Present Value of an Annuity: Meaning, Formula, and Example

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. The higher the discount rate, the lower the present value of the annuity.

Present value (PV) is an important calculation that relies on the concept of 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 refers to how much money would be needed today to fund a series of future annuity payments.
  • Because of the time value of money, a sum of money received today is worth more than the same sum at a future date.
  • You can use a present value calculation to determine whether you'll receive more money by taking a lump sum now or an annuity spread out over a number of years.
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Present Value of an Annuity

Understanding the Present Value of an Annuity

An annuity is a financial product that provides a stream of payments to an individual over a period of time, typically in the form of regular installments. 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.

Because of the time value of money, money received today is worth more than the same amount of money in the future because it can be invested in the meantime. 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 is an important concept for annuities because it allows individuals to compare the value of receiving a series of payments in the future to the value of receiving a lump sum payment 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. Present value calculations can also be used to compare the relative value of different annuity options, such as annuities with different payment amounts or different payment schedules.

Present Value and the Discount Rate

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. 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 an individual could earn a 5% return by investing in a high-quality corporate bond, they might use a 5% discount rate when calculating the present value of an annuity. The smallest discount rate used in these calculations is the risk-free rate of return. 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.

It's important to note that the discount rate used in the present value calculation is not the same as the interest rate that may be applied to the payments in the annuity. 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.

The opposite of present value is future value (FV). The FV of money is also calculated using a discount rate, but extends into the future.

Formula and Calculation of the Present Value of an Annuity

The formula for the present value of an ordinary annuity, is below. An ordinary annuity pays interest at the end of a particular period, rather than at the beginning:

P = PMT × 1 ( 1 ( 1 + r ) n ) r where: P = Present value of an annuity stream PMT = Dollar amount of each annuity payment r = Interest rate (also known as discount rate) n = Number of periods in which payments will be made \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} P=PMT×r1((1+r)n1)where:P=Present value of an annuity streamPMT=Dollar amount of each annuity paymentr=Interest rate (also known as discount rate)n=Number of periods in which payments will be made

Example of the Present Value of an Annuity

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:

Present value = $ 50 , 000 × 1 ( 1 ( 1 + 0.06 ) 25 ) 0.06 = $ 639 , 168 \begin{aligned} \text{Present value} &= \$50,000 \times \frac { 1 - \Big ( \frac { 1 }{ ( 1 + 0.06 ) ^ {25} } \Big ) }{ 0.06 } \\ &= \$639,168 \\ \end{aligned} Present value=$50,000×0.061((1+0.06)251)=$639,168

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.

Annuity vs. Annuity Due

An ordinary annuity makes payments at the end of each time period, while an annuity due 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):

P = PMT × 1 ( 1 ( 1 + r ) n ) r × ( 1 + r ) \begin{aligned} &\text{P} = \text{PMT} \times \frac { 1 - \Big ( \frac { 1 }{ ( 1 + r ) ^ n } \Big ) }{ r } \times ( 1 + r ) \\ \end{aligned} P=PMT×r1((1+r)n1)×(1+r)

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

Present value = $ 50 , 000 × 1 ( 1 ( 1 + 0.06 ) 25 ) 0.06 × ( 1 + . 06 ) = $ 677 , 518 \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} Present value=$50,000×0.061((1+0.06)251)×(1+.06)=$677,518

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 Ordinary Annuity Differ From 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:

P=PMT×1(1(1+r)n)rwhere:P=Present value of an annuity streamPMT=Dollar amount of each annuity paymentr=Interest rate (also known as discount rate)n=Number of periods in which payments will be made\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}P=PMT×r1((1+r)n1)where:P=Present value of an annuity streamPMT=Dollar amount of each annuity paymentr=Interest rate (also known as discount rate)n=Number of periods in which payments will be made

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

With an annuity due, in which payments are made at the beginning of each period, 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):

P=PMT×1(1(1+r)n)r×(1+r)\begin{aligned} &\text{P} = \text{PMT} \times \frac { 1 - \Big ( \frac { 1 }{ ( 1 + r ) ^ n } \Big ) }{ r } \times ( 1 + r ) \\ \end{aligned}P=PMT×r1((1+r)n1)×(1+r)

The Bottom Line

The present value (PV) of an annuity is the current value of future payments from an annuity, given a specified rate of return or discount rate. It is calculated using a formula that takes into account the time value of money and the discount rate, which is an assumed rate of return or interest rate over the same duration as the payments. The present value of an annuity can be used to determine whether it is more beneficial to receive a lump sum payment or an annuity spread out over a number of years.

Article Sources
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  1. George Brown College. "Formula Sheet for Financial Mathematics."

  2. Wai-sum Chan and Yiu-kuen Tse. “Financial Mathematics for Actuaries (Third Edition),” Pages 40-43. World Scientific Publishing Company, 2021.

  3. All Finance Journal. "Time Value of Money: A Case Study on Its Concept and Its Application in Real Life Problems," Pages 18-23.

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