This is “Comparing Projects with Unequal Lives”, section 13.6 from the book Finance for Managers (v. 0.1).
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Ruth is deciding which shingles to put on her roof. Shingle A costs $1 per sq. ft. and is rated to last 10 years. Shingle B costs $1.40 per sq. ft. and is rated to last 15 years. If Ruth intends to stay in her house for the rest of her life, which shingle should Ruth select?
One particularly troublesome comparison that arises often is when two repeatable mutually exclusive projects have different time lengths. For example, we can use a cheaper substitute, but it won’t last as long, so we’ll need to replace it more frequently. How do we know which project is better?
If the projects are either independent or not repeatable, we can use NPV confidently. All positive NPVs should be selected if they are independent, and the highest NPV will indicate the best choice if they aren’t repeatable. But it can be the case that the highest NPV project can be inferior to a shorter project with a lower NPV.
To analyze this problem, we need to calculate the equivalent annual annuity (EAA)The steady cash payment received by an annuity with the same length and NPV as the project., which is the steady cash payment received by an annuity with the same length and NPV as the project. For example, we know that Gator Lover’s Ice Cream Project A lasted for 5 years and had an NPV of $8,861.80 at a rate of 10%. If we solve for the yearly payment of an annuity with a PV of $8,861.80, r = 10%, n = 5 years, and FV = 0, we get an EAA of $2,337.72. Thus, we should be indifferent between receiving the cash flows of Project A and receiving $2,337.72 per year for 5 years (since they both have the same NPV)!
Once EAAs are calculated for all projects being considered, it’s a simple matter of picking the higher one.
Compute and compare the following projects’ NPVs and EAAs at a 10% discount rate.
Project J costs $100,000 and earns $50,000 each year for five years.
Project K costs $200,000 and earns $150,000 in the first year and then $75,000 for each of the next three years.
Project L costs $25,000 and earns $20,000 each year for two years.