1. DADSS: Lecture 4: Internal Rate of Return and other measures

2. Administrative Details Homework #2 due Today, #3 due next week. Questions?

3. Agenda IRR – Internal Rate of Return Comparing Projects EUAC – Effective Uniform Annual Cash Flows

4. 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

5. 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%?

6. Calculating IRR What would IRR be – we want NPV to be 0, so: IRR= 20% I.e., the project will make money (or break even) until the discount rate exceeds 20%.

7. 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 2 periods (highest power = 1)Easy 3 periods (highest power = 2)Quadratic formula 4 periods (highest power = 3)Cubic roots… ¦¦ 6 periods (highest power = 5)Fields medal 7+ periodsNeed computer

8. 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

9. Calculating IRR IRR = 28%IRR = 20%, 50%, 100% Cash Flow Stream #1Cash Flow Stream #2

10. Two Projects Simple question: which project is better? Project A: IRR = 15% Project B: IRR = 29%

11. 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

12. 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

13. 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!

14. 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!

15. 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

16. 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”

17. Problems with Payback It’s not hard to come up with examples that illustrate just how stupid payback is!

18. 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

19. “Fixing” NPV in Excel Use XNPV to be explicit about the timing of cash flows Not equally spaced time intervals

20. 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

21. LCM Repetition Lowest 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.

22. LCM Repetition Accordingly, Project B is preferred Problems: Why assume you will be able to repeat the projects?

23. EUAC Effective 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

24. 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

25. EUAC EUAC = PMT(DiscRate,YearsCashFlow,-NPV) YearAlt 1Alt 2 0 $(100.00) 1 $50.00 2 3 Rate 10% $24.34 NPV 23.38%IRR EUAC calculated using PMT function

26. 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

27. Effective Uniform Annual Cash Flows What is EUAC really doing? All it’s doing is smoothing out the cash flow stream Hence, Uniform

28. 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

29. 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?