Chapter Three: The Time Value of Money and the Concept of Interest

In any economy in which individuals, firm and government have the time preference, the time
value of money is an important concept. The decision to purchase new plant and equipment or to
introduce a new product in the market requires using capital allocating or capital budgeting
techniques. Essentially, we must determine whether future benefits are mathematical tools of the
time value of money as the first step towards making capital allocating decisions.
The principles of time value analysis have many applications, ranging from setting up schedules
for paying off loans to decisions about whether to acquire new equipment. In fact, of all the
concepts used in finance, none is more important than the time value of money, also called
discounted cash flow (DCF) analysis. Since this concept is used throughout the remainder of the
chapter, it is vital that you understand the material in this chapter before you move on to other
topics.
3.1 The concept of time value of money
The value of money depends on time. The value of a given amount of money at one point in time is not
the same as the value of the same face amount at another time. Thus, the value of money is dependent on
the point of time it occurs at payment or receipt.
It is the expression of the fact that, if the rate of interest is positive, the money in hand now is worth more
than money to be received at a date in the future. It refers to the value derived from the use of money over
time as a result of investment and reinvestment. It is the concept that an amount in hand today is worth
more than the same amount that will be received in future year. This is because; the amount could be
deposited in an interest-bearing bank account (or otherwise invested) from now to time "t" and yield
interest. Similarly, it means that, cash paid out later is worth less than a similar sum paid at an earlier
date.
Key Terms
- Amortization- Repayment of principal and interest earned on an interest-bearing loan (e.g.
Residential mortgage) by a series of equal payments made at equal intervals of time.
- Annuity- A series of payments usually equal in size, made at equal periodic time intervals.
In an ordinary annuity, payments or receipts occur at the end of each period; in an annuity
due, payments or receipts occur at the beginning of each period
- Compound interest - Interest paid (earned) on any previous interest earned, as well as on the
principal borrowed (lent). i.e Interest that is calculated on the principal and accumulated
interest. Every time interest is calculated it is added onto the principal.

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- Discounting- Finding the present value (also known as discounted value or proceeds) of a
future sum of money.
- Effective rate of interest -The annual rate of interest that earns the same amount of interest
per year as a nominal rate compounded a number of times per year. (Note: for a given
nominal annual rate of interest the effective rate of interest increases as the number of
conversion periods per year increases).
- Equivalent rates - Interest rates that accumulate a given principal to the same future value
over the same period of time.
- Face value - The amount that must be paid on the legal due date for a note.
- Focal date -A specific time (for example, now) chosen to compare one or more dated values
of money.
- Future value (also called maturity value) - Amount of money that includes the principal
plus the interest earned over a given period of time at a given interest rate.
- Legal due date (also called date of maturity)- The date on which a promissory note is to be
paid.
- Nominal rate of interest - The stated annual (or yearly) rate of interest charged on the
principal.
- Periodic rate of interest - The rate of interest per compounding period.
- Present value -The principal that grows to a specific future value of money over a given
period of time based on a given interest rate.
- Rate money is worth - the prevailing rate of interest.
- Simple Interest – interest that applies only to the principal (and NOT on accumulated
interest).
- Treasury bills (T- bills) – promissory notes issued by the federal government and some
provincial governments. T-bills are issued at a discount from their face value.
3.2. Simple Interest vs. Compound Interest
Interest is Money paid (earned) for the use of money. It is the growth in a principal amount
representing the fee charged for the use of money for specified time period.
A. Simple Interest: is the return on a principal amount. It is based on the assumption that
interest itself does not earn a return, but this kind of situation occurs rarely in the business
world. In other words Simple interest is interest that is paid (earned) on only the original
amount, or principal, borrowed (lent).
SI = P x I x n

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 Where SI = simple interest
P0 = principal, or original amount borrowed (lent) at time period 0
i = interest rate per time period
n = number of time periods
Example 1: assume that you deposit birr 100 in a savings account paying 8 percent simple interest and
keep it there for 10 years. At the end of 10 years, the amount of interest accumulated is determined as
follows:
SI = 100 x 0.08 x 10=80
Example 2: Suppose that Br. 200,000 is invested at 20% simple interest per annum. The
following table shows the state of the investment, year by year.
Cumulative Amount Interest Earned Amount Principal Year
240,000 40,000 (20% of 200,000) 200,000 1
280,000 40,000 (20% of 200,000) 200,000 2
320,000 40,000 (20% of 200,000) 200,000 3

B. Compound Interest: Interest paid (earned) on any previous interest earned, as well as on
the principal borrowed (lent). It is the return on a principal amount for two or more time
periods, assuming that the interest for each time is added to the principal amount at the
end of each period and earns interest in all subsequent periods. Assuming the above case,
the compounded amount would be as indicated on the following table.
Cumulative Amount Interest Earned Amount Principal Year
240,000 40,000 (20% of 200,000) 200,000 1
288,000 48,000 (20% of 240,000) 240,000 2
345,600 57,600 (20% of 288,000) 288,000 3
3.2 The future value (compounding)
Future value (FV) is the value of a current asset at a future date based on an assumed rate of
growth. The future value is important to investors and financial planners, as they use it to estimate
how much an investment made today will be worth in the future.
It is the value at some future time of a present amount of money, or a series of payments, evaluated
at a given interest rate.
3.2.1. Future Value of Single Amount: The accumulated amount of a single amount invested, at
compound interest may be computed period by period by a serious of multiplications.
Example: Suppose your father gives you 10,000 on your eighteenth birthday. You deposited this
amount in a bank at 8 per cent compounded quarterly for one year. How much future sum would

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you receive after one year?
Solution
If n is used to represent the number of periods that interest is to be compounded, `i` is used to
represent the interest per period, and `p` is the principal amount invested, the series of
multiplications to compute the amount is:
a = P (1+i) n
a= 10,000 (1 + 0.02)4
a= 10,824.32
Example: Hana deposited Br. 1,800 in her savings account in her saving account compounded
annually. Her account earns 6% compounded annually. How much will she have After 7 years?
To solve this problem, let’s identify the given items: PV = Br, 1,800; i = 6%; n = 7 (Meskerem
1990-1997).
FVn = PV (1 + i)n
= Br. 1,800 (1.06)7
= Br. 2,706.53
On the other hand the (FVIF i,n) is founded by 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:
FVIFi= the future value of $1 left on deposits for n periods at rate of i percent per period
FVn = PV (FVIFi,n)
FV7 = Br. 1,800 (FVIF6%, 7)
= Br. 1,800 (1.5036)
= Br. 2,706.48
3.2.2. Annuities
A series of equal payments or receipts occurring over a specified number of periods. If payments
occur at the end of each period, as they typically do, the annuity is called an ordinary, or
deferred, annuity. If payments are made at the beginning of each period, the annuity is an annuity
due.
Many measurement situations involve periodic deposits, receipts, withdrawals, or payments (called
rents), with interest at a stated rate compounded at the time that each rent is paid or received.
These situations are considered annuities if all the following conditions met:
─ The periodic rents are equal in amount.
─ The time period between rents is constant, such as a year a quarter of a year, or a month.

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 ─ The interest rate per time period remains constant.
─ The interest is compounded at the end of each time period.
3.2.3. Future Value of Ordinary Annuity
The future value of an ordinary annuity is the aggregate sum of the future value of each individual
payment on the final rent is made. If we assume an investor who wants to know how much is the
value of just birr 1 deposit to be made at the end of each of the coming five years at 10 %
compounded annually, the first payment earns an interest for four years ( n-1) years because, it is
made at the end of the first year. So, it is late to earn anything in the first year. The second
payment earns interest for 3 (n-2) years. And the last payment does not earn anything because it is
made just at the end of the last year. Thus, for ordinary annuity of n payments, there is n-1
compounding periods.
The Future value of Ordinary Annuity is calculated by the following formula:
FVOA= R [(1+i) n -1]
i
Or FVOAn= PMT (FVIFA i, n)
Where:
FVOA = is the future value of ordinary annuity
R = Periodic payments
i = Interest rate
n = Periods for which rent is made
Example: Ato Minyahil wish’s to determine the sum of money he will have in his saving account
at the end of 6 years by depositing birr 1,000 at the end of each year for the next 6 years at an
annual interest rate of 8 per cent .
Required:
1. Determine the amount.
2. Prepare fund accumulation table
Solution: Given
R= 1000
i=8%
n=6
n
1. FVoA = R [(1+i) -1]
i
6
= 1000 [(1+ 0.08) -1] = 7,336
0.08
2. Fund accumulation table

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Fund balance Increase in fund Interest earned Annual Deposit Date
balance December 31
1,000.00 1,000.00 - 1,000 Year 1
2,080.00 1,080.00 80.00 1,000 Year 2
3,246.40 1,166.40 166.40 1,000 Year 3
4,506.10 1,259.70 259.70 1,000 Year 4
5,866.60 1,360.50 360.50 1,000 Year 5
7,336.00 1,469.30 469.33 1,000 Year 6

3.2.4. Future Value of Annuity Due

The future value of annuity due is the total amount on deposit one period after the final rent is
made. It assumes periodic rents occur at the beginning of the period. The future value of annuity
due can be given by either of the following formulas:
FVAD= R [(1+i) n -1](1+i) or
i
FVADn=PMT (FVIFAi,n )(1+i) or
n+1
FVAD= R [(1+i) - 1] - R
i
Where: FVAD- Future Value of Annuity Due
R= Periodic payments
i=Interest rate
n=Periods for which rent is made
Example: Abera deposits birr 5,000 on January 1 of each year for the coming eight years in to an
account paying 9 % compounded annually. How much will be in his account by the end of the year
8?
Solution
FVAD= 5,000 [(1.09)8-1] * (1.09)
0.09
= 5,000 (11.0285) *(1.09) = 60,105.00 or
8+1
FVAD = 5,000 [(1.09) -1] - 5,000
0.09
= 5,000 [13.021] –5,000 = 65,105- 5,000 = 60,105.00
3.3 The present value (Discounting)
Present value is the value right now of some amount of money in the future. For example, if you
are promised $110 in one year, the present value is the current value of that $110 today. It is the
current value of a future sum of money or stream of cash flows given a specified rate of return.

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Present value states that an amount of money today is worth more than the same amount in the
future.
In other words, present value shows that money received in the future is not worth as much as an
equal amount received today. Unspent money today could lose value in the future by an implied
annual rate due to inflation or the rate of return if the money was invested.
Calculating present value involves assuming that a rate of return could be earned on the funds
over the period. Present value is calculated by taking the expected cash flows of an investment
and discounting them to the present day.
3.3.1. Present Value of Single Amount
It is a method of assessing the worth of an investment by investing the compounding process to
give the present value of future cash flows. This process is "discounting". It is obtained as
follows:

P= A or P= A (1+i) -n
(1+i) n
P= present value of the amount
A= Future Value
n= period of payment
i= rate of interest.

Example: Ato Bekalu has been given the opportunity to receive birr 10,000 four years from
now. If he can earn 6 % on his investment, what is the amount that would make him indifferent if
he is to receive the amount as of today?
Solution
P= A = 10,000 = 10,000 = 7,921
(1+i) n (1+0.06)4 1.26428

This means that, if Bekalu deposited birr 7,921 in to the bank at interest rate of 6 per cent, he
will get birr 10,000 at the end of 4 years.
3.3.2. Present Value of Ordinary Annuity
The present value of ordinary annuity is the discounted value of a series of future rents on the
date one period before the first rent is made. It can be calculated by using the following formula:
PVOA = R [1- (1+i)-n]
I
Where: PVOA = Present Value of Ordinary Annuity
R= Periodic Payments

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 i = Interest Rate
n= Periods for which Rent is Made
PVOAn=PMT (PVIFAi,n)
Example: ABC Corporation purchased Machinery on January 1 2001 and agreed to pay for the
purchase payment of birr 5,000 each including principal and interest on December 31 of each of
the next five year beginning December 31, 2001. The agreed interest rate is 6 per cent
compounded annually.
Required:
A. Compute the price of the machinery.
B. Prepare liability table for the purchase showing periodic interest charges
Solution
A. Price of the machinery
PVOA= R [1- (1+i)-n]
i
= 5,000 [1-(1.06)-5] = 21,061.82
0.06
Therefore, the price of the machinery is birr 21,061.82
B. Debt Payment Program/Liability table
Debt Net Reduction in Payment at Interest
Balance Debt End of the year 6 %/Year Date
21,061.82 - - - Jan 1,01
17,325.53 3,736.29 5,000 1,263.71 Dec 31,01
13,365.06 3,960.47 5,000 1,039.53 Dec 31,02
9,166.96 4,198.10 5,000 801.90 Dec 31,03
4,716.98 4,449.98 5,000 550.02 Dec 31,04
0 4,716.98 5,000 283.02 Dec 31,05

3.3.4. Present Value of Annuity Due

The present value of annuity due is the discounted value of a series of future rents on the date the
first rent is received or paid. It can be calculated by using either of the following formulas:
PVAD= R [1- (1+i)-n] (1+i) or
i
PVAD= R [1- (1+i) – (n-1)] +R
i
PVADn=PMT (PVIFAi,n)(1+i)
Where: PVAD= Present Value of annuity Due

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 R= Periodic payments
i = Interest rate
n= Periods for which rent is made
Example -XYZ Company acquired office equipment on January 1, 1999 and agreed to pay for
the purchase with three installments of birr 5,000 each including principal and interest, every
year beginning January 1, 1999. The interest rate was 12 per cent compounded annually.
Required:
A. Compute the acquisition cost of the equipment.
B. Prepare a liability table showing periodic interest charges and payment
A. PVAD= R [1- (1+i)-n] (1+i)
i
= 5,000 [(1- (1.12)-3] (1.12)
0.12
= 13,450.26
B. Liability Table
Ending Dent Interest Balance Accruing Payment at Beginning Debt Lear
Balance Interest Beginning Balance
9,464.29 1,014.03 8,450.26 5,000 13,450.26 1999
5,000 535.71 4,464.29 5,000 9,464.29 2000
0 0 0 5,000 5,000 2001

3.4. Determining the Interest Rate

If birr 1,000 is deposited at compound interest on January 1 year 1, and the amount on deposit on
December 31,year 10, is birr 1,806.11, what is the semiannual interest rate accruing on the
deposit?
Solution
The amount of 1 for 20 periods at the unstated rate of interest is birr 1,806.11
So, I= 1,806.11
1,000
= Birr 1.80611
Referring the future value table, 1.80611 is the amount of 1 for 20 periods at 3 %. Therefore, the
semiannual interest rate is 3%.

THE END