Two mutually exclusive projects with different lives cannot be compared on net present value, because the comparison silently assumes the shorter project's capacity vanishes when it ends. Two repairs exist and they always agree.
The least common multiple method replicates each project until both chains end at the same date, then compares the chains. Honest, transparent and tedious: five and eight years means a forty-year comparison with eight replications of one and five of the other.
How is the equivalent annual annuity calculated?
The equivalent annual annuity converts each project's net present value into the level annual amount with the same present value over its own life, EAA = NPV / AF(r, n). Whichever project has the higher annuity wins, and the answer is identical to the chain answer, because the chain net present value is just the annuity valued over the common horizon. Worked Example 2 shows the identity to nine decimal places, and shows the two criteria disagreeing on the raw comparison: machine B has the higher net present value at $101,530.85 against $91,756.21, and machine A has the higher equivalent annual annuity at $24,205.06 against $19,031.35.
Over forty years, machine A's chain is worth $236,702.49 and machine B's $186,108.54.
Both methods assume replication at the same terms, which is a real assumption and often a bad one for technology that is improving or for a lease that will reprice. Say so when you use them. For costs rather than profits the same device is called the equivalent annual cost, and the rule flips to lowest wins.
Advertisement
What is capital rationing, and how do you handle it?
Capital rationing is the case where positive-net-present-value projects must be refused because the budget is fixed. Ranking by the profitability index is the standard answer and it is exactly right only when projects are divisible, because then the budget can always be spent to the last rupee. Integer projects break it, and the size of the break is easy to measure.
Six projects, a ₹500 crore budget, and a 12% cost of capital give three answers that are worth memorising as a ladder. Ranking by profitability index and taking projects until the budget binds yields ₹150.60 crore of net present value on ₹450 crore of spending, leaving ₹50 crore unspent. The best feasible integer bundle yields ₹157.80 crore on ₹490 crore. And the divisible upper bound, allowing a fractional project, yields ₹170.40 crore. Greedy ranking gives up ₹7.20 crore, 4.5627% of what was available.
The whole of that loss is the ₹50 crore the ranking could not put to work.
The practical rule follows from the ladder. Use the profitability index to rank and to build a shortlist, then enumerate the feasible bundles near the budget and take the best, because with fewer than twenty projects the enumeration is instant and the ranking heuristic can be wrong by more than a rounding error. Report the unspent balance every time, since it is the cost of the constraint and it is often the number that gets the budget raised.