Date post: | 15-Nov-2014 |
Category: |
Data & Analytics |
Upload: | ashish-jaiswal |
View: | 282 times |
Download: | 5 times |
Ashish Jaiswal MBA (Section A) Sem I- 2013-15
2DefinitionWhen the simple interest (not paid as soon as it falls due) is
added to the principal for next period, is called Compound Interest (Abbreviated as C.I.).In other words ,when the simple interest produced after each prefixed period (often called interest period or conversion period) is added to the principal and the whole amount then produces interest for the next period, then the sum by which the original principal is increased at the of all the specified conversion periods is known as Compound Interest for the given period. Thus Compound Interest = Amount of the last period –
Principal of the first period
In case of compound interest the conversion period may be 1 year, 1/2 year, 1/3 year, ¼ year, 1 month etc.
10/17/2013
Ashish Jaiswal MBA (Section A) Sem I- 2013-15
3
Formula Of Compound Interest
The amount of Rs. P at R% per annum for n years is obtained by the formula (often called the formula for compound interest) given below:
Amount = Principal ,Symbolically, A = P
Compound Interest = A-P or P - P or P
where A= Amount, P= Principal, R= Rate, n= Time
TimeRate
1001
nR
1001
n
R
1001
1
1001
nR
10/17/2013
4
Illustration 1
Find compound interest of Rs. 10,000 at rate of 10% for three years.
Solution:
Amount = Principal
= 10,000
= 10,000
=10,000
=10,000 × 11/10 × 11/10 × 11/10 = Rs. 13,310
Compound Interest = Amount – Principal = 13,310 – 10,000 = Rs. 3,310.
TimeRate
1001
3
100
101
3
10
11
3
10
11
10/17/2013Ashish Jaiswal MBA (Section A) Sem I- 2013-15
5
Illustration 2
When Anshu is born Rs. 5,000 is placed by his mother in an account that pays interest at the rate of 10% p.a. Compound interest. What amount will there be to his credit on Anshu’s 18th birthday?
Solution:
Amount = Principal
A = P
A = 5,000
=5,000 on substitute the values
Taking logarithm on both sides
log A = log 5,000 + 18log 1.1 = 3.6990 + 18(.0414) = 3.6990 + 0.7452 = 4.4442A = antilog (4.4442) = 27,810
Amount to Anshu’s credit on his 18th birthday = Rs. 27,810
TimeRate
1001
nR
1001
18
100
101
181.1
10/17/2013Ashish Jaiswal MBA (Section A) Sem I- 2013-15
Ashish Jaiswal MBA (Section A) Sem I- 2013-15
6
To Compute Compound Interest when the number of Conversion. Periods (Years, say) is not an integer:
Method I (i) Calculate Amount and/or compound interest for the whole years by any method. (ii) Assuming this amount as principal, find simple interest for the rest fractional part at the same rate. (iii) Add this interest to the amount obtained in (i) to get the final amount. (iv) Subtract original Principal from this final amount to compute compound interest. ORAdd the two interests [obtained in (i) and (ii) to find the required compound interest and add this compound interest to the original principal to find the required amount.
10/17/2013
Ashish Jaiswal MBA (Section A) Sem I- 2013-15
7
Method II Use the formula,
A = P
Where, k = Number of whole years t = Fraction of the fractional year and n = k + t, time.
Method III Use the usual Formula,
A = P
1001
1001
RtRk
nR
1001
10/17/2013
8
Illustration
Find the compound interest on Rs. 2,500 in 2½ years at 4% p.a. Compounded annually. Find the amount also.
Solution:
First Method- Formula: A + P
Amount for 2 years, A = 2,500
= 2,500
= 2,500 = = Rs.2,704
Compound Interest for 2 years = Amount – Principal = 2,704 – 2,500 = Rs.204
Now, Principal = Rs.2,704 , Rate = 4 , Time = ½ year.
nR
1001
2
100
41
2
100
104
2
25
26
2525
26262500
10/17/2013Ashish Jaiswal MBA (Section A) Sem I- 2013-15
Ashish Jaiswal MBA (Section A) Sem I- 2013-15
9
Simple Interest = = = Rs. 54.08
Compound Interest for 2½ years = 204 + 54.08 = Rs.258.08 and Amount = P + C.I = 2,500 + 258.08 = Rs.2,758.08
Second Method- Formula: A + P
Where k = number of whole years = 2 t = fraction of the fractional year = ½
A = 2,500
= 2,500 = antilog [log 2,500 + 2 log (1.04) + log 1.02] = antilog [3.3979 + 2(.0170) + .0086]
100
PRT100
2/142704
1001
1001
RtRk
100
2/141
100
41
2
02.104.1 2
10/17/2013
Ashish Jaiswal MBA (Section A) Sem I- 2013-15
10
= antilog [3.3979 + .0340 + .0086] A = antilog [3.4405] = 2,757
Compound Interest = A – P = 2,757 – 2,500 = Rs. 257
Third Method:
Formula: A = P
where P = 2,500 , R = 4 , n = 2½ = 5/2
=> A = 2,500 = 2,500
Taking logarithm on both sides, log A = log 2,500 + 5/2 ( log 104 – log 100)
= 3.3979 + 5/2 (2.0170 – 2.0000) = 3.3979 +
nR
1001
2/5
100
41
2/5
100
104
2
0170.5
10/17/2013
Ashish Jaiswal MBA (Section A) Sem I- 2013-15
11
= 3.3979 + .0425 = 3.4404
A = antilog (3.4404) = Rs. 2,757
Compound Interest = A – P = 2,757 – 2,500 = Rs.257
Remark: The difference between the result of method I and method II is due to the use of logarithm. In general, the method II should be
preferred.
10/17/2013
Ashish Jaiswal MBA (Section A) Sem I- 2013-15
12
Computation of Compound Interest when interest is compounded monthly, quarterly, half-
yearly
Let P = Principal
R = Rate of compound interest percent per annum
M= no. of conversion period in a year
N = no. of years
Then Amount, A= P
Here R/m = Rate percent per conversion period
and n×m = no. of conversion periods
mnmR
100
/1
10/17/2013
13
Illustration
Find the compound Interest on Rs.1,200 @ 8% annually for two years if:
1) the interest is calculated annually.2)the interest is calculated half-yearly.3) the interest is calculated quarterly.4) the interest is calculated monthly.
Solution:
1) Interest is compounded annually:
A = P , where P = 1,200 , R = 8, n = 2
= 1,200 = 1,200 × = 1,200 ×
Using logarithm table, log A = log 1,200 + (log 27 – log 25) = 3.0792 + 2[1.4314 – 1.3979]
= 3.0792 + 2(.0335) = 3.0792 + .0670 =
3.1462 A = antilog (3.1462) = Rs. 1,401
that is, Compound Interest = A – P = 1,401 – 1,200
= Rs. 201
nR
1001
2
100
81
2
100
108
2
25
27
10/17/2013Ashish Jaiswal MBA (Section A) Sem I- 2013-15
Ashish Jaiswal MBA (Section A) Sem I- 2013-15
14
2) Interest is compounded half-yearly:
A = P , where Rate = R/2 = 8/2 = 4
Conversion Period = 2n = 2×2 = 4
=> A = 1,200
= 1,200
= 1,200
log A = log 1,200 + 4 log (1.04)1,200 = = 3.0792 + 4(.0170) = 3.0792 + .0680 = 3.1472 A = antilog (3.1472) = Rs. 1,404
Compound Interest = 1,404 – 1,200 = Rs.204 [ if = 1.16985865 then
A = 1,403.830272 = Rs. 1,403.83]
nR
2
100
2/1
4
100
41
404.1
404.1
404.1
10/17/2013
Ashish Jaiswal MBA (Section A) Sem I- 2013-15
15
3) Interest is compounded quarterly
A = 1,200 , where Rate = R/4 = 8/4 = 2
Conversion Period = 4n = 4 × 2 = 8
= 1,200
= 1,200 = 1,200
log A = log 1,200 + 8 log (1.02) = 3.0792 + 8(.0086)
= 3.0792 + .0688 = 3.1480
A = antilog (3.1480) = Rs. 1,406
that is, Compound Interest = A- P = 1,406 – 1,200 = Rs. 206
8
100
21
nR
4
100
4/1
802.1 802.1
10/17/2013
Ashish Jaiswal MBA (Section A) Sem I- 2013-15
16
4) Interest is compounded monthly:
A = P , where Rate = R/12 = 8/12
Conversion Period = 12n = 12 × 2 = 24
=> A = 1,200
= 1,200
= 1,200 = 1,200
log A = log 1,200 + 24 [log 302 – log 300] = 3.0792 + 24 [2.4800 = 2.4771] A = 3.0792 + 24 [0.0029]
= 3.0792 + .0696 = 3.1488 A = antilog (3.1488) = Rs. 1,409
Compound Interest = A – P = 1,409 – 1,200 = Rs. 209
nR
12
100
12/1
24
100
12/81
24
10012
81
24
300
21
24
300
302
10/17/2013
17
Illustration
Find Compound Interest of Rs. 10,000 for 2 years @ 8% per annum compounding monthly.
Solution:
Formula: Amount = Principal
Here Principal = 10,000
Time = 2 years = 2 × 12 months = 24 months
Rate = 8/12 per month
Amount, A = 10,000
= 10,000
= 10,000
log A = log (10,000) + 24 [log 151 – log 150] = 4.0000 + 24[2.1790 – 2.1761]
timeRate
1001
24
100
12/81
24
100
12/81
24
150
151
10/17/2013Ashish Jaiswal MBA (Section A) Sem I- 2013-15
Ashish Jaiswal MBA (Section A) Sem I- 2013-15
18
= 4.0000 + 24 × 0.0029 = 4.0000 + 0.0696 = 4.0696
A = antilog (4.0696) = Rs. 11,740
Compound Interest = Amount – Principal = 11,740 – 10,000
= Rs. 1,740
10/17/2013
19
To Find The Principal and Rate
Illustration
A sum of money given at compound interest becomes Rs. 2,420 in 2 years and Rs. 2,662 in 3 years. Find the money and rate of interest.
Solution:
Amount of 2 years = Rs. 2,420
Amount of 3 years = Rs. 2,662
Interest of third year = 2,662 – 2,420 = Rs. 242
Now Principal = Rs. 2,420 , Interest = Rs. 242, Rate = R, Time = 1 year.
Formula: Rate = Simple Interest × 100/ Principal × Time
= 242 × 100/ 2,420 ×1 = 10
Again Principal = P, Time = 2 years, Rate = 10% p.a., Amount = Rs. 2,420
10/17/2013Ashish Jaiswal MBA (Section A) Sem I- 2013-15
Ashish Jaiswal MBA (Section A) Sem I- 2013-15
20
Unsolved Illustration
A Father desires to distribute Rs. 51,783 amongst his two sons who are respectively 12 and 15 years old, in such a way that the sums invested @5% p.a. compound interest will give the same amount to both of them When they attain the age of 18. How should he divide the sum?
Answer: 27,783
10/17/2013
Ashish Jaiswal MBA (Section A) Sem I- 2013-15
21
Nominal Rate of Interest When compound interest is calculated monthly, quarterly or half-yearly, then the predetermined rate of interest per annum is known as Nominal Rate Of Interest.
Effective Rate of Interest
When compound interest is calculated monthly, quarterly or half-yearly, the interest to the principal each time increases the principal and accordingly the interest rate per annum will be more than the usual rate. This new rate of interest is termed as Effective Rate of Interest. In simple words, when compound interest is calculated monthly, quarterly or half-yearly, then the interest earned on Rs.100 for a year is Effective Rate Of Interest.
Where = 100
eR
1
1001
m
m
R
10/17/2013
Ashish Jaiswal MBA (Section A) Sem I- 2013-15
22
Relationship between Effective and Nominal Rates Let 1 + j = => j = - 1
where, i = nominal rate of interest per rupee per annum m = the number of times interest is compounded in a yearand i + j = Amount of rupee 1 after one year j = effective rate of interest per rupee per annum.
Remark 1: When compound interest is calculated yearly, the concept of effective rate or ‘Nominal Rate’ does not arise.
Remark 2: The two rates of interest are said to be identical if the compound interest is different conversion periods but after one year they
yield the same compound interest.
m
m
i
1
m
m
i
1
10/17/2013
23
Illustration
Solution:
Interest on Rs. 100 for 1 year = Rs.4 .......(i)
Amount = 100 + 4 = Rs. 104
When interest is compounded quarterly, then
Amount = 100
Amount = 100 = 100
Amount = 100 = 100 × 1.041 = 104.10
( Using log tables)
Compound Interest = 104.10 – 100 = Rs. 4.10 .....(ii)
Thus, the effective rate of interest is 4.10% per annum.
If nominal rate of interest is 4% per annum and interest is compounded quarterly, then find the effective rate of interest
41
100
4/41
4
10
11
4
100
101
401.1
10/17/2013Ashish Jaiswal MBA (Section A) Sem I- 2013-15