Monthly Compound Interest Formula

The monthly compound interest formula is used to compute compound interest every month. Compound interest is sometimes known as interest on interest. The first period of compound interest resembles the first period of simple interest, but the second period is distinct. Interest is computed on the interest earned over the previous period of time beginning with the second period, hence the name "interest on interest." Compound interest is interest compounded on a loan or deposit's principal amount. Compound interest refers to the process of interest being added back to the principal sum so that interest can be earned during the next compounding period.

The Monthly Compound Interest Formula is a mathematical tool used to calculate the growth of an investment or debt when interest is compounded monthly. This formula is crucial in finance, as it more accurately reflects how many real-world financial products, such as savings accounts, loans, and investments, accrue interest over time.

Monthly Compound Interest Formula:

A = P(1 + r/12)^(nt)

Where:

• A = Final amount
• P = Principal (initial investment or loan amount)
• r = Annual interest rate (in decimal form)
• n = Number of times interest is compounded per year (12 for monthly compounding)
• t = Time in years

Monthly compound Interest Formula

The monthly compound interest formula is also known as the interest on-interest formula since the interest is computed each month and added back to the principal. The final amount, excluding the principal, is known as total compound interest. Therefore formula of compound interest:

Compound Interest (CI) =  Amount - Principal 

                                               =  P(1 + r/12)^12t - Principal 

Where, 

• P is principal 
• r is rate of interest 
• t is time period

Sample Questions on Monthly Compound Interest Formula

Question 1: If Punit lends Rs 2,500 to his friend at an annual interest rate of 5.3%, compounded per month. What will be the compound interest after the end of the year by using the compound interest formula?

Solution:  

To find : Compound interest accumulated after 1 year.

P = 2500, r = 0.053 (5.3%), n = 12 , and t = 1 (given)

Using monthly compound interest formula,

CI = P(1 + (r/n))^nt - P

Put the values given 

CI = 2500{1 + (0.053/12)}¹² - 2500

CI = 2500{(12 + 0.053 )/12}¹² - 2500

= 2500(12.053/12)¹²  - 2500

= 2500(1.0044)¹² - 2500

= 2500(1.0633) - 2500

= 2658.25 - 2500

= 158.25 

So, the compound interest after 1 year will be Rs 158.25 .  

Question 2: A person borrowed Rs 900 from the Rohit at some rate compounded per month and that amount becomes quadruple in 1 year. What will be the interest rate at which a person borrowed the money by using the monthly compound interest formula?

Solution:

To find : Interest rate (r) = ? 

P = 900, n = 12, and t = 1, Amount = 3600 (given)

Using formula,

CI = Amount - Principal

Put the values,

CI =  3600 - 900 = 2700

Using monthly compound interest formula,

CI = P(1 + (r/12) )^12t - P

Put the given values ,

2700 = 900 {1 + (r/12)}^12×1 - 900

2700 = 900 {1 + (r/12)}¹²

2700 + 900 =  900 {1 + (r/12)}¹²

3600/900 = {1 + (r/1200)}¹²

Reduce the RHS by power of 12 

(4^1/12) x 1200 = 1200 + r 

1.12246 x 1200 = 1200 + r 

1346.954 = 1200 + r 

r = 1346.954 - 1200 

r = 146.95%

Therefore the rate of interest is 146.95 % 

Question 3: If a person lends Rs 3,500 at an annual interest rate of 3 %, compounded per month. What will be the compound interest after the end of the 2 years by using the compound interest formula?

Solution: 

 To find : Compound interest accumulated after 2 year.

P = 3500, r = 0.03 (3%), n = 12 , and t = 2 (given)

Using monthly compound interest formula,

CI = P(1 + (r/n))^nt - P

Put the values given

CI = 3500 {1 + (0.03/12)}^12x2 - 3500

CI = 3500 {(12 + 0.03 )/12}²⁴ - 3500

= 3500(12.03/12)²⁴  - 3500

= 3500(1.0025)²⁴ - 3500

= 3500(1.061757) - 3500

= 3716.125 - 3500

= 216.125

So, the compound interest after 2 year will be Rs 216.125.  

Question 4: If a person lends Rs 5000 at an annual interest rate of 5 %, compounded per month. What will be the compound interest after the end of the 3 years by using the compound interest formula?

Solution: 

 To find : Compound interest accumulated after 3 year.

P = 5000, r = 0.05(5%), n = 12, and t = 3 (given)

Using monthly compound interest formula,

CI = P(1 + (r/n))^nt - P

Put the values given

CI = 5000{1 + (0.05/12)}^12x3 - 5000

CI = 5000{(12 + 0.05 )/12}³⁶ - 5000

= 5000 (12.05/12)³⁶  - 5000

= 5000 (1.00416)³⁶ - 5000

= 5000 ( 1.1611946) - 5000

= 5805.973 - 5000

= 805.973

So, the compound interest after 3 year will be Rs 805.973.

Question 5: What will be the compound interest if the principal is Rs 1758 at the rate of 4 % for 2 years?

Solution: 

To find : Compound interest accumulated after 2 year.

P = 1758, r = 0.04 (4 %), n = 12, and t = 2 (given)

Using monthly compound interest formula,

CI = P(1 + (r/n))^nt - P

Put the values given

CI = 1758 {1 + (0.04/12)}^12x2 - 1758

CI = 1758 {(12 + 0.04)/12}²⁴ - 1758

= 1758(12.04/12)²⁴ - 1758

= 1758(1.0033)²⁴ - 1758

= 1758(1.0822796) - 1758

= 1902.647 - 1758

= 144.647

So, the compound interest after 3 year will be Rs 144.647.

Question 6: What will be the compound interest if the principal is Rs 2000 at the rate of 7 % for 5 years?

Solution: 

To find : Compound interest accumulated after 5 year.

P = 2000 , r = 0.07 (7%), n = 12 , and t = 5 (given)

Using monthly compound interest formula,

CI = P(1 + (r/n))^nt - P

Put the values given

CI = 2000 {1 + (0.07/12)}^12x5 - 2000

CI = 2000{(12 + 0.07)/12}⁶⁰ - 2000

= 2000(12.07/12)⁶⁰ - 2000

= 2000(1.00583)⁶⁰ - 2000

= 2000(1.417625) - 2000

= 2835.250 - 2000

= 835.250

So, the compound interest after 3 year will be Rs 835.250.

Practice Questions on Monthly Compound Interest Formula

1. Calculate the final amount after 3 years on an initial investment of $5,000 at 6% annual interest, compounded monthly.

2. How much will you owe after 5 years on a $10,000 loan at 8% annual interest, compounded monthly?

3. What initial investment is needed to reach $20,000 in 4 years at 5% annual interest, compounded monthly?

4. Calculate the effective annual rate for an investment offering 7% annual interest, compounded monthly.

5. How long will it take for $2,000 to double at 9% annual interest, compounded monthly?

6. What's the difference in final amount between simple interest and monthly compound interest on $15,000 for 2 years at 4% annual rate?

7. If you invest $100 monthly for 10 years at 6% annual interest, compounded monthly, how much will you have?

8. Calculate the monthly payment needed to pay off a $50,000 loan in 15 years at 5% annual interest, compounded monthly.

9. How much interest will you earn in total on a $10,000 investment over 6 years at 3% annual interest, compounded monthly?

10. Compare the final amounts after 8 years for $20,000 invested at 5% compounded annually vs. monthly.

Summary

The Monthly Compound Interest Formula is a powerful tool in financial calculations, allowing for precise computation of growth or debt accumulation when interest is compounded monthly. It takes into account the effect of earning interest on previously earned interest, resulting in exponential growth over time. This formula is essential for accurate financial planning, investment analysis, and understanding the true cost of loans or the potential returns on savings and investments.