Rs 150000 eighteen months ago. Currently, the investment is
worth Rs 168925. Mr. Dhakal knows the investment has paid interest every three
Effective interest
months, but he does not know what the yield on his investment is. Help Mr. Dhakal.
rate
Compute both the annual percentage rate (APR) and the effective annual rate of
interest.
3–17 A mortgage company offers to lend you Rs 85000; the loan calls for payments of
Rs 8273.59 per year for 30 years. What interest rate is the mortgage company
charging you?
Effective rate of
interest
3–18 You are thinking to buy a car , and a local bank is willing to lend you Rs20,000 to buy
the car. Under the terms of the loan, it will be fully amortized over 5 years (60
Amortized loan
months), and the nominal rate of interest will be 12 percent, with interest paid
monthly. What would be the monthly payment on the loan? What would be the
effective rate of interest on the loan?
3–19 Madhyamanchal Inc. just borrowed Rs 25000. Loan is to be repaid in equal
installments at the end of each of the next 5 years, and the interest rate is 10 percent.
a. Set up an amortization schedule for the loan.
b. How large must each annual payment be if the loan is for Rs 50000? Assume
that the interest rate remains at 10 percent and the loan is paid off over 5 years.
c.
How large must each payment be if the loan is for Rs 50000, the interest rate is
10 percent, and the loan is paid off in equal installments at the end of each of the
next 10 years? This loan is for the same amount as the loan in part b, but the
payments are spread out over twice as many periods. Why are these payments
not half as large as the payments on the loan in part b?
Amortization
schedule
3–20 The management of Campaign for Peace in Nepal Limited decided to buy a printing
press by taking a loan of Rs1,500,000 for 4 years from Peace Cooperative Limited.
The loan bears a compound annual interest of 12 percent and calls for equal annual
installment payments at the end of each of the 4 years.
a. What is the amount of annual payments?
b. Prepare a schedule showing the fraction of interest and principal payment for
each year.
c.
What fraction of payment made in year 2 represents the principal?
d. What fraction of payment made in year 4 represents the interest?
Amortization
schedule
3–21 You are planning to borrow Rs 1,000,000 on a 5-year, 12 percent annual payment
fully amortized term loan. What fraction of payment made at the end of second year
will represent the payment of interest? What fraction of payment made at the end of
third year will represent the repayment of principal? What would be the amount of
interest and principal paid in the final year?
Amortized loan
3–22 Assume that it is now January 1, 2006. On January 1, 2007, you will deposit Rs 1000
into a savings account that pays 8 percent.
Future value
TIME VALUE OF MONEY
a.
b.
c.
d.

Chapter 3
97
If the bank compounded interest annually, how much will you have in your
account on January 1, 2010?
What would your January 1, 2010, balance be if the bank used quarterly
compounding rather than annual compounding?
Suppose you deposited the Rs 1000 in 4 payments of Rs 250 each on January 1 of
2007, 2008, 2009, and 2010. How much would you have in your account on
January 1, 2010, based on 8 percent annual compounding?
Suppose you deposited 4 equal installments in your account on January 1 of
2007, 2008, 2009, and 2010. Assuming an 8 percent interest rate, how large
would each of your payments have to be for you to obtain the same ending
balance as you calculated in part a?
3–23 Krishna Nepal is 63 years old and recently retired. He wishes to provide retirement
income for himself and is considering an annuity contract with the National Life
Time value of
Insurance Corporation. Such a contract pays him an equal rupee amount each year
money
that he lives. For this cash flow stream, he must put up a specific amount of money at
the beginning. According to actuary tables, his life expectancy is 15 years, and that is
the duration on which the insurance company bases its calculations regardless of
how long he actually lives.
a. If the Insurance Company uses a compound annual interest rate of 5 percent in
its calculations, what must Mr. Nepal pay at the outset for an annuity to
provide him with Rs10,000 per year? (Assume that the expected annual
payments are at the end of each of the 15 years)
b. What would be the purchase price if the compound annual interest rate is 10
percent?
c.
Mr. Nepal had Rs30,000 to put into an annuity. How much would he receive
each year if the insurance company uses a 5 percent compound annual interest
rate in its calculation?
3–24 You opened an account in Bank of Kathmandu (BOK). The bank pays interest at the
rate of 3 percent per annum and compounds quarterly.
Time value of
money
a.
If you deposit Rs5,000 now, how much shall it grow at the end of 5 years?
b.
What rate will you earn if the money deposited in the bank account doubles in 5
years?
c.
How long will it take to grow Rs 5,000 to Rs 10,955 if the bank pays interest at 4
percent per annum compounded annually?
d.
Assume that you deposit Rs5,000 at the end of each quarter for 4 years. What
will be the balance in your account at the end of fourth year if the bank pays
interest at 4 percent per annum compounded quarterly?
3–25 Assume that it is now January 1, 2006, and you will need Rs 1000 on January 1, 2010.
Your bank compounds interest at an 8 percent annual rate.
Time value of
money
a. How much must you deposit on January 1, 2007, to have a balance of Rs 1000 on
January 1, 2010?
b. If you want to make equal payments on each January 1 from 2007 through 2010
to accumulate the Rs 1000, how large must each of the 4 payments be?
c.
If your father offered either to make the payments calculated in part b or to give
you a lump sum of Rs 750 on January 1, 2007, which would you choose?
d. If you have only Rs 750 on January 1, 2007, what interest rate, compounded
annually, would you have to earn to have necessary Rs 1000 on January 1, 2010?
98
Chapter 3
e.
f.
g.

BUSINESS FINANCE
Suppose you can deposit only Rs 186.29 each January 1 from 2007 through 2010,
but you still need Rs 1000 on January 1, 2010. What interest rate, with annual
compounding, must you seek out to achieve your goal?
To help you reach your Rs 1000 goal, your mother offers to give you Rs 400 on
January 1, 2007. You will get a part time job and make 6 additional payments of
equal amounts each 6 months thereafter, If all this money is deposited in a bank
that pays 8 percent, compounded semiannually, how large must each of the 6
payments be?
What is the effective annual rate being paid by the bank in part f?
3–26 To complete your last year in business school and then go through law school, you
will need Rs 10000 per year for 4 years, starting next years (that is you will need to
withdraw the first Rs 10000 one year from today). Your rich uncle offers to put you
through school, and he will deposit in a bank paying 7 percent interest a sum of
money that is sufficient to provide the four payments of Rs 10000 each. His deposit
will be made today.
a. How large must the deposit be?
b. How much will be in the account immediately after you make the first
withdrawal? After the last withdrawal?
Present value of
annuity
3–27 Mrs. Rita wants a refrigerator that costs Rs 12000. She has arranged to borrow the
total purchase price of refrigerator from a finance company at a simple interest rate
equal to 12 percent. The loan requires quarterly payments for a period of three years.
If the first payment is due three months after purchasing the refrigerator, what will
be the amount of her quarterly payments on the loan?
Solving for payment
3–28 You are the manager of Nepal Bangladesh Bank, Butwal Branch. Mr. Lamsal, a
government official visits you for advice. He needs to have Rs 500000 at the end of 3
years for his daughter’s enrolment in MBA. To accumulate this sum, he has decided
to deposit certain amount at NB bank at the beginning of every year for 3 years. How
much will Mr. Lamsal have to deposit each year at the NB bank that pays interest at
5 percent compounded annually?
Solving for payment
3–29 While Mr. A.K Chhetri was a student at the Tribhuvan University, he borrowed
Rs 12,000 in student loans at an annual interest rate of 9 percent. If Mr. Chhetri
repays
Rs 1500 per year, how long, to the nearest year, will it take him to repay the loan?
Solving for time
3–30 Suppose you had just celebrated your 19th birthday. A rich uncle set up a trust fund
for you that will pay Rs 100,000 when you turn 25 years. If the relevant discount rate
is 11 percent, how much is this fund worth today?
Solving for present
value
3–31 You have just joined the investment-banking firm of Pandey and Pandey Company.
They have offered you two different salary arrangements. You can have Rs 30,000
Evaluation cash flow
per year for next two years or Rs 20,000 for the next two years, along with a Rs 30,000
signing bonus today. If the interest rate is 12% compounded quarterly, which do you
prefer?
3–32 You need to accumulate Rs 10,000. To do so, you plan to make deposits of Rs 1750
per year, with the first payment being made a year from to day, in a bank account
Solving for time
which pays 6 percent annual interest. Your last deposit will be more than Rs 1750 if
and payments
more is needed to round out to Rs 10,000. How many years will it take you to reach
your Rs 10,000 goal, and how large will the last deposit be?
TIME VALUE OF MONEY

Chapter 3
99
3–33 Mr,. Sharma has Rs 42180.53 in brokerage account, and plans to contribute an
additional Rs 5000 every year at an annual interest rate of 12 percent. If Mr. Sharma
has to accumulate Rs 250,000, how many years will it take for him to reach his goal?
Solving for time
3–34 A 15-year security has a price of Rs 340.4689. The security pays Rs 50 at the end of
each of next 5 years, and then it pays a different cash flow amount at the end of each
Solving for payment
of the following 10 years. Interest rates are 9 percent. What is the annual cash flow
amount between 6 and 15 years?
3–35 Miss Sabita has just own the National Lottery and has three award options to choose
from. She can elect to receive a lump sum payment today of Rs 61 million, 10 annual
end-of-year payments of Rs 9.5 million, or 30 annual end-of-year payments of Rs 5.5
million. If she expects to earn an 8 percent annual return on her investment, which
option should she choose?
Evaluating cash flow
3–36 Mr. Hari, like many college students, recently filled out a credit card application. Not
surprisingly, his application was accepted at a nominal interest rate of 24 percent.
Reaching a financial
Upon receiving the credit card, he purchased a new stereo, which costs him Rs
goal
305.44, with the card. Upon receiving his first bill, he was delighted to learn that the
credit card company only requires a minimum payments of Rs 10 per month.
a. If Hari makes the minimum payment every months, how many months will it
be before the account is completely paid off?
b. If Hari continues to make the minimum payment, how much will the final
payment be? Assume that the last payment occurs at the end of the month,
rather than in the middle of the billing period.
MINI CASE
Case 1:
Mr. Ramesh Maharjan has just completed his Bachelor Degree in Business
Administration from Tribhuvan University. Assume that it is now January 1, 2007,
and he is planning to accumulate Rs 150,000 in January 2012 for his post graduate
study in abroad. Today he is thinking for an investment in a security that pays 12
percent annual interest. His only source of income today is his monthly salary of Rs
10,000 from his job in Standard Chartered Bank, main branch, Kathmandu. Out of his
monthly income he spends 60 percent amount for his living. He has also saved Rs
100,000 in his bank account from his job over the years. In adherence to his plan
about further study, you are required to answer the following:
a. How much must he deposit in lump sum on January 1, 2007, to accumulate a
balance of Rs 150,000 on January 1, 2012?
b. If he wants to make equal payments on each January 1 from 2008 through 2012
to accumulate Rs 150,000, how large must each annual payments be?
c.
If he wants to invest his monthly net saving in the security, the first payment
being made one month from now, how much he could accumulate in January 1,
2012? Assume interest is compounded monthly.
d. If he invests his monthly saving into the security, the first payment being made
today, how many months it would take him to accumulate the sum of Rs
150,000?
100
Chapter 3
e.
f.
g.
h.

BUSINESS FINANCE
What is the effective annual rate of monthly compounded interest rate in part
‘c’?
Instead, if he could invest Rs 25,000 every year starting one year from now, at
what annual rate of interest he could accumulate Rs 150,000 on January 1, 2012?
If his bank balance of Rs 100,000 today pays 10 percent annual interest
compounded quarterly, in how many years he could accumulate required sum
of Rs 150,000?
If his bank balance of Rs 100,000 today pays 10 percent annual interest
compounded quarterly, to which value it will grow on January 1, 2012?
