Compound Interest

Ashish Jaiswal MBA (Section A) Sem I- 2013-15

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Compound Interest

10/17/2013

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

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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

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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

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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

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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

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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

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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

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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

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Ashish Jaiswal MBA (Section A) Sem I- 2013-15

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= 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

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Ashish Jaiswal MBA (Section A) Sem I- 2013-15

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= 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

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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

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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

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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

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Ashish Jaiswal MBA (Section A) Sem I- 2013-15

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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

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Ashish Jaiswal MBA (Section A) Sem I- 2013-15

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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

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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

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= 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

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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

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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

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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

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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

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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