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 |