The present value of a deferred ordinary annuity (Pdeferred) is determined on a date two or more periods before the first cash flow in the series.Suppose, for example, that Helen Swain buys an annuity on January 1, 2007 that yields her four annual receipts of
$1,000 each, with the first receipt on January 1, 2011. The interest rate is 6% com- pounded annually. What is the cost of the annuity—that is, what is the present value on January 1, 2007 of the four cash flows of $1,000 each to be received on January 1, 2011, 2012, 2013, and 2014—discounted at 6%? Example M-17 shows the facts of this problem diagrammatically.
$10,000 7.246888
P0 Factor for Pd
n,i
Present Value of a Deferred Ordinary Annuity M21
EXAMPLE M-16 Diagram of the Present Value of an Annuity Due—
Amount of Each Cash Flow to Be Determined
Jan. 1, 2007
Jan. 1, 2016 Interest Rate Is 8% Compounded Annually
$10,000 is the present value of the 10 payments of an unknown amount on this date of the first cash payment
$
$
Jan. 1, 2008
$
Jan. 1, 2009
$
Jan. 1, 2010
$
Jan. 1, 2011
$
Jan. 1, 2012
$
Jan. 1, 2013
$
Jan. 1, 2014
$
Jan. 1, 2015
$ 10 cash flows of an
unknown amount
8 Compute and use the present value of a deferred ordinary annuity.
There are two ways to compute the present value of a deferred annuity. The first method involves a combination of the present value of an ordinary annuity (P0) and the present value of a single sum due in the future (p). For the stated problem it is necessary to determine first the present value of an ordinaryannuity of four cash flows of $1,000 each to find a single present value figure discounted to January 1, 2010. Note that because the present value of an ordinary annuity table is used, the present value of the four cash flows is computed on January 1, 2010, notJanuary 1, 2011. That single sum is discounted for three more periods at 6% to arrive at the present value on January 1, 2007. Using the factors of $1 each, the present value is stated as follows:
PdeferredC[(P0
n,i)(pk,i)]
whereP0
n,ipresent value of the ordinary annuity of the ncash flows of 1 at the given interest rate i
pk,ipresent value of the single sum of 1 for kperiods of deferment
Substituting appropriate factors from Tables 4 and 3, respectively, in this formula, the fol- lowing solution is obtained:
PdeferredC[(P0
n4, i6%)(pk3,i6%)]
$1,000[(3.465106)(0.839619)]
$2,909.37
An alternative approach involves a combination of two ordinary annuities. For example, it is possible to calculate the present value of an ordinary annuity of n kcash flows of 1. From this amount is subtracted the present value of the k(the period of defer- ment, which is 3 in this example) cash flows of 1. This procedure removes the cash flows that were not available to be received; yet the discount factor for the three periods of deferments on the four cash flows that are to be received remains in the calculated factor.
This difference is multiplied by the value of each cash flow to determine the final present value of the deferred annuity. Example M-18 illustrates this approach.
In effect, the present value of an ordinary annuity of n kcash flows, minus the pres- ent value of an ordinary annuity of the kcash flows, becomes a converted factor for the present value of a deferred annuity, as follows:
PdeferredC(Converted Factor for Present Value of Deferred Annuity of 1)
Using the factors from Table 4, the converted factor for the deferred ordinary annuity stated in the preceding problem is determined as follows:
P0
nk7,i6%(5.582381) P0
k3,i6%(2.673012) 2.909369
EXAMPLE M-17 Diagram of the Present Value of a Deferred Ordinary Annuity
Jan. 1, 2007
Jan. 1, 2014
Interest Rate Is 6% Compounded Annually The present value
of the deferred annuity is determined on this date, which
is 2 or more periods before the first cash receipt
$1,000 4 cash flows of $1,000
deferred 3 periods
Jan. 1, 2013
$1,000 Jan. 1,
2012
$1,000 Jan. 1,
2011
$1,000
Jan. 1, 2008
Jan. 1, 2009
Jan. 1, 2010
The present value of the four cash flows of $1,000 each, deferred three periods, is
$2,909.37, calculated as follows:
Pdeferred$1,000(2.909369) $2,909.37
Note that the two methods produce the same present value figure. Also, note that the period of deferment is onlythree periods and not four because the present value of an ordinary annuity table is used (see Example M-18 in the second approach). This assump- tionisrequired if the problem is to be solved by the use of ordinaryannuity factors rather than annuity due factors.
Another Application
Besides determining the present value of a deferred annuity, other types of problems can be solved by using the previous approaches. For example, suppose that David Jones wants to invest $50,000 on January 1, 2007 so that he may withdraw 10 annual cash flows of equal amounts beginning January 1, 2013. If the fund earns 12% annual interest over its life, what will be the amount of each of the 10 withdrawals?
Example M-19 shows the facts of this problem. A simpler method that can be used to solve this problem is a variation of the second suggested solution. Here, the value of C can be determined from the following expression of the present value of a deferred annu- ity formula:
C
Using Table 4, the converted factor for 10 cash flows of 1 each, deferred 5 periods at 12%, is as follows:
Converted Factor P0
nk15,i12%(6.810864) P0
k5,i12%(3.604776) 3.206088
Then the amount of each cash flow is
C $15,595.33
The accuracy of the answer produced by the second approach can be tested using the amount of each cash flow and the solution from the first approach. The present value of 10 cash flows of $15,595.33 deferred 5 periods and discounted at 12% must be $50,000 if the first solution is correct. The proof can be calculated as follows:
Pdeferred$15,595.33[(5.650223)(0.567427)]
$50,000
A slight rounding-error difference may occur with this method because the solution requires the multiplication of two factors, P0
n,iandpk,i, which are rounded.
$50,000 3.206088
Pdeferred
Converted Factor for Present Value of Deferred Annuity of 1
Present Value of a Deferred Ordinary Annuity M23
EXAMPLE M-18 Diagram of Converted Table Factor of Present Value of a Deferred Ordinary Annuity
Interest Rate Is 6% Compounded Annually
$1 EqualsPdeferred
$1
$1
$1 Less
$1
$1
$1
$1 Start with
$1
$1
$1
$1
$1
$1