Which of the following terms refers to a sequence of payments at fixed intervals at compound interest?

Compound Interest: The future value (FV) of an investment of present value (PV) dollars earning interest at an annual rate of r compounded m times per year for a period of t years is:

FV = PV(1 + r/m)mtor

FV = PV(1 + i)n

where i = r/m is the interest per compounding period and n = mt is the number of compounding periods.

One may solve for the present value PV to obtain:

PV = FV/(1 + r/m)mt

Numerical Example: For 4-year investment of $20,000 earning 8.5% per year, with interest re-invested each month, the future value is

FV = PV(1 + r/m)mt   = 20,000(1 + 0.085/12)(12)(4)   = $28,065.30

Notice that the interest earned is $28,065.30 - $20,000 = $8,065.30 -- considerably more than the corresponding simple interest.

Effective Interest Rate: If money is invested at an annual rate r, compounded m times per year, the effective interest rate is:

reff = (1 + r/m)m - 1.

This is the interest rate that would give the same yield if compounded only once per year. In this context r is also called the nominal rate, and is often denoted as rnom.

Numerical Example: A CD paying 9.8% compounded monthly has a nominal rate of rnom = 0.098, and an effective rate of:

r eff =(1 + rnom /m)m   =   (1 + 0.098/12)12 - 1   =  0.1025.

Thus, we get an effective interest rate of 10.25%, since the compounding makes the CD paying 9.8% compounded monthly really pay 10.25% interest over the course of the year.

Mortgage Payments Components: Let where P = principal, r = interest rate per period, n = number of periods, k = number of payments, R = monthly payment, and D = debt balance after K payments, then

R = P � r / [1 - (1 + r)-n]

D = P � (1 + r)k - R � [(1 + r)k - 1)/r]

Accelerating Mortgage Payments Components: Suppose one decides to pay more than the monthly payment, the question is how many months will it take until the mortgage is paid off? The answer is, the rounded-up, where:

n = log[x / (x � P � r)] / log (1 + r)

Future Value (FV) of an Annuity Components: Ler where R = payment, r = rate of interest, and n = number of payments, then

FV = [ R(1 + r)n - 1 ] / r

Future Value for an Increasing Annuity: It is an increasing annuity is an investment that is earning interest, and into which regular payments of a fixed amount are made. Suppose one makes a payment of R at the end of each compounding period into an investment with a present value of PV, paying interest at an annual rate of r compounded m times per year, then the future value after t years will be

FV = PV(1 + i)n + [ R ( (1 + i)n - 1 ) ] / i where i = r/m is the interest paid each period and n = m � t is the total number of periods.

Numerical Example: You deposit $100 per month into an account that now contains $5,000 and earns 5% interest per year compounded monthly. After 10 years, the amount of money in the account is:

FV = PV(1 + i)n + [ R(1 + i)n - 1 ] / i =
5,000(1+0.05/12)120 + [100(1+0.05/12)120 - 1 ] / (0.05/12) = $23,763.28

Value of a Bond:

V is the sum of the value of the dividends and the final payment.

You may like to perform some sensitivity analysis for the "what-if" scenarios by entering different numerical value(s), to make your "good" strategic decision.

Replace the existing numerical example, with your own case-information, and then click one the Calculate.

Most of us have had the experience of making a series of fixed payments over a period of time—such as rent or car payments—or receiving a series of payments for a period of time, such as interest from a bond or certificate of deposit (CD). These recurring or ongoing payments are technically referred to as "annuities" (not to be confused with the financial product called an annuity, though the two are related).

There are several ways to measure the cost of making such payments or what they're ultimately worth. Here's what you need to know about calculating the present value (PV) or future value (FV) of an annuity.

Key Takeaways

• Recurring payments, such as the rent on an apartment or interest on a bond, are sometimes referred to as "annuities."
• In ordinary annuities, payments are made at the end of each period. With annuities due, they're made at the beginning of the period.
• The future value of an annuity is the total value of payments at a specific point in time.
• The present value is how much money would be required now to produce those future payments.

Two Types of Annuities

Annuities, in this sense of the word, break down into two basic types: ordinary annuities and annuities due.

• Ordinary annuities: An ordinary annuity makes (or requires) payments at the end of each period. For example, bonds generally pay interest at the end of every six months.
• Annuities due: With an annuity due, by contrast, payments come at the beginning of each period. Rent, which landlords typically require at the beginning of each month, is a common example.

You can calculate the present or future value for an ordinary annuity or an annuity due using the following formulas.

Calculating the Future Value of an Ordinary Annuity

Future value (FV) is a measure of how much a series of regular payments will be worth at some point in the future, given a specified interest rate. So, for example, if you plan to invest a certain amount each month or year, it will tell you how much you'll have accumulated as of a future date. If you are making regular payments on a loan, the future value is useful in determining the total cost of the loan.

Consider, for example, a series of five $1,000 payments made at regular intervals.

Because of the 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. So, let's assume that you invest $1,000 every year for the next five years, at 5% interest. Below is how much you would have at the end of the five-year period.

Rather than calculating each payment individually and then adding them all up, however, you can use the following formula, which will tell you how much money you'd have in the end:

FV Ordinary Annuity = C × [ ( 1 + i ) n − 1 i ] where: C = cash flow per period i = interest rate n = number of payments \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 example above, here's how it would work:

FV Ordinary Annuity = $ 1 , 0 0 0 × [ ( 1 + 0 . 0 5 ) 5 − 1 0 . 0 5 ] = $ 1 , 0 0 0 × 5 . 5 3 = $ 5 , 5 2 5 . 6 3 \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​

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

Calculating the Present Value of an Ordinary Annuity

In contrast to the future value calculation, a present value (PV) calculation tells you how much money would be 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 a period of five years, here is how a present value calculation would look. It shows that $4,329.58, invested at 5% interest, would be sufficient to produce those five $1,000 payments.

This is the applicable formula:

PV Ordinary Annuity = C × [ 1 − ( 1 + i ) − n i ] \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 equation, here is the result:

PV Ordinary Annuity = $ 1 , 0 0 0 × [ 1 − ( 1 + 0 . 0 5 ) − 5 0 . 0 5 ] = $ 1 , 0 0 0 × 4 . 3 3 = $ 4 , 3 2 9 . 4 8 \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​

Calculating the Future Value of an Annuity Due

​An annuity due, you may recall, differs from an ordinary annuity in that 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, it requires a slight modification to the formula used to calculate the future value of an ordinary annuity and results in higher values, as 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 than January 31 it would have an additional month to grow.

The formula for the future value of an annuity due is as follows:

FV Annuity Due = C × [ ( 1 + i ) n − 1 i ] × ( 1 + i ) \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:

FV Annuity Due = $ 1 , 0 0 0 × [ ( 1 + 0 . 0 5 ) 5 − 1 0 . 0 5 ] × ( 1 + 0 . 0 5 ) = $ 1 , 0 0 0 × 5 . 5 3 × 1 . 0 5 = $ 5 , 8 0 1 . 9 1 \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​

Again, please note that the one-cent difference in these results, $5,801.92 vs. $5,801.91, is due to rounding in the first calculation.

Calculating the Present Value of an Annuity Due

Similarly, the formula for calculating the present value of an annuity due takes into account the fact that payments are made at the beginning rather than the end of each period.

For example, you could use this formula to calculate the present value 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 would cost you, in terms of present value, assuming you kept your money in an account earning 5% interest.

This is the formula for calculating the present value of an annuity due:

PV Annuity Due = C × [ 1 − ( 1 + i ) − n i ] × ( 1 + i ) \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:

PV Annuity Due = $ 1 , 0 0 0 × [ ( 1 − ( 1 + 0 . 0 5 ) − 5 0 . 0 5 ] × ( 1 + 0 . 0 5 ) = $ 1 , 0 0 0 × 4 . 3 3 × 1 . 0 5 = $ 4 , 5 4 5 . 9 5 \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​

Present Value of an Annuity

The Bottom Line

The formulas described above make it possible—and relatively easy, if you don't mind the math—to determine the present or future value of either an ordinary annuity or an annuity due. Financial calculators (you can find them online) also have the ability to calculate these for you with the correct inputs.

What is a sequence of payments made at fixed intervals?

Which of the following refers to a series of payments made at fixed intervals *?

What kind of payment is a sequence of payments made at equal fixed intervals or period of time primarily used as an income stream for retirees?

What sequence of periodic payments or deposits made at the end of each payment interval?