Though we had learnt about Simple and Compound Interests at school, because of the technological advantages and new gadgets over the years we have forgotten how to calculate it. This is my sincere effort to refresh the minds of interested persons about its concepts and how to calculate mannually.
2. What is Interest?
When you borrow Money from someone
Or use somebody else’s Money
You have to pay a service charge to him.
This amount is paid back to the Lender
along with the original amount borrowed.
This is sometimes known as the cost of
Money which doesn’t belong to you, but
you have used it.
31-Jul-2013 2(C) Himansu S M
3. What is Interest?
This extra amount is called the
“INTEREST”
The original amount borrowed is known as
the “PRINCIPAL” or “CAPITAL” in different
situations
The sum of both Principal and the interest
is known as “AMOUNT”
31-Jul-2013 3(C) Himansu S M
4. Types of Interest
There are basically TWO types of Interest
They are:
SIMPLE INTEREST
COMPOUND INTEREST
31-Jul-2013 4(C) Himansu S M
5. Interest Calculation
To estimate or calculate the Interest we
must have the following parameters as
input:
A rate known as the Rate of Interest (RI or
RoI) which is expressed in Percent per
Year
A time period expressed in Years or
Months or Days
31-Jul-2013 5(C) Himansu S M
6. Interest Calculation
The Principal on which the Interest is to be
calculated
And finally the Type of Interest (Methods
of calculation are different)
For advance Business Applications the
“Number of times of Interest Accrual in a
year” is required
This is known as Compounding
31-Jul-2013 6(C) Himansu S M
7. Simple Interest
Simple Interest is dependent on:
Rate of Interest
Time Period
Principal
And the Principal remains the same at the
beginning of all the Periods
It means that the accrual of Interest is
linear
31-Jul-2013 7(C) Himansu S M
8. Compound Interest
Compound Interest is dependent on:
Rate of Interest
Time Period
Principal
And the Principal increases by the interest
amount at the end of each Period
Interest for the next period is calculated on
this increased Principal
31-Jul-2013 8(C) Himansu S M
9. Compound Interest
It means that the Principal plus Interest of
one period becomes the Principal for the
next period
This goes on till the total time period for
which the compound interest is calculated
This Period is called the period of
compounding or the compounding interval
31-Jul-2013 9(C) Himansu S M
11. Compound Interest
It means that the accrual of Interest is
NOT linear, but exponential
The compounding may be
Yearly, Half-Yearly,
Quarterly, Monthly,
Weekly, Daily,
Continuous (Infinitely Compounded)
31-Jul-2013 11(C) Himansu S M
12. Comparison [ @ 10% pa ]
Simple Interest
Year Principal Interest
1 100 10
2 100 10
3 100 10
4 100 10
5 100 10
6 100 10
7 100 10
8 100 10
TOTAL 100 80
Compound Interest
Year Principal Interest
1 100 10
2 110 11
3 121 12.1
4 133.1 13.3
5 146.4 14.6
6 161.1 16.1
7 177.2 17.7
8 194.9 19.5
TOTAL 100 114.5
31-Jul-2013 12(C) Himansu S M
15. Formula for Interest Calculation
Let’s assume:
Principal = P
Amount = A
Total Interest = I
Interest Rate = i expressed in % pa
Time Period = t expressed in Years
Frequency of Compounding = n expressed
in no. Of times in a Year
31-Jul-2013 15(C) Himansu S M
16. Formula for Simple Interest
A = P + I
Example:
If P = 100,
I = 50
Then A = 100 + 50 = 150
31-Jul-2013 16(C) Himansu S M
17. Formula for Simple Interest
I = P * t * i / 100
Example:
If P = 150
i = 12 % pa,
And t = 3 Yrs
Then I = 150 * 3 * 12/100 = 54
31-Jul-2013 17(C) Himansu S M
18. Formula for Simple Interest
A = P * (1 + t * i / 100)
Example:
If P = 150
i = 12 % pa,
And t = 3 Yrs
Then A = 150 * (1 + 3 * 12 / 100)
150 * 1.36 = 204
31-Jul-2013 18(C) Himansu S M
19. Formula for Simple Interest
Out of the Five Basic Variables:
Principal
Amount
Interest
Time Period and
Rate of Interest,
If we know any Three, then rest can be
calculated by manipulating the formula
31-Jul-2013 19(C) Himansu S M
20. Formula for Compound Interest
Pls note that the “Simple Interest” CAN be
directly calculated, but the “Compound
Interest” CAN’T be directly calculated.
First the Amount is calculated and then the
difference of Amount & Principal is the
“Interest”
A = P + I
I = A – P
31-Jul-2013 20(C) Himansu S M
21. Formula for Compound Interest
A = P * ( 1 + i / 100 / n ) ^ ( t * n )
[ the symbol ^ denotes “to the power of” or
“raised to” ]
i = Rate of interest
t = Time period
n = Compounding frequency
P = Principal
A = Amount
31-Jul-2013 21(C) Himansu S M
22. Formula for Compound Interest
Example: let’s take the same example as
our previous slide – the graph of
comparison
P = 100
t = 8 yrs
i = 10 % pa.
n = 1 time every Year
See next slide-
31-Jul-2013 22(C) Himansu S M
23. Formula for Compound Interest
A = 100 * ( 1 + 10 / 100 / 1 ) ^ ( 8 * 1 )
= 100 * ( 1.10 ) ^ 8
= 100 * 2.144
= 214.4
So I = A – P = 214.4 – 100 = 114.4
Which matches our result.
31-Jul-2013 23(C) Himansu S M
24. Formula for Compound Interest
How to find Rate of Interest:
If A, P, t are given
For simplicity let’s assume n=1
Then the formula is:
i = [ { ( A / P ) ^ (1 / t ) } – 1 ] * 100
31-Jul-2013 24(C) Himansu S M
25. Formula for Compound Interest
Example: Let’s take the Last example
A = 214.4
P = 100
t = 8
n = 1
i = [{( 214.4 / 100 ) ^ ( 1 / 8 )} – 1 ] * 100
= ( 2.144 ^ 0.125 – 1 ) * 100
= ( 1.10 -1 ) * 100 = 0.10 * 100 = 10 % pa
31-Jul-2013 25(C) Himansu S M
26. Some Norms
Simple Interest is rarely used in today’s
world
Business, Banks, Statistics, Finance, Dem
-ography, Population, Accounting, every-
where the Compounding Interest / Growth
/ Increase are used.
31-Jul-2013 26(C) Himansu S M
27. Some Norms
If the compounding interval is not
mentioned then it is assumed to be
“Yearly”
The compounding interval is NEVER more
than a Year, it means the value “n” is
never less than 1
So mentioning only the rate of interest
without the compounding interval is
incomlete information
31-Jul-2013 27(C) Himansu S M
28. Compounding Interval
The more the Compounding Frequency,
Or the less the Compounding Interval,
The more is the Effective Annual Interest.
The Formula for Calculation is:
A = P * ( 1 + i / 100 / n ) ^ ( t * n )
And the Effective Annual Interest is:
I = A - P
31-Jul-2013 28(C) Himansu S M
29. Compounding Interval
Example: Let’s Say:
P = 100
i = 12 % pa
t = 1 Year
n = 1 (Yearly), 2 (Half-Yearly), 4
(Quarterly), 12 (Monthly), 52
(Weekly), 365 (Daily), etc.
Let’s Calculate:
31-Jul-2013 29(C) Himansu S M
36. Continuous Compounding
What if we keep increasing value of “n”
further to say very high no. or infinity,
That means the interval getting smaller
and smaller to say zero.
This is known as Infinitely
compounding, for which the formula is:
A = P * e ^ ( i * t )
31-Jul-2013 36(C) Himansu S M
37. Continuous Compounding
Where e = base of natural logarithm =
2.71828
But there are a few tricks:
The “ I ” is expressed in decimal –
Example: 12 % is 0.12
The “ t ” is expressed in multiples of the
period of interest rate –
Example: if RoI is per annum then 3 yrs 6
months shall be 3.5 yrs.
31-Jul-2013 37(C) Himansu S M
38. Continuous Compounding
Calculation Example:
Let’s take same example of P = 100 & i =
12 % pa. If t = 1 yr what is I?
A = P * e ^ ( i * t )
= 100 * 2.71828 ^ (0.12 * 1 )
= 100 * 2.71828 ^ 0.12
= 100 * 1.127497 = 112.7497
So I = 112.7497 – 100 = 12.7497
31-Jul-2013 38(C) Himansu S M
39. Continuous Compounding
We see here as the compounding
frequency increases or interval decreases
the effective annual rate increases
So, in the limiting case it is the highest
yielding of all the other frequencies of
compounding.
Theoretically it is the highest compound
interest.
31-Jul-2013 39(C) Himansu S M