# Interest Rate Calculator

Calculate the interest rate for a loan given the loan amount and monthly payment.



## Annual Interest Rate:

%

 Total Payments: $Total Interest:$
Learn how we calculated this below

## How to Calculate an Interest Rate

Interest rates are used as a method of paying lenders for borrowers using their money, and they are typically quoted in annual terms. The interest rate signals how risky a potential loan is, and the financial market determines interest rates.

If you are a lender, then the higher the interest rate the better because this means you are receiving more money. But, if you are a borrower, then you will want to receive the lowest interest rate possible because this means you are paying less money to borrow.

### Simple and Compound Interest Rates

You can calculate interest using either simple interest or compound interest. With simple interest, the interest is accrued each period on the original loan amount.

But with compound interest, the interest is calculated on the prior periods’ accumulated interest and the original principal amount.

The interest rate calculator calculates the interest rate for loans and investments that accrue simple or compound interest given the loan amount, monthly payment, and loan term.

The formulas for calculating the interest rate using simple interest or compound interest are defined below.

#### Simple Interest Rate Formula

The simple interest rate formula is as follows:

r = \frac{i}{pn}

Where:
r = interest rate
i = total interest paid
p = loan principal
n = loan term in years

##### Example

Let’s try an example. If the bank says you will pay $100 in interest on a$1,000, two-year loan, what would your simple interest rate be?

Using the simple interest formula above, we can calculate:

r = \frac{\$100}{\$1,000 \times 2}
r = \frac{\$100}{\$2,000}
r = 5\%

#### Compound Interest Rate Formula

The compound interest rate formula is as follows:

r = k \times \left ( \sqrt[nk]{\frac{i + p}{p}} − 1 \right )

Where:
r = interest rate
k = compounding periods per year
i = total interest paid
p = loan principal
n = loan term in years

##### Example

Let’s now find what the compound interest rate would be with the same assumptions of $100 in interest on a$1,000, two-year loan that compounds monthly.

r = 12 \times \left ( \sqrt[2 \times 12]{\frac{\$100 + \$1,000}{\$1,000}} − 1 \right ) [formula may scroll beyond screen] r = 12 \times \left ( \sqrt[24]{\frac{\$1,100}{\\$1,000}} − 1 \right )
[formula may scroll beyond screen]
r = 12 \times \left ( 1.003979 − 1 \right )
r = 12 \times 0.003979
r = 4.77\%

In this instance, the compounding rate is less than the simple interest rate. This is due to the compounding rate needing to “catch up” to be equal to the simple interest rate.

This formula will calculate compound interest assuming a single interest payment at the end of each compounding period.

To find the interest rate for a loan with regular interest payments, you need to use an iterative algorithm, such as the Newton-Raphson method.[1] This method is much more complex than a simple formula and requires making multiple guesses to narrow down the interest rate over various iterations.

### Nominal Interest Rate

The nominal interest rate is the rate most lenders, banks, etc., advertise and it does not include compounding. If the effective annual interest rate is known, then the nominal interest rate can be calculated.

See the next section below for a definition of effective annual interest rate and how it compares to the nominal interest rate.

#### Nominal Interest Rate Formula

The formula for the nominal interest rate is:

r = n \times \left [ \left ( 1 + i \right )^{1/n} – 1 \right ]

Where:
r = nominal interest rate
i = effective annual rate
n = number of times interest is compounded per period

##### Example

Let’s put numbers to this to get a better understanding. What is the nominal interest rate for an 8% effective annual rate compounded daily (n = 365)?

r = 365 \times \left [ \left ( 1 + 0.08 \right )^{1/365} – 1 \right ]
r = 365 \times \left [ 1.08^{0.0027397} – 1 \right ]
r = 365 \times 0.0210874
r = 7.7\%

Note that the nominal interest rate is less than the effective annual interest rate because there is no compounding of the nominal interest rate. You would earn less on an investment with a nominal interest rate than an investment using an effective annual interest rate.

### Productive Annual Rate

The effective annual rate (EAR) is the real cost of borrowing money. It differs from the nominal interest rate because it is a compounded interest rate. With compounding, the more frequent the compounding, the more interest is paid. Let’s look at a few examples.

#### Productive Annual Rate Formula

The effective annual rate formula is:

r = \left ( 1 + \frac{i}{n} \right )^{n} − 1

Where:
r = effective annual rate
i = annual rate
n = compounding periods

##### Example

Let’s say you have an opportunity to lend money at 5% annual interest (nominal interest rate). What is the EAR of compounding monthly and compounding semiannually?

Would you prefer compounding monthly (n = 12) or semiannually (n = 2)?

We can insert these figures into our formula.

###### Monthly Compounding
r = \left ( 1 + \frac{0.05}{12} \right )^{12} − 1
r = 1.004167^{12} − 1
r = 1.051162 – 1
r = 5.1162\%
###### Semiannual Compounding
r = \left ( 1 + \frac{0.05}{2} \right )^{2} − 1
r = 1.025^{2} − 1
r = 1.05063 – 1
r = 5.0625\%

Since you are a lender in this example, you would want interest compounded monthly because that will give you a higher EAR and you would collect more interest.

If the roles were reversed and you were borrowing money, then you would prefer interest to be compounded semiannually because you will pay less interest over the life of the loan. You can also find these results using our APY calculator.

In both these examples, the nominal interest rate is less than the effective annual interest rates.

Productive annual rates are generally higher than the nominal rate because effective interest rates are forward-looking rates, they tell you what you will earn on these investments in the future. Nominal rates are the annual interest rates regardless of compounding, while increasing the compounding increases the effective annual rate when compared to the nominal rate.

### What is an interest rate?

An interest rate is the rate that a lender charges a borrower to use its money. It is a percentage of the principal of the loan.

Interest rates are a function of the riskiness of the loan – the riskier the loan for the lender, the higher the rate they generally charge.

There are two ways to show an interest rate: the nominal interest rate has no compounding and the effective interest rate compounds a loan’s interest.

### Perform you want a low or high interest rate?

Low interest rates are generally better when borrowing money since they result in lower amounts of interest paid on a loan. Higher interest rates are better when lending money since they result in higher amounts of interest paid.

So, if you’re taking out a loan it is often better to seek a lower interest rate, and if you’re lending money, for instance with a savings account, then you want a higher interest rate.

### Can you reduce an interest rate?

There are multiple ways that an interest rate can be reduced. The interest rate is basically a function of the borrower’s risk level. The higher the risk, the higher the interest rate will be.

Improve Credit Score:
First, an interest rate can be reduced by improving an individual’s credit score or a business’s bond credit rating, which is an indicator that the borrower is not as risky.

Append Collateral:
Another way to lower the interest rate is by posting collateral to a loan to reduce the risk to the lender.

Mortgage loans always have a lower interest rate than credit cards because if the borrower stopped paying their mortgage, the lender will take the home and sell it. However, if a credit card user stopped making payments on their credit card, the lender has no asset that it can sell.

Refinance Debt:
Refinancing debt is a third way to lower an interest rate, since high amounts of debt are seen as a risk factor for lenders. A smaller loan amount can also lower the rate because the payment will be a smaller percentage of your monthly income and will be less risky for the lender.

Shorten Loan Term:
Shortening the term of the loan is another way to reduce the rate since a shorter term means the lender will receive their money back sooner, reducing the risk. For example, a 15-year mortgage will always have a lower rate than a 30-year mortgage.

## References

1. Garrett, S. J., Newton-Raphson Method, ScienceGuide, 2015, https://www.sciencedirect.com/topics/mathematics/newton-raphson-method