How to Calculate an Annual Payment on a Loan
Familiarize yourself with the formula for calculating annual payments on a loan., Understand the variables in the equation., Plug the values into the formula., Solve for the numerator of the equation., Solve for the denominator., Solve for the...
Step-by-Step Guide
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Step 1: Familiarize yourself with the formula for calculating annual payments on a loan.
Assuming a fixed interest rate and evenly spaced payments, the annual payment amount for an annuity (anything that must be paid in yearly increments) can be determined using the following formula:
AnnualPayment=(r(P))(1−(1+r)−n){\displaystyle AnnualPayment={\frac {(r(P))}{(1-(1+r)^{-n})}}}, The first step of finding annual payments on a loan is to understand what each of the letters means.
Fortunately, each letter simply represents one of the elements of a loan.
This information can be easily found on your loan agreement.If you don't have a copy of your loan agreement, contact your lender. r represents the interest rate per period.
Because this represents an annual interest rate in this case, this number may be referred to as an APR (annual percentage rate).
P represents the principal, or the amount borrowed.
This can also be referred to as the present value.
N represents the number of periods in the loan.
In this case, periods equals years, and would just be the number of years on your loan agreement. , Once you know the terms of your loan, you can plug them into the formula above to determine the annual payment.
For example, consider a $10,000 loan with an annual interest rate of 9%, for a period of two years.
AnnualPayment=(0.09($10,000))(1−(1+0.09)−2){\displaystyle AnnualPayment={\frac {(0.09(\$10,000))}{(1-(1+0.09)^{-2})}}} Note that when inputting a percent (9% in this case), it must be input as a decimal. 9% therefore becomes .09. , The first step in calculating annual payments on a loan is to solve for the numerator (the top part of the equation).
Multiply .09 x $10,000 to get $900.
This completes the left side of the equation.
Your equation should now look like this:
AnnualPayment=$900(1−(1+0.09)−2){\displaystyle AnnualPayment={\frac {\$900}{(1-(1+0.09)^{-2})}}} , The next step is to solve the denominator (the bottom part of the equation).
This will be done in three steps.
First, add 1 to .09, to give
1.09.
Your equation should now look like this:
AnnualPayment=$900(1−(1.09)−2){\displaystyle AnnualPayment={\frac {\$900}{(1-(1.09)^{-2})}}} , Raise
1.09 to the power of
-2 (the term).
The result will be
0.8417.
Recall that when solving an equation, brackets are always solved first, followed by exponents (the
-2).
Your equation should now look like this:
AnnualPayment=$9001−(0.8417){\displaystyle AnnualPayment={\frac {\$900}{1-(0.8417)}}} , Subtract
0.8417 from 1, to obtain
0.1583.
This would complete the bottom part of the equation.
Remember to keep as many decimal places present in calculation as possible.
This will ensure accuracy, especially in larger loan amounts.
Your equation should now look like this:
AnnualPayment=$9000.1583{\displaystyle AnnualPayment={\frac {\$900}{0.1583}}} , Divide the top of your equation by the bottom to get the annual payment on your loan.
Solving the example equation, you get
5685.41.
Therefore, your annual payment would be $5,685.41. , An amortization table allows you to see each payment you will make for the remainder of your loan, broken down into how much is principal, how much is interest, and what the remaining balance on the loan is.
This allows you to see exactly what your monthly (or annual) payment is, and less and less of the payment goes to interest over time as the amount owed decreases.Simply input the amount, interest rate, and term into the calculator, and the amortization table will show every monthly payment from the current point to the end of the loan. -
Step 2: Understand the variables in the equation.
-
Step 3: Plug the values into the formula.
-
Step 4: Solve for the numerator of the equation.
-
Step 5: Solve for the denominator.
-
Step 6: Solve for the exponent.
-
Step 7: Finish solving for the denominator.
-
Step 8: Complete your calculation.
-
Step 9: Use online resources to construct an amortization table to understand the annual payments.
Detailed Guide
Assuming a fixed interest rate and evenly spaced payments, the annual payment amount for an annuity (anything that must be paid in yearly increments) can be determined using the following formula:
AnnualPayment=(r(P))(1−(1+r)−n){\displaystyle AnnualPayment={\frac {(r(P))}{(1-(1+r)^{-n})}}}, The first step of finding annual payments on a loan is to understand what each of the letters means.
Fortunately, each letter simply represents one of the elements of a loan.
This information can be easily found on your loan agreement.If you don't have a copy of your loan agreement, contact your lender. r represents the interest rate per period.
Because this represents an annual interest rate in this case, this number may be referred to as an APR (annual percentage rate).
P represents the principal, or the amount borrowed.
This can also be referred to as the present value.
N represents the number of periods in the loan.
In this case, periods equals years, and would just be the number of years on your loan agreement. , Once you know the terms of your loan, you can plug them into the formula above to determine the annual payment.
For example, consider a $10,000 loan with an annual interest rate of 9%, for a period of two years.
AnnualPayment=(0.09($10,000))(1−(1+0.09)−2){\displaystyle AnnualPayment={\frac {(0.09(\$10,000))}{(1-(1+0.09)^{-2})}}} Note that when inputting a percent (9% in this case), it must be input as a decimal. 9% therefore becomes .09. , The first step in calculating annual payments on a loan is to solve for the numerator (the top part of the equation).
Multiply .09 x $10,000 to get $900.
This completes the left side of the equation.
Your equation should now look like this:
AnnualPayment=$900(1−(1+0.09)−2){\displaystyle AnnualPayment={\frac {\$900}{(1-(1+0.09)^{-2})}}} , The next step is to solve the denominator (the bottom part of the equation).
This will be done in three steps.
First, add 1 to .09, to give
1.09.
Your equation should now look like this:
AnnualPayment=$900(1−(1.09)−2){\displaystyle AnnualPayment={\frac {\$900}{(1-(1.09)^{-2})}}} , Raise
1.09 to the power of
-2 (the term).
The result will be
0.8417.
Recall that when solving an equation, brackets are always solved first, followed by exponents (the
-2).
Your equation should now look like this:
AnnualPayment=$9001−(0.8417){\displaystyle AnnualPayment={\frac {\$900}{1-(0.8417)}}} , Subtract
0.8417 from 1, to obtain
0.1583.
This would complete the bottom part of the equation.
Remember to keep as many decimal places present in calculation as possible.
This will ensure accuracy, especially in larger loan amounts.
Your equation should now look like this:
AnnualPayment=$9000.1583{\displaystyle AnnualPayment={\frac {\$900}{0.1583}}} , Divide the top of your equation by the bottom to get the annual payment on your loan.
Solving the example equation, you get
5685.41.
Therefore, your annual payment would be $5,685.41. , An amortization table allows you to see each payment you will make for the remainder of your loan, broken down into how much is principal, how much is interest, and what the remaining balance on the loan is.
This allows you to see exactly what your monthly (or annual) payment is, and less and less of the payment goes to interest over time as the amount owed decreases.Simply input the amount, interest rate, and term into the calculator, and the amortization table will show every monthly payment from the current point to the end of the loan.
About the Author
Frank Young
Creates helpful guides on lifestyle to inspire and educate readers.
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