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官方开奖结果体彩网:present value (PV) or 澳洲幸运5官方开奖结果体彩网:future value (FV) of an annuity.

Types of Annuities

Annuities as ongoing payments can be defined as ordinary an𒁃nuities or annuities due.

• Ordinary Annuities: An 澳洲幸运5官方开奖结果体彩网: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官方开奖结果体彩网: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 helps determine the total cost oꦯf the loan.

Consider a s🌱eries of five $1,000 payments made at regular intervals.

Because of the 澳洲幸运5官方开奖结果体彩网: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.

Suppose you invest $1,000 annually for five years at 5% interest. Below i🥀s how much you would have at the end of the five 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​

U🍷sing the example 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. $🔯5,525.6ཧ3, is due to rounding in the first calculation.

Present Value of an Ordinary Annuity

In contrast to the FV calculation, the PV calculation tells you how much money is required now💦 to produce a series of payments in the future, again assuming a set interest rate.

Using the same 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 int♋o the equation, 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 paܫyments 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 are 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 th🌠an 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-cenඣt difference in these results, $5,801.92 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 a🍸n annuity due considers that payments are made at the beginning rather than the end of each 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.

This is the formula f💙or 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. Such calculations and their results help with financial planning and investment decision-making.