## Compound Interest Formula With Examples

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

The formula for compound interest is **P (1 + r/n)^(nt)**, where P is the initial 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

### 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…

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.

### Interactive compound interest formula

I have created the calculator below to show you the formula and resulting accrued investment/loan value (A) for the figures that you enter. Note that this calculator requires JavaScript to be enabled in your browser.

### Formulae to find compound interest rate, time and principal

It may be that you want to manipulate the compound interest formula to work out the interest rate for IRR or CAGR, or a principal investment/loan figure. Here are the formulae you need.

### Formula for interest rate (r)

Should you wish to work out the yearly interest rate you’re getting on your savings, investment, personal loan or car loan, this formula can help. Note that you should multiply your result by 100 to get a percentage figure (%)

### Formula for 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.

**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 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

### Compound interest formula (with 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 Compound Interest?

Compound interest (or compounding interest) is the interest on a loan or deposit calculated based on both the initial principal and the accumulated interest from previous periods. Thought to have originated in 17th-century Italy, compound interest can be thought of as “interest on interest,” and will make a sum grow at a faster rate than simple interest, which is calculated only on the principal amount.

The rate at which compound interest accrues depends on the frequency of compounding, such that the higher the number of compounding periods, the greater the compound interest. Thus, the amount of compound interest accrued on $100 compounded at 10% annually will be lower than that on $100 compounded at 5% semi-annually over the same time period. Because the interest-on-interest effect can generate increasingly positive returns based on the initial principal amount, compounding has sometimes been referred to as the “miracle of compound interest.”

### KEY TAKEAWAYS

- Compound interest (or compounding interest) is interest calculated on the initial principal, which also includes all of the accumulated interest from previous periods on a deposit or loan.
- Compound interest is calculated by multiplying the initial principal amount by one plus the annual interest rate raised to the number of compound periods minus one.
- Interest can be compounded on any given frequency schedule, from continuous to daily to annually.
- When calculating compound interest, the number of compounding periods makes a significant difference.

### How Compound Interest Works

Compound interest is calculated by multiplying the initial principal amount by one plus the annual interest rate raised to the number of compound periods minus one. The total initial amount of the loan is then subtracted from the resulting value.

The formula for calculating the amount of compound interest is as follows:

**Compound interest**= total amount of principal and interest in future (or future value)*less*principal amount at present (or present value)

= [P (1 +i)^{n}] – P

= P [(1 +i)^{n }– 1]

Where:

P = principal

i= nominal annual interest rate in percentage terms

n = number of compounding periods

Take a three-year loan of $10,000 at an interest rate of 5% that compounds annually. What would be the amount of interest? In this case, it would be:

$10,000 [(1 + 0.05)

^{3}– 1] = $10,000 [1.157625 – 1] = $1,576.25

### Compound Interest Schedules

Interest can be compounded on any given frequency schedule, from daily to annually. There are standard compounding frequency schedules that are usually applied to financial instruments.

The commonly used compounding schedule for savings accounts at banks is daily. For a certificate of deposit (CD), typical compounding frequency schedules are daily, monthly, or semiannually; for money market accounts, it’s often daily. For home mortgage loans, home equity loans, personal business loans, or credit card accounts, the most commonly applied compounding schedule is monthly.

There can also be variations in the time frame in which the accrued interest is actually credited to the existing balance. Interest on an account may be compounded daily but only credited monthly. It is only when the interest is actually credited, or added to the existing balance, that it begins to earn additional interest in the account.

Some banks also offer something called continuously compounding interest, which adds interest to the principal at every possible instant. For practical purposes, it doesn’t accrue that much more than daily compounding interest unless you want to put money in and take it out the same day.

More frequent compounding of interest is beneficial to the investor or creditor. For a borrower, the opposite is true.

### Compounding Periods

When calculating compound interest, the number of compounding periods makes a significant difference. The basic rule is that the higher the number of compounding periods, the greater the amount of compound interest.

The following table demonstrates the difference that the number of compounding periods can make for a $10,000 loan with an annual 10% interest rate over a 10-year period.

### Special Considerations

Compound interest is closely tied to the time value of money and the Rule of 72, both important concepts in investing.

### Time Value of Money Consideration

Understanding the time value of money and the exponential growth created by compounding is essential for investors looking to optimize their income and wealth allocation.

The formula for obtaining the future value (FV) and present value (PV) are as follows:

FV = PV (1 +i)^{n }and PV = FV / (1 + i)^{ n}

For example, the future value of $10,000 compounded at 5% annually for three years:

= $10,000 (1 + 0.05)

^{3}

= $10,000 (1.157625)

= $11,576.25

The present value of $11,576.25 discounted at 5% for three years:

= $11,576.25 / (1 + 0.05)

^{3}

= $11,576.25 / 1.157625

= $10,000

The reciprocal of 1.157625, which equals 0.8638376, is the discount factor in this instance.

### Rule of 72 Consideration

The so-called Rule of 72 calculates the approximate time over which an investment will double at a given rate of return or interest “i,” and is given by (72/i). It can only be used for annual compounding.

As an example, an investment that has a 6% annual rate of return will double in 12 years. An investment with an 8% annual rate of return will thus double in nine years.

### Compound Annual Growth Rate (CAGR)

The compound annual growth rate (CAGR) is used for most financial applications that require the calculation of a single growth rate over a period of time.

Let’s say your investment portfolio has grown from $10,000 to $16,000 over five years; what is the CAGR? Essentially, this means that PV = -$10,000, FV = $16,000, and t = 5, so the variable “i” has to be calculated. Using a financial calculator or Excel, it can be shown that i = 9.86%.

**FAST FACT**

According to the cash-flow convention, your initial investment (PV) of $10,000 is shown with a negative sign because it represents an outflow of funds. PV and FV must necessarily have opposite signs to solve for “i” in the above equation.

### CAGR Real-Life Applications

The CAGR is extensively used to calculate returns over periods of time for stock, mutual funds, and investment portfolios. The CAGR is also used to ascertain whether a mutual fund manager or portfolio manager has exceeded the market’s rate of return over a period of time. If, for example, a market index has provided total returns of 10% over a five-year period, but a fund manager has only generated annual returns of 9% over the same period, the manager has underperformed the market.

The CAGR can also be used to calculate the expected growth rate of investment portfolios over long periods of time, which is useful for purposes such as saving for retirement. Consider the following examples:

**Example 1:** A risk-averse investor is happy with a modest 3% annual rate of return on her portfolio. Her present $100,000 portfolio would, therefore, grow to $180,611 after 20 years. In contrast, a risk-tolerant investor who expects an annual return of 6% on her portfolio would see $100,000 grow to $320,714 after 20 years.

**Example 2:** The CAGR can be used to estimate how much needs to be stowed away to save for a specific objective. A couple who would like to save $50,000 over 10 years toward a down payment on a condo would need to save $4,165 per year if they assume an annual return (CAGR) of 4% on their savings. If they are prepared to take a little extra risk and expect a CAGR of 5%, they would need to save $3,975 annually.

**Example 3:** The CAGR can also demonstrate the virtues of investing earlier rather than later in life. If the objective is to save $1 million by retirement at age 65, based on a CAGR of 6%, a 25-year old would need to save $6,462 per year to attain this goal. A 40-year old, on the other hand, would need to save $18,227, or almost three times that amount, to attain the same goal.

### **FAST FACT**

- CAGRs also crop up frequently in economic data. Here is an example: China’s per capita GDP increased from $193 in 1980 to $6,091 in 2012. What is the annual growth in per capita GDP over this 32-year period? The growth rate “i” in this case works out to be an impressive 11.4%.

### Pros and Cons of Compounding

Though the miracle of compounding has led to the apocryphal story of Albert Einstein calling it the eighth wonder of the world or man’s greatest invention, compounding can also work against consumers who have loans that carry very high interest rates, such as credit card debt. A credit card balance of $20,000 carried at an interest rate of 20% compounded monthly would result in a total compound interest of $4,388 over one year or about $365 per month.

On the positive side, compounding can work to your advantage when it comes to your investments and can be a potent factor in wealth creation. Exponential growth from compounding interest is also important in mitigating wealth-eroding factors, such as increases in the cost of living, inflation, and reduced purchasing power.

Mutual funds offer one of the easiest ways for investors to reap the benefits of compound interest. Opting to reinvest dividends derived from the mutual fund results in purchasing more shares of the fund. More compound interest accumulates over time, and the cycle of purchasing more shares will continue to help the investment in the fund grow in value.

Consider a mutual fund investment opened with an initial $5,000 and an annual addition of $2,400. With an average annual return of 12% over 30 years, the future value of the fund is $798,500. The compound interest is the difference between the cash contributed to an investment and the actual future value of the investment. In this case, by contributing $77,000, or a cumulative contribution of just $200 per month, over 30 years, compound interest is $721,500 of the future balance.

Of course, earnings from compound interest are taxable, unless the money is in a tax-sheltered account; it’s ordinarily taxed at the standard rate associated with the taxpayer’s tax bracket.

### Compound Interest Investments

An investor who opts for a reinvestment plan within a brokerage account is essentially using the power of compounding in whatever they invest.

Investors can also experience compounding interest with the purchase of a zero-coupon bond. Traditional bond issues provide investors with periodic interest payments based on the original terms of the bond issue, and because these are paid out to the investor in the form of a check, interest does not compound. Zero-coupon bonds do not send interest checks to investors; instead, this type of bond is purchased at a discount to its original value and grows over time. Zero-coupon bond issuers use the power of compounding to increase the value of the bond so it reaches its full price at maturity.

Compounding can also work for you when making loan repayments. Making half your mortgage payment twice a month, for example, rather than making the full payment once a month, will end up cutting down your amortization period and saving you a substantial amount of interest.

## How to Calculate Compound Interest

If it’s been a while since your math class days, fear not: There are handy tools for figuring out compounding. Many calculators (both handheld and computer-based) have exponent functions you can utilize for these purposes.

### Calculating Compound Interest in Excel

If more complicated compounding tasks arise, you can perform them in Microsoft Excel—in three different ways.

- The first way to calculate compound interest is to multiply each year’s new balance by the interest rate. Suppose you deposit $1,000 into a savings account with a 5% interest rate that compounds annually, and you want to calculate the balance in five years. In Microsoft Excel, enter “Year” into cell A1 and “Balance” into cell B1. Enter years 0 to 5 into cells A2 through A7. The balance for year 0 is $1,000, so you would enter “1000” into cell B2. Next, enter “=B2*1.05” into cell B3. Then enter “=B3*1.05” into cell B4 and continue to do this until you get to cell B7. In cell B7, the calculation is “=B6*1.05”. Finally, the calculated value in cell B7—$1,276.28—is the balance in your savings account after five years. To find the compound interest value, subtract $1,000 from $1,276.28; this gives you a value of $276.28.
- The second way to calculate compound interest is to use a fixed formula. The compound interest formula is ((P*(1+i)^n) – P), where P is the principal, i is the annual interest rate, and n is the number of periods. Using the same information above, enter “Principal value” into cell A1 and 1000 into cell B1. Next, enter “Interest rate” into cell A2 and “.05” into cell B2. Enter “Compound periods” into cell A3 and “5” into cell B3. Now you can calculate the compound interest in cell B4 by entering “=(B1*(1+B2)^B3)-B1”, which gives you $276.28.
- A third way to calculate compound interest is to create a macro function. First start the Visual Basic Editor, which is located in the developer tab. Click the Insert menu, and click on Module. Then type “Function Compound_Interest (P As Double, I As Double, N As Double) As Double” in the first line. On the second line, hit the tab key and type in “Compound_Interest = (P*(1+i)^n) – P.” On the third line of the module, enter “End Function.” You have created a function macro to calculate the compound interest rate. Continuing from the same Excel worksheet above, enter “Compound interest” into cell A6 and enter “=Compound_Interest(B1, B2, B3).” This gives you a value of $276.28, which is consistent with the first two values.

### Other Compound Interest Calculators

A number of free compound interest calculators are offered online, and many handheld calculators can carry out these tasks as well.

- The free compound interest calculator offered through Financial-Calculators.com is simple to operate and offers to compound frequency choices from daily through annually. It includes an option to select continuous compounding and also allows input of actual calendar start and end dates. After inputting the necessary calculation data, the results show interest earned, future value, annual percentage yield or APY) (a measure that includes compounding), and daily interest.
- Investor.gov, a website operated by the U.S. Securities and Exchange Commission (SEC), offers a free online compound interest calculator. The calculator is fairly simple, but it does allow inputs of monthly additional deposits to the principal, which is helpful for calculating earnings where additional monthly savings are being deposited.
- A free online interest calculator with a few more features is available at TheCalculatorSite.com. This calculator allows calculations for different currencies, the ability to factor in monthly deposits or withdrawals, and the option to have inflation-adjusted increases to monthly deposits or withdrawals automatically calculated as well.

### How Can I Tell if Interest Is Compounded?

The Truth in Lending Act (TILA) requires that lenders disclose loan terms to potential borrowers, including the total dollar amount of interest to be repaid over the life of the loan and whether interest accrues simply or is compounded.

Another method is to compare a loan’s interest rate to its annual percentage rate (APR), which the TILA also requires lenders to disclose. The APR converts the finance charges of your loan, which include all interest and fees, to a simple interest rate. A substantial difference between the interest rate and APR means one or both of two scenarios: Your loan uses compound interest, or it includes hefty loan fees in addition to interest. Even when it comes to the same type of loan, the APR range can vary wildly between lenders depending on the financial institution’s fees and other costs.

You’ll note that the interest rate you are charged also depends on your credit. Loans offered to those with excellent credit carry significantly lower interest rates than those charged to borrowers with poor credit.

### What is a Simple Definition of Compound Interest?

Compound interest refers to the phenomenon whereby the interest associated with a bank account, loan, or investment increases exponentially—rather than linearly—over time. The key to understanding the concept is the word “compound.”

Suppose you make a $100 investment in a business that pays you a 10% dividend every year. You have the choice of either pocketing those dividend payments like cash or reinvesting those payments into additional shares. If you choose the second option, reinvesting the dividends and compounding them together with your initial $100 investment, then the returns you generate will start to grow over time.

### Who Benefits From Compound Interest?

Simply put, compound interest benefits investors, but the meaning of “investors” can be quite broad. Banks, for instance, benefit from compound interest when they lend money and reinvest the interest they receive into giving out additional loans. Depositors also benefit from compound interest when they receive interest on their bank accounts, bonds, or other investments.

It is important to note that although the term “compound interest” includes the word “interest,” the concept applies beyond situations for which the word interest is typically used, such as bank accounts and loans.

### Can Compound Interest Make You Rich?

Yes. In fact, compound interest is arguably the most powerful force for generating wealth ever conceived. There are records of merchants, lenders, and various businesspeople using compound interest to become rich for literally thousands of years. In the ancient city of Babylon, for example, clay tablets were used over 4,000 years ago to instruct students on the mathematics of compound interest.

In modern times, Warren Buffett became one of the richest people in the world through a business strategy that involved diligently and patiently compounding his investment returns over long periods of time. It is likely that, in one form or another, people will be using compound interest to generate wealth for the foreseeable future.

### 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):

Your Answer: R = 3.813% per year

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 = Pe^{rt} 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:

**Derivation of Compound Interest Formula**

The formula for compound interest can be derived from the formula for simple interest. The formula for simple interest is the product of the principal, time period, and rate of interest (SI = ptr/100). Before looking into to derivation of the formula for compound interest, let us understand the basic difference between simple interest, compound interest computation. The principal remains constant over a period of time, for simple internet computation, but for compound interest computation the interest is added to the principal, for compound interest computation.

**Derivation:**

Let the principal is P and the rate of interest be r. At the end of the first compounding period, the simple interest on the principal is P × r/100. And hence, the amount is P + P × r/100 = P(1 + r/100). The amount is taken as the principal for the second computation period.

At the end of the second compounding period, the simple interest on the principal is: P(1 + r/100) × r/100, and hence the amount is: P(1 + r/100) × r/100 + P(1 + r/100) × r/100 = P(1 + r/100)^{2}.

Continuing in this manner for n compounding periods, the amount at the end of the n^{th }compounding period is A = P(1 + r/100)^{n}.

From the above formulas and computations, you can observe that the compound interest is the same as simple interest for the first interval. But, over a period of time, there is a remarkable difference in returns.

The simple interest value for each of the years is the same, as the principal on which it is calculated is constant. But the compound interest is varying and increasing across the years. Because the principal on which the compound interest is calculated is increasing. The principal for a particular year is equal to the sum of the initial principal value, and the accumulated interest of the past years.

For example, a sum of $10,000 is deposited at a rate of 10%. The below table explains the difference between simple interest and compound interest computation on this principal:

Simple Interest Calculation (r = 10%) | Compound Interest Calculation(r = 10%) |

P = 10,000 Time = 1 year Interest = 1000 |
P = 10,000 Time = 1 year Interest = 1000 |

P = 10,000 Time = 1 year Interest = 1000 |
P = 11000 Time = 1 year Interest = 1100 |

P = 10,000 Time = 1 year Interest = 1000 |
P = 12100 Time = 1 year Interest = 1210 |

P = 10,000 Time = 1 year Interest = 1000 |
P = 13310 Time = 1 year Interest = 1331 |

P = 10,000 Time = 1 year Interest = 1000 |
P = 14641 Time = 1 year Interest = 1464.1 |

Total Simple Interest = 5000 | Total Compount Interest = 6105.1 |

Total Amount = 1000 + 5000 = 6000 | Total Amount = 1000 + 6105.1 = 7105.1 |

**Compound Interest Formula for Different Time Periods**

Compound interest for a given principal can be calculated for different time periods using different formulas.

**Compound Interest Formula – Half Yearly**

The interest in the case of compound interest varies based on the period of computation. If the time period for the calculation of interest is half-yearly, the interest is calculated every six months, and the amount is compounded twice a year.

The formula to calculate the compound interest when the principal is compounded semi-annually or half-yearly is given as:

Here the compound interest is calculated for the half-yearly period, and hence the rate of interest r, is divided by 2 and the time period is doubled. The formula to calculate the amount when the principal is compounded semi-annually or half-yearly is given by:

In the above expression,

- A is the amount at the end of the time period
- P is the initial principal value, r is the rate of interest per annum
- t is the time period
- C.I. is the compound interest.

**Compound Interest Formula – Quarterly**

If the time period for the calculation of interest is quarterly, the interest is calculated for every three months, and the amount is compounded 4 times a year. The formula to calculate the compound interest when the principal is compounded quarterly is given as:

Here the compound interest is calculated for the quarterly time period, and hence the rate of interest r, is divided by 4 and the time period is quadrupled. The formula to calculate the amount when the principal is compounded quarterly is given by:

In the above expression,

- A is the amount at the end of the time period
- P is the initial principal value, r is the rate of interest per annum
- t is the time period
- C.I. is the compound interest.

**Monthly Compound Interest Formula**

The monthly compound interest formula is also known as the interest calculated per month i.e., n = 12. Total compound interest is the final amount excluding the principal amount. The monthly compound interest formula is expressed as:**CI = P (1 + r/12) ^{12t} – P**

**Daily Compound Interest Formula**

When the amount compounds daily, it means that the amount compounds 365 times in a year. i.e., n = 365. The daily compound interest formula is expressed as:**CI = P (1 + r/365) ^{365t} – P**

**Important Notes**

- Compound interest depends on the amount accumulated at the end of the previous tenure but not on the original principal.
- Banks, insurance companies, etc. generally levy compound interest.
- If the interest is compounded quarterly, the formula of amount is given by:

While calculating the compound interest, the rate of interest, and each time period must be of the same duration.

**Tips & Tricks**

**The rule of 72:**It is a quick method to know how long it will take for your money to double. Doubling Time = 72/Interest Rate

Using the rule of 72, we can find the number of years to double your money by simply dividing 72 by the rate of interest. For example, at an 8% compounded interest rate your money will double in 72 ÷ 8 = 9- The time duration over which an interest rate is applicable is referred to in many different terms. Sometimes it is called “per annum” or “annual” or “per year”. All of these mean you’ll get the given rate of interest over a period of 1 year. Semi-annual is 6 months. While quarterly is 3 months duration.

**Solved Examples on Compound Interest**

**Example 1:** Noah lends $4000 to Emma at an interest rate of 10% per annum, compounded half-yearly for a period of 2 years. Can you help him find out how much amount he gets after a period of 2 years from Emma?

**Solution:**

Let us identify the data given to us: The principal amount ‘P’ is $4000. The rate of interest, r’ is 10% per annum. Conversion period = Half-year, Rate of interest per half-year = 10/2 % = 5%. The time period ‘t’ is 2 years.

Time Period | Amount Calculation |

1^{st} half year | Principal = $4000 Interest = 5% × $4000 = (5/100) × 4000 = $200 Amount = $4000 + $200 = $4200 |

2^{nd} half year | Principal = $4200 Interest = 5% × $4200 = 5/100 × 4200 = $210 Amount = $4200 + $210 = $4410 |

3^{rd} half year | Principal = $4410 Interest = 5% × $4410 = $220.5 Amount = $4410 + $220.5 = $4630.5 |

4^{th} half year | Principal = $4630.5 Interest = 5% × $4630.5 = $231.53 Amount = $4630.5 + $231.53 = $4862.03 |

The total interest to be paid over 2 years 200 + $210 + $220.5 + $231.53 = $862.03. Total Amount = P + I=$4000 + $862.03 = $4862.03. Therefore the total amount is $4862.03.

**Example 2: **Solve the above-given problem using the compound interest formula.

**Solution:**

The principal amount ‘P’ is $4000. The rate of interest ‘r’ is 10% per annum. Conversion period = Half-year, Rate of interest per half-year = 10/2% = 5%. The time period ‘t’ is 2 years. The compounding frequency ‘n’ is 2.

Let us substitute the given data in the compound interest formula: A = P(1+{r / 2}/100)^{2n}= 4000(1+{10 / 2}/100)^{2(2)}= $4862.03

Therefore the final amount is $4862.03, and the compound interest formula makes the solution simple.

### FAQs on Compound Interest

#### How to Calculate Compound Interest?

The formula used to calculate compound interest is CI = P( 1 + r/100)^{n} – P. Here in this formula the amount is calculated and then the principal is subtracted from it, to obtain the compound interest value.

#### What Is the Difference Between Simple and Compound Interest?

Simple interest is the interest paid only on the principal, whereas, compound interest is the interest paid on both principal and interest compounded at regular intervals.

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

There is a direct formula for the calculation of compound interest. A = P(1 + r/100)^{n}. Here we need to define the rate of interest and the time interval at which the compound interest is calculated.

#### How To Calculate Amount Using Compound Interest Formula?

There is a general compound interest formula for the calculation of compound interest i.e.,

CI = Amount – Principal

where, Amount = P(1 + r/100)^{t}. By substituting the given parameters such as P (principal amount), r (rate of interest), and t (time) amount can be easily calculated.

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

The monthly compound interest formula is given as

CI = P(1 + (r/12) )^{12t }– P. Where, P is the principal amount, r is the interest rate in decimal form, n = 12 (it means that the amount compounded 12 times in a year), and t is the time.

#### What Is the Daily Compound Interest Formula?

The daily compound interest formula is given as

A = P (1 + r / 365)^{365 t}, where P is the principal amount, r is the interest rate of interest in decimal form, n = 365 (it means that the amount compounded 365 times in a year), and t is the time.

#### What Is the Future Value Compound Interest Formula?

The future value compound interest formula is expressed as FV = PV (1 + r / n)^{n t}. Here, PV = Present Value (Initial investment), r = rate of interest, n = number of times the amount is compounding, and t = time in years.

#### Is Interest Compounded Daily Better than Monthly?

The interest compounded daily has 365 compounding cycles a year. It will generate more money compared to interest compounded monthly, which has only 12 compounding cycles per year.

#### What Are the Main Disadvantages of Compound Interest?

If we miss a payment by a day also, towards the end of tenure it may incur a huge loss. The interest calculation is for the next cycle and for a higher value. Compound interest is actually designed to help the lenders but not the borrowers.

#### How Does Compound Interest Depend on Time Period?

The compound interest depends on the time interval of calculation of interest. The time interval for the calculation of interest can be a day, a week, month, quarterly, half-yearly. For the shorter time period of calculation, the net accumulated compound interest is higher.

#### How Much is Compound Interest Greater than Simple Interest?

The compound interest can be greater than the simple interest. The compound interest value varies and increases for successive time periods. An initial principal of $100 invested over a period of time would give a simple interest of $10, $10, %10… over successive time periods of 1 year, but would give a compound interest of $10, $11, $12.1, $13.31….. Thus the compound interest is greater than the simple interest. Only for the first year, or for the first cycle of calculation, the compound interest, and the simple interest values are equal.

#### Can Compound Interest be Greater than Principal?

The compound interest can be greater than the principal. The compound interest value varies and increases for successive time periods. An initial principal of $100 invested over a period of time would give a compound interest of $10, $11, $12.1, $13.31….over successive time periods of 1 year each. Thus the compound interest increases over a period of time and can be greater than the initial principal value.

#### How Do you Calculate Compound Interest for Half Year?

The formula for calculation of compound interest for half year is CI = p(1 + {r/2}/100)^{2t}.- p. Here in this formula ‘A’ is the final amount, ‘p’ is the principal, and ‘t’ is the time in years. In the formula we can observe that the rate of interest is halved and the time is doubled, to account for the calculation of compound interest for half a year.

#### What Is the Information Required to Calculate Compound Interest?

The calculation of compound interest requires us to know the principal, rate of interest, and the time period. Also, we need to know the time interval for which the interest is to be calculated.

#### What Are the Units of Compound Interest?

The units of compound interest are the unit of currency and are the same as the unit used for the principal value. If the principal is in dollars, or yen, the compound interest would also be in dollars or yen.

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