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 Annuity Formula
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What is the Annuity Formula?
The term “annuity” refers to the series of periodic payments to be received either at the beginning of each period or at the end of the period in the future. If the payment is received at the end of each period then it is called ordinary annuity while in the other case it is called annuity due.
The equation for annuity payment based on PV of an ordinary annuity is calculated based on PV of an ordinary annuity, effective interest rate and a number of periods.
Mathematically, the equation for an ordinary annuity is represented as,
where,
 PVA _{Ordinary} = Present value of an ordinary annuity
 r = Effective interest rate
 n = Number of periods
The equation for annuity payment an annuity due is calculated based on PV of an annuity due, effective interest rate and a number of periods.
Mathematically, the equation for annuity due is represented as,
where,
 PVA _{Due} = Present value of an annuity due
 r = Effective interest rate
 n = Number of periods
Explanation of the Annuity Formula
The formula for the calculation of annuity payment can be derived by using the PV of ordinary annuity in the following steps:
Step 1: Firstly, determine the PV of the annuity and confirm that the payment will be done at the end of each period. It is denoted by PVA _{Ordinary}.
Step 2: Next, determine the interest rate on the basis of the current market return. Then, the effective rate of interest is computed by dividing the annualized interest rate by the number of periodic payments in a year and it is denoted by r.
r = Annualized interest rate / Number periodic payments in a year
Step 3: Next, determine the number of periods by multiplying the number of periodic payments in a year and the number of years and it is denoted by n.
n = Number of periodic payments in a year * Number of years
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Step 4: Finally, the formula for annuity payment based on PV of an ordinary annuity is calculated based on PV of ordinary annuity (step 1), effective interest rate (step 2) and a number of periods (step 3) as shown above.
The formula for the calculation of annuity payment can also be derived by using the PV of an annuity due in the following steps:
Step 1: Firstly, determine the PV of the annuity and confirm that the payment will be done at the beginning of each period. It is denoted by PVA _{Due}.
Step 2: Next, determine the interest rate on the basis of the current market return. Then, the effective rate of interest is computed by dividing the annualized interest rate by the number of periodic payments in a year and it is denoted by r.
r = Annualized interest rate / Number periodic payments in a year
Step 3: Next, determine the number of periods by multiplying the number of periodic payments in a year and the number of years and it is denoted by n.
n = Number of periodic payments in a year * Number of years
Step 4: Finally, the equation for annuity payment based on PV of an annuity due is calculated based on PV of annuity due (step 1), effective interest rate (step 2) and a number of periods (step 3) as shown above.
Examples of Annuity Formula (with Excel Template)
Let’s see some simple to advanced examples of annuity formula to understand it better.
Example #1
Let us take the example of David who won a lottery worth $10,000,000. He has opted for an annuity payment at the end of each year for the next 20 years as a payout option. Determine the amount that David will be paid as annuity payment if the ongoing rate of interest in the market is 5%.
Given below is the data used for the calculation of annuity payments.
PVA _{Ordinary} = $10,000,000 (since the annuity to be paid at the end of each year)
Therefore, the calculation of annuity payment can be done using the above formula as,
 Annuity = 5% * $10,000,000 / [1 – (1 + 5%)^{20}]
Calculation of Annuity Payment will be –
 Annuity = $802,425.87 ~ $802,426
Therefore, David will pay annuity payments of $802,426 for the next 20 years in case of ordinary annuity.
Example #2
Let us take the above example of David and determine the annuity payment if it is paid at the beginning of each year with all other conditions the same.
We will use the same data as the above example for the calculation of Annuity payments.
Therefore, the calculation of annuity payment can be done using the above formula as,
 Annuity = r * PVA _{Due} / [{1 – (1 + r)^{n}} * (1 + r)]
 Annuity = 5% * $10,000,000 / [{1 – (1 + 5%)^{20}} * (1 + 5%)]
Calculation of Annuity Payment will be –
 Annuity = $764,215.12 ~ $764,215
Therefore, David will pay annuity payments of $764,215 for the next 20 years in case of annuity due.
Annuity Formula
You can use the following Annuity Formula Calculator.
PVA Ordinary  
r  
n  
Annuity Formula =  
Annuity Formula =  r * 
 
0 * 

Relevance and Uses
The annuity payment is one of the applications of the time value of money which is further indicated by the difference between annuity payments based on ordinary annuity and annuity due. The reason for lower annuity payment for an annuity due is that the money is received at the start of each period and as such, it is believed that the money will be invested in the market and interest will be earned during that period.
The equation for annuity payment finds application in the calculation of income annuities, amortized loans, lottery payouts, structured settlements and any other type of fixed periodic payments.
Recommended Articles
This has been a guide to Annuity Formula. Here we discuss how to calculate Annuity Payments in Excel for Ordinary and due annuity using its formula along with practical examples and downloadable excel template. You can learn more about financial analysis from the following articles –
 Types of Financial Analysis  Definition
 Definition of Financial Analysis
 Top 4 Most Common Financial Analysis Tools
 Excel Rate Formula
 Annuity vs Perpetuity – Compare
 Amortization Schedule of Mortgage
 Effective Annual Rate Formula
 Geometric Mean Return Formula
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