# APR Calculator

By Tibor Pal, PhD candidate and Dariusz Walczy-Antkowicz
Last updated: Aug 27, 2021

APR Calculator is an advanced device that helps you to compute the Annual Percentage Rate (APR), that is, the annual rate charged for the credit. APR then represents the total cost of the borrowed money. By computing the APR rate, you can easily compare different loan offers so that you can have a better understanding of the real cost of borrowing. In the following text, you can learn what APR is and how it differs from other interest rates. You can also read about what APR means in multiple scenarios, so that you can apply our calculator as a credit card APR calculator, mortgage loan APR calculator or APR calculator for car loans. We also provide a method of how to calculate APR and other relevant interest rates if you are looking to do these calculations by hand. Lastly, we will give you some practical hints about what is a good APR for a credit card.

## What is APR - Introducing the Annual Percentage Rate

When you are considering taking a loan, it is always beneficial to consider different offers from multiple sources and compare various loan constructions. Hence, if you are able to figure out how much a loan would cost you it not only gives you insight into whether you can afford the payments based on your income, but also provides a guideline for the comparison, so you get the best loan possible. There are two ways to express the cost of a loan: in nominal term (for example dollar amount) or as a percentage of the loan amount. More precisely, these two concepts are the Finance Charge and the Annual Percentage Rate respectively.

The Annual Percentage Rate, or, in its more common form, APR, is a particular type of interest rate often published by banks or financial institutions. While the primary purpose of APR is to inform potential borrowers about the cost of a particular loan, its computation is highly variable, depending on the type of loan and the general jurisdiction. If you read on, you will learn what forms APR can take and what their advantages are by surveying the most common interest rates and their peculiarities.

## APR in the context of interest rates

There are various forms of interest rates in the financial world, so being familiar with their characteristics not only helps you to distinguish them but also gives you practical guidance when you are facing a situation involving a financial transaction.

Perhaps the best way to demonstrate the strengths of APR and illustrate the capabilities of this APR Calculator is to give a few examples of APR in the most common interest rates, so we can demonstrate their peculiarities from a borrowing point of view. The most frequently used interest rates are:

• Nominal Interest Rate (r)
• Periodic Rate (i)
• Effective Annual Rate (EAR)
• Annual Percentage Rate (APR)

In the following, we delineate these rates so you will have a solid foundation of knowledge from which you can fully understand the power of our APR calculator.

1. Nominal Interest Rate

The most basic interest rate is a nominal interest rate, also known as simple interest (or the headline or quoted interest rate). If you are unsure what rate someone is referring to in the media or an financial discussion, it is probably this one. It is also the number that banks usually advertise as the interest rate on a particular loan. The nominal interest rate represents the borrowing cost of the loan over a year, i.e., as a percentage of the loan amount. However, there are two important reasons that might cause you to turn to other types of interest rates if you are about to measure the real yearly cost of your loan:

• Nominal interest rate doesn't account for the effect of compound interests
• It also doesn't cover any additional cost beyond the interest
1. Periodic Rate

Before we go into detail about the other interest rates, it is worth familiarizing yourself with how an interest rate is applied over a payment period. In most of the cases, you do not refinance your loan on a yearly bases, so you may want to know what the interest rate is for each period. This is the Periodic Rate, which is often how banks or financial institutions announce their rate, not as an annual term but, for example, on a monthly bases. To be able to compare different loan offers, it is logical to make the period of the interest rate the same across all offers. This is also help as financial formulas are typically based on the periodic interest rate. The obvious way to get the Periodic interest rate is the following:

`Periodic rate = Nominal interest rate / Number of payments in a year.`

1. Effective Annual Rate (EAR)

Coming back to the shortcomings of the nominal interest rate, if we take into account the effect of compounding interest we arrive at the Effective Annual Rate (EAR or EFF%). The concept of EAR is analogous to the Annual Percentage Yield (APY), however, the latter form is applied mainly on investments. To understand the advantage of this rate, let's assume that you need some quick money to fix your car. You have two options: either you take a loan where the interest is compounding yearly, or use your credit card which compounds monthly. Let's assume that, in both scenarios, the nominal interest rate is 6 percent without any additional financial cost. Notice that when compounding occurs yearly, the Periodic rate and the Effective Annual Rate will be identical to the nominal interest rate. In the case of monthly compounding frequency, however, the Effective Annual Rate will be higher as interest is charged more often on your remaining loan amount. To see the exact value we need to apply the following financial formula for the EAR:

`EAR = ((1 + Periodic rate) ^ Number of payments - 1) * 100`

`Periodic rate = 6 / 12 = 0.5% = 0.005`

`EAR = ((1 + 0.005) ^ 12 - 1) * 100 = 6.17%`

As you can see, the yearly interest rate is 0.17 percentage point higher than the stated nominal interest rate if you choose your credit card for the purchase. While the difference does not seem to be much, if you borrow a considerable amount with an extended loan term, the difference might become ample.

1. Annual Percentage Rate (APR)

Continuing on, you may find yourself in a situation where the second point is true: there are additional cost connected to the loan besides interest that increase your final expense. Since banks are profit-oriented, they aim to maximize their financial gain by obtaining low-cost funds (savings) and lending out as expensive as possible (loans). Highly simplifying their operation, the difference between the two transactions is their profit. To acquire more income, however, they might provide other services that they additionally charge to the borrower.

APR is aimed at incorporating and revealing these costs and expressing them in the yearly rate. In this way, APR might be a better measure when you are about to evaluate the real cost of borrowing or want to compare different loan offers.

In the United States, according to the Truth-in-Lending Act, banks are obliged to quote an APR, which is the multiplication of the percentage payment for a payment period by the number of payments in the year. In some other countries, the jurisdiction may differ. For instance, in the European Union, APRs are calculated in a slightly different way: the yearly rate also includes the part of the interest that results from the compounding factor.

In any case, but especially if you live in the United States, it is worth estimating the APR that includes all cost factors including the effect of compounding, that is, the Effective Annual Percentage Rate (EAPR). It is often the case that banks, with a legal loophole, announce an APR rate that doesn't include specific expenses included in the loan contract. Accordingly, we designed this calculator in such a way so you can estimate the Effective APR to reflect all of the arising financial costs of borrowing.

To conclude, the table below demonstrates the essential peculiarities of the most prevailing interest rates.

Considerations Nominal Interest Rate (r) Effective Annual Rate (EAR) Annual Percentage Rate (APR) Effective APR (EAPR)
Alternative terms Headline, Quoted interest APY, EFF% n/a n/a
Effect of compounding No Yes No Yes
Inclusion of additional costs No No Yes Yes

## Using the APR calculator - Variables and financial terms

Now that you have some information on the background of different type of interest rates from the previous section, it is time to get familiar with our APR calculator so you can analyze a loan construction from multiple angles to ensure you incorporate all emerging financial costs.

As a starting point, let's go through the parameters and terms you may encounter on this page. To be consistent, we have grouped the variables according to their roles: first are the ones that you need to provide in order to specify the loan construction (input variables), followed by the ones that result from these previously stated parameters and provide a base for the evaluation or comparison of the particular loans (output variables). Lastly, you can check out the supplementary variables that are the by-product of the computational process, which can be found in the advanced mode.

1. Loan Specification

To make this calculator work, you need to provide the following values:

• Loan amount (A) - the amount of loan under consideration.

• Interest rate (r) - the annual nominal interest rate as a percentage. Note that percentage rates are generally converted to decimals for complex computations (for example `6% = 0.06`).

• Loan term (t) - the interval over which you need to repay the Loan Amount and all connected cost (interest and other additional fees).

• Compounding frequency (m) - the number of times interest compounding occurs. For example, when compounding is applied annually, m=1, when quarterly, m=4, monthly, m=12, etc.

• Payment frequency (q) - the regularity that part of the loan is repaid.

• Fees rolled into loan - all additional fees that are paid during the loan term. Since it is attached to the loan amount, banks generally charge interest on it.

• Fees paid separately - fees that are payable in advance (Prepaid Finance Charge) or at the time the loan is consummated. Interest is not charged on these fees, but it still raises the APR of a loan.

1. Main Results

The main and additional results are the immediate output of the previously specified loan construction section:

• Effective Annual Rate (EAR) - an estimate of the yearly rate adjusted by the compounding effect. As it was mentioned, this indicator doesn't account for any additional costs attached to the loan.

• Annual Percentage Rate (APR) - estimates the cost of borrowing per year as a percentage of the Loan Amount. It takes into consideration all additional costs without incorporating the compounding factor.

• Effective Annual Percentage Rate (Effective APR) - the APR adjusted by the effect of compounding - the ultimate indicator for the cost of borrowing in the context of this calculator.

• Total additional fees - the sum of all costs connected to the loan (Fees rolled into loan plus fees paid separately).

• Installment or Payment amount (PMT) - the amount of money that needs to be paid over each payment period set by the Payment frequency.

• Total finance charge or Cost of borrowing - the total expenses of the loan. In other words, this is the total amount of money you pay to use the credit (Interest plus all additional fees).

• Total Payments - the sum of the Loan amount and Total finance charge; thus, this is the sum of money you need to pay back after signing the loan contract.

• Total Interest Payment - the sum of interest that comes from borrowing.

• Principal or Present Value (PV) - the total amount of loan including the rolled in fees (Loan amount plus fees rolled into loan).

1. Supplementary Information

• Number of periods (t) - the life span of the loan in years converted from the previously given loan term.

• Equivalent interest rate (eq_r) and Periodic equivalent interest rate (eq_i) - that interest rates that are computed when the payments and compounding occur with a different frequency. In other words, the equivalent rate is aimed at converting the nominal interest rate from one compounding frequency to another while keeping the Effective Rate unchanged.

For example, assume you have a loan with annual rate of 6 percent (`r=6`), and you have to pay it back quarterly (`q=4`), but compounding occurs monthly (`m=12`). As the payment frequency is quarterly, the interest will be charged on your loan on quarterly bases as well. Hence, to determine the nominal interest rate of a loan paid once a quarter but compounding monthly, you need to find the equivalent interest rate.

The general formula of the equivalent rate and its periodic form are the following.

`eq_r = (q * ((1 + r / m) ^ (m / q) - 1))`

`eq_i = eq_r / q`

After substituting the values from our example, we need to solve the following equation:

`eq_r = (4 * ((1 + 0.06 / 12) ^ (12 / 4) - 1)) = 0.0603005 ≈ 6.03%`

`eq_i = 6.03% / 4 = 1.5075%`

Note, that the equivalent interest rate only harmonizes the payments and compounding of different frequencies, thus, except the case of annual payment frequency, it is not equal to the Effective Annual Rate (EAR).

• Approximate APR is a proxy for the Annual Percentage Rate.

Since estimating APR involves complex mathematics, we've decided to present to you a simplified formula that gives you an approximate value for the APR:

`Approximate APR = (2 * q * Total Finance Charge) / (Loan Amount * (n + 1))`

## How to calculate APR - An example with the applied formulas

Now that you know some financial terms, let's go through an example of how to calculate APR of a loan.

Let's assume you are considering taking a mortgage of 200,000 dollars and you would like to pay back monthly. You get an offer from a bank with 30 years loan term and a 6 percent interest rate, compounding monthly. Not including the interest, the total amount of fees that are rolled into the loan is 5,000 dollars.

The following steps lead us to the Effective APR:

1. Define monthly payment

`i = r / q = 0.06 / 12 = 0.005`

`n = t * q = 30 * 12 = 360`

`PV = Loan amount + Fees rolled into the loan = 205,000`

`PMT = (PV * i * (1 + i) ^ n) / ((1 + i) ^ n - 1)`

`PMT = (205,000 * 0.005 * (1 + 0.005) ^ 360 / ((1 + 0.005) ^ 360 = 1,229.08`

Thus your monthly payment would be 1,229.08 dollars if you accept the offer.

1. Estimate APR

To precisely define APR, we need to apply the same equation that we use to estimate the Payment amount, but instead of Present value (Principal) we use Loan amount. Why? When the bank charges you additional fees, you pay interest not only on the original Loan amount but also on the additional fees rolled into the loan. Therefore, the 6 percent interest rate is applied to the amount of 205,000 dollars. In reality, however, you receive only 200,000 dollars; thus, your Payment amount (PMT) should be compared to the original Loan amount.

Substituting these values into the Loan payment formula, we get the following equation for the monthly APR, that is, the periodic Annual Percentage Rate (APR_i):

`1,229.08 = (200,000 * APR_i * (1 + APR_i) ^ 360 / ((1 + APR_i) ^ 360`

Solving the equation for APR_i requires an intricate analytical procedure, the so-called Newton-Raphson Method. To conserve space, we won't elaborate on this subject but will instead give you the final value:

`APR_i = 0.519 %`

By multiplying the periodic APR by the payment frequency, we receive the annual percentage for the APR:

`APR = APR_i * q = 0.50933 * 12 = 6.232 %`

1. Estimate Effective APR

However, the above figure isn't the real cost of your credit as we have so far omitted the effect of compounding interest. The formula required to obtain the Effective APR that includes the compounding factor is identical to the previously introduced equation for the Effective Annual Rate, but the only difference is that APR is used instead of the nominal interest rate:

`Effective APR = (1 + APR / m) ^ m - 1 = (1 + 0.06232 / 12) ^ 12 - 1 = 0.06413 = 6.413 %`

Thus, the annual rate charged for the borrowing of 200,000 dollars, including all fees, plus the compounding interest, is 6.413 percent.

For simplicity, the above example had equal compounding and payment frequencies and also had no additional fees paid separately. Because of the particular technique required for the internal rate estimation, differing from these conditions further complicate and extend the computation procedure.

As the above review suggests, you can efficiently utilize our device as a mortgage APR calculator, but similarly, you can apply as an APR calculator for a car purchase as well.

## What is a good APR for a credit card

Credit cards are a widely used financial product. If you are about to open a credit account or are just wondering what cost you have to bear when you purchase something using your credit card, you can use this device as a credit card APR calculator.

Still, you may need some benchmark figure to compare the estimated APR to. So what is a good APR for a credit card? There is no simple answer to this question as APRs are highly variable according to its type and the services or benefits attached to the card. Besides, the actual Federal Reserve's policy rates and your credit scores also affect the credit card APR.

According to CreditCards.com, the average credit card APR is 17.73 percent as of May 22, 2019. It means that a good APR should be at least this rate or below it.

• BrownMath.com
• CreditCards.com
• Eugene F. Brigham and Michael C. Ehrhardt: Financial Management - Theory & Practice (15th edition) - 2017, Cengage Learning
• Stephen A. Ross: Focus on Personal Finance - An Active Approach to Help You Achieve Financial Literacy (5th edition) - 2016, McGraw-Hill Education
• S. J. Garrett: An Introduction to the Mathematics of Finance - A Deterministic Approach (2nd edition) - 2013, Institute and Faculty of Actuaries
Tibor Pal, PhD candidate and Dariusz Walczy-Antkowicz
Loan Specification
Loan amount
\$
Interest rate (Nominal)
%
Loan term
yrs
mos
Compounding frequency
monthly
Fees rolled into loan
\$
Fees paid separately
\$
Main Results
Effective Annual Rate (EAR)
5.116
%
Annual Percentage Rate (APR)
5.273
%
Effective APR
5.403
%
4,500
\$
Installment
2,000.71
\$
Total finance charge
111,627.42
\$
Total payments
361,627.42
\$
Total interest payment
107,127.42
\$
Principal
253,000
\$
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