Recurring or ongoing payments are technically annuities. Whether making a series of fixed payments over a period, such as rent or car loan, or receiving periodic income from a bond or certificate of deposit (CD), you can calculate the 澳洲幸运5开奖号码历史查询:present value (PV) or 澳洲幸运5开奖号码历史查询:future value (FV) of an annuity.

Types of Annuities

Annuities꧃ as ongoing payments can be defined as ordinary annuities o✅r annuities due.

• Ordinary Annuities: An 澳洲幸运5开奖号码历史查询:ordinary annuity makes (or requires) payments at the end of a particular period. For example, bonds generally pay interest at the end of every six months.
• Annuities Due: An 澳洲幸运5开奖号码历史查询:annuity due, by contrast, involves payments made at the beginning of each period. Rent or loan payments required at the beginning of each month are examples.
  

Future Value of an Ordinary Annuity

FV measures how much a series of regular payments will be worth at some point in the future, given a specified interest rate. If you plan to invest a certain amount each month or year, FV will tell you how much you will accumulate. If you are making regular payments on a loan, the FV heꦜlps determine th🐎e total cost of the loan.

Conside൲r a series of 𒁃five $1,000 payments made at regular intervals.

Because of the 澳洲幸运5开奖号码历史查询:time value of money—the concept that any given sum is worth more🧸 now than it will be in the future because it can be invested in the meantime—the first $1,000 payment is worth more than the second, and so on.

S🍷uppose you invest $1,000 annually for five years at 5% interest. Below is how much you would have at the end of the f꧙ive years.

Or use the Future Value formula:

$\begin{aligned} &\text{FV}_{\text{Ordinary~Annuity}} = \text{C} \times \left [\frac { (1 + i) ^ n - 1 }{ i } \right] \\ &\textbf{where:} \\ &\text{C} = \text{cash flow per period} \\ &i = \text{interest rate} \\ &n = \text{number of payments} \\ \end{aligned}$​FVOrdinary Annuity​=C×[i(1+i)n−1​]where:C=cash flow per periodi=interest raten=number of payments​

Using the exampl༺e above, here's how it would work:

$\begin{aligned} \text{FV}_{\text{Ordinary~Annuity}} &= \$1,000 \times \left [\frac { (1 + 0.05) ^ 5 -1 }{ 0.05 } \right ] \\ &= \$1,000 \times 5.53 \\ &= \$5,525.63 \\ \end{aligned}$FVOrdinary Annuity​​=$1,000×[0.05(1+0.05)5−1​]=$1,000×5.53=$5,525.63​

The one-cent difference in these꧂ results, $5,525.64 vs. $5,525.63, is due 🏅to rounding in the first calculation.

Present Value of an Ordinary Annuity

In contrast to the FV calculation, the PV calculation tells y🐻ou how much money is required now to produce a series of payments in the future, again assumin𒉰g a set interest rate.

Using the sam𒐪e example of five $1,000 payments made over five years, here is how a PV calculation would look. It shows that $4,329.48, invested at 5% interest, would be sufficient to produce those five $1,000 payments.

This is the applicable formula:

$\begin{aligned} &\text{PV}_{\text{Ordinary~Annuity}} = \text{C} \times \left [ \frac { 1 - (1 + i) ^ { -n }}{ i } \right ] \\ \end{aligned}$​PVOrdinary Annuity​=C×[i1−(1+i)−n​]​

If we plug the same numbers as above into the e♑quation, here ༒is the result:

$\begin{aligned} \text{PV}_{\text{Ordinary~Annuity}} &= \$1,000 \times \left [ \frac {1 - (1 + 0.05) ^ { -5 } }{ 0.05 } \right ] \\ &=\$1,000 \times 4.33 \\ &=\$4,329.48 \\ \end{aligned}$PVOrdinary Annuity​​=$1,000×[0.051−(1+0.05)−5​]=$1,000×4.33=$4,329.48​

Future Value of an Annuity Due

​The annuity due's payments are made at the beginning, rather than the end, of each period.

To account for payments occurring at the beginning of each 🔴period, the ordinary annuity FV formula above requires a slight modification. It then results in the higher values shown below.

The reason the values ꦍare higher is that payments made at the beginning of the period have more time to earn interest. For example, if the $1,000 was invested on January 1 rather than January 31, it would have an additional month to grow.

The formula for the FV of an annuity due is:

$\begin{aligned} \text{FV}_{\text{Annuity Due}} &= \text{C} \times \left [ \frac{ (1 + i) ^ n - 1}{ i } \right ] \times (1 + i) \\ \end{aligned}$FVAnnuity Due​​=C×[i(1+i)n−1​]×(1+i)​

Here, we use the same numbers as in our previous🐽 examples:

$\begin{aligned} \text{FV}_{\text{Annuity Due}} &= \$1,000 \times \left [ \frac{ (1 + 0.05)^5 - 1}{ 0.05 } \right ] \times (1 + 0.05) \\ &= \$1,000 \times 5.53 \times 1.05 \\ &= \$5,801.91 \\ \end{aligned}$FVAnnuity Due​​=$1,000×[0.05(1+0.05)5−1​]×(1+0.05)=$1,000×5.53×1.05=$5,801.91​

The one-cent difference in these results, $5,801ꦏ.92 vs. $💫5,801.91, is due to rounding in the first calculation.

Present Value of an Annuity Due

Similarly, the formula for calculating the PV of an annuity due considers that payments are made at the beginning rather than the end of e🧔ach period.

For example, you could use this formula to calculate the PV of your future rent payments as specified in your lease. Let's say you pay $1,000 a month in rent. Below, we can see what the next five months cost at present value, assuming you kept your money in an account earning 5% interest.

Thisꦕ is the formula for calculating the🅠 PV of an annuity due:

$\begin{aligned} \text{PV}_{\text{Annuity Due}} = \text{C} \times \left [ \frac{1 - (1 + i) ^ { -n } }{ i } \right ] \times (1 + i) \\ \end{aligned}$PVAnnuity Due​=C×[i1−(1+i)−n​]×(1+i)​

So, in this example:

$\begin{aligned} \text{PV}_{\text{Annuity Due}} &= \$1,000 \times \left [ \tfrac{ (1 - (1 + 0.05) ^{ -5 } }{ 0.05 } \right] \times (1 + 0.05) \\ &= \$1,000 \times 4.33 \times1.05 \\ &= \$4,545.95 \\ \end{aligned}$PVAnnuity Due​​=$1,000×[0.05(1−(1+0.05)−5​]×(1+0.05)=$1,000×4.33×1.05=$4,545.95​

The Bottom Line

Present value and future value formulas help individuals determine what an ordinary annuity🎶 or an annuity due is worth now or later. Sﷺuch calculations and their results help with financial planning and investment decision-making.