The Foundation
4.5 Present Value of an Annuity
86 THEFOUNDATION
Steps 3 through 6 look like the following on the BA II PLUS® calculator:
Again, the answer is $13,431.22.
Concept Check 4.4
1 What steps are involved in solving for the future value of an annuity?
2 When using the future value of an annuity formula (Equation 4.3), at what point in time is the future value computed?
We use the present value of an ordinary annuity formula, Equation 4.4:
PV PMT r
r
n
( )
= −
+
⎡
⎣
⎢⎢
⎢
⎤
⎦
⎥⎥
⎥
1 1
1
Substituting PMT = $2,500, r = 0.06, and n = 5, we get:
PV $ ,
( . )
. $ , .
= ⎡ −
⎣
⎢⎢
⎢
⎤
⎦
⎥⎥
⎥ = 2 500
1 1
1 06
0 06 5 10 530 91
If we had not used the present value of an annuity formula, we would have needed to solve for five individual present value amounts and added them together. Imagine the work that would be required to compute the present value of the 360 payments in a 30-year loan with monthly payments.
We can also solve for the present value of an ordinary annuity using the BA II PLUS® calculator. For the above example, we would enter the following keystrokes:
1 2nd CLR TVM 2 5 N
3 6 I/YR
4 2,500 +/− PMT 5 CPT PV
These steps look like the following on your BA II PLUS® calculator:
Again, the present value of this ordinary annuity is $10,530.91.
Remember two important points when solving for the present value of an ordinary annuity.
• The value for n is the number of payments or receipts in the annuity. This definition for n differs from its definition when solving future value and present value calculations using Equations 4.1 and 4.2. When compounding or discounting with Equations 4.1 or 4.2, n refers to the compounding or discounting interval of time. This point will be clear by reviewing Examples 4.1 and 4.3. Contrast this with Example 4.7 where we used the number of payments in the annuity as our value for n.
5 N
6 CPT
I/Y PV
2,500 +/−
PMT FV
(last step)
88 THEFOUNDATION
• When using the present value of an annuity formula, the present value is obtained one period before the annuity begins. This convention follows from the derivation of the present value of an ordinary annuity formula.5 End-of-year annuity payments and the present value at time 0 are often referred to as an ordinary annuity.
Thus, with an ordinary annuity, we have end-of-year payments with a time diagram as shown below, and the present value is obtained one period before where the annuity starts.
Years Amount
What happens if we need to compute a present value when the payments do not conform to those in the figure above? An adjustment must be made. We illustrate two such adjust- ments in Examples 4.8 and 4.9.
Example 4.8 Present Value of an Annuity Due
Using the data in Example 4.7, let’s compute the present value of the five loan payments assuming beginning of year payments, with the first payment occurring today.
Years Amount
Solution: We can begin by computing the present value of the five payments in the annuity:
PV $ ,
( . )
. $ , .
= ⎡ −
⎣
⎢⎢
⎢
⎤
⎦
⎥⎥
⎥ = 2 500
1 1
1 06
0 06 5 10 530 91
0 1 2 3 4 5
PMT PMT PMT PMT PMT
5 This derivation is beyond the scope of our discussion, but can be found in many corporate finance texts.
0 1 2 3 4
PMT PMT PMT PMT PMT
Of course, we get the same result as in Example 4.7. But since the first payment in the annuity occurs today, the present value obtained at time period t = −1. To calculate the present value at time 0, we need to take the future value of this present amount for 1 year:
FV0= PV−1(1 + r)1
FV = $10,530.91(1.06)1= $11,162.76
If we combine these two steps, the present value of an annuity with beginning of year payments that begin today can be computed:
PV PMT r
r n r
( ) ( )
= −
+
⎡
⎣
⎢⎢
⎢
⎤
⎦
⎥⎥
⎥ +
1 1
1 1
This annuity scheme with beginning of year payments is called an annuity due. In Exam- ple 4.9, we illustrate a similar adjustment needed when computing the present value of an annuity when the payments begin in the second period or later.
We can also solve for the present value of an annuity due using the BA II PLUS® calcu- lator. In this situation, each payment occurs at the beginning of the year. Therefore, we must set the calculator to the BGN mode ([2nd] [BGN] [2nd] [SET] [2nd] [QUIT]) before entering the relevant data and computing PV. For the above example, we would enter the following keystrokes:
1 2nd CLR TVM
2 2nd BGN 2nd SET 2nd QUIT (Note: BGN should appear in the upper right corner of the display window)
3 5 N 4 6 I/YR
5 2,500 +/− PMT 6 CPT PV
Steps 3 through 6 look like the following on your BA II PLUS® calculator:
The present value of this annuity due is $11,162.76.
5 N
6 CPT
I/Y PV
2,500 +/−
PMT FV
(last step)
90 THEFOUNDATION
Example 4.9 Present Value of an Annuity Due – Extended
Compute the present value of five payments of $2,500 with the first payment occurring 5 years from today. Assume an annual interest rate of 6 percent.
Year
Cash flows
Solution: When we compute the present value of the five payments, we get a “present value” in year 4 (one period before where the annuity starts). This present value amount is still a future value with respect to time 0. Thus, we must discount this amount for 4 years by taking the present value of this “future amount” for 4 years. If we combine these two steps, we get:
PV PMT r
r r
n
n
( )
( )
= −
+
⎡
⎣
⎢⎢
⎢
⎤
⎦
⎥⎥
⎥ +
⎡
⎣⎢ ⎤
⎦⎥
1 1
1 1
1
PV $ ,
( . )
. ( . ) $ , .
= ⎡ −
⎣
⎢⎢
⎢
⎤
⎦
⎥⎥
⎥
⎡
⎣⎢ ⎤
⎦⎥ = 2 500
1 1
1 06 0 06
1
1 06 8 341 47
5
4
1 4 2 4 3 1 2 3
Computes the “PV in year 4” of the five payments in
the annuity
Discounts the
“PV in year 4”
back to year 0
0 1 2 3 4 5 6 7 8 9
PV $2,500 $2,500 $2,500 $2,500 $2,500
Practical Financial Tip 4.3
After working annuity due problems using the BA II PLUS® calculator, reset the calculator to the end-of-period, END, mode. To switch between the BGN and END modes, press [2nd] [BGN] [2nd] [SET]. If the display indicates the desired mode, which in this case would be END, press [2nd] [QUIT]. If nothing appears in the upper right corner of the display window, the BA II PLUS® calculator is set in the END mode.
Concept Check 4.5
1 What are the steps involved in solving for the present value of an annuity?
2 What is the difference between an ordinary annuity and an annuity due?