4.7 Compound Interest
We have seen earlier the formula for SI as
Simple Interest (SI) = P*N*(R/100)
Where
P = Principal
N = Period
R = Rate of Interest
4.7 Example 1 : If a person deposits
Rs 10000 in a Bank as an FD for one year, how much interest does he
get after one year and after 2 years?
Workings:
In this Problem P =10000 R=6 and N=1
Simple Interest for one year = P*N*(R/100) = 10000*1*6/100 =
600
If the period is 2 years then N=2
Simple Interest for 2 years = P*N*(R/100) = 10000*2*6/100 =
1200
Assume in the above case, the depositor chooses not
to receive the interest after one year
but requests the bank to pay him interest at the time of maturity of deposit
(I.e. after 2 years). In such a case should bank pay him more interest than Rs
1200?
Bank does pay him little more. It pays interest on
the first year interest (Rs 600) at the same rate of 6%. Why does bank pay
extra? This is because, Bank has used the interest amount for one year for it’s
activities and hence bank
is bound to give interest on interest. This is called ‘compound interest’.
In case of
simple interest, the principal amount remains constant throughout, whereas in
the case of compound interest, the principal amount goes on increasing at the
end of the period (term).
Let us calculate
simple and compound interest for an initial deposit of Rs 10,000.
N =1, R =6
|
|
I year |
II year |
|
Principal in the beginning of the year
(P) |
P=10000 |
P=10600 (Amount at the end of I year becomes
new principal) |
|
Interest for one year (SI) |
PNR/100 = 10000*1*6/100 = 600 |
PNR/100 = 10600*1*6/100 = 636 |
|
Amount at the end of year (P+SI) |
10600(=10000+600) |
11236(=10600+636) |
Total interest will be 1236 (=600+636)
Thus, the depositor gets Rs 36 extra interest in
the compound interest when compared with simple interest
The formula used for calculating compound interest
is given below
Maturity Amount = P*(1+(R/100)) N
Compound Interest (CI) =
Maturity Amount – Initial deposit =P*(1+(R/100)) N-P
4.7 Exercise : Use the above formula to verify that CI on Rs 10000 for two years @ 6% is indeed
1236
Let us find
the difference between Simple interest and Compound interest for 5 years on a
deposit of Rs 10000 at 9%
We will be
using the above formulae for SI and CI with P =10000, R= 9 and N = 1 to 9
Compound
interest for 5 years = P*(1+(R/100)) N-P = 10000*(1+(9/100)) 5-P
= 15386.24 -10000 =5386.24
Table:
Comparison of SI and CI On a Principal of Rs 10000 @R=9% for 1 year to 9
years ( Calculator was used for working)
|
Number
of Years --ŕ |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
|
Compound
Interest(CI) |
900 |
1881 |
2950.29 |
4115.82 |
5386.24 |
6771 |
8280.39 |
9925.63 |
11718.93 |
|
Simple Interest(SI) |
900 |
1800 |
2700.00 |
3600.00 |
4500.00 |
5400 |
6300.00 |
7200 |
8100 |
|
Extra interest |
0 |
81 |
250.29 |
515.82 |
886.24 |
1371 |
1980.39 |
2725.63 |
3618.93 |
The above table can be represented in a bar chart
as given below.
(Colors of numbers in the table match with Colors of
bars in the chart: CI, SI and Extra interest)
|
|
We clearly
see the benefit of compound interest on deposits. The benefit increases with
the increase in the term of the deposit.
Note that Compound
interest is paid to the depositor in the case Fixed and Cumulative Term
Deposits (FD,CTD)
Do you
observe that the initial deposit nearly doubles?
With compound
interest @ 9%, interest paid in 8 th year is equal to initial deposit and hence the original amount doubles in 8 years
Exercise Using the formula for CI, check that the principal amount doubles
(Interest =Principal) with the rate and approximate period (number of years) as
mentioned below:
Table : Rate of interest and period combination for
the principal amount to double
|
Rate% ----> |
7 |
8 |
9 |
10 |
11 |
12 |
|
Approximate
years required for doubling of
Principal |
10 Years 3 Months |
9 years |
8 Years |
7 Years 3 Months |
6 Years 9 Months |
6 Years 2 Months |
Normally
compound interest is calculated quarterly and hence the initial deposit doubles
in a lesser period than mentioned above.
The Banks
also use pre calculated table of interest called Ready Reckoner for calculating
compound interests for different periods and different rates.
Note that
Banks always charge compound interest on any type of loan taken by borrowers
4.7 Problem 1: Let us take the case of
Ram (4.5 Example 1) wherein he decides not to take interest from the bank
yearly. He chooses the option of
investing Rs. 5000 in the bank
for 6 years as a cumulative deposit at the rate of 8% compounded
interest. Let us calculate how much does he get at the end of 6 years
Solution :
P= 5000
R =8
N=6Years
Maturity Amount = 5000(1+8/100)6 =
5000*1.08*1.08…(in all 6 times) =7934.37
Compound interest = maturity amount -deposit amount
=7934.37-5000=2934.37
Thus in all he gets Rs. 2934.37 as cumulative
interest. Compare this with Rs. 2400 he gets as total interest in simple
interest method (4.5 Example 1). In compound interest method he gets Rs 534.37
extra.
Thus cumulative fixed deposit is useful for people
who do not need interest money often and who are willing to wait for the total
amount to be received at the end of maturity period.
In the above Problem we had calculated compound
interest yearly (interest on interest was calculated once a year). However
Banks calculate the compound interest quarterly (i.e. once in three months).
Since a year has 4 quarters, Banks calculate Compound interest four times in a
year.
4.7 Problem 2: Simple interest on a
sum of money for 2 years at 6.5% per annum is Rs5200. What will be the compound
interest on that sum at the same rate for the same period?
Solution:
We need to find
the principal in order to calculate CI
Let P be the
principal
We know
SI = (P*n*R) /100
= P*2*(13/2)/100 = 13P/100
It is given
that SI= 5200
5200 = 13P/100
P = 5200*100/13 =
40000
Maturity Amount= P*(1+(R/100)) N = 40000*(1+13/200)2
= 40000*(213/200)*(213/200) = 213*213 = 45369
CI = Maturity amount –
principal = 45369-40000 = 5369
Verification:
SI = (P*n*R) /100 = 40000*2*(13/2)/100 = 40000*13/100 = 5200 which is the Si
given in the problem, hence our value for Principal is correct.
4.7 Problem 3: The difference between
Compound interest and simple interest on a certain sum for 2 years at 7.5% per
annum is Rs 360. Find the sum
Solution:
We need to
find the principal in order to calculate CI
Let P Be the principal amount
SI = (P*n*R)
/100 = P*2*(15/2)/100 = 15P/100
Maturity Amount= P*(1+(R/100)) N =
P*(1+15/200)2 = P*(215/200)*(215/200) = P*46225/40000
CI = Maturity amount –
Principal = 46225P/40000 –P
It is given that CI-SI = 360
360 = 13325P/40000 –P
– 15P/100 = (46225P -40000P -6000P)/40000 = 225P/40000
P = 360*40000/225
= 64000
Verification:
SI = (P*n*R) /100 = 64000*2*(15/2)/100 = 64000*15/100 = 9600
Maturity Amount= P*(1+(R/100)) N = 64000*(1+15/200)2
= 64000*(215/200)*(215/200) = 64000*46225/40000 = 73960
CI = Maturity amount – Principal =73960 -64000 =
9960
CI-SI = 9960-9600 =360
which is as given in the problem and hence our value for P is correct.
4.7 Problem 4: Rekha invested a sum of
Rs 12000 at 5% Interest compounded yearly. If she receives an amount of Rs
13230 at the end of n years find the period
Solution:
Let n be the period
Maturity Amount== P*(1+(R/100)) n = 12000*(1+(5/100))
n = 12000*(1+1/20)n =
12000*(21/20)n
It is given that maturity amount is 13230
13230 = 12000*(21/10)n
(21/10)n=13230/12000
= 411/400 = 21*21/(20*20) = (21/20)2
n =2
Verification: By substituting value
of n and others in the formula for CI, find out that amount received is 13230.
4.7 Problem 5: At what rate per cent
compound interest, does a sum of money become 2.25 times itself in 2 years?
Solution:
Here N=2. Since we are given that amount becomes
2.25 times in 2 years, A =2.25P.
Let P be the Principal and R be the rate to be
found
A = P*(1+(R/100)) N= P*(1+(R/100))2
Since A =2.25
2.25P = P*(1+(R/100))2
2.25 =9/4 =(1+(R/100))2
Since 9/4 =(3/2)2
We have 3/2 = (1+(R/100)
On simplification we get R/100 = 1/2
R = 50
Verification: We have N=2, R=50
and Let P be the principal amount
A = P*(1+(R/100)) N= P*(1+(50/100))2
=P*(150/100)2 = P*(3/2)2 = P*9/4 = 2.25P which is as
given in the problem.
Application
of Compound interest formula other than in Banking:
Normally, companies buy machinery and equipments
for their use. Because of the usage, the value of the machine reduces over a
period.
This is the reason why second hand machine or vehicle
is available at a lower price.
This is reduction in value is called ‘depreciation’. The rate at which the value reduces
is called ‘rate of depreciation’.
If the cost of machine or equipment depreciates at
the rate of R% every year its value after N years is given by the formula
Value after N years = (Present value)*(1-(R/100))
N
Conversely
The present value of machine = (It’s value N years
ago) *(1-(R/100)) N
4.7 Problem 6 : The current population of a town is 16,000. It is estimated that the
population of the town to grow as follows:
First 6 years
@ 5%
Next 4 years
@8%
Find out the
population after 10 years
Solution:
We use the formula for CI for finding out population
after 10 years :
Population after N years = (Present population)*(1+(R/100))
N
Conversely
Present population = (Population N years ago) *(1+(R/100))
N
Step1: First
find out population at the end of 6 years. (Here P=16000, N=6, R=5)
Population at the end of 6 years
= P*(1+(R/100)) N
=
16000(1+5/100)6
= 21445
Thus at the
end of first 6 years population is likely to be 21,500
Step 2: Find
out population at the end of 4 years. (Here P=21500, N=4, R=8)
Population at
the end of 4 years
= P*(1+(R/100)) N
=
21500(1+8/100)4
= 29250
Thus 29,250
is the likely population of the town after 10 years.
4.7 Problem 7: Have you not observed a rubber ball losing height on each bounce? Let us
say that each time a rubber ball bounces , it raises only to 90% of its
previous height. If it is dropped from the top of 25 meter tall building, to
what height would ir rise after bouncing the ground 2 times
Hint : (As in depreciation)
Since ball raises only
90% of its previous height on the next bounce, we could say it loses(depreciates)
10% of its previous value
Thus P=25, R =10
Hence the formula
Height raised after two
bounce = 25((1-(10/100))
2 = 20.25m
While calculating the compound interest,
when the interest is compounded at different periodicity other than every year,
the formula for compound interest calculation changes slightly.
|
When
interest is compounded |
Yearly |
The
Principal changes |
Every year |
Interest is calculated once
a year(t=1) |
|
Half yearly |
The
Principal changes |
Every half year |
Interest is calculated twice a year(t=2) |
|
|
Quarterly |
The
Principal changes |
Every quarter |
Interest is calculated four times a year(t=4) |
|
|
Monthly |
The
Principal changes |
Every month |
Interest is calculated twelve times a year(t=12) |
Let R be the rate of interest per annum and N be
the number of years for which the interest is calculated and t be the periodicity(year, half
yearly, quarterly, monthly) with which compound interest is calculated.
Then the formula for maturity amount changes to
A = P*(1+(R/t*100)) N*t
Note:
The above change in formula is due to the fact
that, the rate per year is converted to rate per half year(R/2), rate per
quarter(R/4), rate per month(R/12) if the interest is calculated half yearly(2
times), quarterly(4 times) or monthly(12 times) respectively. Also, note that
in such cases N changes to 2N, 4N and 12N respectively.
4.7 Problem 8: What is
the maturity amount on Rs 50,000 placed with the bank if it pays 6% compounded
interest for the first three years and 7% for the next two years with interest compounded
every quarter.
Hint: As in 4.7 Problem 6, solve the problem in two steps as shown below by
using the formula A = P*(1+(R/t*100)) N*t.
1. Calculate the maturity
amount after 3 years (12 quarters) @ 6% for three years on principal of Rs
50,000 (N=3, t=4, R=6)
2. With the maturity
amount as obtained in step 1 as principal, calculate the maturity amount for
next two years (8 quarters) @7% (N=2, t=4, R=7).
4.7 Summary of learning
|
No |
Points to remember |
|
1 |
Maturity
Value= P*(1+(R/100)) N |
|
2 |
Compound
interest = P*(1+(R/100)) N-P |
