The present value of an ordinary annuity is determined 1 period before the first cash flow in the series is made.For example, assume that Kyle Vasby wants to calculate the present value on January 1, 2007 of four future withdrawals (cash flows) of $1,000, with the first withdrawal being made on December 31, 2007, 1 year after the determi- nation of the present value. The applicable interest rate is 6% compounded annually.
Example M-12 shows this information graphically.
5 Compute and use the future value of an annuity due.
6 Compute and use the present value of an ordinary annuity.
2. An alternative approach is to multiply the future value of an ordinary annuity factor by 1 plus the interest rate. Thus, the future value in this example would be computed as $1,000 (4.374616 1.06) $4,637.09.
Solving by Determining the Present Value of a Series of Single Sums
The solution to this problem can be determined by using the present value of a single sum. For instance, the answer can be calculated in the following two steps: (1) determine the present value of four individual cash flows of 1 each for one, two, three, and four years, as we show in Example M-13; and (2) multiply the final results of the summation by $1,000.
Step 1 The present value of four cash flows of 1 for one, two, three, and four years discounted at 6% is determined in Example M-13.
Step 2 Now it is possible to determine the present value of the four cash flows of
$1,000 each by multiplying the $1,000 by 3.465105:
$1,000 3.465105 $3,465.11
The present value on January 1, 2007 is $3,465.11; or we can say that $3,465.11 must be invested on January 1, 2007 to provide for four withdrawals of $1,000 each starting on December 31, 2007, given an interest rate of 6%.
Present Value of an Ordinary Annuity M15
EXAMPLE M-12 Diagram of Present Value of an Ordinary Annuity
Dec. 31, 2010 Jan. 1,
2007
Interest Rate Is 6% Compounded Annually The present value of an
ordinary annuity is determined one period before the
first withdrawal
$1,000 Dec. 31,
2007
$1,000
Dec. 31, 2008
$1,000
Dec. 31, 2009
$1,000 4 withdrawals
of $1,000 each
3.465105* Dec. 31,
2010 Jan. 1,
2007
Dec. 31, 2007
Dec. 31, 2008
Dec. 31, 2009 Present value of 1
on January 1, 2007 (from Table 3) 0.943396 0.889996 0.839619 0.792094
*The value of 3.465105 is slightly smaller than the factor for P0n = 4, i = 6% of 3.465106 in Table 4 discussed later in this section; this is the result of rounding each of the four factors for Pn, i.
$1
$1
$1
$1 EXAMPLE M-13 Present Value of Four Cash Flows of 1 for One, Two, Three,
and Four Years at 6%
Shortcut Approaches
Formula Approach
Even though the preceding approach can be used, it is time-consuming for calculations involving a large number of cash flows. The formula for the present value of an ordinary annuity of any amount is:
whereP0 present value of an ordinary annuity of a series of cash flows of any amount C amount of each cash flow
n number of cash flows (not the number of time periods) i interest rate for each of the stated time periods
In the example, the present value of an ordinary annuity of four cash flows of $1,000 each at 6% compounded annually can be calculated as follows:
Based on these calculations and formula observe that:
1. The results are the same as those produced in the first approach, $3,465.11.
2. The formula is developed from the formulas for both the future value of 1(f) and the present value of 1(p):
(1i)nf p
3. Thus the formula can be restated as follows:
P0C
The formula for the present value of an ordinary annuity can be converted to that for a series of cash flows of 1 each as follows:
where P0
n,iis the present value of an ordinary annuity of n cash flows of 1 each at interest rate i. This formula can be expressed for the present value of an ordinary annuity of cash flows of any sizeas:
P0C(P0
n,i)
P i
i
n i
n 0
1 1
1
, =
- + 1p
( i )
1 (1i)n P0
4
1 000
1 1
1.06
0.06 3 465
=
-
=
$ , $ , .111
P C i
i
n 0
1 1
= 1
- +
In a two-step approach the present value of four future withdrawals (cash flows) of
$1,000 each discounted at 6% is recalculated as follows:
Step 1
Step 2 P0$1,000(3.46511) $3,465.11
This calculation is exactly the same as that of the first formula, except that the process is divided into two steps. The two-step approach is the one used when tables of the present value of an ordinary annuity of 1 are available.
Table Approach The formula for P0
n,ican be used to construct a table of the present value of any series of cash flows of 1 each for any interest rate. All that is necessary is to substitute in the formula the desired number of cash flows for the various required interest rates. Table 4 at the end of the Module shows the factors for P0
n,i. Turning to Table 4, observe the following:
1. The numbers in the first column (n) represent the number of cash flows of 1 each.
In this calculation the number of cash flows and time periods are equal.
2. The present value amounts are always smaller than the number of cash flows of 1.
For example, the present value of three cash flows of 1 at 2% is 2.883883.
Since Table 4 shows the precalculation of P0
n,ior
the generalized table approach is as follows:
P0C(Factor for P0
n,i)
Thus, to calculate the present value on January 1, 2007 of four future withdrawals (cash flows) of $1,000 discounted at 6%, with the first cash flow being withdrawn on December 31, 2007, it is necessary to look up the P0
n4,i6%value in the present value of an ordinary annuity of 1 table (Table 4); it is 3.465106. This factor is then multiplied by
$1,000 to determine the present value figure of $3,465.11:
P0$1,000(3.465106) $3,465.11
Over the 4 periods, the annuity yields interest each period as follows:
Beginning Cash Ending
Period Balance Interest Flow Balance
1 $3,465.11 $207.91 $(1,000) $2,673.02
2 2,673.02 160.38 (1,000) 1,833.40
3 1,833.40 110.00 (1,000) 943.40
4 943.40 56.60 (1,000) 0
Summary and Illustration
You can solve several kinds of problems by using the present value of an ordinary annu- ity of 1 table. We present one additional example: a problem involving the calculation of the periodic cash flows when the present value and interest rate are known.
1 1
1 - +i
i
n
P0n=4,i=6%
1 1 4
1 06
0 06 3 4
= -
. =
. . 66511
Present Value of an Ordinary Annuity M17
Example: Determining the Value of Periodic Cash Flows When the Present Value Is Known Suppose that on January 1, 2007 Rex Company borrows $100,000 to finance a plant expansion project. It plans to pay this amount back with interest at 12% in equal annual payments over a 10-year period, with the first payment due on December 31, 2007. What is the amount of each payment?
Example M-14 shows the facts of the problem. The present value and the compound interest rate are known. The amount of each of the 10 cash flows is the unknown item and is $17,698.42, calculated as follows:
C
$17,698.42
Remember that each of these payments of $17,698.42 includes (1) a payment of annual interest, and (2) a retirement of debt principal. For example, the interest for 2007 is $12,000 (12% $100,000). Thus the amount of the payment on principal is $5,698.42 ($17,698.42 $12,000). For the year 2008 the interest is $11,316.19 [12% ($100,000
$5,698.42)], and the retirement of principal is $6,382.23 ($17,698.42 $11,316.19). The lastpayment of $17,698.42 on December 31, 2016, will be sufficient to retire the remaining principal and to pay the interest for the tenth year. ♦
$100,000 5.605223
P0 Factor for P0
n10,i12%
P0 Factor for P0
n,i