1. Copyright © 2008 Lumina Decision Systems, Inc. Net Present Value (NPV), Internal Rate of Return (IRR) and Modified IRR (MIRR) Lonnie Chrisman, Ph.D. Lumina Decision Systems Analytica User Group Part 1 : 20 Nov 2008 Part 2: 4 Dec 2008

2. Copyright © 2008 Lumina Decision Systems, Inc. Uses NPV and IRR Capital budget planning, e.g. Choosing between investments Deciding whether to fund new projects NPV and IRR are metrics used to compare the valuation of alternative cash flows over time.

3. Copyright © 2008 Lumina Decision Systems, Inc. Present Value Question: Which is worth more? $10,000 to be received with certainty one year from today, or: $10,000 received right now? $9,500 received now? $9,000 received now? $8,000 received now? Estimated present value: __________

4. Copyright © 2008 Lumina Decision Systems, Inc. Factors influencing time-value of money Inflation. Risk-free rate of return. Opportunities to put money to work. Alternative investments. Your remaining lifespan. Financing rate. Uncertainty (?)

5. Copyright © 2008 Lumina Decision Systems, Inc. Discount Rate Quantifies your time-value of money, as a % per time-period (often %/year). Exercise 1: Using your present value of $10,000 received in one year, compute your implied discount rate. If you felt $10,000 in one year is worth $9,300 now, your discount rate would be: r = 10000/9300 – 1 = 7.5%

6. Copyright © 2008 Lumina Decision Systems, Inc. Present Value Exercises Exercise 2: Using a discount rate of 10%/yr, what is the present value of $10,000 received in: 2 years? 10 years? 6 months? 3 months?

7. Copyright © 2008 Lumina Decision Systems, Inc. Cash Flow & NPV Exercise 3: A $10,000 investment (a possible project) will return $2,000 annually for 10 years. Represent this cash flow in Analytica. (There are 11 cash flow “events”) Using a 12% discount rate: Compute the present value (in Analytica) of each cache flow event. Compute the net present value for this investment. Should you make the investment?

8. Copyright © 2008 Lumina Decision Systems, Inc. Interpreting NPV Assuming perfect cash-flow knowledge: NPV > 0  Investment adds value. Proceed with project. NPV < 0  Reject project Projects (investments) are often mututally exclusive. Pursue option with max(NPV) as long as its NPV>0. NPV = maximum you are willing to pay today to guarantee the future cash flow. But… NPV is only one consideration. Organizations also have non-monetary objectives to consider.

9. Copyright © 2008 Lumina Decision Systems, Inc. NPV function’s offset The NPV (in both Analytica and Excel) assumes the first point is 1 time period in the future. Present value is: NPV(r,cashFlow,T) * (1+r) or cashFlow 0 + NPV(r,cashFlow,futureT)

10. Copyright © 2008 Lumina Decision Systems, Inc. NPV Exercises Exercise 4: A real estate investment of $200K will result in monthly rental earnings of $1K for 8 years, and will then be sold (with certainty) for $250K. Compute the NPV for this investment. Assume annual discount rate = 8% Graph the NPV-curve for all discount rates from 0% to 20%. Compare NPV-curve if you increase rental earnings 2% each year.

11. Copyright © 2008 Lumina Decision Systems, Inc. More NPV exercises Exercise 5: A $1.5B nuclear power plant with a 30 year lifetime will generate $200M per year for 30 years (realized at the end of each year), and then cost $5B to decommission on the 31 st year. Compute the NPV using an 8% discount rate. Graph the NPV for discount rates ranging from 0% to 20%.

12. Copyright © 2008 Lumina Decision Systems, Inc. XNPV – unequal time periods Exercise 6: A $100K investment on 4 Dec 2008 pays a $2K dividend on 1 Jan 2009, another on 1 June 2009, and then returns your $100K on 1 Aug 2009. Using discount rate = 10%, use XNPV to compute the net present value.

13. Copyright © 2008 Lumina Decision Systems, Inc. Investment Decisions using NPV Exercise 7: A $10,000 treasury note with 3.5% coupon rate, matures on 15 Dec 2009. If you require a 2% return, how much are you willing to pay on 4 Dec 2008? Actual quote (4 Dec 2008): $10,459.06 Should you buy it?

14. Copyright © 2008 Lumina Decision Systems, Inc. Uncertain Cash Flows (ENPV) A proposed product: Requires Poisson(9) months to develop, at LogNormal(µ:10K,σ:3K) per month. Will launch if successful, P(success)=60% After launch, monthly earnings for Poisson(24) months: Earnings m ~Normal(10K,8K) Exercise 8: (use monthly discount_rate=1%) Build a model of this cash flow. Graph: Bands(cash_flow) Compute NPV. Compute: E[NPV], Graph: PDF(NPV)

15. Copyright © 2008 Lumina Decision Systems, Inc. Interpreting NPV with Uncertainty What is the criteria for pursuing a project/investment? ENPV>0? getFract(NPV,10%)>0? ENPV>0 and getFract(NPV,1%)>-100K? Expected return / risk tolerance tradeoff…

16. Copyright © 2008 Lumina Decision Systems, Inc. Net Present Utility Motivations: Incorporating risk-tolerance or risk-adversity. Non-monetary considerations, e.g.: Community good will Economic development Strategic fit Staff skill development U(earnings,cgw,ed,sf,skd) – utility is a non-linear function of earnings and other factors. NPV(r,U,T) – can be used to compute and base decisions on NPU. When U captures risk-tolerance, ENPV>0 serves as a go/no-go decision criteria.

17. Copyright © 2008 Lumina Decision Systems, Inc. Obtaining Corporate Discount Rate from Stock Price From Capital Asset Pricing Model (CAPM) Parameters: d = corporate discount rate (rate shareholders expect from investments). β = stock price “beta”=Cov(stock,market)/Var(market) r f = Risk-free rate of interest E[R m ] = Expected return from the stock market.

18. Copyright © 2008 Lumina Decision Systems, Inc. CAPM Exercise Exercise 9: Find the shareholder-implied discount rate for AAPL, using β=1.93, E[R m ]=8% (ave. return from NASDAQ 1980- 2008), r f =1.5%.

19. Copyright © 2008 Lumina Decision Systems, Inc. Part II Internal Rate of Return (IRR)

20. Copyright © 2008 Lumina Decision Systems, Inc. Rate of Return Exercise 10: A $10,000 investment today pays $15,000 in 3 years. What is the annual rate of return?

21. Copyright © 2008 Lumina Decision Systems, Inc. Internal Rate of Return (IRR) Exercise 11: A $1M investment returns $200K per year for 10 years. What is the rate of return? Plot the NPV of this investment as a function of discount rate (from 0-20%) Use IRR function to find rate or return. Compare to an annual return of $180K or $220K per year. Graph IRR as annual return varies from $100K to $300K.

22. Copyright © 2008 Lumina Decision Systems, Inc. IRR in perspective IRR measures the quality or efficiency of an investment, but not the magnitude. Has intuitive appeal, extremely widely used in practice. Does not require a discount rate. Many downsides – can be extremely misleading. Highly non-robust. Shunned by textbooks and academics. Prevailing wisdom: Use NPV, avoid IRR!

23. Copyright © 2008 Lumina Decision Systems, Inc. IRR has many downsides Exercise 12: Return to NPV graph for our nuclear power plant in Exercise 5 ($1.5B initial cost, $5B decommission cost at 30y, $200M earnings per year). What is the rate of return for this project?

24. Copyright © 2008 Lumina Decision Systems, Inc. IRR is not uniquely determined IRR may have multiple solutions: Up to as many solutions as there are sign changes in the cash flow. IRR may also have zero solutions: E.g., Zero sign changes. Common example: Project never reaches profitability.

25. Copyright © 2008 Lumina Decision Systems, Inc. IRR can mislead Contradictory results when costs come after revenue. Lower IRR can be better. Exercise 13: Graph the NPV-curve (0-10%) for these two cash flows, and compute IRR: Year  0123456 Proj 1-200200 -900 Proj 2-200195 -900

26. Copyright © 2008 Lumina Decision Systems, Inc. IRR distorts apparent value of intermediate returns Appears as if your positive intermediate earnings can be re-invested at IRR rate. Makes options with exceptionally high IRR look too good. Appears as if your negative intermediate expenses are financed from the start at IRR rate. Makes bad projects look too good, good projects with late cash flow look bad. In practice, these distortions are often huge, and highly relevant – compensating for them drastically changes relative merits. Note: MIRR helps here (discussed later).

27. Copyright © 2008 Lumina Decision Systems, Inc. IRR and Uncertainty Expected Rate of Return E[IRR] is non-sense! Is usually does not exist. (e.g., non-zero probability that project never turns profit).  Will usually be NaN in a Monte-Carlo simulation When it does exist, its results don’t make sense. Median may exist when Mean doesn’t, but is also nonsense. Exercise 14: A $1000 investment returns $1100 with 50% or $900 with 50% after T years. Compute the EIRR for T=[0.5, 1, 2, 4] years. Note: The expected rate of return is “obviously” 0% in all four cases, right? What is the EIRR?

28. Copyright © 2008 Lumina Decision Systems, Inc. Expected Rate of Return “Expected rate of return” more accurately describes the discount rate where the ENPV-curve crosses zero. This is not the same as E[Irr(v,T)]! It is equal to Irr(E[v],T) Technically called “IRR of the Expected Cash Flow”. Behaves more intuitively as expected. Still suffers from same drawbacks as IRR with certain cash flows, of course.

29. Copyright © 2008 Lumina Decision Systems, Inc. Exercise: Expected Rate of Return A proposed product (from Exercise 8): Requires Poisson(9) months to develop, at LogNormal(µ:10K,σ:3K) per month. Will launch if successful, P(success)=70% After launch, monthly earnings: Earningsm~Normal(10K,8K) Exercise 15: Compute Irr of expected cash flow. Compare to graph computed during NPV section. Attempt to compute E[Irr]. Does it work?

30. Copyright © 2008 Lumina Decision Systems, Inc. Modified IRR (MIRR) MIRR is a variation of NPV that expressed as a rate-of-return. Robustness of NPV Intuitive appeal of IRR Avoids IRR distortions Requires: reinvestRate (rrate): Interest rate we can expect to receive on positive flows. financeRate (frate): Interest rate we expect to pay to finance negative flows.

31. Copyright © 2008 Lumina Decision Systems, Inc. MIRR details Idea: (index T := 0..n) All positive flows are re-invested at rrate until project end (Time=T n ). PV equivalent to all gains arriving on T n. gain := Npv(rrate,posFlows,T) * (1+rrate) n+1 All negative flows financed at frate is equivalent to all expenses occurring at T 0 (index T:=0..n) cost := Npv(frate,negFlows,T)*(1+frate) MIRR = (gain/-cost) 1/n -1

32. Copyright © 2008 Lumina Decision Systems, Inc. MIRR Exercises Exercise 16: Create a User-Defined Function to compute MIRR. Exercise 17: Compute the MIRR for these cash flows from the earlier “IRR can mislead” exercise using rrate=frate=6%. Year  0123456 Proj 1-200200 -900 Proj 2-200195 -900

33. Copyright © 2008 Lumina Decision Systems, Inc. Summary It matters when a cash flow occurs. “Time value of money”. Discount_rate quantifies our own time-value of money. Inflation, risk-free investment, other opportunities, remaining lifespan, financing rate, etc. NPV captures the magnitude of a return. IRR and MIRR capture the efficiency (quality) of an investment. IRR is heavily used in practice, but often highly misleading. Its upside is that it does not require any discount rate estimates. The Monte-Carlo simulation of IRR is nonsense. IRR is poorly suited for uncertain cash flows. NPV and MIRR are robust and meaningful when used with Monte-Carlo simulation.