# Present value of an annuity Definition

## What is the present value of an annuity?

The present value of an annuity is the present value of future payments of an annuity, given a specified rate of return or a discount rate. The higher the discount rate, the lower the present value of the annuity.

Key points to remember

- The present value of an annuity refers to the amount of money that 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 will receive more money by taking a lump sum now or an annuity spread over a number of years.

Present value of an annuity

## Understanding the present value of an annuity

Due to the time value of money, money received today is worth more than the same amount in the future because it can be invested in the meantime. Similarly, $ 5,000 received today is worth more than the same amount spread over five annual payments of $ 1,000 each.

The future value of money is calculated using a discount rate. The discount rate refers to an interest rate or assumed rate of return on other investments over the same term as the payments. The smallest discount rate used in these calculations is the risk-free rate of return. US Treasury bonds are generally considered to be the closest thing to a risk-free investment, so their yield is often used for this purpose.

## Example of the present value of an annuity

The formula for the present value of an ordinary annuity, as opposed to an annuity owed, is below. (A regular annuity pays interest at the end of a specified period, rather than at the beginning, as is the case with an annuity due.)

$$

P

=

PMT

Ã—

1

–

(

1

(

1

+

r

)

m

)

r

or:

P

=

Present value of an annuity stream

PMT

=

Dollar amount of each annuity payment

r

=

Interest rate (also called the discount rate)

m

=

Number of periods during 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 {Current value of an annuity stream} & text {PMT} = text {Dollar amount of each annuity payment} & r = text {Interest rate (also called discount rate)} & n = text {Number of periods during which payments will be made} end {aligned}

P=PMTÃ—r1–((1+r)m1)or:P=Present value of an annuity streamPMT=Dollar amount of each annuity paymentr=Interest rate (also called the discount rate)m=Number of periods during which payments will be made

Suppose a person has the option of receiving a regular annuity that pays $ 50,000 per year for the next 25 years, with a discount rate of 6%, or a lump sum payment of $ 650,000. What is the best option? Using the above formula, the present value of the annuity is:

$$

Current value

=

$

50

,

000

Ã—

1

–

(

1

(

1

+

0.06

)

25

)

0.06

=

$

639

,

168

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

Current value=$50,000Ã—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 gain by choosing the lump sum payment over the annuity.

A regular annuity makes payments at the end of each period, while an owed annuity makes them at the beginning. All other things being equal, the annuity due will be worth more in the present.

With an annuity due, in which payments are made at the start 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

Ã—

1

–

(

1

(

1

+

r

)

m

)

r

Ã—

(

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=PMTÃ—r1–((1+r)m1)Ã—(1+r)

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

$$

Current value

=

$

50

,

000

Ã—

1

–

(

1

(

1

+

0.06

)

25

)

0.06

Ã—

(

1

+

.

06

)

=

$

677

,

518

begin {aligned} text {Current value} & = $ 50,000 times frac {1 – Big ( frac {1} {(1 + 0.06) ^ {25}} Big)} {0.06} times (1 + .06) & = $ 677.518 end {aligned}

Current value=$50,000Ã—0.061–((1+0.06)251)Ã—(1+.06)=$677,518

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

## 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 growth rate. It is important for investors because they can use it to estimate the value of an investment made today in the future. This would help them make sound investment decisions based on their anticipated needs. However, external economic factors, such as inflation, can negatively affect the future value of the asset by eroding its value.

## How does the ordinary annuity differ from the annuity due?

An ordinary annuity is a series of equal payments made at the end of consecutive periods over a specified period. An example of a regular annuity includes loans, such as mortgages. Payment of an annuity due is made at the start of each period. A common example of paying an annuity due is rent. This discrepancy in the timing of payments results in different present and future value calculations.

## What is the formula for the present value of an ordinary annuity?

The present value formula of an ordinary annuity is as follows:

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

P=PMTÃ—r1–((1+r)m1)or:P=Present value of an annuity streamPMT=Dollar amount of each annuity paymentr=Interest rate (also called the discount rate)m=Number of periods during 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 start of each period, the formula is slightly different from that of a regular annuity. To find the value of an annuity due, simply multiply the above formula by a factor of (1 + r):

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

P=PMTÃ—r1–((1+r)m1)Ã—(1+r)