CHAPTER 3
TIME VALUE OF MONEY
3.1 Introduction to Time Value of Money
Ignoring the effects of inflation, a dollar to day is worth more than a dollar to be received a year
from now. In other words, we would all prefer to receive a specific amount of money now rather
than some future date. This preference rests on the time value of money. The term interest is
used to describe the price charged for using money over time. When payments for the time value
of money are made or accrued, interest expense is incurred. When payments for time value of
money are received or accrued, interest revenue is realized. Business decisions often involve
receiving cash or other assets now in exchange for a promise to make payments after one or
more periods.
Time value of money refers to the fact that a dollar in hand today is worth more than a dollar
promised at some time in the future. One reason for this is that you could invest the money
received today and earn interest while you waited; so a dollar today would grow to more than a
dollar later. The tradeoff between money now and money later thus depends on, among other
things, the rate you can earn by investing. This is the interest on your money which is the price
paid for the use of money over time.
3.2 Reason for time value of Money
Why do you think should money have time value?
The following four factors make money to have time value:
(i) Consumption Preference: Individuals prefer current consumption to future consumptions.
So people would have to be offered more in the future to give up current consumption.
(ii) Inflation: Inflation is a general increase in prices that erodes the purchasing power of
money. Hence, the value of money decreases with time when there is inflation.
(iii) Uncertainty (Risk): As compared to today’s money future cash flows have risks (default
risk). Hence, delaying cash collection means assuming greater risks. Individuals want to be
rewarded for this additional risk assumed in relation to uncertainty of future cash flows.
(iv) Investment Opportunities: Cash received today could be invested and fetch additional
income.
Therefore, as money has time value, the same cash flows in different periods have different
values. This makes aggregation and comparison of cash flows at different times illogical unless
adjusted for the above factors. This adjustment is made using discount rates.
To understand future value, we need to recall the idea of compounding. 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.
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 (FVIF i, n)
Where:
(FVIF i, 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: Alem deposited Br. 1,800 in her savings account in September 2008. Her account
earns 6 percent compounded annually. How much will she have in September 2015?
To solve this problem, let us identify the given items: PV = Br, 1,800; i = 6%; n = 7 (September
2008 – September 2015).
FVn = PV (1 + i)n
= Br. 1,800 (1.06)7
= Br. 2,706.53
The (FVIF i, n) can be found by using a scientific calculator or using interest tables given at the
end of financial management books. From such tables, by looking down the first column to
period 7, and then looking across that row to the 6% column, we see that FVIF 6%, 7 = 1.5036.
Then, the value of Br. 1,800 after 7 years is found as follows:
FVn = PV (FVIF i, n)
FV7 = Br. 1,800 (FVIF6%, 7)
= Br. 1,800 (1.5036) = Br. 2,706.48
3.2. Present Value of Money
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.
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 (PVIF i, n)
Where:
(PVIF i, n) = The present value interest factor for i and n = 1/ (1 + i)n
Example: Waliya Company owes Br. 50,000 to Adugna Co. at the end of 5 years. Adugna Co.
could earn 12% on its money. How much should Adugna Co. accept from Waliya Company 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
3.3. Future Value of an Annuity
An annuity is a series of equal periodic rents (receipts, payments, withdrawals, or 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. Last but not least, interest should be compounded in same manner 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
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 ----------------------- PMT n
The future value is computed at point n where PMT n is made.
[ ]
n
( 1+ i ) − 1
FVA n = PMT i
Where:
FVA n = Future value of an ordinary annuity
PMT = Periodic payment
i = Interest rate per period
n = Number of periods
Or
FVA n = PMT (FVIFA i, n)
Where:
(FVIFA i, n) = the future value interest factor for an annuity
( 1+i )n −1
= i
Example 1: JT Company has planned to acquire machinery after five years. To that end, the
company deposits Birr 3,000.00 at the end of each year at a deposit rate of 12%. How much is
the terminal (future value) of the deposits at the end of the fifth year?
Given: FVA n =? i = 12% n = 5; PMT = 3,000
FVA n = PMT (FVIFA i, n)
FVA 5 = 3,000 (FVIFA, 12 %, 5)
FVA 5= 3,000 (6.35284736)
FVA 5 = Birr 19,058.54
Example 2: You need to accumulate Br. 250,000 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: FVA n = Br. 250,000; i = 12% 12 = 1%; n = 5 x 12 = 60 months; PMT =?
FVA n = PMT (FVIFA i, n)
Br. 250,000 = PMT (FVIFA, %, 60)
Br. 250,000 = PMT (81.670)
PMT = Br. 250,000/81.670
PMT = Birr 3,061
(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 PMT2 PMT3 ----------------------- PMT n + 1
The future value of an annuity due is computed at point n where PMT n + 1 is made
FVA n (Annuity due) = PMT (FVIFA i, n) (1 + i)
Or
= PMT
[
( 1+ i )n − 1
i (1 + i)
]
Example: Assume example 1 for ordinary annuity except that the payments are made at the
beginning instead of end of each year. How much is the terminal (future value) of the deposits at
the end of the fifth year?
FVA n (Annuity due) = PMT (FVIFA i, n) (1 + i)
FVA 5 = 3,000 (FVIFA 12%, 5) (1 + 12%)
FVA 5 = 3,000 (6.35284736) (1.12)
FVA 5 = Birr 21, 345.57
(iii) Deferred Annuities
A Delayed Annuity One of the tricks in working with annuities or perpetuities is getting the
timing exactly right. This is particularly true when an annuity or perpetuity begins at a date
many periods in the future. This is called deferred annuity
3.4. 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−
PVA n = PMT
( 1+i )n
i
= PMT [
1− ( 1 + i )−n
i ]
= PMT (PVIFA i, n)
Where:
PVA n = the present value of an ordinary annuity
(PVIFA i, n) = the present value interest factor for an annuity
1− ( 1 + i )−n
= i
Example 1: On January 1, 2008, Tutu Company has borrowed Birr 500,000 by issuing an 8%
note compounded annually to one of its local banks, which is payable Br. 100,000 a year for five
years starting on December 31, year 1. What is the present value of this debt on January 1, year
1? Prepare a loan amortization schedule.
Solution: The present value of the debt on January 1, year 1, is equal to the present value of an
ordinary annuity of five rents reported as Br. 399,271 (Br. 100,000 x 3.99271) in the accounting
records on January 1, year 1.
The repayment program (loan amortization table) for this debt is summarized below:
Tutu Company
Repayment program for Debt of Br. 399,271 at 8% interest
Interest Expense Repayment at Net reduction Debt balance
Date at 8% a year end of year in debt
Jan. 1, year 1 ----- ------ ----- Br. 399,271
Dec. 31, year 1 Br. 31,942 Br. 100,000 Br. 68,058 331, 213
Dec. 31, year 2 26,497 100,000 73,503 257,710
Dec. 31, year 3 20,617 100,000 79,383 178,327
Dec. 31, year 4 14,266 100,000 83,734 92,593
Dec. 31, year 5 7,407 100,000 92,593 –0-
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 ---------------- PMT n
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.
PVA n = (Annuity due) = PMT
[
1 −(1+i)−n
i ]
(1 + i) = PMT (PVIFA i, 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%; PVA n (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.
3.5. Special Case Annuities
3.5.1. Perpetuities
Perpetuity is an annuity with indefinite cash flows. In perpetuity payments are made
continuously forever.
Present Value of Perpetuity
The present value of perpetuity is found by using the following formula:
[
Payment
PV (Perpetuity) = Interest
=
]
PMT
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) =?
[
Payment
PV (Perpetuity) = Interest
=
]
PMT
i =
=
Birr 7 , 000
7% = Br. 100,000.
This means that receiving Br. 7,000 every year forever is equal to receiving Br. 100,000 now,
given that the market discount rate is 7%.
3.6. Uneven Cash Flow Streams
Uneven cash flow stream is a series of cash flows in which the amount varies from one period to
another.
3.6.1. Future Value of Uneven Cash Flows
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. 4,000, Br. 1,200, 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.
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
3.6.2. 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.
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. 213.5
Br. 650.42
