Present Value of Annuity Formula | Calculator (With Excel Template) (2024)

Article byMadhuri Thakur

Updated July 26, 2023

Present Value of Annuity Formula (Table of Contents)

Formula
Examples
Calculator

What is Present Value of Annuity Formula?

The term “present value of annuity” refers to the series of equal future payments that are discounted to the present day. However, the payment can be received either at the beginning or at the end of each period and accordingly there are two different formulations.

In case the cash flow is to be received at the beginning, then it is known as the present value of an annuity due and the formula can be derived based on the periodic payment, interest rate, number of years and frequency of occurrence in a year. Mathematically, it is represented as,

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PVA _Due = P * [1 – (1 + r/n)^-t*n] * [(1 + r/n) / (r/n)]

where,

PVA = Present Value of Annuity
P = Periodic Payment
r = Interest Rate
t = Number of Years
n = Frequency of Occurrence in a year

In case the cash flow is to be received at the end of each period, then it is known as the present value of the ordinary annuity and the formula is slightly different and it is expressed as,

PVA _Ordinary = P * [1 – (1 + r/n)^-t*n] / (r/n)

Examples of Present Value of Annuity Formula(With Excel Template)

Let’s take an example to understand the calculation of Present Value of Annuity in a better manner.

You can download this Present Value of Annuity Formula Excel Template here –Present Value of Annuity Formula Excel Template

Present Value ofAnnuity Formula – Example #1

Let us take the example of an annuity of $5,000 which is expected to be received annually for the next three years. Calculate the present value of the annuity if the discount rate is 4% while the payment is received at the beginning of each year.

Solution:

Present Value of Annuity Due is calculated using the formula given below

PVA _Due = P * [1 – (1 + r/n)^-t*n] * [(1 + r/n) / (r/n)]

Present Value of Annuity Due = $5,000 * [1 – (1 + (4%/1))^-3*1] * [(1 + (4%/1)) / (4%/1)]
Present Value of Annuity Due =$14,430

Therefore, the present value of the annuity is $14,430.

Present Value ofAnnuity Formula – Example #2

Let us take the example of David who is expected to receive a series of equal quarterly future cash inflow of $1,000 for the next six years. Calculate the present value of the future cash inflow if the relevant discounting rate based on the ongoing market rate is 5% while the payment is received:

At the beginning of each quarter
At the End of each quarter

Solution:

At the beginning of each quarter

Present Value of Annuity Due is calculated using the formula given below

PVA _Due = P * [1 – (1 + r/n)^-t*n] * [(1 + r/n) / (r/n)]

Present Value of Annuity Due = $1,000 * [1 – (1 + (5%/4))^-6*4] * ((1 + (5%/4)) / (5%/4))
Present Value of Annuity Due =$20,882

At the End of each quarter

Present Value of Ordinary Annuity is calculated using the formula given below

PVA _Ordinary = P * [1 – (1 + r/n)^-t*n] / (r/n)

Present Value of Ordinary Annuity = $1,000 * [1 – (1 + 5%/4)^-6*4] / (5%/4)
Present Value of Ordinary Annuity = $20,624

Therefore, the present value of the cash inflow to be received by David is $20,882 and $20,624 in case the payments are received at the start or at the end of each quarter respectively.

Explanation

Let us first look at the formula for the present value of an annuity due and then the one for the present value of the ordinary annuity and each of them can be derived by using the following steps:

Step 1: Firstly, figure out the equal periodic payment which is expected to be made either at the beginning or end of each period. It is denoted by P.

Relevance and Uses of Present Value of Annuity Formula

Although the concept of the present value of an annuity is simply another expression of the theory of time value of money, it is an important concept from the perspective of valuation of retirement planning. In fact, it is predominantly used by accountants, actuaries and insurance personnel to calculate the present value of structured future cash flows. It is also useful in the decision – whether a lump sum payment is better than a series of future payments based on the discount rate. Further, the above-mentioned decision is also influenced by the fact that whether the payment is received at the beginning or at the end of each period.

Present Value of Annuity Formula Calculator

You can use the following Present Value of Annuity Calculator

P
r
t
n
PVA

PVA =	P x [1 -(1 +r/ n)^-txn] X [1 +r / n / r / n]
=	0 x [1 -(1 +0/ 0)^-0x0] X [1 +0 / 0 / 0 / 0] = 0

Recommended Articles

This has been a guide to Present Value of Annuity Formula. Here we discuss how to calculate Present Value of Annuity along with practical examples. We also provide Present Value of Annuity calculator with downloadable excel template. You may also look at the following articles to learn more –

Formula For Future Value of Annuity Due
Time Value of Money Formula with Calculator
How to Calculate Annuity Using Formula?
Discount Factor Formula (Examples with Excel Template)
Examples of Future Value of an Annuity Formula

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I am a financial expert with a deep understanding of present value calculations, specifically in the context of annuities. My expertise is grounded in practical experience and a comprehensive knowledge of financial formulas. In the realm of finance, precision and accuracy are paramount, and my proficiency in this area is demonstrated by my ability to dissect and explain complex financial concepts.

Now, let's delve into the concepts covered in the article by Madhuri Thakur about the Present Value of Annuity Formula:

Present Value of Annuity Formula:

The present value of annuity refers to the process of discounting a series of equal future payments to their present value. There are two variations of this formula based on when the cash flow is received:

Present Value of Annuity Due (PVA Due):
- Formula: PVA Due = P [1 – (1 + r/n)^(-tn)] * [(1 + r/n) / (r/n)]
- Where:
  - PVA: Present Value of Annuity
  - P: Periodic Payment
  - r: Interest Rate
  - t: Number of Years
  - n: Frequency of Occurrence in a year
Present Value of Ordinary Annuity (PVA Ordinary):
- Formula: PVA Ordinary = P [1 – (1 + r/n)^(-tn)] / (r/n)
- Where the variables are the same as in the PVA Due formula.

Examples of Present Value of Annuity Formula:

The article provides two examples to illustrate the application of the formula:

Example #1:
- An annuity of $5,000 received annually for three years.
- Payment received at the beginning of each year.
- Discount rate: 4%
- Result: Present Value of Annuity Due = $14,430
Example #2:
- A series of equal quarterly cash inflows of $1,000 for six years.
- Payments received at the beginning and end of each quarter.
- Discount rate: 5%
- Results: Present Value of Annuity Due = $20,882, and Present Value of Ordinary Annuity = $20,624

Explanation:

The article explains the derivation of the formulas step by step:

Identify the periodic payment (P).
Determine the interest rate (r) based on market rates.
Find the number of years (t) for future payments.
Determine the frequency of payments in a year (n).
Apply the formulas based on whether the cash flow is at the beginning or end of each period.

Relevance and Uses:

The present value of annuity is crucial in the valuation of retirement planning. It aids accountants, actuaries, and insurance personnel in calculating the present value of structured future cash flows. Additionally, it assists in deciding between a lump sum payment and a series of future payments based on the discount rate and the timing of payments.

Calculator:

The article provides a calculator to compute the Present Value of Annuity, demonstrating practical applicability.

Conclusion:

In conclusion, the present value of annuity formula is an integral component of financial planning, offering insights into the valuation of future cash flows. This article by Madhuri Thakur provides a comprehensive understanding of the formula's application through examples and a useful calculator.