Learning Objectives of this
article:

How alternative proposals are compared when
proposals have unequal lives?
Proposals with Unequal Useful Lives:
There are various methods to evaluate capital
investment proposals. Some use present values of the
cash flows and some ignore present value concept. If
you want to read these methods, click on a link
below:

Average rate of return method

Cash payback method

Net present value method

Internal rate of return
In these articles, we have assumed that all the
alternative proposals have equal useful lives. But
in practice, however, alternative proposals may have
unequal lives. For example, one proposal may have a
useful life of 5 years and the other 7 years.
Choosing a proposal from various alternatives
require more consideration when they have unequal
lives. This
article explains how to compare alternative
proposals when proposals have different useful lives.
The following example uses the net present value
method to compare the two alternative proposals with
different useful lives:
Example:
Assume that alternative proposals X and Y are
being compared. Each proposal requires an initial
investment of $100,000 and has the following
expected cash flow and useful life:
Year 
Proposal X 
Proposal Y 
1 
$30,000 
$30,000 
2 
$30,000 
$30,000 
3 
$25,000 
$30,000 
4 
$20,000 
$30,000 
5 
$15,000 
$30,000 
6 
$15,000 
 
7 
$10,000 
 
8 
$10,000 
 

if the desired rate of return is 10%, each
proposals net present value could be determined as
follows:
Net Present Value Analysis:
Proposal X 

Proposal Y 
Year 
Present Value
of $1 at 10% 
Net Cash Flow 
Present Value of Net Cash Flow 
1 
.909 
$30,000 
$27,270 
2 
.826 
$30,000 
$24,780 
3 
.751 
$25,000 
$18,775 
4 
.683 
$20,000 
$13,660 
5 
.621 
$15,000 
$9,315 
6 
.564 
$15,000 
$8,460 
7 
.513 
$10,000 
$5,130 
8 
.467 
$10,000 
$4,670 






$155,000 
$112,060 




Amount to be invested 
100,000 




Net
present value 
$12,060 






Year 
Present Value
of Annuity of $1 at 10% 
Net Cash Flow 
Present Value of Net Cash Flow 
15 
3.791 
$30,000 
$113,730 





Amount to be
invested 
100,000 




Net
present value 
$13,730 




Because of the unequal useful lives of the two
proposals, the net present values determined above
are not comparable. To make them comparable for the
analysis, the proposals can be adjusted to end at
the same time. This can be done by assuming that
proposal X is to be terminated at the end of five
years and the asset sold. This assumption require
that the residual value of Proposal X be estimated
at the end of five years and that this value be
included as a cash flow at that date. Both proposals
will then cover five years, and net present value
analysis can be used to compare the two proposals
over the same fiveyear period. To illustrate,
assume the Proposal X has an estimated residual
value of $40,000 at the end of year 5. For Proposal
X, the excess of the present value over the amount
to be invested is $18,640 for a 5year life, as
shown below:
Year 
Present Value
of $1 at 10% 

Net Cash Flow 
Present Value of Net Cash Flow 
1 
.909 

$30,000 
$27,270 
2 
.826 

$30,000 
$24,780 
3 
.751 

$25,000 
$18,775 
4 
.683 

$20,000 
$13,660 
5 
.621 

$15,000 
$9,315 
5 
.621 
R.V 
$40,000 
24,840 








$160,000 
$118,640 





Amount to be invested 
100,000 





Net
present value 
$18,640 






If a five year live is used for
both proposals, the net present value for
Proposal X exceeds the net present value for
Proposal Y by $4,910 ($18,640  $13,730).
Therefore, Proposal X may be viewed as the more
attractive of the two proposals.
