Formula for Calculating Interest on a Loan

Calculating interest on a loan involves several key formulas, each serving different types of loans and interest calculations. The most common formulas include Simple Interest, Compound Interest, and Amortized Interest. Understanding these can help borrowers and lenders make informed financial decisions. Below is a detailed guide on how to calculate interest for each type.

1. Simple Interest

Simple Interest is calculated using a straightforward formula. It’s used for loans where interest is not compounded, meaning it is only calculated on the principal amount.

Formula:

$\text{Simple Interest}=P\times r\times t$Simple Interest=P×r×t

Where:

• P = Principal amount (the initial sum of money)
• r = Annual interest rate (decimal form)
• t = Time the money is borrowed for (in years)

Example Calculation:

Suppose you borrow $5,000 at an annual interest rate of 6% for 3 years.

• Principal (P) = $5,000
• Interest Rate (r) = 6% = 0.06
• Time (t) = 3 years

Using the formula:

$\text{Simple Interest}=5000\times 0.06\times 3=900$Simple Interest=5000×0.06×3=900

The interest accrued over 3 years would be $900. The total amount to be repaid is $5,900.

2. Compound Interest

Compound Interest is calculated on the principal amount and also on the interest that has been added to the principal. This type of interest is more complex and is used for loans where interest is compounded periodically.

Formula:

$A=P\times {(1+\frac{r}{n})}^{n\times t}$A=P×(1+nr​)n×t

Where:

• A = Amount of money accumulated after n years, including interest.
• P = Principal amount
• r = Annual interest rate (decimal form)
• n = Number of times interest is compounded per year
• t = Time the money is borrowed for (in years)

Example Calculation:

Consider a loan of $5,000 with an annual interest rate of 6% compounded quarterly for 3 years.

• Principal (P) = $5,000
• Interest Rate (r) = 6% = 0.06
• Number of Compounding Periods per Year (n) = 4
• Time (t) = 3 years

Using the formula:

$A=5000\times {(1+\frac{0.06}{4})}^{4\times 3}$A=5000×(1+40.06​)4×3

$A=5000\times {(1+0.015)}^{12}$A=5000×(1+0.015)12

$A=5000\times {\left(1.015\right)}^{12}$A=5000×(1.015)12

$A=5000\times 1.1956\approx 5978$A=5000×1.1956≈5978

The total amount to be repaid would be approximately $5,978. The interest accrued is $978.

3. Amortized Interest

Amortized Interest is used for loans where payments are made in regular intervals, and each payment includes both principal and interest. This is typical for mortgages and car loans.

Formula:

$M=\frac{P\times \frac{r}{n}}{1-{(1+\frac{r}{n})}^{-n\times t}}$M=1−(1+nr​)−n×tP×nr​​

Where:

• M = Monthly payment
• P = Principal amount
• r = Annual interest rate (decimal form)
• n = Number of payments per year
• t = Time the money is borrowed for (in years)

Example Calculation:

For a $5,000 loan with an annual interest rate of 6% to be paid off monthly over 3 years:

• Principal (P) = $5,000
• Annual Interest Rate (r) = 6% = 0.06
• Number of Payments per Year (n) = 12
• Time (t) = 3 years

Using the formula:

$M=\frac{5000\times \frac{0.06}{12}}{1-{(1+\frac{0.06}{12})}^{-12\times 3}}$M=1−(1+120.06​)−12×35000×120.06​​

$M=\frac{5000\times 0.005}{1-{\left(1.005\right)}^{-36}}$M=1−(1.005)−365000×0.005​

$M=\frac{25}{1-0.837}$M=1−0.83725​

$M=\frac{25}{0.163}\approx 153.40$M=0.16325​≈153.40

The monthly payment would be approximately $153.40. Over the course of 3 years, the total payment would be $5,534.40. The interest paid would be $534.40.

Summary

• Simple Interest: Calculated on the principal amount only, straightforward and easy to compute.
• Compound Interest: Includes interest on interest, making it more complex but often used for savings and investments.
• Amortized Interest: Used for loans with regular payments, including both principal and interest, suitable for mortgages and car loans.

Table of Example Calculations

Loan Type	Principal	Rate	Time	Compounding/Payments	Total Amount	Interest Paid
Simple Interest	$5,000	6%	3 years	N/A	$5,900	$900
Compound Interest	$5,000	6%	3 years	Quarterly	$5,978	$978
Amortized Interest	$5,000	6%	3 years	Monthly	$5,534.40	$534.40

Each formula has its application depending on the type of loan and payment structure. Understanding these calculations helps in planning and managing finances more effectively.