# Calculating Present Value of an Annuity: Formula and Practical Examples

## What Is the Present Value of an Annuity?

The present value of an annuity is the current value of future payments from an annuity, given a specified rate of return, or [discount rate](https://www.investopedia.com/terms/d/discountrate.asp). Understanding this concept is important for making informed financial decisions, such as comparing lump-sum options versus annuity payments.

The higher the discount rate, the lower the present value of the annuity. [Present value (PV)](https://www.investopedia.com/terms/p/presentvalue.asp) relies on the time value of money, whereby a dollar today is relatively more "valuable" in terms of its purchasing power than a dollar in the future.

### Key Takeaways

*   The present value of an annuity is a critical financial concept that helps investors assess the value of future annuity payments by calculating how much money would be needed at present to achieve those future payments at a specified rate of return.
*   Understanding the time value of money is essential, as it highlights that money available today can be invested, making it more valuable than the same amount received in the future.
*   The choice between taking a lump sum today versus annuity payments over time can significantly impact financial planning, and calculating the present value of an annuity allows for an informed and beneficial decision.
*   The discount rate plays a pivotal role in computing present value; a higher discount rate reduces the present value of future annuity payments, while a lower rate results in a higher present value.
*   Annuities come in different forms, such as ordinary annuities and annuities due, with the latter being more valuable in present value terms because payments are received sooner.

![Present Value of an Annuity](https://www.investopedia.com/thmb/fZ08jt3Q36mvEpQHQve3N1JyLKg=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/present-value-annuity.asp-Final-7d2ae860c2b044069d77d5e626c5a6f3-72cf6d2c9f9041608b7cd343cbc3f2f4.png)

*— Investopedia / Dennis Madamba*

## Breaking Down the Present Value of an Annuity

An [annuity](https://www.investopedia.com/ask/answers/12/what-is-an-annuity.asp) is a financial product that pays regular installments to a person over time. Annuities can be either immediate or deferred, depending on when the payments begin. Immediate annuities start paying out right away, while deferred annuities have [a delay before payments begin](https://www.investopedia.com/terms/d/delayed-perpetuity.asp).

Due to the [time value of money](https://www.investopedia.com/terms/t/timevalueofmoney.asp), receiving money today is better than receiving the same amount later, as it can be invested. By the same logic, $5,000 received today is worth more than the same amount spread over five annual installments of $1,000 each.

Present value helps compare getting future payments against a lump sum today. By calculating the present value of an annuity, individuals can determine whether it is more beneficial for them to receive a lump sum payment or to receive an annuity spread out over a number of years. This can be particularly important when making financial decisions, such as whether to take a lump sum payment from a pension plan or to receive a series of payments from an annuity.

The pension provider will [determine the commuted value](https://www.investopedia.com/terms/c/commuted-value.asp) of the payment due to the [beneficiary](https://www.investopedia.com/terms/b/beneficiary.asp). They do this to ensure they are able to meet future payment obligations.

Present value can compare different annuities by payment amounts or schedules.

### How the Discount Rate Affects Present Value

The discount rate is a key factor in calculating the present value of an annuity. The discount rate is an assumed rate of return or interest rate that is used to determine the present value of future payments.

The discount rate reflects the time value of money, which means that a dollar today is worth more than a dollar in the future because it can be [invested and potentially earn a return](https://www.investopedia.com/ask/answers/033015/why-time-value-money-tvm-important-concept-investors.asp). The higher the discount rate, the lower the present value of the annuity, because the future payments are discounted more heavily. Conversely, a lower discount rate results in a higher present value for the annuity, because the future payments are discounted less heavily.

In general, the discount rate used to calculate the present value of an annuity should reflect the individual's opportunity cost of capital, or the return they could expect to earn by investing in other financial instruments. For example, if someone can earn a 5% return from a corporate bond, they may use 5% as the discount rate. The smallest discount rate used in these calculations is the [risk-free rate of return](https://www.investopedia.com/terms/r/risk-freerate.asp). U.S. Treasury bonds are generally considered to be the closest thing to a risk-free investment, so their return is often used for this purpose.

Note that the discount rate for present value is different from the annuity's interest rate. The discount rate reflects the time value of money, while the interest rate applied to the annuity payments reflects the cost of borrowing or the return earned on the investment.

> **Fast Fact**
>
> The opposite of present value is future value (FV), which also uses a discount rate but projects into the future.

## Calculating the Present Value of an Annuity

The [formula for the present value](https://www.investopedia.com/retirement/calculating-present-and-future-value-of-annuities/) of an [ordinary annuity](https://www.investopedia.com/terms/o/ordinaryannuity.asp) is below. An ordinary annuity pays interest at the end of a particular period, rather than at the beginning:

$\begin{aligned} &\text{P} = \text{PMT} \times \frac { 1 - \Big ( \frac { 1 }{ ( 1 + r ) ^ n } \Big ) }{ r } \\ &\textbf{where:} \\ &\text{P} = \text{Present value of an annuity stream} \\ &\text{PMT} = \text{Dollar amount of each annuity payment} \\ &r = \text{Interest rate (also known as discount rate)} \\ &n = \text{Number of periods in which payments will be made} \\ \end{aligned}$

## Real-World Example: Assessing Annuity Present Value

Assume a person has the opportunity to receive an ordinary annuity that pays $50,000 per year for the next 25 years, with a 6% discount rate, or take a $650,000 lump-sum payment. Which is the better option? Using the above formula, the present value of the annuity is:

$\begin{aligned} \text{Present value} &= \$50,000 \times \frac { 1 - \Big ( \frac { 1 }{ ( 1 + 0.06 ) ^ {25} } \Big ) }{ 0.06 } \\ &= \$639,168 \\ \end{aligned}$

Given this information, the annuity is worth $10,832 less on a time-adjusted basis, so the person would come out ahead by choosing the lump-sum payment over the annuity.

## Comparing Ordinary Annuities and Annuities Due

An ordinary annuity makes payments at the end of each time period, while an [annuity due](https://www.investopedia.com/terms/a/annuitydue.asp) makes them at the beginning. All else being equal, the annuity due will be worth more in the present. In the case of an annuity due, since payments are made at the beginning of each period, the formula is slightly different. To find the value of an annuity due, simply multiply the above formula by a factor of (1 + r):

$\begin{aligned} &\text{P} = \text{PMT} \times \frac { 1 - \Big ( \frac { 1 }{ ( 1 + r ) ^ n } \Big ) }{ r } \times ( 1 + r ) \\ \end{aligned}$

So, if the example above referred to an annuity due rather than an ordinary annuity, its value would be as follows:

$\begin{aligned} \text{Present value} &= \$50,000 \times \frac { 1 - \Big ( \frac { 1 }{ ( 1 + 0.06 ) ^ {25} } \Big ) }{ 0.06 } \times ( 1 + .06 ) \\ &= \$677,518 \\ \end{aligned}$

In this case, the person should choose the annuity due option because it is worth $27,518 more than the $650,000 lump sum.

## Why Is Future Value (FV) Important to Investors?

Future value (FV) is the value of a current asset at a future date based on an assumed rate of growth. It is important to investors as they can use it to estimate how much an investment made today will be worth in the future. This would aid them in making sound investment decisions based on their anticipated needs. However, external economic factors, such as inflation, can adversely affect the future value of the asset by eroding its value.

## How Does an Ordinary Annuity Differ From an Annuity Due?

An ordinary annuity is a series of equal payments made at the end of consecutive periods over a fixed length of time. An example of an ordinary annuity includes loans, such as mortgages. The payment for an annuity due is made at the beginning of each period. A common example of an annuity due payment is rent. This variance in when the payments are made results in different present and future value calculations.

## What Is the Formula for the Present Value of an Ordinary Annuity?

The formula for the present value of an ordinary annuity is:

$\begin{aligned} &\text{P} = \text{PMT} \times \frac { 1 - \Big ( \frac { 1 }{ ( 1 + r ) ^ n } \Big ) }{ r } \\ &\textbf{where:} \\ &\text{P} = \text{Present value of an annuity stream} \\ &\text{PMT} = \text{Dollar amount of each annuity payment} \\ &r = \text{Interest rate (also known as discount rate)} \\ &n = \text{Number of periods in which payments will be made} \\ \end{aligned}$

## What Is the Formula for the Present Value of an Annuity Due?

With an annuity due, payments are made at the beginning of each period. So the formula is slightly different than that of an ordinary annuity. To find the value of an annuity due, simply multiply the above formula by a factor of (1 + r):

$\begin{aligned} &\text{P} = \text{PMT} \times \frac { 1 - \Big ( \frac { 1 }{ ( 1 + r ) ^ n } \Big ) }{ r } \times ( 1 + r ) \\ \end{aligned}$

## The Bottom Line

The present value (PV) of an annuity is the current worth of future annuity payments, calculated using a specified discount rate that reflects the time value of money, where money today is worth more than the same amount in the future due to its earning potential.

PV helps individuals compare annuity payout structures or decide between taking a lump sum versus regular payments. A higher discount rate lowers PV, while a lower rate increases it, making the choice of discount rate essential. Understanding PV empowers people to factor in opportunity costs and make more informed financial decisions.

> **Article Sources**
>
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