Monthly Compound Interest Formula

Interest is the extra amount of fees paid to borrow money or any other assets. Interest is paid on a monthly, quarterly, or annual basis. The interest earned depends on the time for which money is invested, the amount of money invested, and the rate at which money is invested.

What is Principal?

The amount borrowed by a person/group of persons is called the principal.

What is Interest?

Interest is the percentage of principal that lenders charge for the use of their money. In simple words, The interest rate is the amount that the lender charges the borrower for a given time interval. The interest rate on a loan is usually recorded on an annual basis called the annual percentage rate (APR).

Types of Interest 

Two common types of interest  are :

1.  Simple Interest 
2.  Compound Interest

What is Simple Interest?

Simple Interest (S.I.)  is a method of calculating the amount of interest on a given principal amount.

The formula to calculate Simple Interest is

S.I. = (P×R×T )/100 

P = Principal

R = Rate of Interest

T = Time period

Amount

Amount (A) is the total amount to be repaid at the end of the time period for which it was borrowed.

The formula for the total amount in the case of simple interest can also be written as :

Amount (A) = S.I.+P

A = P×(1+ R×T /100)

A = Amount

P = Principal

R = Rate of Interest

T = Time period

What is Compound Interest? 

Compound interest is the interest charged on the amount of a loan or deposit. It is the most commonly used concept in our daily life. The compound interest of an amount depends on both the principal and the interest earned over time.

Compound interest for the first period is similar to simple interest but the difference occurs in the second period of time. From the second period, the interest is also calculated on the interest thus earned in the previous period of time, and thus it is known as interest on interest. 

Derivation of Compound Interest Formula

We can derive the compound interest formula with the help of simple interest formula

For year ‘1’

The S.I. of one year = C.I. of one year

The S.I. for first is  

S.I.(for first year)  = (P×R×T)/100

we know that, Amount =S.I.+P

Thus the Amount after first year =S.I.(for first year) +P

This implies Amount =P+(P×R×T) /100 = P(1+R×T/100) = P(1+R/100)

Here T=1 since we are calculating C.I. for one year.

For year ‘2’

The total amount of year 1 will now be considered as the Principal for second year 

Hence P(for second year) = P(1+R/100)

Hence, amount = P(1+R/100)(1+R/100) 

Amount(after second year) = P(1+R/100)²

For year ‘n’

Amount(after n years) = P(1+R/100)ⁿ 

Now, 

Formula to Calculate Compound Interest = Amount – Principal

C.I. = P(1+R/100)ⁿ – P

C.I. = P[(1+R/100)ⁿ – 1]

where,  

 P = Principal

R = Rate of Interest

n = Number of times interest is compounded per year

Monthly Compound Interest Formula

Monthly compound interest refers to the compounding of interest on a monthly basis, which implies that the compounding interest is charged both on the principal as well as the accumulated interest on a monthly basis.

Derivation of Monthly Compound Interest Formula

The formula for calculating the compound interest is as,

C.I.= P(1+R/100)ⁿ – P

If the time period for calculating interest is monthly, the interest is calculated for each month, since there are 12 months in a year therefore, the amount is compounded 12 times a year. This implies, that n = 12. 

And hence, the formula to calculate monthly compound interest is given as

Monthly Compound Interest Formula = P×(1+(R/12))^12×T − P

where,  

P = Principal

R = Rate

T = Time 

Sample Questions

Question 1: A sum of Rs.15000 is borrowed and the rate is 8%. What is the monthly compound interest for 3 years?

Solution:

Given,

Principal(P) = Rs 15000

Rate(R) = 8%

Time(T) = 3 years

Formula for monthly compound interest = Amount – Principal

Amount = P(1+(R/12))^12×T

hence,  Amount = 15000[1+ 8/12×100]^12×3

Amount = 15000[  151/150  ]³⁶                                                                                                                                                                                                                   

Amount = 15000×1.27023 = 19053.355

Therefore, Compound Interest = 19053.355-15000 = 4053.555

The monthly Compound Interest for 3 years = Rs 4053.555

Question 2: Rohan borrowed a sum of Rs 10000 at the rate of 5%. Calculate the monthly compounded interest for 2 years?

Solution:

Given,

Principal(P) = Rs. 10000

Rate(R) = 5%

Time(T) = 2 years

Formula for monthly compound interest = P[1+(R/12)]^12×T – P

Hence,  Compound Interest   = 10000[1+(5/12×100)]^12×2 – 10000 

= 10000[  241/240 ]²⁴ – 10000                                                                                                                                                                                                                 

= 10000(1.104941) – 10000

= 11049.413-10000

= 1049.413

Hence, The monthly compound interest = Rs 1049.413

Question 3: Derive the formula for the Rate of interest from the monthly compound interest formula?

Solution:

Interest formulas are used to calculate the amount of money you need to repay on a loan and the interest you earn on time deposits, trusts, and other investments.

Formula for calculating the rate of interest for a given amount is :

We Know that,

 A = P[1+(R/12)]^12×T

A/P  = (1+R/12)^12×T                                                                                                                                                                                                   

(A/P)^1/12×T = 1 +R/12

(A/P)^1/12×T  – 1 = R/12 

Rate = 12×[ (A/P)^1/12×T ] 

Question 4: Mr. Nitin invested Rs 1000 during the year 2010. After 10 years, he sold the investment for Rs 1500 in the year 2020. Calculate the rate of the investment if compounded monthly.

Solution:

Given,

Principal(P) = Rs 1000

Time(T) = 10 years     

Amount = Rs 1500

Since, Amount = P[1+(R/12)]^12×T

Hence, 

1500 = 1000[1+R/12]^12×10 

1500/1000   = [1+ R /12 ]¹²⁰

(1.5)^1/120 =  1+R/12

1.003 = 1 +R/12

1.003-1 = R/12

12×0.00338 = R

Hence, Rate = 0.0406 OR 4.06%

Question 5: Find the Compound Interest when Principal = Rs.5000, Rate = 5% per annum ,Time = 4years.

Solution:

Given,

Principal (P) = Rs. 5000

Rate of interest (R) = 5% per annum

Time = 4years 

Calculating Compound Interest from given information 

Interest for first year = (5000×5×1) /100 = 250

Amount after first year = 5000+250 = Rs. 5250

Principal(for second year) = 5250

Interest for second year = (5250×5×1) /100 = 262.5

Amount after second year =5250+262.5 = Rs. 5512.5

Principal(for third year) = 5512.5

Interest for third year = (5512.5×5×1) /100 = 275.625

Amount after third year = 5512.5+275.625 = Rs. 5788.125

Principal(for fourth year) =5788.125

Interest for fourth year = (5788.125×5×1) /100 = 289.40625

Hence, the Amount after fourth year = 5788.125+289.40625 = Rs. 6077.53125

Now, Compound Interest = Amount – Principal = Rs.1077.53125

Question 6: Find the  Compound Interest for Principal=Rs 12000, Rate = 3%, Time =2 years.

Solution:

Given,

Principal(P) = Rs 12000

Rate of interest = 3%

Time = 2 years 

Formula for compound interest = Amount – Principal

Where, Amount = P(1+R/100)ⁿ

Amount = 12000(1+3/100)²

Amount = 12000(1.0609)

Hence, Amount = Rs 12730.8

Compound Interest = 12730.8 – 12000 =Rs 730.8