Equivalent Annual Annuity

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 foreseeablefuture. 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 that has life length of years, real required rate of return and net present value of , the equivalent annual annuity in real terms is

= (1)

where stands for equivalent annuity (in real terms) and is the present value annuity factor for the given rate and life length . This annuity factor is given in the second displayed expression and in footnote 5 on page 41 in (BMA). Using this expression, (1) becomes

= .(2)

Going back to the original problem, for each alternative for supplying the service we calculate its and we choose the alternative that has the largest . It is that 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 = 150,000 and = 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

= = = $23,208.

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

= = $464,165.

The same computation for the new airplane is

= = = $22,067.

Therefore, the true value added of the new airplane alternative is the present value of a perpetuity of $22,067 or

= = $441,334.

Using the used airplane to supply the expanded air freight capacity for the indefinite future adds the greatest value. It maximizes the . The present value annuity factors of 6.4632 and 12.4622 were calculated with a hand calculator. You can see these factors in (BMA) Appendix Table 3, page 970.

In Section 6.3 in (BMA) the concept of equivalent annual cost is used to rank alternatives for supplying some service where the alternatives for supplying the service differ in terms of life length and costs but do not differ in terms of revenues they generate. For such problems, can be just the present value of costs of each alternative and in this case, (2) gives the equivalent annual cost of each alternative. The decision that maximizes value added is then the alternative that has the smallest equivalent annual cost.

On the bottom of page 127 and the top of page 128, (BMA) caution about calculating equivalent annual cost using nominal cash flows and this caution is appropriate. They envision calculating a nominal annuity using one life length as in (2) above. In the case of nominal cash flows you can no longer use the for one life length like we did in (1) and (2) above. You must allow to grow at the rate of inflation. This requires you to look at the infinite sequence of replacements. Let the inflation rate be denoted by , let denote the nominal required rate of return, and recall the relationship between real and nominal rates of return

= .(3)

See page 50 and footnote 10, page 128, in (BMA).

Again, let denote the net present value of one alternative for providing the needed service that has a life length of years. At the end of the first years, if the alternative is repeated all costs and benefits will have risen by the inflation rate and the net present value of the replacement will be . At the end of another years, if the alternative is replaced, its replacement will have net present value of , etc. The net present value, in nominal terms, of the infinite sequence of replacements is

=

= (4)

=

= (5)

= .(6)

To get (4) we used the relationship in (3) to get that

= .

To get (5) we used the formula for the sum of a power series

=

for any ( see footnotes 3, 4 page 40 in (BMA)), where here = . That is a lot of work but the result (6) is what we want, the net present value in nominal terms of the entire sequence of replacements.

We can choose alternatives on the basis of this net present value. The one with the largest such net present value is the one that maximizes value added. We can convert this net present value to an equivalent annual nominal annuity. It is the nominal amount that has present value in perpetuity that is the same net present value, where the value of the nominal perpetuity is computed using the nominal required rate of return. From (6), letting denote this equivalent annual nominal annuity, we have that by definition

= = .(7)

Solving (7) for gives

= .(8)

So choosing the that has the largest is the same as choosing that has the largest , so long as all s have the same required rate of return.

It is also the case that choosing the with the largest is the same as choosing the with the largest . To see this, look at equation (2) again.Using (2) and (8) gives us that

= =

= .(9)

So long as the ratio is positive, ranking s on the basis of equivalent annual annuity will give the same ranking whether you express this in nominal or real terms. The ratio may be greater than one if the inflation rate is positive or less than one if the inflation rate is negative.

Notice that by (7) the value of the nominal perpetuity is the of the . Dividing both sides of (9) by gives the satisfying result that

= = .(10)

The value added, the net present value, of the is the same, the present value of the perpetuity of equivalent annual annuity, whether that annuity is in real or nominal terms.

Look back at the express mail service example and assume that the inflation rate is 3%. Using (3) above gives that = (1.05)(1.03) = 1.0815 so that is 8.15%. In this case we can use our previous calculations of for each alternative to calculate for each. For the used airplane this calculation is

= = 23,208 = 23,208(1.63) = $37,829.

For the new airplane, the calculation is

= = 22,067(1.63) = $35,969.

Again, the used airplane alternative has the largest equivalent annual annuity in nominal terms because this equivalent annual annuity is 1.63 times the equivalent annual annuity in real terms.

References:

(BMA) R. A. Brealey, S. C. Myers, and F. Allen, Principles of Corporate Finance, 8th ed., McGraw- Hill, Inc., 2006. In particular, see section 6.3 for a presentation of the equivalent annual cost concept with all the appropriate cautions about its use extending to the broader concept of equivalent annual annuity.

The material here on the concept of equivalent annual annuity is taken from

N. Seitz and M. Ellison, Capital Budgeting and Long-Term Financing Decisions, 3rd ed., Dryden Press, 1999, Chapters 7 and 10.

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