Compound interest is a fundamental math concept that plays an important part in most U.S. competitive examinations, including the SAT, ACT, GRE, GMAT, and even financial-related certifications like the CFA and CPA. Being familiar with compound interest formulas can allow students to get a better score on the mathematics section of these tests, as most of the tests examine students on complex financial situations, rates of interest, and growth patterns. 

• For prospective SAT takers, compound interest problems are typical components of the data analysis and problem-solving area of the exam and hence of key importance.
• This detailed article goes deep into the compound interest world, discussing its formula, derivations, and applications for various time periods, including yearly, half-yearly, quarterly, and monthly. 

Read on to discover important concepts, learn practical applications, and sharpen your problem-solving skills to achieve success in your future U.S. exams!

What is Compound Interest?

Interest in mathematics as well as in statistics is the extra or additional money spent by organisations like banks/post offices on money deposited with them at the same time interest is also returned by people if they borrow money.  C.I. is generally the addition of interest to the principal sum of a loan/deposit, or in other words, it is also identified as of interest on interest. C.I. is standard in finance and economics.

For example;

Ram’s father deposited some money in the post office for 4 years. Every year the money grows more than the earlier year.

Similarly, Ankur has some money in the bank and every year some interest is computed to it, which is displayed in the passbook. This interest is not the same, each year it increases.

Commonly, the interest given/charged is never simple. The interest is determined by the amount of the previous year. This is recognized as interest compounded/C.I.

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Compound Interest Formula

When there is a situation where the amount of the first year becomes principal for the second year and the amount of the second year becomes the principal for the third year and so on. Then this is called compound interest. For better understanding let’s have a look at the formulas and their meaning.

Compound Interest = Amount – Principal

Or

CI = A – P

Here;

Amount(A) is given by the formula;

\(A=P\left(1+\frac{r}{n}\right)^{nt}\)

Where;

‘A’ stands for the amount.

‘P’ is the principal.

‘r’ denotes the rate of interest.

‘n’ is the number of times interest is compounded yearly.

‘t’ is the time in years.

Substituting these values in the CI formula we obtain:

CI = A – P

\(CI=P\left(1+\frac{r}{n}\right)^{nt}-P\)

The above formula is the general formula when the principal is compounded n times in a year. If in case the interest is compounded annually/yearly/per year, the amount and CI is given by the formula:

\(A=P\left(1+\frac{R}{100}\right)^T\)

Therefore CI is calculated by the formula;

\(C.I.=P\left(1+\frac{R}{100}\right)^T-P\)

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Derivation of Compound Interest Formula

The compound interest equation/formula can be derived with the help of simple interest formulas as shown below.

The formula for SI is:

\(S.I.=\frac{\left(P\times R\times T\right)}{100}\)

Where; P is the principal amount, R is the rate of interest and T denotes the time.

The simple interest= CI for one year

The SI for the first year is;

\(SI\text{ (first year)}=\frac{\left(P\times R\times T\right)}{100}\)

Amount= SI+P

Hence, the amount after the 1st year = P+SI(first year)

Amount= \(P+\frac{P\times R\times T}{100}=P\left(1+\frac{R\times T}{100}\right)=P\left(1+\frac{R}{100}\right)\)

Here T=1 as we are calculating for one year.

This total amount is now the principal for second year as per the CI concept:

Hence P( for second year)=\(P\left(1+\frac{R}{100}\right)\)

The SI for the second year is=\(\frac{\left(P\times R\times T\right)}{100}\)

Therefore, the amount after the 2nd year is again= SI+P=\(P\left(1+\frac{R}{100}\right)\)

But here P=\(P\left(1+\frac{R}{100}\right)\)

Hence, amount=\(P\left(1+\frac{R}{100}\right)\left(1+\frac{R}{100}\right)\)

Amount(after second year)=\(P\left(1+\frac{R}{100}\right)^2\)

Similarly for n years,

Amount(A)=\(P\left(1+\frac{R}{100}\right)^n\)

Now, CI = A – P

\(CI=P\left(1+\frac{R}{100}\right)^n-P\)

After simplification we get:

\(CI=P\left[\left(1+\frac{R}{100}\right)^n-1\right]\)

Hence proved.

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Compound Interest Formula for Different Time Periods

So far in the article, we read about the CI definition and formula along with the derivation on the yearly basis. The compound interest can also be calculated for half-yearly, quarterly, monthly and so on. Let us drive through these formulas as well:

Compound Interest Formula Half Yearly

When the interest is compounded half-yearly i.e. the interest is determined every six months or we can say the amount is compounded twice in a given year. The formula is as follows:

\(A=P\left(1+\frac{\left(\frac{R}{2}\right)}{100}\right)^{2T}\)

And CI = A – P therefore;

\(C.I=P\left(1+\frac{\left(\frac{R}{2}\right)}{100}\right)^{2T}-P\)

The point to note here is while calculating for half-yearly; in the actual form the rate of interest is divided by 2 and the time is multiplied by 2 or is doubled.

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Compound Interest Quarterly

In the last section, we learn how to calculate the CI for half-yearly or semi-annually, in the continuation let us learn the formula on a quarterly basis.

Similar to half-yearly; the rate of interest r in the quarterly format is divided by 4 and the time is multiplied by 4. The formulas are listed below:

\(A=P\left(1+\frac{\left(\frac{R}{4}\right)}{100}\right)^{4T}\)

CI = A – P

\(C.I=P\left(1+\frac{\left(\frac{R}{4}\right)}{100}\right)^{4T}-P\)

Monthly Compound Interest Formula

Similar to half-yearly and quarterly calculations, we can compound the data monthly as well. The formula for the same is as follows:

\(C.I=P\left(1+\frac{\left(\frac{R}{12}\right)}{100}\right)^{12T}-P\)

For the monthly compound interest calculation, we divide the rate by 12 and multiply the time by 12 as per the month as n=12.

How to Calculate Compound Interest?

We can understand C.I. as the outcome of reinvesting interest, rather than spending it out so that interest in the succeeding period is then received on the principal sum plus previously accumulated interest. Until now we are clear with the various formulas relating to the CI calculation varying from yearly, half-yearly, quarterly and monthly as well. Let us now understand the compound interest formula with a solved example.

An amount of 25000 is deposited in ICICI Bank for 2 years, obtaining the interest compounded annually at the rate of 10%.

Given:

P = 25000

R = 10%

T = 2 years

According to formula;

\(C.I.=P\left(1+\frac{R}{100}\right)^T-P\)

Substituting the value of P, R and T in the formula:

\(C.I.=25000\left(1+\frac{10}{100}\right)^2-25000\)

C.I.=30250-25000=5250

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Key Points on Compound Interest

Compounded interest is determined on Principal + Accumulated Interest periodically. Some of the key points relating to the topic are as follows:

• The principal amount on compounded interest continues to change during the tenure.
• Returns on C.I. are relatively high.
• The calculation for compound interest is more complex as compared to simple interest.
• The CI relies on the amount collected at the end of the earlier tenure and not on the initial principal when compared to SI. 
• The interest rate for the first year in C.I. as well as in simple interest is the same.
• Leaving the first year calculations, the interest compounded annually > simple interest for the given data.

Applications of Compound Interest

There are some conditions where we could use the compound interest formula. Three of them are listed below.

• Demographic Dynamics: Increase or decrease in population. Compound interest finds application in demography when assessing changes in population over time. It aids in understanding population growth or decline, considering factors like birth rates, death rates, and migration patterns. The formula proves beneficial in projecting future population figures, contributing to strategic urban planning and resource allocation.
• Biological Growth and Microorganisms: The growth of a bacteria if the speed of growth is identified. In microbiology, compound interest is employed to model the growth of bacterial populations. Understanding the speed of bacterial growth is crucial in various fields, including medicine and environmental science. The compound interest formula facilitates predicting bacterial population sizes at different time points, guiding researchers in optimizing conditions for experiments or addressing public health concerns.
• Economic and Financial Dynamics: The value of an item, if its price increases/decreases in the intermediate years. Compound interest plays a pivotal role in economic contexts, particularly when evaluating the changing value of commodities or assets. If an item's price experiences fluctuations over intermediate years, the compound interest formula becomes a valuable tool. It helps calculate the compounded impact of price changes, assisting investors, economists, and policymakers in comprehending the real economic impact and making informed decisions.

Challenges in Compound Interest Calculation

Despite its widespread use, compound interest calculations come with certain challenges and considerations that should be acknowledged:

1. Complexity in Mental Calculation: Compound interest calculations can become intricate, especially when considering multiple compounding periods and varying interest rates. Mental calculations for compound interest may pose challenges, and individuals often rely on financial calculators or spreadsheet tools for accuracy.
2. Variable Interest Rates: In real-world scenarios, interest rates may fluctuate over time. Calculating compound interest with changing rates requires a dynamic approach, and tools like the compound interest formula may need to be adapted to accommodate variable rates.
3. Influence of Economic Factors: Economic conditions can impact interest rates and compounding frequencies. Understanding how economic factors influence compound interest is vital for individuals and investors navigating financial landscapes.
4. Practical Application in Investments: Applying compound interest concepts to investment decisions requires a nuanced understanding of risk, market conditions, and the time horizon. Practical application involves considerations beyond the basic formula, such as market volatility and potential returns.

Solved Examples on Compound Interest

In any type of interest whether it be simple or compound apart from the definition and related concepts, formulas associated with examples play a major role. We have been through various types of formulas, now let’s practice some solved examples for the same.

Solved Examples 1: A invested Rs. 3000 on compound interest at a rate of interest 10% for 2 years and B invested Rs. 3200 on compound interest at a rate of interest 15% for 3 years. Find total C.I . (compounded annually).

Solution:

Given:

Sum of Rs. 3000 invested at rate = 10% for 2 years

Sum of Rs. 3200 invested at rate = 15% for 3 years

Formula:

Let P = Principal, R = rate of interest and T = time period

\(C.I.=P\left(1+\frac{R}{100}\right)^T-P\)

Calculation:

C.I after 2 years = \(3000\left(1+\frac{10}{100}\right)^2-3000\)= Rs. 630

C.I after 3 years = \(3200\left(1+\frac{15}{100}\right)^3-3200\)= Rs. 1666.8

Total C.I = 630 + 1666.8 = Rs. 2296.8

∴ Total C.I. is Rs. 2296.8

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Solved Examples 2: Find the C. I at the rate of 20% for 3 years on that principal which in 2 years at the rate of 10% per annum gives Rs.10500 as compound interest. (when compounded annually)

Solution:

Let P = Principal, R = rate of interest and T = time period

Annual compound interest formula is:

\(C.I.=P\left(1+\frac{R}{100}\right)^T-P\)

Given,

R = 10% and T = 2

⇒ \(10500=P\left(1+\frac{10}{100}\right)^2-P\)

⇒ 10500 = 0.21P

⇒ P = 50000

⇒ Principal = Rs. 50000

Then,

R = 20% and T = 3

C.I.

= \(50000\left(1+\frac{20}{100}\right)^3-50000\)

= \(50000\left(1.2\right)^3-50000\)

= 36400

Solved Examples 3: Calculate the compound interest/CI on 10000 rupees, for 2 years duration when the rate of 4% is given, and the interest is being compounded half-yearly.

Solution:

P = 10000

R = 4%

T = 2 years

Being compounded half-yearly the rate will get divided by 2, and time will get multiplied by 2, by a formula

\(A=P\left(1+\frac{\left(\frac{R}{2}\right)}{100}\right)^{2T}\)

\(A=10000\left(1+\frac{2}{100}\right)^4=10824.32\\\)

\(C.I.=10824.32-10000=824.32\)

Questions to Ace Your Exams

Q1. How to calculate compound interest in Excel? How do you find compound interest?
To calculate compound interest in Excel, use the formula: A = P(1 r/n)^(nt), where A is the future value, P is the principal amount, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the number of years. 
Therefore, Input these values into Excel cells, replacing them with your specific data. To manually find compound interest, use the same formula without Excel. Subtract the principal amount from the future value obtained, providing the accrued interest. Excel automates this calculation, making it efficient for various scenarios and financial analyses.

Q2. How does compound interest work? How to calculate compound interest per month?
Compound interest is calculated on both the initial principal and accumulated interest from previous periods. To calculate monthly compound interest, use the formula A = P(1 r/n)^(nt), where A is the final amount, P is the principal, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the number of years. To break it down monthly, divide the annual interest rate by 12 and set n to 12. This formula ensures interest is recalculated each month, incorporating the previous interest, resulting in a more accurate reflection of the growing investment over time.

Q3. What is compounding interest?
Compound interest refers to the method of calculating interest on both the initial principal and the accumulated interest from previous periods. Unlike simple interest, which is calculated solely on the principal amount, compounding interest enables the interest to grow exponentially over time. Therefore, this means that each interest calculation is based on the continuously increasing total amount, leading to accelerated growth of the investment or debt. Hence, it's a powerful financial concept that plays a pivotal role in savings accounts, loans, and investments, allowing for the compounding of earnings or debts over various periods, ultimately influencing the overall financial outcome.

Q4. What is the difference between simple and compound interest?
Simple interest is calculated solely on the principal amount, resulting in a fixed interest earned each period. In contrast, compound interest considers both the principal and accumulated interest, leading to interest being calculated on the updated total. However, simple interest is linear, providing constant returns, while compound interest yields exponential growth due to interest compounding over time. 
Furthermore, while simple interest suits straightforward calculations, compound interest, with its compounding effect, often results in higher returns, making it a preferred choice for long-term investments where interest is reinvested into the amount, contributing to the overall growth of the investment.

Mastery of compound interest is critical for anyone studying for U.S.-based competitive exams such as the SAT, ACT, GRE, GMAT, and financial certifications like the CFA and CPA. As these exams often test with questions based on intricate financial situations and growth patterns, proficiency in compound interest formulas can significantly improve problem-solving speed and accuracy. This article has offered a complete manual that encompasses basic principles, formulae, derivation, and day-to-day usage of compound interest over various duration periods such as yearly, half-yearly, quarterly, and monthly. Practicing these basics well and implementing them with diversified examples, the students can get better at calculations, improve performance in tests, and confidently solve any question that is related to compound interest that could be set in their exam.