Chapter 3
Time Value of Money & the Concept of Interest
3.0 Aims and Objective
After completing this unit, students should be able to:
appreciate the concept of the time value of money,
understand the meaning of interest and differentiate between simple and compound interests,
differentiate among a single cash flow, an annuity and uneven cash flows,
understand future and present values of various types of cash flows, and
Understand the techniques and applications of future and present values.
3.1 Introduction
Many decisions in finance involve choices of receiving or paying cash at different time periods. As you
know the goal of a firm is wealth maximization. This goal recognizes the difference in the value of equal
cash flows received at different time periods. So the concept of the time value of money is that money
received now is generally better than the same amount of money received some time later. This is because
there is an opportunity to invest the money we have now and earn a return on it. The time value of money
is very important for taking any financial decision since in a business we are investing huge amounts of
money today in anticipation of uncertain future returns or revenues because capital is not only scarce but
also has cost and cost in simple terms is nothing but the interest.
Suppose you would like to borrow Birr 1, 000 today and return the same after a month without
any interest. Do you think someone is going to lend you Birr 1, 000? Definitely not. If you are
prepared to pay interest of 3% for a month on the borrowed money, people will come forward to
lend you money. The reason is simple money is not available freely and it is capable of earning
interest i.e., Birr 30. It is evident that today’s Birr 1, 000 is equivalent to Birr 1, 030 after a
month. Here Birr 30 is called cost of capital in financial management. Hence, the time value of
money is a very important concept in financial management. It has many applications in
financial decisions like loan settlements, investing in bonds and stocks of other entities,
acquisition of plant and equipment. Therefore, understanding the time value of money concept is
essential to achieve the wealth maximization goal of a firm.
3.2 Time Value of Money
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By experience, we all know that the value of a sum of money received today is more than its value
received after some time this is called time value of money. Conversely, the sum of money received in
future is less valuable than it is today. The present worth of birr received after some time will be less than
a birr received today. Since, a birr received today has more value, individuals, as rational human beings
would naturally prefer current receipt to future receipts. The time value of money is also known as time
preference for money. The time preference for money in business unit normally expressed in terms of rate
of return or more popularly as a discount rate. In a business revenues are spread over a period of time i.e.,
the life of the project. It is nothing but we are trying to calculate the present value versus future value.
The first basic point in the concept of the time value of money is to understand the meaning of
interest. Interest is the cost of using money (capital) over a specified time period. There are two
basic types of interest: simple interest and compound. Simple interest can be understood in two
different ways. One is that simple interest is an interest computed for just a period. If interest is
computed for one period only, the interest is always simple interest. Another way to understand
simple interest is that it is an interest computed for two or more periods whereby only the
principal (original) value would earn interest. In simple interest the previously earned interests
do not produce another interest.
Compound interest, on the other hand, is an interest computed for a minimum of two periods
whereby the previous interests produce another interest for subsequent or next periods. Here both
the principal and previous interests bring additional interest. Though both simple and compound
interest are there, in financial management we are largely interested on compound interest.
3.3 Techniques of present value Vs future values in the concept of Time value of money
3.3.1. Future value
To understand future value, we need to understand compounding first. Compounding is a
mathematical process of determining the value of a cash flow or cash flows at the final period.
The cash flow(s) could be a single cash flow, an annuity or uneven cash flows. Future value (FV)
is the amount to which a cash flow or cash flows will grow over a given period of time when
compounded at a given interest rate. Future value is always a direct result of the compounding
process.
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3.3.1.1.Future Value of a Single Amount
This is the amount to which a specified single cash flow will grow over a given period of time
when compounded at a given interest rate. The formula for computing future value of a single
cash flow is given as:
FVn = PV (1 + i) n
Where:
FVn = Future value at the end of n periods
PV = Present Value, or the principal amount
i = Interest rate per period
n= Number of periods
Or
FVn = PV (FVIFi,n)
Where:
(FVIFi, n) = The future value interest factor for i and n
The future value interest factor for i and n is defined as (1 + i)n and it is the future value of 1 Birr
for n periods at a rate of i percent per period.
Example: Hana deposited Br. 1,800 in her savings account in September 2000. Her account
earns 6 percent (6%) compounded annually. How much will she have in September 2007? To
solve this problem, let’s identify the given items: PV = Br, 1,800; i = 6%; n = 7 (September
2000– September 2007).
FVn = PV (1 + i)n
= Br. 1,800 (1.06)7
= Br. 2,706.53
The (FVIFi,n) can be found by using a scientific calculator or using interest tables. From the first
table by looking down the first column to period 7, and then looking across that row to the 6%
column, we see that FVIF6%,7 = 1.5036. Then, the value of Br. 1,800 after 7 years is found as
follows:
FVn = PV (FVIFi,n)
FV7 = Br. 1,800 (FVIF6%, 7)
= Br. 1,800 (1.5036) = Br. 2,706.48
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3.3.1.2. Future Value of an Annuity
An annuity is a series of equal periodic rents (receipts, payments, withdrawals, and deposits)
made at fixed intervals for a specified number of periods. For a series of cash flows to be an
annuity four conditions should be fulfilled. First, the cash flows must be equal. Second, the
interval between any two cash flows must be fixed. Third, the interest rate applied for each
period must be constant. Lastbut not least, interest should be compounded during each period. If
any one of these conditions is missing, the cash flows cannot be an annuity.
Basically, there are two types of annuities namely ordinary annuity and annuity due. Broadly
speaking, however, annuities are classified into three types:
i) Ordinary Annuity,
ii) Annuity Due, and
iii) Deferred Annuity
i) Future value of an Ordinary Annuity – An ordinary annuity is an annuity for which the cash
flows occur at the end of each period. Therefore, the future value of an ordinary annuity is the
amount computed at the period when exactly the final (n th) cash flow is made. Graphically,
future value of an ordinary annuity can be represented as follows:
0 1 2 ------------------ n
PMT1 PMT2 ---------------PMTn
The future value is computed at point n where PMTn is made.
[ ]
n
(1+i) − 1
FVAn = PMT i
Where:
FVAn = Future value of an ordinary annuity
PMT = Periodic payments
i = Interest rate per period
n = Number of periods
Or
FVAn = PMT (FVIFAi,n)
Where:(FVIFAi, n) = the future value interest factor for an annuity
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(1+i)n −1
= i
Example: You need to accumulate Br. 25,000 to acquire a car. To do so, you plan to make equal
monthly deposits for 5 years. The first payment is made a month from today, in a bank account
which pays 12 percent interest, compounded monthly. How much should you deposit every
month to reach your goal?
Given: FVAn = Br. 25,000; i = 12% 12 = 1%; n = 5 x 12 = 60 months; PMT= ?
FVAn = PMT (FVIFAi, n)
Br. 25,000 = PMT (FVIFA, %, 60)
Br. 25,000 = PMT (81.670)
PMT = Br. 25,000/81.670
PMT = Br. 306.11
ii) Future value of an Annuity Due. An annuity due is an annuity for which the payments occur
at the beginning of each period. Therefore, the future value of an annuity due is computed
exactly one period after the final payment is made. Graphically, this can be depicted as:
0 1 2 --------------------- n
PMT1 PMT2PMT3 -----------------------PMTn + 1
The future value of an annuity due is computed at point n where PMTn + 1 is made
FVAn (Annuity due) = PMT (FVIFAi, n) (1 + i)
Or
= PMT
[
(1+i)n − 1
i ]
( 1 + i)
Example: Assume that pervious example except that the first payment is made today instead of a
month from today. How much should your monthly deposit be to accumulate Br. 25,000 after 60
months?
FVAn (Annuity due) = PMT (FVIFAi, n) (1 + i)
Br. 25,000 = PMT (FVIFAi, n) (1 + i)
Br. 25,000 = PMT (81.670) (1.01)
PMT = Br. 25,000/82.487
PMT = Br. 303.08
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iii) Future value of Deferred Annuity is an annuity for which the amount is computed two or
more period after the final payment is made.
0 1 2 ------------------n --------------n + x
PMT1 PMT2PMTn
The future value of a deferred annuity is computed at point n + x
FVAn (Deferred annuity) = PMT (FVIFAi, n) (1 + i)x
= PMT
[
(1+i)n − 1
i ]
(1 + i)x
Where x = the number of periods after the final payment; and X 2.
Example: Bright man has a savings account, which he had been depositing Br. 3,000 every year
on January 1, starting in 1990. His account earns 10% interest compounded annually. The last
deposit Bright man made was on January 1, 1999. How much money will he have on December
31, 2003? (No deposits are made after January 1999).
Jan.
1990 1991 1992 93 94 95 96 97 98 99 2000 2001 02 03 2004
3,000 3,000 3,000 3,000 3,000 3,000 3,000 3,0003,000 3,000
The future value is computed on December 31, 2003 (or January 1, 2004).
Given: PMT = Br. 3,000; i = 10%; n = 10; x = 5
FVAn (Deferred annuity) = PMT (FVIFAi, n) (1 + i)x
= Br. 3,000 (FVIFA 10%, 10) (1.10)5
= Br. 3,000 (15.937) (1.6105)
= Br. 76, 999.62
3.3.2 Future Value of Uneven Cash Flows
Uneven cash flow stream is a series of cash flows in which the amount varies from one period to
another. The future value of an uneven cash flow stream is computed by summing up the future
value of each payment.
Example: Find the future value of Br. 1,000, Br. 3,000, Br. 4000, Br. 1200, and Br. 900
deposited at the end of every year starting year 1 through year 5. The appropriate interest rate is
8% compounded annually. Assume the future value is computed at the end of year 5.
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0 1 2 3 4 5
1,000 3,000 4,000 1,200 900
FVIF8%, 4 Br. 1,000 (1, 3605) = Br. 1,360.50
FVIF8%, 3 Br. 3,000 (1.2597) = 3,779.10
FVIF8%,2 Br. 4,000 (1.1664) = 4,665.60
FVIF8%, 1 Br. 1,200 (1.0800) = 1,296.00
Br. 900 (1.0000) = 900.00
FV = Br. 12,001.20
3.4 Present Value
Present value is the exact reversal of future value. It is the value today of a single cash flow, an
annuity or uneven cash flows. In other words, a present value is the amount of money that should
be invested today at a given interest rate over a specified period so that we can have the future
value. The process of computing the present value is called discounting.
3.4.1 Present Value of a Single Amount
It is the amount that should be invested now at a given interest rate in order to equal the future
value of a single amount.
( )
n
FVn 1
= FVn
PV = ( 1+i )
n 1+i
Where:
PV = Present Value
FVn = Future value at the end of n periods
i = Interest rate per period
n = Number of periods
Or
PV = FVn (PVIFi, n)
Where:
(PVIFi, n) = The present value interest factor for i and n = 1/ (1 + i)n
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Example:Finfine PLC owes Br. 50,000 to Sebeta Company at the end of 5 years. Sebeta
Company could earn 12% on its money. How much should Sebeta Company accept from Finfine
PLC as of today? Given: FV5 = Br. 50,000; n = 5 years; i = 12%; PV =?
PV = FV5 (PVIF12%, 5) = Br. 50,000 (0.5674) = Br. 28,370
3.4.2 Present Value of an Annuity
i) Present value of an Ordinary Annuity is a single amount of money that should be invested
now at a given interest rate in order to provide for an annuity for a certain number of future
periods.
[ ]
1
1−
PVAn = PMT
( 1+i )n
i
= PMT [
1− ( 1 + i )−n
i ] = PMT (PVIFAi, n)
Where:
PVAn = The present value of an ordinary annuity
(PVIFAi, n) = The present value interest factor for an annuity
1− ( 1 + i )−n
= i
Example:Mr Samuel retired as general manager of Adonai Foods Company. But he is currently
involved in a consulting contract for Br. 35,000 per year for the next 10 years. What is the
present value of Samuel’s consulting contract if his opportunity costs are 10%?
Given: PMT = Br. 35,000; n = 10 years; i = 10%; PVAn=?
PVA10 = Br. 35,000 (PVIFA10%, 10)
= Br. 35,000 (6.1446) = Br. 215,061. This means if the required rate of return is
10%, receiving Br. 35,000 per year for the next 10 years is equal to receiving Br. 215,061 today.
ii) Present value of an Annuity Due – is the present value computed where exactly the first
payment is to be made. Graphically, this is shown below:
0 1 2 3 ---------------- n
PMT1 PMT2 PMTn
The present value of an annuity due is computed at point 1 while the present value of an ordinary
annuity is computed at point 0.
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PVAn = (Annuity due) = PMT i [
1 −(1+i)−n
]
(1 + i) = PMT (PVIFAi, n) (1 + i)
Example: Ruth Corporation bought a new machine and agreed to pay for it in equal installments
of Br. 5,000 for 10years. The first payment is made on the date of purchase, and the prevailing
interest rate that applies for the transaction is 8%. Compute the purchase price of the machinery.
Given: PMT = Br. 5,000; n = 10 years; i = 8%; PVAn (Annuity due) =?
PVA (Annuity due) = Br. 5,000 (PVIFA 8%, 10) (1.08)
= Br. 5,000 (6.7101) (1.08) = Br. 36,234.54. So the cost of the
machinery for Ruth is Br. 36,234.54. We have identified the case as an annuity due rather than
ordinary annuity because the first payment is made today, not after one period.
iii) Present value of a Deferred Annuity is computed two or more periods before the first
payment is made.
PVAn (Deferred annuity) = PMT i [
1 −(1+i)−n
]
(1 + i)-x = PMT (PVIFAi, n) (1 + i)-x
Where x is the number of periods between the date when he first payment is made and the date
the present value is computed.
Example:Sifan Chartered Accountants has developed and copyrighted an accounting software
program. Sifanagreed to sell the copyright to Steel Company for 6 annual payments of Br. 5,000
each. The payments are to begin 5 years from today. If the annual interest rate is 8%, what is the
present value of the six payments?
0 1 2 3 4 5 6 7 8 9 10
PVAn= ? 5,000 5,000 5,000 5,000 5,000 5,000
X
Given: n = 6; PMT = Br. 5,000; X = 4; PVA6 (Deferred annuity) =?
i = 8% PVA6 (Deferred annuity) = Br. 5,000 (PVIFA8%, 6) (1.08)-4
= Br. 5,000 (4.6229) (0.7350) = Br. 16,989.16
3.4.3 Present Value of Uneven Cash Flows
The present value of an uneven cash flow stream is found by summing the present values of
individual cash flows of the stream.
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Example: Suppose you are given the following cash flow stream where the appropriate interest
rate is 12% compounded annually. What is the present value of the cash flows?
Year 1 2 3
Cash flow Br. 400 Br. 100 Br.300
Br. 400 (0.8929) PVIF12%, 1
= Br. 357.16
Br. 100 (0.7972) PVIF12%, 2
= Br. 79.72
Br. 300 (0.7118) PVIF12%, 3
= Br. 213.54
Br. 650.42
3.4.5. Present Value of a Perpetuity
Perpetuity is an annuity with indefinite cash flows. In perpetuity payments are made
continuously forever. The present value of perpetuity is found by using the following formula:
PV (Perpetuity) = Payment = PMT
Interest rate i
Example: What is the present value of perpetuity of Br. 7,000 per year if the appropriate
discount rate is 7%?
Given: PMT = Br. 7,000; i = 7%;, PV (Perpetuity) = ?
PV (Perpetuity) = PMT = Br. 7,000 = Br. 100,000.
i 7%
This means that receiving Br. 7,000 every year forever is equal to receiving Br. 100,000 now.
3.5. Summary
In our day-to-day life we prefer possession of a given amount of cash now, rather than the same
amount at future time. This is time value of money or time preference for money, which arises
because of (a) uncertainty of cash flows (b) subjective preference for consumption and (c)
availability of investments. The last justification is the most sensible justification for the time
value of money. Interest rate or time preference rate gives money its value and facilitates the
comparison of cash flows accounting at different time periods. Two alternative procedures can
be used to find the value of cash flows: compounding and discounting. In compounding, future
values of cash flows at a given interest rate at the end of a given period of time are found. So,
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future value is direct result of compounding process. Future value can be computed for a single cash
flow, an annuity, and an uneven cash flow stream. Present value is a direct result of the discounting
process. Present value can be computed for a single cash flow, an annuity, uneven cash flows,
and perpetuity. Formulas are available for computing both present values and future values of
various types of given cash flows. A scientific calculator or interest tables can be applied to solve
time value of money questions. The longer time period and the higher the interest rate, the larger
the future value. But the opposite is true for present values. The concepts and techniques of the
time value of money have many applications in financial management.
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