# Monthly Compound Interest Formula

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## Monthly Compound Interest Formula

The monthly compound interest formula is used to find the compound interest per month. Compound interest is widely known as interest on interest. Compound interest for the first period is similar to the simple interest but the difference occurs in and from the second period of time. From the second period, the interest is also calculated on the interest thus earned on the previous period of time, that is why it is known as interest on interest. Let us learn more about the monthly compound interest formula along with solved examples.

## What Is the Monthly Compound Interest Formula?

The monthly compound interest formula is also known as the formula of interest on interest calculated per month, the interest is added back to the principal each month. Total compound interest is the final amount excluding the principal amount.

### Monthly Compound Interest Formula

The formula for the compound interest is derived from the difference between the final amount and the principal, which is: CI = Amount – Principal. The formula of monthly compound interest is:

CI = P(1 + (r/12) )12t – P

Where,

• P is the principal amount,
• r is the interest rate in decimal form,
• t is the time.

## Derivation of Monthly Compound Interest Formula

The formula for calculating the compound interest is as,

CI = P (1 + r/100)n

• P is the principal amount
• r is the rate of interest
• n is frequency or no. of  times the interest is compounded annually
• t is the overall tenure.

If the time period for the calculation of interest is monthly, the interest is calculated for each month, and the amount is compounded 12 times a year as there are 12 months in a year. The formula to calculate the compound interest when the principal is compounded monthly is given as:

CI = P(1 + (r/12) )12t – P

Here the compound interest is calculated for a month (time period). Thus, the rate of interest r, is divided by 12 and the time period is 12 times.

## Examples Using Monthly Compound Interest Formula

Example1: If Sam lends \$1,500 to his friend at an annual interest rate of 4.3%, compounded per month. Calculate the interest after the end of the year by using the compound interest formula.

Solution:

To find: Compound interest accumulated after 1 year.

P = 1500, r = 0.043 (4.3%), n = 12 , and t = 1 (given)

Using monthly compound interest formula,

CI = P(1 + (r/n) )nt – P

Put the values,

CI = 1500(1 + (0.043/12))12 – 1500

CI = 65.786

Answer: The compound interest after 1 year will be \$65.786.

Example 2: James borrowed \$600 from the bank at some rate compounded per month and that amount becomes quadruple in 2 years. Calculate the rate at which James borrowed the money by using the monthly compound interest formula.

Solution:

To find: Interest rate

P = 600, n = 12, and t = 2, Amount = 2400 (given)

Using formula,

CI = Amount – Principal

Put the values,

CI = 2400 – 600 = 1800

Using monthly compound interest formula,

CI = P(1 + (r/12) )12t – P

Put the values,

1800 = 600(1+ (r/12))12×2 – 600

4 = (1+ (r/12))24

r = 71.4

Answer: The Interest rate on the given amount of money is 71.4%.

Example 3: Calculate the monthly compound interest on the sum of \$6000 borrowed at the rate of 10% for 2 years.

Solution:

To find: Monthly compound interest

P=\$6000, r=10%, t=2years (given)

CI = P(1 + (r/12) )12t – P

Put the values,

= 6000(1+10/12)12×2 – 6000

= 7322.35 – 6000 = 1322.35

Answer: The monthly compound interest for 2 years is \$1322.35

## Examples Using Monthly Compound Interest Formula

Example1: If Sam lends \$1,500 to his friend at an annual interest rate of 4.3%, compounded per month. Calculate the interest after the end of the year by using the compound interest formula.

Solution:

To find: Compound interest accumulated after 1 year.

P = 1500, r = 0.043 (4.3%), n = 12 , and t = 1 (given)

Using monthly compound interest formula,

CI = P(1 + (r/n) )nt – P

Put the values,

CI = 1500(1 + (0.043/12))12 – 1500

CI = 65.786

Answer: The compound interest after 1 year will be \$65.786.

Example 2: James borrowed \$600 from the bank at some rate compounded per month and that amount becomes quadruple in 2 years. Calculate the rate at which James borrowed the money by using the monthly compound interest formula.

Solution:

To find: Interest rate

P = 600, n = 12, and t = 2, Amount = 2400 (given)

Using formula,

CI = Amount – Principal

Put the values,

CI = 2400 – 600 = 1800

Using monthly compound interest formula,

CI = P(1 + (r/12) )12t – P

Put the values,

1800 = 600(1+ (r/12))12×2 – 600

4 = (1+ (r/12))24

r = 71.4

Answer: The Interest rate on the given amount of money is 71.4%.

Example 3: Calculate the monthly compound interest on the sum of \$6000 borrowed at the rate of 10% for 2 years.

Solution:

To find: Monthly compound interest

P=\$6000, r=10%, t=2years (given)

CI = P(1 + (r/12) )12t – P

Put the values,

= 6000(1+10/12)12×2 – 6000

= 7322.35 – 6000 = 1322.35

Answer: The monthly compound interest for 2 years is \$1322.35

## FAQs on Monthly Compound Interest Formula

### What Is the Monthly Compound Interest Formula in Math?

The monthly compound interest formula is used to find the compound interest per month. The formula of monthly compound interest is: CI = P(1 + (r/12) )12t – P where, P is the principal amount, r is the interest rate in decimal form, and t is the time.

### How to Calculate Amount Using Monthly Compound Interest Formula?

There is a direct formula for the calculation of monthly compound interest. A = CI = P(1 + (r/12) )12t

• Step 1: Here we need to define the principal and the rate of interest at which the compound interest is calculated so check for the values of P, r and t.
• Step: Put the values in the formula, A = CI = P(1 + (r/12) )12t

### What Is r In the Monthly Compound Interest Formula?

In the monthly compound interest formula, CI = P(1 + (r/12) )12t – P, r refers to the interest rate on the principal.

### What Are the Components of the Monthly Compound Interest Formula?

The calculation of monthly compound interest requires us to know the principal, rate of interest, and the time period.

## Calculator Use

The compound interest calculator lets you see how your money can grow using interest compounding.

Calculate compound interest on an investment, 401K or savings account with annual, quarterly, daily or continuous compounding.

We provide answers to your compound interest calculations and show you the steps to find the answer. You can also experiment with the calculator to see how different interest rates or loan lengths can affect how much you’ll pay in compounded interest on a loan.

Read further below for additional compound interest formulas to find principal, interest rates or final investment value. We also show you how to calculate continuous compounding with the formula A = Pe^rt.

## The Compound Interest Formula

This calculator uses the compound interest formula to find principal plus interest. It uses this same formula to solve for principal, rate or time given the other known values. You can also use this formula to set up a compound interest calculator in Excel®1.

A = P(1 + r/n)nt

In the formula

• A = Accrued amount (principal + interest)
• P = Principal amount
• r = Annual nominal interest rate as a decimal
• R = Annual nominal interest rate as a percent
• r = R/100
• n = number of compounding periods per unit of time
• t = time in decimal years; e.g., 6 months is calculated as 0.5 years. Divide your partial year number of months by 12 to get the decimal years.
• I = Interest amount
• ln = natural logarithm, used in formulas below

### Compound Interest Formulas Used in This Calculator

The basic compound interest formula A = P(1 + r/n)nt can be used to find any of the other variables. The tables below show the compound interest formula rewritten so the unknown variable is isolated on the left side of the equation.

### How to Use the Compound Interest Calculator: Example

Say you have an investment account that increased from \$30,000 to \$33,000 over 30 months. If your local bank offers a savings account with daily compounding (365 times per year), what annual interest rate do you need to get to match the rate of return in your investment account?

In the calculator above select “Calculate Rate (R)”. The calculator will use the equations: r = n((A/P)1/nt – 1) and R = r*100.

Enter:

• Total P+I (A): \$33,000
• Principal (P): \$30,000
• Compound (n): Daily (365)
• Time (t in years): 2.5 years (30 months equals 2.5 years)

Showing the work with the formula r = n((A/P)1/nt – 1):

So you’d need to put \$30,000 into a savings account that pays a rate of 3.813% per year and compounds interest daily in order to get the same return as the investment account.

### How to Derive A = Pert the Continuous Compound Interest Formula

A common definition of the constant e is that:

With continuous compounding, the number of times compounding occurs per period approaches infinity or n → ∞. Then using our original equation to solve for A as n → ∞ we want to solve:

This equation looks a little like the equation for e. To make it look more similar so we can do a substitution we introduce a variable m such that m = n/r then we also have n = mr. Note that as n approaches infinity so does m.

Replacing n in our equation with mr and cancelling r in the numerator of r/n we get:

## Compound Interest Formula With Examples

Compound interest, or ‘interest on interest’, is calculated with the compound interest formula.

The formula for compound interest is A = P(1 + r/n)(nt), where P is the principal balance, r is the interest rate, n is the number of times interest is compounded per time period and t is the number of time periods.

The concept of compound interest is that interest is added back to the principal sum so that interest is gained on that already-accumulated interest during the next compounding period. How important is it? Just ask Warren Buffett, one of the world’s most successful investors:

“My wealth has come from a combination of living in America, some lucky genes, and compound interest.”

Warren Buffett

In this article, we’ll take a look at the compound interest formula in more depth. We’ll also go through an example and discuss other variations of the formula that can help you to calculate the interest rate and time factor or to incorporate regular contributions. Should you wish to try some calculations using your own figures, you can use our popular compound interest calculator.

## How to use the compound interest formula

To use the compound interest formula you will need figures for principal amount, annual interest rate, time factor and the number of compound periods. Once you have those, you can go through the process of calculating compound interest.The formula for compound interest, including principal sum, is:
A = P(1 + r/n)(nt)

Where:

• A = the future value of the investment/loan, including interest
• P = the principal investment amount (the initial deposit or loan amount)
• r = the annual interest rate (decimal)
• n = the number of times that interest is compounded per unit t
• t = the time the money is invested or borrowed for

It’s worth noting that this formula gives you the future value of an investment or loan, which is compound interest plus the principal. Should you wish to calculate the compound interest only, you need to deduct the principal from the result. So, your formula looks like this:

Compounded interest only (without principal): P (1 + r/n) (nt) – P

### Let’s look at an example

If an amount of \$5,000 is deposited into a savings account at an annual interest rate of 5%, compounded monthly, the value of the investment after 10 years can be calculated as follows…

P = 5000.
r = 5/100 = 0.05 (decimal).
n = 12.
t = 10.

If we plug those figures into the formula, we get the following:

A = 5000 (1 + 0.05 / 12) (12 * 10) = 8235.05.

So, the investment balance after 10 years is \$8,235.05.

### Methodology

A few people have written to me asking me to explain step-by-step how we get the 8235.05. This all revolves around BODMAS / PEMDAS and the order of operations. Let’s go through it:

A = 5000 (1 + 0.05 / 12) ^ (12(10))

(note that ^ means ‘to the power of’)

Using the order of operations we work out the totals in the brackets first. Within the first set of brackets, you need to do the division first and then the addition (division and multiplication should be carried out before addition and subtraction). We can also work out the 12(10). This gives us…

A = 5000 (1 + 0.00416) ^ 120

(note that the over-line in the calculation signifies a decimal that repeats to infinity. So, 0.00416666666…)

Then:

A = 5000 (1.00416) ^ 120

The exponent goes next. So, we calculate (1.00416) ^ 120.

This means we end up with:

5000 × 1.6470095042509848

= 8235.0475.

You may have seen some examples giving a formula of A = P ( 1+r ) t . This simplified formula assumes that interest is compounded once per period, rather than multiple times per period.

### The benefit of compound interest

I think it’s worth taking a moment to examine the benefit of compound interest using our example. The benefit hopefully becomes clear when I tell you that without compound interest, your investment balance in the above example would be only \$7,500 (\$250 per year for 10 years, plus the original \$5000) by the end of the term. So, thanks to the wonder of compound interest, you stand to gain an additional \$735.05.

To give a graphical example, the graph below shows the result of \$1000 invested over 20 years at an interest rate of 10%. The principal figure is in green. The blue part of the graph shows the result of 10% interest without compounding. Finally, the purple part demonstrates the benefit of compound interest over those 20 years.

## Formula for calculating interest rate (r)

r = n[(A/P)(1/nt)-1]

Where:

• r = the interest rate (decimal)
• A = the future value of the investment
• P = the principal investment amount
• n = the number of times that interest is compounded per unit t
• t = the time the money is invested for

## Formula for calculating principal (P)

This formula is useful if you want to work backwards and find out how much you would need to start with in order to achieve a chosen future value.P = A / (1 + r/n)nt

Where:

• P = the principal investment amount
• A = the future value of the investment
• r = the interest rate (decimal)
• n = the number of times that interest is compounded per unit t
• t = the time the money is invested for

Example: Let’s say your goal is to end up with \$10,000 in 5 years, and you can get an 8% interest rate on your savings, compounded monthly. Your calculation would be: P = 10000 / (1 + 0.08/12)(12×5) = \$6712.10. So, you would need to start off with \$6712.10 to achieve your goal.

## Formula for calculating time (t)

This variation of the formula works for calculating time (t), by using natural logarithms.

t = ln(A/P) / n[ln(1 + r/n)]

Where:

• A = the value of the accrued investment/loan
• P = the principal amount
• r = the annual interest rate (decimal)
• n = the number of times that interest is compounded per unit t
• t = the time the money is invested or borrowed for

## Formula for regular contributions

A lot of people have asked me to include a single formula for compound interest with monthly additions. Believe me when I tell you that it isn’t quite as simple as it sounds. In order to work out calculations involving monthly additions, you will need to use two formulae – our original one, listed above, plus the ‘future value of a series‘ formula for the monthly additions.

At the request of readers, I’ve adapted the formula explanation to allow you to calculate periodic additions, not just monthly (added May 2016). These formulae assume that your frequency of compounding is the same as the periodic payment interval (monthly compounding, monthly contributions, etc). If you would like to try a version of the formula that allows you to have a different periodic payment interval to the compounding frequency, please see the ‘ periodic payments’ section below.

If the additional deposits are made at the END of the period (end of month, year, etc), here are the two formulae you will need:

Compound interest for principal:

P(1+r/n)(nt)

Future value of a series:

PMT × {[(1 + r/n)(nt) – 1] / (r/n)}

If the additional deposits are made at the BEGINNING of the period (beginning of year, etc), here are the two formulae you will need:

Compound interest for principal:

P(1+r/n)(nt)

Future value of a series:

PMT × {[(1 + r/n)(nt) – 1] / (r/n)} × (1+r/n)

Where:

• A = the future value of the investment/loan, including interest
• P = the principal investment amount (the initial deposit or loan amount)
• PMT = the monthly payment
• r = the annual interest rate (decimal)
• n = the number of times that interest is compounded per unit t
• t = the time (months, years, etc) the money is invested or borrowed for

Example

If an amount of \$5,000 is deposited into a savings account at an annual interest rate of 5%, compounded monthly, with additional deposits of \$100 per month (made at the end of each month). The value of the investment after 10 years can be calculated as follows…

P = 5000. PMT = 100. r = 5/100 = 0.05 (decimal). n = 12. t = 10.

If we plug those figures into the formulae, we get:

• Total = [ Compound interest for principal ] + [ Future value of a series ]
• Total = [ P(1+r/n)^(nt) ] + [ PMT × (((1 + r/n)^(nt) – 1) / (r/n)) ]
• Total = [ 5000 (1 + 0.05 / 12) ^ (12 × 10) ] + [ 100 × (((1 + 0.00416)^(12 × 10) – 1) / (0.00416)) ]
• Total = [ 5000 (1.00416) ^ (120) ] + [ 100 × (((1.00416^120) – 1) / 0.00416) ]
• Total = [ 8235.05 ] + [ 100 × (0.647009497690848 / 0.00416) ]
• Total = [ 8235.05 ] + [ 15528.23 ]
• Total = [ \$23,763.28 ]

So, the investment balance after 10 years is \$23,763.28.

## Different periodic payments

A few people have requested a version of the above formula that takes into account the number of periodic payments (both formulae above assume your periodic payments match the frequency of compounding). For example, your money may be compounded quarterly but you’re making contributions monthly. In this case, you may wish to try this version of the formula, originally suggested by Darinth Douglas, and then expanded upon by Jean-Baptiste Delaroche. I’m most grateful for their input. This formula assumes that regular deposits are paid at the beginning rather than at the end of the period.

Compound interest for principal:

P(1+r/n)(nt)

Future value of a series:

PMT × p {[(1 + r/n)(nt) – 1] / (r/n)}

(With ‘p’ being the number of periodic payments in the compounding period, divided by n)

Example

An amount of \$100 is deposited quarterly into a savings account at an annual interest rate of 10%, compounded monthly. The value of the investment after 12 months can be calculated as follows…

PMT = 100. r = 0.1 (decimal). n = 12. p = 4/n = 4/12 = 0.3333333.

If we plug those figures into the formula, we get the following:

• Total = PMT × p {[(1 + r/n)(nt) – 1] / (r/n)}
• Total = 100 × 0.3333333 × {[(1 + 0.1 / 12) ^ (12 × 1) – 1] / (0.1 / 12)}
• Total = 100 × 0.3333333 × {[1.008333 ^ (12) – 1] / 0.008333}
• Total = 100 × 0.3333333 × {0.104709 / 0.008333}
• Total = 100 × 0.3333333 × 12.565583
• Total = 418.85

So, the investment balance after 12 months is \$418.85 (or \$418.84 if you round the numbers during the calculation).

## What is the Monthly Compound Interest?

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. Monthly compounding is calculated by principal amount multiplied by one plus rate of interest divided by a number of periods whole raise to the power of the number of periods and that whole is subtracted from the principal amount which gives the interest amount.

### Monthly Compound Interest Formula

The equation for calculating it is represented as follows,

A= (P (1+r/n)nt) – P

Where

• A= Monthly compound rate
• P= Principal amount
• R= Rate of interest
• N= Time period

Generally, when someone deposits money in the bank, the bank pays interest to the investor in the form of quarterly interest. But when someone lends money from the banks, the banks charge the interest from the person who has taken the loan in the form of monthly compounding interest. The higher the frequency, the more the interest charged or paid on the principal. For example, the interest amount for monthly compounding will be higher than the amount for quarterly compounding. This is the business model of a bank in a broader way where they make money in the differential of the interest paid for the deposits, and the interest receives for the loan disbursed.

Examples

Example #1

A sum of \$4000 is borrowed from the bank where the interest rate is 8%, and the amount is borrowed for a period of 2 years. Let us find out how much will be monthly compounded interest charged by the bank on loan provided.

Below is the given data for the calculation

Example #2

A sum of \$35000 is borrowed from the bank as a car loan where the interest rate is 7% per annum, and the amount is borrowed for a period of 5 years. Let us find out how much will be monthly compounded interest charged by the bank on loan provided.

Below is the given data for the calculation

Example #3

A sum of \$1 00,000 is borrowed from the bank as a home loan where the interest rate is 5% per annum, and the amount is borrowed for a period of 15 years. Let us find out how much will be monthly compounded interest charged by the bank on loan provided.

Below is the given data for the calculation

= \$29435

So the monthly interest will be \$ 29,435.

## Calculating annual compound interest

Mathematically challenged? Don’t worry, just head right on over to our savings calculator page to crunch the numbers. But if you’re determined to put your maths skills to the test here’s how you calculate compound interest:

2. This value is put to the power of the amount of years you want to leave your savings for

3. Multiply the result by the ‘principal’, which is the current balance of your account

4. You’ll have the new balance of your account

Here’s an example: If you put \$10,000 away in a savings account to earn 3% p.a. for 2 years, the calculations to work out your compound interest might look like this:

1. Divide your interest rate by 12 (interest rates are expressed annually, so to get a monthly figure, you have to divide it by the number of months in a year.)

2. Add 1 to this to account for the effects of compounding

3. Put all this to the power of the number of months your savings will be put away

4. Multiply it all by the ‘principal’, which is the current balance of your account

5. You’ll have the new balance of your account

A little confused? Let’s go back to our example: If you took \$10,000 at 3% p.a., and put it away for the same two years, but with interest compounding monthly, you’d calculate the interest like this:

And if you look at the example above, you can see that by compounding monthly, you’ve made an extra \$8.57.

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