This page covers the following topics regarding the
timing of annuity payments:
 Distinction between an Ordinary Annuity and an AnnuityDue
 Calculating the Value of an Annuity Due
 Visual Comparison of Cash Flows
 Example Problems
 Excel
 HP12C
1. Distinction between an Ordinary
Annuity and an AnnuityDue
Each payment of an ordinary annuity belongs to
the payment period preceding its date, while the payment of
an annuitydue 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 annuitydue 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 annuitydue.
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 annuitydue).
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 annuitydue,
we simply calculate the value of
the comparable ordinary annuity
and multiply the result by a factor of (1 + i)
as shown below...
Annuity_{Due} = Annuity_{Ordinary} 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 annuitydue 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:
AnnuityDue. With an annuitydue
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
annuitydue is 140.40
(calculated as 131.22 x 1.07).
The fact that the value of the annuitydue
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...
AnnuityDue. With an annuitydue
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
annuitydue is 89.33
(calculated as 81.95 x 1.09).
The fact that the value of the annuitydue
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.
4. Example Problems
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 HP12C calculator ([g][BEG]).
5. Excel
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
nonzero 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...
6. HP12C
The following image illustrates how the value
of an annuity due can be calculated using the
HP12C calculator's builtin 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 HP12C as 9.19 rather than 9.18.
