| Annuity Due vs Ordinary Annuity |
This page covers the following topics regarding the timing of annuity payments:
- Distinction between an Ordinary Annuity and an Annuity-Due
- Calculating the Value of an Annuity Due
- Visual Comparison of Cash Flows
- Example Problems
- Excel
- HP-12C
1. Distinction between an Ordinary Annuity and an Annuity-Due
Each payment of an ordinary annuity belongs to the payment period preceding its date, while the payment of an annuity-due refers to a payment period following its date.
The meaning of the above statement may not be immediately obvious until we look at it graphically...
A more simplistic way of expressing the distinction is to say that payments made under an ordinary annuity occur at the end of the period while payments made under an annuity due occur at the beginning of the period.
A third possibility is to define an annuity due in terms of an ordinary annuity: an annuity-due is an ordinary annuity that has its term beginning and ending one period earlier than an ordinary annuity. This definition is useful because this is how we will compute an annuity due; i.e., in relation to an ordinary annuity (discussed further in "Calculating the Value of an Annuity Due" below).
Most annuities are ordinary annuities. Installment loans and coupon bearing bonds are examples of ordinary annuities. Rent payments, which are typically due on the day commencing with the rental period, are an example of an annuity-due.
Note that an ordinary annuity is sometimes referred to as an immediate annuity, which is unfortunate because it implies that the payments are made immediately (i.e., at the beginning of the period, which would be the case with an annuity-due). However, ordinary annuity is the more widely used term.
2. Calculating the Value of an Annuity Due
An annuity due is calculated in reference to an ordinary annuity. In other words, to calculate either the present value (PV) or future value (FV) of an annuity-due, we simply calculate the value of the comparable ordinary annuity and multiply the result by a factor of (1 + i) as shown below...
AnnuityDue = AnnuityOrdinary x (1 + i)
This makes sense because if we go back to our earlier definitions we see that the difference between the ordinary annuity and the annuity due is one compounding period.
Note also that the above formula implies that both the PV and the FV of an annuity due will be greater than their comparable ordinary annuity values. This is illustrated graphically in the section that follows, "Visual Comparison of Cash Flows." It can also be clearly seen in the discount and accumulation schedules constructed in the "Excel" section.
The following examples illustrate the mechanics of the ordinary annuity calculation and subsequent annuity due calculation.
a. Present Value of an Annuity
Using the example problem from the Present Value of an Annuity page, we calculate the PV of an ordinary annuity of $50 per year over 3 years at 7% as...
...and the present value of an annuity due under the same terms is calculated as...
...and just as we thought, the PV of the annuity due is greater than the PV of the ordinary annuity; by 9.18 in this example.
b. Future Value of an Annuity
Using the example problem from the Future Value of an Annuity page, we calclate the FV of an ordinary annuity of $25 per year over 3 years at 9% as...
...and the future value of an annuity due under the same terms is calculated as...
...and again the FV of the annuity due is greater than the FV of the ordinary annuity; in this example by 7.38.
3. Visual Comparison of Cash Flows
The distinction between an ordinary annuity and an annuity-due can be easily grasped by visualizing the timing of the payments.
a. Present Value of an Annuity:
Ordinary Annuity. Continuing with the same example from the Present Value of an Annuity page, the following illustration shows how payments are applied in the case of an ordinary annuity:
Annuity-Due. With an annuity-due the payments are made at the beginning rather than the end of the period...
Note that the PV of the ordinary annuity is 131.22 and the PV of the annuity-due is 140.40 (calculated as 131.22 x 1.07).
The fact that the value of the annuity-due is greater makes sense because all the payments are being shifted back (closer to the start) by one period. This means the PV should be larger under the annuity due because all the payments are made earlier. In other words, they are all closer to the "present" so they are subject to less discounting. Note that there is no need to discount the first payment under the annuity due at all; since it is made at the very outset, its PV is its face value.
b. Future Value of an Annuity:
Continuing with the same example from the Future Value of an Annuity page, the following illustration shows how payments are applied in the case of an ordinary annuity...
Annuity-Due. With an annuity-due the payments are made at the beginning rather than the end of the period.
Note that the FV of the ordinary annuity is 81.95 and the FV of the annuity-due is 89.33 (calculated as 81.95 x 1.09).
The fact that the value of the annuity-due is greater makes sense because all the payments are being shifted back (closer to the start) by one period. Moving the payments back means there is an additional period available for compounding. Note the under the annuity due the first payment compounds for 3 periods while under the ordinary annuity it compounds for only 2 periods. Likewise for the second and third payments; they all have an additional compounding period under the annuity due. The additional compounding generates a larger FV.
The following solved problems illustrate the distinction between an ordinary annuity and an annuity due.
QID 7. At 5% annual interest, what is the difference in the present value of $100 paid at the end of each year for 10 years and $100 paid at the beginning of each year?
This problem calculates the difference between the present value (PV) of an ordinary annuity and an annuity due. The timing difference in the payments is illustrated in an Excel schedule.
QID 32. You plan to deposit $100 into a savings account at the end of each month for the next 5 years. a) At 3% compounded monthly, how much will you have accumulated at the end of 5 years? b) How much difference would it make if the payments were made at the beginning of the month rather than at the end?
This problem calculates the amount to which a monthly payment will grow over time (i.e., the FV) assuming payments are made 1) at the end of each month; and 2) the beginning of each month. The discussion includes an Excel accumulation schedule and graphics showing how the annuity due calculation is specified in the Excel FV function and the HP-12C calculator ([g][BEG]).
There are two ways to value an annuity in Excel: use of a financial function or construction of a discount or accumulation schedule.
a. Financial Functions
Excel provides a PV function and a FV function to compute the present or future value of an annuity.
These functions can be used to compute the value of either an ordinary annuity or an annuity due. An annuity due is calculated when the "type" parameter is set to 1. An ordinary annuity is calculated when the "type" parameter is set to 0 or if it is omitted.
These functions are briefly illustrated below and discussed in more detail in the Present Value of an Annuity page and the Future Value of an Annuity page.
a.1. PV Function
The ordinary annuity and annuity due values for our previous example are computed with the PV function below...
...where the cell formulas look like this...
a.2. FV Function
The ordinary annuity and annuity due values for our previous example are computed with the FV function below...
...where the cell formulas look like this...
b. Discount and Accumulation Schedules
The value of an ordinary annuity or annuity due can be computed in Excel without the use of special functions. To do so we simply evaluate each payment period one at a time and carry forward the accumulated or discounted value to the next period in a manner similar to a chain calculation. Each period's beginning and ending values together with the payment and interest amounts are recorded in a schedule. The ending balance for the last period is the PV or FV of the annuity.
The PV of an annuity is computed with a discount schedule and the FV of an annuity is computed with an accumulation schedule.
In addition to providing us with the PV or FV of the annuity, the discount or accumulation schedule allows us to observe the value of the annuity at the end of any period in the term.
Comparing the same schedule for both an ordinary annuity and an annuity due as presented below, makes it easy grasp the fundamental difference between the two. Looking at the "int" column in the schedules we can see that they always differ by the value of one compounding period.
b.1. Discount Schedule
In a discount schedule we simply discount each payment back to its PV.
For example, for an ordinary annuity we take the first payment made at the end of the first period and discount it back to the start date.
We then add to this amount the payment made at the end of the second period discounted back to the start date. And likewise for subsequent payments.
Note that for an annuity due payments are made at the beginning of the period and therefore are not discounted in the payment period to which they apply.
Completed discount schedules for both types of annuities look like this...
...where the cell formulas look like this...
Note that a discount schedule is not the same as an amortization schedule. With an amortization schedule we start with a non-zero PV amount which is paid down to zero by application of a portion of each payment to principal over the term. An amortization schedule is typically provided with a mortgage to show the break out of principal and interest for each payment. With a discount schedule the PV is zero and we are simply valuing the stream of payments back to their present value.
b.2. Accumulation Schedule
The FV of an annuity can be calculated by constructing an accumulation schedule in which each payment is "accumulated" or compounded in sequence over the term of the annuity.
...where the cell formulas look like this...
The following image illustrates how the value of an annuity due can be calculated using the HP-12C calculator's built-in TVOM functions.
Here we use the same values as the PV of an annuity problem above to calculate PV when the payments are made at the end of the period (ordinary annuity) and at the beginning of the period (annuity due).
The assumption of when payments are made (at the end or the beginning of the period) is made by setting the "Payment Mode."
By default the Payment Mode is set to the end of the period (ordinary annuity). It can be changed to the beginning of the period (annuity due) by pressing [g][BEG] at which point the status indicator in the display shows "BEGIN" and all payments entered into the calculation are assumed to be made at the beginning of the period.
Note that due to rounding, the difference above will actually display in the HP-12C as -9.19 rather than -9.18.
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Copyright © 2002-2008 by David R. Frick & Co., CPA
This page was last updated on 11/30/07. |
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