Future Value of an Annuity 71
Time Line:
05%1 2 3
100 100 100
105
110.25
FVA 3 315.25
Here we show the regular time line as the top portion of the diagram, but we also show
how each cash flow is compounded to produce the value FVAnin the lower portion of
the diagram.
Equation:
(2-4)
.
The first line of Equation 2-4 represents the application of Equation 2-1 to each indi-
vidual payment of the annuity. In other words, each term is the compounded amount
of a single payment, with the superscript in each term indicating the number of peri-
ods during which the payment earns interest. For example, because the first annuity
payment was made at the end of Period 1, interest would be earned in Periods 2
through n only, so compounding would be for n 1 periods rather than n periods.
Compounding for the second payment would be for Period 3 through Period n, or
n 2 periods, and so on. The last payment is made at the end of the annuity’s life, so
there is no time for interest to be earned.
The second line of Equation 2-4 is just a shorthand version of the first form, but
the third line is different—it is found by applying the algebra of geometric progres-
sions. This form of Equation 2-4 is especially useful when no financial calculator is
available. Finally, the fourth line shows the payment multiplied by the Future Value
Interest Factor for an Annuity (FVIFAi,n),which is the shorthand version of the
formula.
- NUMERICAL SOLUTION:
The lower section of the time line shows the numerical solution, which involves using
the first line of Equation 2-4. The future value of each cash flow is found, and those
FVs are summed to find the FV of the annuity, $315.25. If a long annuity were being
evaluated, this process would be quite tedious, and in that case you probably would use
the form of Equation 2-4 found on the third line:
.
(2-4)
$100a
(10.05)^3 1
0.05
b$100(3.1525)$315.25
FVAnPMTa
(1i)n 1
i
b
PMT(FVIFAi,n)
PMTa
(1i)n 1
i
b
PMT a
n
t 1
(1i)nt
FVAnPMT(1i)n^1 PMT(1i)n^2 PMT(1i)n^3 PMT(1i)^0
↑
↑
Time Value of Money 69
