# 2.10 Capital Budgeting for Mutually Exclusive Projects with Unequal Lives: The Annual Annuity Approach

There is a much simpler manner, rather than using the RCA, in which this problem may be resolved. This method goes by three different names: the “Annual Annuity Approach,” (“AAA”), the “Equivalent Annual Annuity (“EAA”),” or the “Annualized Net Present Value Approach” (“ANPV”). In this method, we first calculate the NPV for each project, given the respective lives. We’ll just call it “AAA.”

The next step requires calculating the annual annuity cash flow equivalent of the NPV. It is as if one is saying that the project is, instead, an annuity – rather than uneven cash flows as will virtually always be the case – with the same NPV. You will note that this calculation is reminiscent of a mortgage. One simply takes the NPV and divides it by the relevant present value annuity factor. Here, again, is the mortgage formula:

Principal (NPV) = (Annuity Cash Flow) (PVAF)

So, we have two projects with uneven cash flows, and let us say that we have already calculated the respective NPVs, which we shall refer to as “Principal” in the mortgage formula. Please solve for AAA, given the two competing projects’ lives and NPVs, and using 12% for the AAA calculation. Take note that, at first glance, you may prefer “B” due to its higher NPV.

 Project A Project B Life 3 Years 6 years NPV \$5,000 \$6,500 Discount Rate .12 Solution: AAA \$5,000/2.4018= \$2,082 \$6,500/4.1114= \$1,581

In order to arrive at the AAA, we, once again, employed (above) the mortgage formula:

Project A:        \$5,000 = (x) (2.4018)                             Project B:       \$6,500 = (x) (4.1114)

Normally one may assume that the longer the term of the project, the greater the NPV will be. This may often, but not necessarily always, be true. Thus, both methods will be consistent with one another in that only one project will be preferred.

Analysis: In this case, even though “B” has a higher NPV, “A” provides greater annual equivalent free cash flow. “A” is the one to go with – based on AAA. Again, in a manner similar to a mortgage, the “annual equivalent payment” times the PVAF equals the NPV! The project with the higher AAA must, by virtue of mathematical necessity, also have the higher NPV when extended out to a common life. In other words, \$5,000 + (\$5,000 ÷ 1.123) = \$8,559 > \$6,500. This solution represents the present value of the project in its first iteration (\$5,000) plus the present value of its replicated NPV in the second iteration (\$5,000 ÷ 1.123).

At this point, you may wonder “what use is the RCA approach”? The RCA may be simpler to convey to the managers / decision makers who are not well-informed regarding TVM – even though there will be more number-crunching work in having to come up with more future periods and more cash flows.