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# Compound Interest - Calculator, Definition, Formula and Derivation

One of the most often utilized notions in our everyday lives is compound interest. However, the concept of compound interest is a little hard to grasp. Our banking and finance sector use this concept to perform various tasks like implementing interest, calculating the total amount after certain years, etc.

## Definition:

Compound interest (CI) can be defined as the addition of the interest and the interest on the main amount of the loan/deposit. It is the result of money being reinvested rather than being reimbursed. The principal value plus the prior interest earned in the next cycle is paid for this interest. Cash is linked to compound interest. The number of compounding periods has a major impact on the calculation of the compound interest. The basic principle is that the longer the period, the greater the aggregate interest.

It contrasts with Simple Interest (SI), which does not compound since previously earned interest is not applied to the current period's principal amount. Compound interest is customary in finance and economics.

## Formula:

A=P (1 + r/n) nt is the formula for compound interest.

Here P represents the principal amount, and r is the yearly nominal interest rate in decimal notation in this equation.

A = Amount Accrued (principal + interest).

R/100 = r, where R is the annual nominal interest rate represented as a percentage.

n = the number of compounding periods per unit of time.

t = time in decimal years; 3 months equals 0.25 years, for example. To get decimal years, multiply the number of months in your half-year by 12.

The above formula is the generalized formula for compound interest. If someone wants to calculate the compound interest annually, the formula changes to

A = P (1 + R/100) t

Here ‘t’ refers to the time in years. For instance, if you want to calculate the compound interest for two years then CI => A = P (1+r/100)2.

## Derivation:

Let us consider, a principal amount of P is deposited in ‘XYZ’ bank, for ‘n’ number of years and the interest at the rate of R for each year.

First, calculating the Simple interest (SI) for 1st year

SI = Principal x Rate x Time(years) / 100

Therefore, Amount (A) = P + SI

= P + (P x R x T) / 100

Taking P common we get

(SI)1 = P {(1 + R) / 100} = P1 …... (i)

Now, interest for upcoming second year

(SI)2 = P1 x R x T / 100

Therefore, Amount for second year (A2) = P1 + (SI)2

P1 + {P1 x R x T / 100}

Taking P1 common we find

P1 (1 + R/100)

From equation (i) we substitute the value of P1

P (1 + R/100) (1 + R/100)

P (1 + R/100)2

If we continue to do the same step for ‘n’ number of years, then the equation will change to –

The total amount comes out to be A = P (1 + R/100) n

We know that compound interest or CI = Amount – Principal

So, CI = P (1 + R/100) n – P

Or CI = P {(1 + R/100) n – 1}       Talk to our expert
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