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# present value of annuity due formula

## Present Value of Annuity Due Formula

PV = Present Value

Pmt = Periodic payment

i = Discount rate

n = Number of periods

The present value of annuity due formula shows the value today of series of regular payments. The payments are made at the start of each period for n periods, and a discount rate i is applied.

The formula discounts the value of each payment back to its value at the start of period 1 (present value).

The Excel PV function can be used instead of the present value of annuity formula, and has the syntax shown below.

*The FV argument is not used when using the Excel present value of an annuity due function.

## Example Using the Present Value of Annuity Due Formula

If a payment of 6,000 is received at the start of each period for 9 periods, and the discount rate is 6%, then the value of the payments today is given by the present value of annuity due formula as follows:

The same answer can be obtained using the Excel PV function as follows:

The present value of annuity due formula is one of many annuity formulas used in time value of money calculations, discover another at the link below.

The formula for the present value of an annuity due, sometimes referred to as an immediate annuity, is used to calculate a series of periodic payments, or cash flows, that start immediately.

### How is the Present Value of an Annuity Due Derived?

The present value of an annuity due formula uses the same formula as an ordinary annuity, except that the immediate cash flow is added to the present value of the future periodic cash flows remaining. The number of future periodic cash flows remaining is equal to ** n - 1**, as

**includes the first cash flow.**

*n*To show this visually, the extended version of the present value of annuity due formula of

shows that the first cash flow is not discounted and that the discounted cash flows start at period 2. After factoring out the first immediate payment, the additional payments consist of an ordinary annuity with ** n - 1** payments remaining.

The formula shown on the top of the page can be shown as P + PV of ordinary

#### Alternative Formula for the Present Value of an Annuity Due

The present value of an annuity due formula can also be stated as

which is ** (1+r)** times the present value of an ordinary annuity. This can be shown by looking again at the extended version of the present value of an annuity due formula of

This formula shows that if the present value of an annuity due is divided by ** (1+r)**, the result would be the extended version of the present value of an ordinary annuity of

If dividing an annuity due by ** (1+r)** equals the present value of an ordinary annuity, then multiplying the present value of an ordinary annuity by

**will result in the alternative formula shown for the present value of an annuity due.**

*(1+r)*## Present Value of an Annuity Due

Present Value of an annuity due is used to determine the present value of a stream of equal payments where the payment occurs at the beginning of each period. The present value of an annuity due formula can also be used to determine the number of payments, the interest rate, and the amount of the recurring payments. Use the present value of an annuity due calculator below to solve the formula.

### Present Value of an Annuity Due Definition

Present Value of an Annuity Due is the present value of a stream of equal payments, where the payment occurs at the beginning of each period.

## Present and Future Value of Annuities

At some point in your life, you may have had to make a series of fixed payments over a period of time – such as rent or car payments – or have received a series of payments over a period of time, such as interest from bonds or CDs. These are called annuities (a more generic use of the word – not to be confused with the specific financial product called an annuity, though the two are related). If you understand the time value of money, you're ready to learn about annuities and how their present and future values are calculated.

Annuities are essentially a series of fixed payments required from you, or paid to you, at a specified frequency over the course of a fixed time period. Payment frequencies can be yearly, semi-annually (twice a year), quarterly and monthly. There are two basic types of annuities: ordinary annuities and annuities due.

- Ordinary annuity: Payments are required at the end of each period. For example, straight bonds usually make coupon payments at the end of every six months until the bond's maturity date.
- Annuity due: Payments are required at the beginning of each period. Rent is an example of annuity due. You are usually required to pay rent when you first move in at the beginning of the month, and then on the first of each month thereafter.

Since the present and future value calculations for ordinary annuities and annuities due are slightly different, we will discuss them separately.

If you know how much you can invest per period for a certain time period, the future value (FV) of an ordinary annuity formula is useful for finding out how much you would have in the future. If you are making payments on a loan, the future value is useful in determining the total cost of the loan. If you know how much you plan to invest each year and the fixed rate of return your annuity guarantees – or, for loans, the amount of your payments and the given interest rate – you can easily determine the value of your account at any point in the future.

Let's now run through Example 1. Consider the following annuity cash flow schedule:

To calculate the future value of the annuity, we have to calculate the future value of each cash flow. Let's assume that you are receiving $1,000 every year for the next five years, and you invest each payment at 5% interest. The following diagram shows how much you would have at the end of the five-year period:

Since we have to add the future value of each payment, you may have noticed that if you have an ordinary annuity with many cash flows, it would take a long time to calculate all the future values and then add them together. Fortunately, mathematics provides a formula that serves as a shortcut for finding the accumulated value of all cash flows received from an ordinary annuity:

where c = cash flow per period

i = interest rate

n = number of payments

Using the above formula for Example 1 above, this is the result:

Note that the one-cent difference between $5,525.64 and $5,525.63 is due to a rounding error in the first calculation. Each value of the first calculation must be rounded to the nearest penny – the more you have to round numbers in a calculation, the more likely rounding errors will occur. So, the above formula not only provides a shortcut to finding the FV of an ordinary annuity, but also gives a more accurate result.

The present value of an annuity is simply the current value of all the income generated by that investment in the future. This calculation is predicated on the concept of the time value of money, which states that a dollar now is worth more than a dollar earned in the future. Because of this, present value calculations use the number of time periods over which income is generated to discount the value of future payments.

If you would like to determine today's value of a future payment series, you need to use the formula that calculates the present value (PV) of an ordinary annuity. This is the formula you would use as part of a bond pricing calculation. The PV of an ordinary annuity calculates the present value of the coupon payments that you will receive in the future.

For Example 2, we'll use the same annuity cash flow schedule as we did in Example 1. To obtain the total discounted value, we need to take the present value of each future payment and, as we did in Example 1, add the cash flows together.

Again, calculating and adding all these values will take a considerable amount of time, especially if we expect many future payments. Though numerous online calculators can determine the present value of an annuity, the formula for a regular annuity is not overly complicated to calculate by hand, if we use a mathematical shortcut for PV of an ordinary annuity.

where c = cash flow per period

i = interest rate

n = number of payments

The formula provides us with the PV in a few easy steps. Here is the calculation of the annuity represented in the diagram for Example 2:

****When you are receiving or paying cash flows for an annuity due, your cash flow schedule would appear as follows:

Since each payment in the series is made one period sooner, we need to discount the formula one period back. A slight modification to the FV-of-an-ordinary-annuity formula accounts for payments occurring at the beginning of each period. In Example 3, let's illustrate why this modification is needed when each $1,000 payment is made at the beginning of the period rather than at the end (interest rate is still 5%):

Notice that when payments are made at the beginning of the period, each amount is held longer at the end of the period. For example, if the $1,000 was invested on January 1 rather than December 31 each year, the last payment before we value our investment at the end of five years (on December 31) would have been made a year prior (January 1) rather than the same day on which it is valued. The future value of annuity formula would then read:

## The Future Value and Present Value of an Annuity

Understanding annuities is crucial for understanding loans, and investments that require or yield periodic payments. For instance, how much of a mortgage can I afford if I can only pay $1,000 monthly? How much money will I have in my IRA account if I deposit $2,000 at the beginning of each year for 30 years, and earns an annual interest rate of 5%, but is compounded daily?

An **annuity** is a series of equal payments in equal time periods. Usually, the time period is 1 year, which is why it is called an annuity, but the time period can be shorter, or even longer. These equal payments are called the **periodic rent**. The **amount of the annuity** is the sum of all payments.

An **annuity due** is an annuity where the payments are made at the beginning of each time period; for an **ordinary annuity**, payments are made at the end of the time period. Most annuities are ordinary annuities.

Analogous to the future value and present value of a dollar, which is the future value and present value of a lump-sum payment, the **future value of an annuity** is the value of equally spaced payments at some point in the future. The **present value of an annuity** is the present value of equally spaced payments in the future.

## The Future Value of an Annuity

The future value of an annuity is simply the sum of the future value of each payment. The equation for the future value of an annuity due is the sum of the geometric sequence:

The equation for the future value of an ordinary annuity is the sum of the geometric sequence:

Without going through an extensive derivation, just note that the future value of an annuity is the sum of the geometric sequences shown above, and these sums can be simplified to the following formulas, where **A = the annuity payment** or periodic rent, **r = the interest rate per time period**, and **n = the number of time periods**.

The **future value of an ordinary annuity** (**FVOA** ) is:

And the **future value of an annuity due** (**FVAD** ) is:

Due (FVAD) Formula

Note that the difference between FVAD and FVOA is:

FVAD = 0 + A(1 + r) 1 + A(1 + r) 2 + . + A(1 + r) n-1 + A(1 + r) n .

FVOA = A(1 + r) 0 + A(1 + r) 1 + A(1 + r) 2 +. + A(1 + r) n-1 + 0.

In other words, the difference is merely the interest earned in the last compounding period. Because payments of an ordinary annuity are made at the end of the period, the last payment earns no interest, while the last payment of an annuity due earns interest during the last compounding period.

### Example — Calculating the Amount of an Ordinary Annuity

If at the end of each month, a saver deposited $100 into a savings account that paid 6% compounded monthly , how much would he have at the end of 10 years ?

r = 6% per year compounded monthly, which = .5% interest per month = .005

n = the number of compounding time periods = 120 in 10 years.

Substituting these values into the equation for the future value of an ordinary annuity:

100 * ((1+ .005 ) 120 -1)/ .005 = **$16,387.93**

### Example — Calculating the Amount of an Annuity Due

If the saver deposited the money at the beginning of the month instead of the end, then there will be an additional amount of money = A (1 + r ) n - A = 100 (1 .005 ) 120 - 100 = $81.94 , which is the difference in this example between an annuity due and an ordinary annuity.

### Example — Calculating the Annuity Payment, or the Periodic Rent

A 20 year old wants to retire as a millionaire by the time she turns 70. (With life spans increasing, and the social security fund being depleted by baby boomers, the retirement age will have invariably risen by the time she reaches 65 years of age, probably to something even higher than 70, actually.) How much will she have to save at the end of each month if she can earn 5% compounded annually , tax-free, to have $1,000,000 by the time she is 70?

**Solution:** Note that the equation for the future value of an annuity consists of 3 independent variables, and 1 dependent variable. In other words, if we know the value of 3 of the variables, then we can determine the remaining variable.

Since r = 5% = .05 , and n = 50 , the interest factor (1 + r ) n - 1)/ r = (1 .05 50 - 1)/ .05 = 209.35 , rounded to 2 decimal places. To find A , we divide both sides of the equation for the future value of an annuity by this interest factor, which yields 1,000,000 / 209.35 = $4,776.69 . So she would have to save $4,776.69 dollars per year, or $398.06 per month, to have $1,000,000 in 50 years—assuming, of course, that she could save it tax-free!

Of course, using the formula for the present value of a dollar, we find that in 50 years , assuming 3% inflation , $1,000,000 will be worth about 1,000,000 /1 .03 50 = $228,107.08 ! Ouch!

Since the current limit for IRA contributions is $2,000 per year for a young person, how much will this earn after 50 years, assuming that the $2,000 is deposited at the end of the year? **FVOA** = 2,000 * (1 .05 50 - 1)/ .05 = **$418,695.99**.

What's that in today's dollars, assuming 3% inflation ? 418,695.99 /1 .03 50 = $95,507.52 ! Clearly, the IRA contribution limits must be raised substantially. Of course, you can save all of the money at the beginning of each year instead of at the end, and this annuity due will yield an extra (using the Annuity Difference Formula above) 2,000 * 1 .05 50 - 2,000 = $20,934.80 which, in today's dollars, again assuming a 3% inflation rate , = $20,934.80 / 1 .03 50 = $4,775.38 more money in today's dollars over the ordinary annuity, but clearly, you'll still be eating dog food when you retire with this amount of cash, unless you are planning to die early! With the limitations on IRAs, stocks are the only viable choice for investments that could possibly yield anything decent to retire on!