-simple-interest-and-simple-discount

Simple Interest
and Simple Discount

Learning Objectives
Money is invested or borrowed in thousands of transactions every day. When an investment
is cashed in or when borrowed money is repaid, there is a fee that is collected or charged.
This fee is called interest.
In this chapter, you will learn how to calculate interest using simple interest. Although
most financial transactions use compound interest (introduced in chapter 2), simple interest
is still used in many short-term transactions. Many of the concepts introduced in this chapter
will be used throughout the rest of this book and are applicable to compound interest.
The chapter starts off with some fundamental relationships for calculating interest and
how to determine the future, or accumulated, value of a single sum of money invested
today. You will also learn how to calculate the time between dates. Section 1.2 introduces the
concept of discounting a future sum of money to determine its value today. This “present
value of a future cash flow” is one of the fundamental calculations underlying the
mathematics of finance. Section 1.3 introduces the concept of the time value of money.
This section also intro- duces equations of value, which allow you to accumulate or
discount a series of financial transactions and are used to solve many problems in financial
mathematics. Section 1.4 intro- duces two methods that are used to pay off a loan through a
series of partial payments. The chapter ends with section 1.5 in which a less common
form of simple interest — simple
discount — is introduced along with the concept of discounted loans.




-simple-interest- Simple Interest
and-simple-
In any financial transaction, there are two parties involved: an investor, who is
discount lending money to someone, and a debtor, who is borrowing money from the
Section 1.1 investor. The debtor must pay back the money originally borrowed, and also
the fee charged for the use of the money, called interest. From the investor’s
point of view, interest is income from invested capital. The capital originally
invested in an interest transaction is called the principal. The sum of the
principal and interest due is called the amount or accumulated value. Any
interest transac- tion can be described by the rate of interest, which is the
ratio of the interest earned in one time unit on the principal.

,2 MATHEMATICS OF FINANCE




In early times, the principal lent and the interest paid might be tangible
goods (e.g., grain). Now, they are most commonly in the form of money.
The practice of charging interest is as old as the earliest written records of
humanity. Four thousand years ago, the laws of Babylon referred to interest
payments on debts.
At simple interest, the interest is computed on the original principal
during the whole time, or term of the loan, at the stated annual rate of interest.
We shall use the following notation:
P  the principal, or the present value of S, or the discounted value of S, or the
proceeds.
I  simple interest.
S  the amount, or the accumulated value of P, or the future value of P, or the
maturity value of P.
r  annual rate of simple interest.
t  time in years.
Simple interest is calculated by means of the formula

I  Prt (1)

From the definition of the amount S we have
SPI
By substituting for I  Prt, we obtain S in terms of P, r, and t:
S  P  Prt

S  P(1 + rt) (2)


The factor (1  rt) in formula (2) is called an accumulation factor at a
simple interest rate r and the process of calculating S from P by formula
(2) is called accumulation at a simple interest rate r.


We can display the relationship between S and P on a time diagram.
P
SP
d
It
…
0
Alternatively,
S  P(1  rt)
P da
t
…
0
The time t must be in years. When the time is given in months, then

t number of months
 12

, CHAPTER 1 • SIMPLE INTEREST AND SIMPLE DISCOUNT 3


When the time is given in days, there are two different varieties of simple
interest in use:
number of days
1. Exact interest, where t  365
i.e., the year is taken as 365 days (leap year or not).
number of days
2. Ordinary interest, where t  360
i.e., the year is taken as 360 days.

CALCULATION TIP:
The general practice in Canada is to use exact interest, whereas the general practice in the United S




OBSERVATION:
When calculating the number of days between two dates, the most common practice in Canada is to
However, it is easier to assume the opposite when using the table on the inside back cover. That is,




EXAMPLE 1 A loan of $15 000 is taken out. If the interest rate on the loan is 7%, how much
interest is due and what is the amount repaid if
a) The loan is due in seven months;
b) The loan was taken out on April 7 and is due in seven months?
Solution a We have P  15 000, r  0.07 and since the actual date the loan was taken
7
out is not given, we use12t  .

, 4 MATHEMATICS OF FINANCE




on
$15 000 ed S?

…
0 7 months


Interest due, I  Prt  $15 000  0.07 127  $612.50
Amount repaid  Future or accumulated value,
S  P  I  $15 000  $612.50  $15 612.50
Alternatively, we can obtain the above answer in one calculation:
S  P 1  rt   $15 000[1  0.07(
12
7 )]  $15 612.50
Solution b Since a date is given when the loan was actually taken out, we must use days.
Seven months after April 7 is November 7. Using the table on the inside back
cover, we find that April 7 is day 97 and November 7 is day 311. The exact
number of days between the two dates is 311  97  214. Thus, t .

d
$15 000 ed S?

…
0 214 days

  $615.52
Interest due, I  Prt  $15 000  0.07  
Future value, S  P  I  $15 000  $615.52  $15
615.52
Alternatively,
S  P 1  rt   $15 000[1  0.07( 214 )]  $15 615.52
365




EXAMPLE 2 Determine the exact and ordinary simple interest on a 90-day loan of $8000
at 8 21 %.
Solution We have P  8000, r  0.085, numerator of t  90 days.

Exact interest, I  Prt  $8000  0.085  
 $167.67
 
Ordinary Interest, I  Prt  $8000  0.085  
$170.00


OBSERVATION:
Notice that ordinary interest is always greater than the exact interest and thus it brings increased r