EquivalentAnnualAnnuity

2024年3月27日发(作者：大众迈腾2015款报价)

Equivalent Annual Annuity

Mutually exclusive alternatives for performing some service typically have unequal lives.

For example, suppose an express mail service provider has to add to its fleet of planes to

meet an increase in demand for air freight. For simplicity, assume it has two choices, a

new airplane that will last 20 years and a used airplane that will last 8 years. In reality,

the firm needs this expanded air freight capacity for the foreseeable future. A simple net

present value analysis of the two alternatives will miss the point that the investment

project is \"providing expanded air freight capacity for the indefinite future.\" The

equivalent annual annuity is the appropriate tool for this problem.

The equivalent annual annuity for an investment is the level annuity over the

investment\'s life that has a present value equal to the investment\'s net present value. For

an alternative

alt

that has life length of

n

years, real required rate of return

r

real

and net

present value of

NPV

?

alt

?

, the equivalent annual annuity in real terms is

EA

real

=

NPV

?

alt

?

PVAF

?

r

real

,n

?

(1)

where

EA

real

stands for equivalent annuity (in real terms) and

PVAF

?

r

real

,n

?

is the

present value annuity factor for the given rate

r

real

and life length

n

. This annuity factor

is given in the second displayed expression and in footnote 5 on page 41 in (BMA).

Using this expression, (1) becomes

EA

real

=

NPV

?

alt

??

r

real

??

1

?

r

real

?

n

?

1

?

r

real

?

n

?

1

. (2)

Going back to the original problem, for each alternative for supplying the service

we calculate its

EA

real

and we choose the alternative that has the largest

EA

real

. It is that

Cases and Readings in Corporate Finance. 1 Professor D. C. Nachman

alternative that maximizes the net present value of supplying the service indefinitely.

Suppose in the express mail service example that the appropriate real rate of return is 5%

and the net present value of the used airplane is $150,000, but because of its longer life

the net present value of the new airplane is $275,000. Here we have

NPV

?

used

?

=

150,000 and

NPV

?

new

?

= 275,000. A simple net present value analysis would indicate

that the new plane would maximize net present value.

But remember, the express mail service firm needs this expanded air freight

capacity for the foreseeable future. So we should consider using the used airplane for 8

years and replacing it with another used airplane for another 8 years, etc. Similarly, we

should consider using the new airplane for 20 years and replacing it with another new

airplane for another 20 years, etc. The true value added of each alternative is then the

present value of each infinite sequence of replacements. This value added is the same as

the present value of the perpetuity of the equivalent annual annuity given by (2).

This computation for the used airplane is

EA

real

=

150,000

?

.05

??

1.05

?

8

?

1.05

?

8

?1

=

150,000

= $23,208.

6.4632

The true value added of this alternative is the present value of a perpetuity of $23,208 or

NPV

=

23,208.05

= $464,165.

The same computation for the new airplane is

EA

real

=

275,000

?

.05

??

1.05

?

20

?

1.05

?

20

?1

=

275,000

= $22,067.

12.4622

Therefore, the true value added of the new airplane alternative is the present value of a

perpetuity of $22,067 or

Cases and Readings in Corporate Finance. 2 Professor D. C. Nachman