DADSS Engineering Economics: Internal Rate of Return (IRR) and Other Metrics
Administrative Details • Homework #2 due Today, #3 due next week. • Questions?
Agenda • IRR – Internal Rate of Return • Comparing Projects • EUAC – Effective Uniform Annual Cash Flows
Internal Rate of Return • IRR is the “break-even” discount rate • First, look at calculating NPV • For some series of cash flows (CF) and some discount rate (r), we arrive at the net present value of the cash flow stream: NPV is some number
Inverting NPV • For IRR, we want to solve for the r that will make the NPV = 0 • Consider a shorter cash flow stream: • Year 0: -$10 • Year 1: $12 • What is the NPV if the discount rate is 10%?
Calculating IRR • Now, what would the IRR be? We want NPV = 0… • The IRR is 20% • That is, this project will at least break even (have a positive or zero NPV) until the discount rate exceeds 20%
But wait… • There’s a complication • What happens with larger cash flow streams? • The IRR is the root of the discounted cash flow equation, which is a large polynomial
Numerical Methods • Excel uses an algorithm (Newton-Raphson) to quickly find the correct IRR – given a guess • Another complication: Descartes’ Rule of Signs • The number of possible roots (or zeroes) of a polynomial is equal to the number of sign changes • Cash Flow Stream #1: 1 change in the cash flow • Year 0 1 2 3 4 5 • Cash Flow -10 +2 +2 +5 +5 +10 • Cash Flow Stream #2: 3 changes, 3 possible IRRs • Year 0 1 2 3 • Cash Flow -10 +47 –72 +36 • Use Excel’s MIRR function (modified IRR) to deal with this problem in a sophisticated way
Calculating IRR Cash Flow Stream #1 Cash Flow Stream #2 IRR = 28% MIRR = 19% (with 10% reinvestment rate) IRR = 20%, 50%, 100% MIRR = 10% (with 10% reinvestment rate)
Comparing Projects • NPV • IRR • EUAC • Payback • Other…
Two Projects • Simple question: which project is better? • Project A: IRR = 15% • Project B: IRR = 29%
Two Project – with Cash Flows • Project A has a lower IRR, but would you really rather have B? • With NPV and a 10% discount rate: • Project A: $13,724 • Project B: $3 • What happened with IRR? • We’re comparing projects of differing scale • using a scale-free metric
Another Two Projects • Forget IRR – we’ll go back to NPV • Which of the following two projects is better (r = 10%)? • Project A: NPV = $120,921 • Project B: NPV = $111,364 • Clearly, Project A, which has a higher NPV, is the better project
Using NPV • Project A may have a higher NPV, but it’s tying up my capital 5x longer than Project B • With Project B, I get my capital back in Year 1 and can go invest in something else • So is Project A really better? • What are we missing? Our tools aren’t working!
Comparing Projects • Apples and Oranges • In the IRR example, we compared projects of differing scale using a scale-free metric • In the NPV example, we compared projects of differing time horizons using a time-neutral metric • That doesn’t work!
What is the “Magic Metric”? • There is no one “correct” measure • IRR has several problems • Multiple roots • Scale insensitivity • But IRR is popular in the business world – it’s easy for people to think in “returns” • NPV is almost always a better measure of project performance, but must nevertheless be used carefully
Another Popular Measure • Popular, but stupid: “Payback” • Payback = number of periods until original capital is returned • In this case, Project B has the shorter (“quicker”) payback, so it would be considered “better”
Problems with Payback • It’s not hard to come up with examples that illustrate just how stupid payback is!
Drawing a Conclusion • NPV, used properly, will never produce a misleading answer • IRR and Payback may produce misleading answers • What does “used correctly” mean? • As we’ve seen, NPV’s main problem is dealing with differing terms or non-synchronous cash flow streams
Not equally spaced time intervals “Fixing” NPV -- Excel • Use XNPV to be explicit about the timing of cash flows
Different Terms • There are three common solutions to the “differing terms” problem • Assume least common multiple repetition (the “common service period” approach) • Use a forward reinvestment rate • Calculate the EUAC
LCM RepetitionLowest Common Multiplier • These projects can’t really be compared with NPV because they have unequal cash flow terms • One common solution is to assume that the projects can be repeated up to a total period determined by the least common multiple of their terms • LCM[2,3] = 6 • Project A gets repeated 3 times • Project B gets repeated 2 times • Then, compare with NPV, IRR, etc.
LCM Repetition • Accordingly, Project B is preferred • Problems: Why assume you will be able to repeat the projects?
EUACEffective Uniform Annual Cash Flows • The easiest way to think of EUAC is as a more sophisticated version of the LCM repetition method • EUAC “annuitizes” uneven cash flow streams, turning NPV from a static number into what is effectively a rate (an amount per period) • Note: This does not completely address the “differing terms” issue because it doesn’t fully treat reinvestment opportunities! • EUAC is most useful for providing an easily-understandable means of comparing projects with highly variable cash flows
Effective Uniform Annual Cash Flows • To transform the present value number into a rate, we need a way of averaging all of the cash flows received through time that actually recognizes the time value of money • Conceptually, it’s an annuity function • In Excel, the PMT function
EUAC • EUAC = PMT(DiscRate,YearsCashFlow,-NPV) EUAC calculated using PMT function
Effective Uniform Annual Cash Flows • NPV indicates Project B is optimal • EUAC indicates Project A is optimal • Both don’t fully treat the 2-year reinvestment possibility in Project A! • This is why they produce different optimal projects
Effective Uniform Annual Cash Flows • What is EUAC really doing? • All it’s doing is smoothing out the cash flow stream (hence the “uniform”)
Effective Uniform Annual Cash Flows • If the cash flows are already uniform, then EUAC doesn’t add any new information – which tells us something: • The less uniform the cash flows, the more useful EUAC becomes
Sensitivity Analysis • The fundamental lesson is all of this is that there is no one over-arching “perfect” metric • Robustness & Convergent Validity • How sensitive is the ranking of outcomes to the inputs? • Is there a consensus across metrics? • In other words: does it matter?