# Compound Interest Calculator

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## What is Compound Interest?

Compound interest is a type of interest where the interest earned in one period is added to the principal, so that in the next period, interest is earned on both the principal and the interest from the previous period. This can cause the interest to grow at an accelerating rate over time.

For example, if you deposit $1000 in a savings account that pays 5% annual interest, at the end of the first year you will have earned$50 in interest, for a total balance of $1050. However, in the second year, you will earn interest not only on the original$1000 principal, but also on the $50 in interest from the first year. This means that your interest earned in the second year will be$52.50, for a total balance of $1102.50. As you can see, the interest earned in each year is growing, because it is calculated on a larger amount. The more frequently the interest is compounded, the more quickly the interest will grow. For example, if the interest is compounded daily, it will grow more quickly than if it is compounded annually. Compound interest can be used to your advantage when investing in things like savings accounts, CDs and bonds, but it can also work against you when you have to pay interest on loans, credit cards and other debts. ## How To Calculate Compound Interest Compound interest is typically calculated using the following formula: $A = P\times (1 + \frac r n)^{(n\times t)}$ Where: • A is the final amount (principal + interest) • P is the initial principal or deposit • r is the annual interest rate (expressed as a decimal) • n is the number of times per year the interest is compounded (e.g. annually, semi-annually, quarterly, monthly, daily) • t is the number of years the money is invested or borrowed for For example, if you deposit$1000 into a savings account that pays 5% annual interest, compounded quarterly, the compound interest calculation would be as follows:

$A = 1000 \times (1 + \frac {0.05} 4)^{(4\times5)} = 1276.28$

This means that after 5 years, your balance would be $1276.28, which includes the$1000 deposit and $276.28 in interest. It's also possible to calculate the Interest separately: $I = A - P$ Where I = Interest earned over the time t It's important to note that this formula assumes that the interest is compounded continuously and not periodically, in the latter case, the formula is a bit different. ## How To Calculate Monthly Compound Interest To calculate the monthly compound interest, you can use the following formula: $A = P\times(1 + \frac r {12})^{(12\times t)}$ Where: • A is the final amount (principal + interest) • P is the initial principal or deposit • r is the annual interest rate (expressed as a decimal) • t is the number of years the money is invested or borrowed for For example, if you deposit$1000 into a savings account that pays 5% annual interest, compounded monthly, the compound interest calculation would be as follows:

$A = 1000\times (1 + \frac {0.05} {12})^{(12\times 5)} = 1291.67$

This means that after 5 years, your balance would be $1291.67, which includes the$1000 deposit and $291.67 in interest. It's also possible to calculate the Interest separately: $I = A - P$ Where I = Interest earned over the time t Notice that in this case, we divided the annual interest rate by 12 (number of months in a year) to obtain the monthly interest rate and also multiplied the number of years by 12 to reflect the number of months. ## How To Calculate Continuous Compound Interest Continuous compound interest is a powerful concept in finance where interest is calculated and added to the principal continuously, rather than at specific intervals like annually, quarterly, or monthly. The formula for calculating continuous compound interest is given by: $A = P \times e^{rt}$ Where: • A is the future value of the investment/loan, including interest. • P is the principal amount (initial amount of money). • r is the annual interest rate (in decimal form). • t is the time the money is invested/borrowed for, in years. • e is the mathematical constant approximately equal to 2.71828. To calculate the continuous compound interest, you would substitute the values of P, r, and t into the formula. The result (A) represents the total amount of money accumulated after the specified time period. It's important to note that continuous compounding often leads to higher returns compared to compounding interest at discrete intervals because the interest is constantly being added to the principal, allowing for interest-on-interest to accumulate more frequently. ## How To Calculate Compound Interest With Additional Deposits To calculate compound interest with additional deposits, you can use the same formula as before: $A = P\times (1 + \frac r n)^{(n\times t)}$ Where: • A is the final amount (principal + interest) • P is the initial principal or deposit • r is the annual interest rate (expressed as a decimal) • n is the number of times per year the interest is compounded (e.g. annually, semi-annually, quarterly, monthly, daily) • t is the number of years the money is invested But you need to take into account the additional deposits made during the time of investment. For example, if you deposit$1000 into a savings account that pays 5% annual interest, compounded monthly, and then you make additional $500 deposit every year for 5 years, the compound interest calculation would be as follows: A = ($1000 + $5001)(1 + 0.05/12)^(121) + ($1000 + $5002)(1 + 0.05/12)^(122) + ($1000 + $5003)(1 + 0.05/12)^(123) + ($1000 + $5004)(1 + 0.05/12)^(124) + ($1000 + $5005)(1 + 0.05/12)^(125) =$13,932.72

This means that after 5 years, your balance would be $13,932.72, which includes the$1000 deposit, $500 in additional deposit for 5 years and$3932.72 in interest.

It's also possible to calculate the Interest separately:

$I = A - P$

Where:

I = Interest earned over the time t

It's important to note that the additional deposit must be made at the end of each year, so that the interest is calculated on the principal and the additional deposit for that year.

It's also important to note that the formula above is a simple case, in real-world scenarios the deposits and interest calculations can be more complex, for example, you might want to calculate the interest for each deposit separately, or the deposit frequency might not be annual.