Introduction to Finance

Interest Rate (%)
Time (years)
5
36.7
10
25.2
15
19.8
20
16.7
With this approach, the time required, particularly at higher 
interest rates, is more feasible. Compounding at 5 percent would 
still require making annual investments for nearly 37 years to 
accumulate $1 million. At 10 percent, it would take a little more 
than 25 years to attain that goal. Of course, the most critical factor, 
which might be easier said than done, is the ability to come up 
with $10,000 per year out of disposable personal income. Good 
luck!
Personal Financial Planning

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9.6 Present Value of an Annuity
235
Further examination of Table 9.3 shows how a $1 annuity grows, or increases, with various 
combinations of interest rates and time periods. For example, if $1,000 is invested at the end 
of each year (beginning with year one) for ten years at an 8 percent interest rate, the future 
value of the annuity would be $14,487 ($1,000 × 14.487). If the interest rate is 10 percent for 
ten years, the future value of the annuity would be $15,937 ($1,000 × 15.937). These results 
demonstrate the benefi ts of higher interest rates on the future values of annuities.
9.6
Present Value of an Annuity
Many present value problems also involve cash fl ow annuities. Usually, these are ordinary 
annuities. Let’s assume that we will receive $1,000 per year beginning one year from now for a 
period of three years at an 8 percent compound interest rate. How much would you be willing 
to pay now for this stream of future cash fl ows? Since we are concerned with the value now, 
this becomes a present value problem.
We can illustrate this problem using a time line as follows: 
Begin:
Year
0
1
2
3
End:
$794
857
926
1
2
3
1 ÷ 1.08 × $1,000 
1 ÷ 1.08 × 1 ÷ 1.08 × $1,000 
1 ÷ 1.08 × 1 ÷ 1.08 × 1 ÷ 1.08 × $1,000 
Payments
$2,577 = PV annuity
Notice that to calculate the present value of this ordinary annuity we must sum the present 
values of the fi rst payment ($794), the second payment ($857), and the third payment ($926). 
This results in a present value of $2,577.
We also can fi nd the present value of this annuity by making the following computations:
PV ordinary annuity = {$1,000[1 ÷ (1.08)
1
]} + {$1,000[1 ÷ (1.08)
2
]}
+ {$1,000[1 ÷ (1.08)
3
]}
= [$1,000(0.926)] + [$1,000(0.857)]
+ 
[$1,000(0.794)]
= 
$1,000(2.577)
= 
$2,577
While the computational process was relatively easy for the preceding three-year ordinary 
annuity example, the required calculations become much more cumbersome as the time period 
is lengthened. As a result, the following equation was derived for fi nding the present value of 
an ordinary annuity (PVA):
PVA
n
= PMT{[1 – (1 ÷ (1 + 
r
)
n
)] ÷ 
r
} (9.7)
The various inputs are the same as previously defi ned. Inserting the data from the preceding 
three-year annuity example results in the following answer:
PVA
3
= $1,000{[1 – (1 ÷ (1 + 0.08)
3
)] ÷ 0.08}
= $1,000[(1 – 0.7938) ÷ 0.08]
= $1,000(0.2062 ÷ 0.08)
= 
$1,000(2.577)
= 
$2,577

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236
C H A PT E R 9 Time Value of Money
Most fi nancial calculators are programmed to readily fi nd present values of annuities. The 
result for the three-year present value of an ordinary annuity problem can be verifi ed with a 
fi nancial calculator. First, clear the calculator. Next, enter 1000 for a TI calculator (or –1000 
for an HP calculator) and press the payments (PMT) key. Then, enter 8 and press the %i key, 
and enter 3 and press the N key. Finally, press the CPT key followed by the PV key to calculate 
the PVA of 2577.10, which rounds to $2,577.