Compound Interest

Compound interest is an interest calculated on the principal and the existing interest together over a given time period. The interest accumulated on a principal over a period of time is also added to the principal and becomes the new principal amount for the next time period. Again, the interest for the next time period is calculated on the accumulated principal value. Compound interest is the method of calculation of interest used for all financial and business transactions across the world. The power of compounding is that it is always greater than or equal to the other methods like simple interest.

An amount of $1000 invested over a period of time at 10% rate would give a simple interest of $100, $100, $100... over successive time periods of 1 year, but would give a compound interest of $100, $210, $331, $464.10... Let us understand more about this, and the calculations of compound interest in the below content.

1.	What is Compound Interest?
2.	Compound Interest Formula
3.	Derivation of Compound Interest Formula
4.	Simple Interest and Compound Interest
5.	Compound Interest Formula for Different Time Periods
6.	Continuous Compound Interest Formula
7.	How to Calculate Compound Interest?
8.	FAQs on Compound Interest

Compound interest is the interest paid on both principal and existing interest. Hence, it is usually termed "interest over the interest". Here, the interest so far accumulated is added to the principal and the resulting amount becomes the new principal for the next interval. i.e., Compound Interest = Interest on principal + Interest over existing interest.

The compound interest is calculated at regular intervals like annually(yearly), semi-annually (half-yearly), quarterly (4 times in a year), monthly (12 times in a year), etc; In case of compound interest, interest income from an investment makes the money grow faster over time! It is exactly what is done by the compound interest to money. Banks or any financial organization calculate the amount based on compound interest only.

Compound interest is the interest that is earned on an initial principal amount as well as the accumulated interest from previous periods. The compound interest is found after calculating the compounded amount over a period of time, based on the rate of interest, and the initial principal. Here are the formulas to find the compounded amount and compound interest.

Formula of Compound Amount

For an initial principal of P, rate of interest per annum of r (r%), time period t in years, frequency of the number of times the interest is compounded annually n, the formula to calculate the total compounded amount is as follows:

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

Formula of Compound Interest

The compound interest is obtained by subtracting the principal amount from the compound amount. Hence, the formula to find just the compound interest is as follows: CI = P (1 + r/n)^nt - P.

In the above expression,

• P is the principal amount
• r is the rate of interest(decimal obtained by dividing rate by 100)
• n is the number of times the interest is compounded annually
• t is the overall tenure.

If the given principal is compounded annually, then we have n = 1 and in this case, the above formulas turn into the following:

Compound amount, A = P(1 + r)^t

Compound interest, C.I = P(1 + r)^t - P .

The compound interest formula is derived from the simple interest formula. 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 and compound interest computation. The principal is constant over a period of time in case of simple interest computation, but in compound interest computation, the interest is added to the principal after every time period.

Derivation:

The compound interest formula is derived as follows:

• Let the principal be P and the rate of interest be R% per annum. Here, the interest is compounded annually, so the compounding period is 1 year. Note that the principal (P) will change after every 1 year. 
• Assume that the interest for the first year is I₁. I₁ = R% of P = R/100 × P
• Then, the amount at the end of the first year is A₁ = P + I₁ = P + (R/100 × P). This will give A₁ = P (1 + R/100)
• Now, let us do the interest calculation for the second year. It is to be noted that the amount (principal + interest of the first year) of the first year will become the principal of the second year. Let this principal be P₂
• Now, P₂ = A₁ = P (1 + R/100)
• Now, the interest for the second year is I₂ = R% of P₂ = R/100 × P₂ = R/100 × P(1 + R/100)
• Now, the amount at the end of the second year will be A₂ = P₂ + I₂ = P (1 + R/100) + R/100 × P(1 + R/100)
• This expression can be written as A₂ = P (1 + R/100) (1 + R/100) = P (1 + R/100)²
• Continuing in this manner for n compounding periods, the amount at the end of t years A = P(1 + R/100)^t.

Instead of writing R/100 every time, we usually convert the rate into decimals by dividing by 100 to get r and substitute it in the formula P (1 + r)^t.

From the above formulas and computations, we can observe that the compound interest is the same as the simple interest for the first interval. But, after a period of time, there is a noticeable difference in the total interest obtained.

The simple interest value for each time period is the same because the principal on which it is calculated is constant. But the compound interest varies and increases across the years. This is because the principal on which the compound interest is calculated each year is increasing. The principal for a particular year in case of compound interest is equal to the sum of the initial principal value, and the accumulated interest of the past years.

Example: Assume that an amount 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 for a period of 5 years.

Simple Interest	Compound Interest
For 1st year: P = 10,000 Time = 1 year Interest = 1000	For 1st year: P = 10,000 Time = 1 year Interest = 1000
For 2nd year: P = 10,000 Time = 1 year Interest = 1000	For 2nd year: P = 10000 + 1000 = 11000 Time = 1 year Interest = 1100
For 3rd year: P = 10,000 Time = 1 year Interest = 1000	For 3rd year: P = 11000 + 1100 = 12100 Time = 1 year Interest = 1210
For 4th year: P = 10,000 Time = 1 year Interest = 1000	For 4th year: P = 12100 + 1210 = 13310 Time = 1 year Interest = 1331
For 5th year: P = 10,000 Time = 1 year Interest = 1000	For 5th year: P = 13310 + 1331 = 14641 Time = 1 year Interest = 1464.1
Total Simple Interest = 5000	Total Compound Interest = 6105.1
Total Amount = 1000 + 5000 = 6000	Total Amount = 1000 + 6105.1 = 7105.1

From the above table, we can understand the power of compounding. Compound interest is more profitable than simple interest if the amounts invested for more than 1 year. For more differences between simple and compound interests, click here.

If you want to calculate the compound interest for a different time period, you can adjust the values of n and t accordingly. The CI formulas are tabulated in the following table for different time periods. In all these formulas, P is the principal amount, r is the rate/100, and t is the number of years.

Compounded Annually Formula	A = P (1 + r)t
Compounded Semi-Annually Formula	A = P (1 + r/2)2t
Compounded Quarterly Formula	A = P (1 + r/4)4t
Compounded Monthly Formula	A = P (1 + r/12)12t
Compounded Weekly Formula	A = P (1 + r/52)52t
Compounded Daily Formula	A = P (1 + r/365)365t

In these formulas, A is the total amount that includes both the compound interest and the principal. If we want to find just the compound interest then we need to subtract P from the formula. For example, the compound interest formula for compounded monthly would be CI = P (1 + r/12)^12t - P.

In all the above formulas of compound interest, the number of times the amount is compounded is finite. But if it is infinite, the compound interest formula turns into

A = lim_n→∞ P (1 + r/n)^nt

= lim_n→∞ P [(1 + r/n)ⁿ]^t (by properties of exponents)

= P (e^r)^t (by limit formulas)

= Pe^rt

This formula is known as the continuous compound interest formula and this gives the total amount after t years. Just the interest amount is calculated using the formula Pe^rt - P as usual. Here is an example to understand this.

Example: If $5000 is invested in a bank where the amount is compounded continuously at a rate of 7% per year, then what is the resultant amount after 3 years?

Solution: Here, P = $5000, r = 7% = 0.07, and t = 3. Substituting these values in the continuous compound interest formula:

A = Pe^rt = 5000(e^{0.07 × 3}) = $6168.39.

The compound interest is the total compounded amount minus the initial amount. Here are the steps to find the compound interest:

• Identify the principal amount (P), i.e., the amount that is invested.
• Identify the rate of interest (r%). Make sure to divide it by 100 while substituting it into the formula for the variable r.
• Identify the number of times the investment is compounded (n).
• Identify the number of years (t).

To find the compound interest:

• if the amount is compounded annually/half-yearly/quarterly/monthly/weekly/daily, then substitute all these values into the formula P(1+r/n)^nt - P.
• if the amount is compounded continuously, then substitute these values in the formula Pe^rt - P.

Important Notes on Compound Interest:

• Compound interest depends on the amount accumulated at the end of the previous tenure, not just on the original principal.
• Banks, insurance companies, etc. generally levy compound interest.
• The compound interest formula is A = P (1 + r/n)^not. Here, if the amount is compounded
  annually, then n = 1
  half-yearly, then n = 2
  quarterly, then n = 4
  monthly, then n = 12
  daily, then n = 365
• If the amount is compounded continuously then we use the formula A = Pe^rt.
• Note that while finding compound interest, each time period and the rate of interest 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 when the amount is compounded annually. It says two things:
  Doubling Time = 72/Interest Rate
  Interest Rate = 72/Doubling Time
  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
• “Per annum” or “annual” or “per year” - all mean the same. 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 in duration.

☛Related Topics:

• Compound Interest Calculator
• Daily Compound Interest Calculator
• Monthly Compound Interest Calculator
• Compound Interest Calculator Quarterly

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What are Compound Interest Formulas?

Here are the list of compound interest formulas when the amount is compounded

• annually: A = P (1 + r)^t
• half-yearly: A = P (1 + r/2)^2t
• quarterly: A = P (1 + r/4)^4t
• monthly: A = P (1 + r/12)^12t
• weekly: A = P (1 + r/52)^52t
• daily: A = P (1 + r/365)^365t

In all these formulas, P is the initial amount, 'r' is the rate of interest, and 't' is the time period.

How to Compute Compound Interest?

The compound interest is found using the formula: CI = P( 1 + r/n)^nt - P. In this formula,

• P( 1 + r/n)^nt represents the compounded amount.
• the initial investment P should be subtracted from the compounded amount to get the compound interest.

What is the Important Difference Between Simple Interest and Compound Interest?

Simple interest is the interest calculated only on the principal (initial investment), but compound interest is the interest calculated on both principal and interest together. Thus, compound interest is more beneficial as compared to simple interest.

How to Calculate Compound Interest?

To calculate the compound interest, we just need to substitute the principal (P), rate r% (r/100), time (t), and the number of times the amount is compounded (n) in the formula P(1 + r/n)^nt - P.

What Is the Monthly Compound Interest Formula?

The monthly compound interest formula is given as CI = P(1 + (r/12) )^12t - P. Here, P is the principal (initial amount), r is the interest rate (for example if the rate is 12% then r = 12/100=0.12), n = 12 (as there are 12 months in a year), and t is the time.

What Is the Daily CI Formula?

The daily CI 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. Here A gives the total amount (principal + interest).

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?

If the amount is compounded daily then it gets compounded 365 a year. It will generate more money compared to interest compounded monthly, which has only 12 compounding cycles per year.

How Does Compound Interest Depend on Time Period?

The compound interest depends on the time period for which the amount is invested/borrowed. The time interval for the calculation of interest can be a day, a week, a month, quarterly, or half-yearly. The more the time interval is the less the compound interest. For example, we get more compound interest if the amount is compounded daily than it is compounded annually.

Which Interest is Greater? Compound Interest or Simple Interest?

Obviously, compound interest is greater than simple interest. This is because simple interest is calculated only on the principal in every tenure, whereas compound interest is calculated on the principal amount + interest so far.

Can Compound Interest be Greater than Principal?

The compound interest can be greater than the principal over a period of time.

How Do you Calculate Compound Interest for Half Year?

The formula for the calculation of compound interest for half year is CI = p(1 + r/2)^2t.- p. Here in this formula 'A' is the final amount, 'p' is the principal, and 't' is the time in years. In this formula, we have divided r by "2" as there are two half-years in a full year.

What Information is Required to Calculate Compound Interest?

To find the compound interest, we should know the principal (P), rate of interest (r%), time period (t), and the number of times the amount gets compounded in a year (n). 

What Are the Units of Compound Interest?

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