Simple and Compound Interest

SIMPLE AND
COMPOUND
INTEREST
A Presentation
By Himansu S M / 31-Jul-2013

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

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”

Types of Interest
 There are basically TWO types of Interest
 They are:
 SIMPLE INTEREST
 COMPOUND INTEREST

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

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

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

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

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

Compound Interest
 At the end of each such period the
accrued interest is added to the Principal
and this becomes the Principal for the next
interval
 In other words, the interest earns interest
 Some call it reinvesting or cumulative
31-Jul-2013 © Himansu S M 10

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)

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

Interests Graph
0
5
10
15
20
25
30
35
40
45
Year1
Year2
Year3
Year4
Year5
Year6
Year7
Year8
Year9
Year10
Year11
Year12
Year13
Year14
Year15
Year16
Simple
Compound

Amount Graph
0
50
100
150
200
250
300
350
400
450
500
Year0
Year1
Year2
Year3
Year4
Year5
Year6
Year7
Year8
Year9
Year10
Year11
Year12
Year13
Year14
Year15
Year16
Simple
Compound

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

Formula for Simple Interest
 A = P + I
 Example:
 If P = 100,
 I = 50
 Then A = 100 + 50 = 150

 I = P * t * i / 100
 Example:
 If P = 150
 i = 12 % pa,
 And t = 3 Yrs
 Then I = 150 * 3 * 12/100 = 54

 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

 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

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

 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

 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-

 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.

 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

 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

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.

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

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

 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:

 For n = 1, Yearly Compounding
 A = 100 * ( 1 + 12 / 100 / 1 ) ^ ( 1 * 1 )
 = 100 * ( 1.12 ^ 1 ) = 112
 Effective i = ( 112 – 100 ) / 100
 = 12 / 100 = 12 % pa.

 For n = 2, Half-Yearly Compounding
 A = 100 * ( 1 + 12 / 100 / 2 ) ^ ( 1 * 2 )
 = 100 * ( 1.06 ^ 2 ) = 112.36
 Effective i = ( 112.36 – 100 ) / 100
 = 12.36 / 100 = 12.36 % pa.

 For n = 4, Quarterly Compounding
 A = 100 * ( 1 + 12 / 100 / 4 ) ^ ( 1 * 4 )
 = 100 * ( 1.03 ^ 4 ) = 112.55
 Effective i = ( 112.55 – 100 ) / 100
 = 12.55 / 100 = 12.55 % pa.

 For n = 12, Monthly Compounding
 A = 100 * ( 1 + 12 / 100 / 12 ) ^ ( 1 * 12 )
 = 100 * ( 1.01 ^ 12 ) = 112.68
 Effective i = ( 112.68 – 100 ) / 100
 = 12.68 / 100 = 12.68 % pa.

 For n = 52, Weekly Compounding
 A = 100 * ( 1 + 12 / 100 / 52 ) ^ ( 1 * 52 )
 = 100 * ( 1.0023 ^ 52 ) = 112.734
 Effective i = ( 112.68 – 100 ) / 100
 = 12.734 / 100 = 12.734 % pa.

 For n = 365, Daily Compounding
 A = 100 * ( 1 + 12 / 100 / 365 ) ^ ( 1 * 365 )
 = 100 * ( 1.00033 ^ 365 ) = 112.747
 Effective i = ( 112.68 – 100 ) / 100
 = 12.747 / 100 = 12.747 % pa.

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 )

 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.

 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

 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.

Interval “ n ” Value RoI (Say) Effective Annual RoI (%)
Yearly 1 12 % pa 12.0000
Half-Yearly 2 12 % pa 12.3600
Quarterly 4 12 % pa 12.5509
Monthly 12 12 % pa 12.6825
Weekly 52 12 % pa 12.7341
Daily 365 12 % pa 12.7475
Continuous Infinity 12 % pa 12.7497

Exercise
 You might have seen Bank Ads depicting
14 – 17 % effective annualised yield after
say 5-7-10 yrs. Let’s calculate that as an
exercise and see how they do it.
 Let’s say ABCD Bank gives 10% pa to
Senior citizens for Fixed Deposits (all bank
interests are quarterly compounded). Now
let’s calculate the effective annualised
yield for 5-7-10 yrs:

Exercise
 t = 5 yrs
 n = 4 Quarterly Compounded
 i = 10 % pa.
 So A = ( 1 + 10% / 100 / 4 ) ^ ( 5 * 4 )
 = ( 1 + 0.025 ) ^ 20 = 1.025 ^ 20
 = 1.6386 So I = 0.6386 in 5 yrs.
 Effective i = 0.6386 / 5
 = 12.772 % pa

Exercise
 t = 7 yrs
 i = 10 % pa.
 So A = ( 1 + 10% / 100 / 4 ) ^ ( 7 * 4 )
 = ( 1 + 0.025 ) ^ 20 = 1.025 ^ 28
 = 1.9965 So I = 0.9965 in 7 yrs.
 Effective i = 0.9965 / 7
 = 14.236 % pa

Exercise
 t = 10 yrs
 i = 10 % pa.
 So A = ( 1 + 10% / 100 / 4 ) ^ ( 10 * 4 )
 = ( 1 + 0.025 ) ^ 20 = 1.025 ^ 40
 = 2.685 So I = 1.685 in 10 yrs.
 Effective i = 1.685 / 10
 = 16.85 % pa

Conclusion
 With the help of Compound Interest one
can trick others.
 Compound interest can give very high
returns after longer periods.
 Example: Say a young man gets a
lumpsum from his father at the age of 25.
If he puts it for 12 % pa till he retires at
60, let’s compare the Simple & Compound
Interest

Conclusion
 Simple Interest:
 12 % pa * 35 yrs = 420 % or 4.2 times
 The Amount will be 1 + 4.2 = 5.2 times
 Compound Interest:
 ( 1 + 12 / 100 ) ^ 35 = 52.8 times
 Hence the return is 10 times in 35 yrs.
 Such is the POWER OF COMPOUNDING!

Disclaimer!
 This is my sincere effort to guide the
interested persons how to calculate the
compound interest.
 Anyone can use a scientific calculator or
any calculator where “x^y” function is built-
in.
 One can use a PC with excel worksheet to
do complex calculations.

THANK YOU
Himansu S M / 31-Jul-2013
31-Jul-2013 (C) Himansu S M 48

Simple and Compound Interest

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