1. Financial Analysis, Planning and Forecasting Theory and Application By Alice C. Lee San Francisco State University John C. Lee J.P. Morgan Chase Cheng F. Lee Rutgers University Chapter 13 Capital Budgeting Under Uncertainty

2. Outline  13.1 Introduction  13.2 Risk-adjustment discount-rate method  13.3 Certainty equivalent method  13.4 The relationship of the risk-adjustment discount rate method to the certainty equivalent method  13.5 Three other related stochastic approaches to capital budgeting  13.6 Inflationary effects in the capital-budgeting procedure  13.7 Multi-period capital budgeting  13.8 Summary and concluding remarks  Appendix 13A. The use of the time-state preference and the option-pricing models in capital budgeting under uncertainty  Appendix 13B. Real Option approach for capital budgeting decision.

3. 13.2Risk-adjustment discount-rate method (13.1) where I 0 = Initial outlay of the capital budgeting project; = A location measure such as the median (or the mean) of the expected risky cash-flow distribution X t in period t; r t = Risk-adjusted discount rate appropriate to the riskiness of the uncertain cash flow X t ; N = Life of the project.

4. 13.3Certainty equivalent method (13.2) where C t = Certainty-equivalent cash flow at period t, I = Riskless interest rate, N = Life of the project.

5. 13.3Certainty equivalent method (13.3) (13.2’)

6. 13.3Certainty equivalent method (13.4) (13.4’) (13.2”)

7. 13.4The relationship of the risk-adjustment discount rate method to the certainty equivalent method (13.5) (13.6) (13.7)

8. 13.4The relationship of the risk-adjustment discount rate method to the certainty equivalent method (13.8) (13.9) (13.10)

9. 13.4The relationship of the risk-adjustment discount rate method to the certainty equivalent method

10.  Example 13.1 We assume that investors retain the same attitude toward risk over time, that is, α1 = α2 = α3 = 0.8. Then the risk- adjusted discount rate for the three periods is: Therefore the risk-adjusted rate decreases over time.

11. 13.4The relationship of the risk-adjustment discount rate method to the certainty equivalent method  Example 13.2 In the capital-budgeting process, we usually apply a constant risk- adjusted discount rate to each period’s cash flows, r1 = r2 = r3. Assuming that this constant value is 0.185, we have: and we see that the value of the certainty equivalents will decrease over time.

12. 13.5Three other related stochastic approaches to capital budgeting  The statistical distribution method  The decision tree method  Simulation analysis  Comparison of the three alternative stochastic methods

13. 13.5.1 The Statistical Distribution Method (13.11) (13.12a) (13.12b)

14. 13.5.1 The Statistical Distribution Method Cov(C t, C τ ) = ρ τt σ τ σ t, (13.12b’) (13.13)

15. 13.5.1 The Statistical Distribution Method (13.14)

16. 13.5.1 The Statistical Distribution Method Table 13.1 Expected cash flow for new product Reprinted by permission of Hillier, F., “The derivation of probabilistic information for the evaluation of risky investments,” Management Science 9 (April 1963). Copyright 1963 by The Institute of Management Sciences. YearSourceExpected Value of Net Cash Flow (in thousands) Standard Deviation (in thousands) 0Initial Investment-$600$50 1Production Cash Outflow-25020 2Production Cash Outflow-20010 3Production Cash Outflow-20010 4Production Cash Outflow-20010 5 Production Outflow – salvage Value -10015 1Marketing30050 2Marketing600100 3Marketing500100 4Marketing400100 5Marketing300100

17. 13.5.1 The Statistical Distribution Method Table 13.2a Illustration of conditional-probability distribution approach Initial Outlay Period 0 (1) Initial Probability P(1) (2) Net Cash Flow (3) Conditional Probability P(3|2,1) (4) Net Cash Flow (5) Conditional Probability P(3|2,1) (6) Cash Flow (7) Joint Probability (8) 0.210000.015 0.2520000.520000.0375 0.330000.0225 0.220000.03 10,0000.320000.530000.530000.075 0.340000.045 0.230000.015 0.2540000.540000.0375 0.350000.0225 0.320000.045 0.330000.440000.06 0.360000.045 0.340000.06

18. 13.5.1 The Statistical Distribution Method Table 13.2a Illustration of conditional-probability distribution approach (Cont’d) Initial Outlay Period 0 (1) Initial Probability P(1) (2) Net Cash Flow (3) Conditional Probability P(3|2,1) (4) Net Cash Flow (5) Conditional Probability P(3|2,1) (6) Cash Flow (7) Joint Probability (8) 0.540000.450000.450000.08 0.360000.06 0.350000.045 0.370000.470000.06 0.390000.045 0.2540000.0125 0.2550000.560000.025 0.2580000.0125 0.2550000.025 0.260000.570000.570000.05 0.2590000.025 0.2570000.0125 0.2580000.590000.025 0.25110000.0125

19. 13.5.1 The Statistical Distribution Method Table 13.2b NPV and joint probability Discount Rate = 4 percent NPV = $2,517.182 Variance = $20,359,090.9093 Standard Deviation = $4,512.105 PVANPVProbability 4,661.2-5,338.80.015 5,550.2-4,449.80.0375 6,439.2-3,560.80.0225 6,474.76-3,525.240.03 7,363.76-2,636.240.075 8,252.76-1,747.240.045 8,288.32-1,711.680.015 9,177.32-822.680.0375 10,066.3266.320.0225 8,297.84-1,602.160.045 10,175.84175.840.06 11,953.841,953.840.045 PVANPVProbability 12,024.962,024.960.06 12,913.962,913.960.08 13,802.963,802.960.06 14,763.084,763.080.045 16,541.086,541.080.06 18,319.088,319.080.045 13,948.043,948.040.0125 15,726.045,726.040.025 17,504.047,504.040.0125 16,686.166,686.160.025 18,464.168,464.160.05 20242.1610,242.160.025 19,388.729,388.720.0125 21,166.7211,166.720.025 22,944.7212,944.720.0125

20. 13.5.2 The Decision-Tree Method Fig. 13.1 Decision tree. Numbers in parentheses are probabilities; numbers without parentheses are NPV.

21. 13.5.2 The Decision-Tree Method NPV = 0.7(1,248.84) + 0.2(814.73) + 0.1(-14.54) = $1,035.68. Decision Variables For the First Stage Net Present Value* Standard Deviation* Coefficient Of Variation Regional$610.50$666.981.093 National767.271067.201.3909 International629.711533.802.4357 Expressed in thousands of dollars.

22. 13.5.2 The Decision-Tree Method Table 13.3 Expected cash flows for various branches of the decision tree 0123456NPV Regional Distribution throughout: High -2000250500600900600500893.70 Medium-20000400550800550400310.80 Low-2000-250200450600450200-614.61 National Distribution throughout: High -30003507001100140010006001,454.47 Medium-300005009001200800700498.90 Low-3000-3501007001000600300-1,036.73 International Distribution Throughout: High -40004509001600220014008002,350.71 Medium-40000600140020001200600969.44 Low-4000-4501007001500700400-1,544.26

23. 13.5.2 The Decision-Tree Method Table 13.3 Expected cash flows for various branches of the decision tree (Cont’d) 0123456NPV High National – High Inter. -3000350-3001600220014008002,145.08 High National – Medium Inter. -3000350-3001400220012006001,473.88 High National – Low Inter. -3000350-3007001500700400-144.85 Medium National – High Inter. -30000-5001600220014008001,623.63 Medium National – Medium Inter -30000-500140020001200600952.43 Medium National – Low Inter -30000-5007001500700400-666.30 Low National – High Inter. -3000-350-900160022001400800917.27 Low National – Medium Inter -3000-350-900140020001200600246.07 Low National – Low Inter -3000-350-9007001500700400-1,372.66

24. 13.5.2 The Decision-Tree Method Table 13.3 Expected cash flows for various branches of the decision tree (Cont’d) 0123456NPV High Region – High National -2000250-5001100140010006001,248.84 High Region – Medium National -2000250-5009001200800700814.73 High Region – Low National -2000250-5007001000600300-14.54 Medium Region – High National -20000-600110014001000600916.00 Medium Region – Medium National -20000-6009001200800700481.89 Medium Region – Low National -20000-6007001000600300-347.38 Low Region – High National -2000-250-800110014001000600490.71 Low Region – Medium National -2000-250-800900120080070056.59 Low Region – Low National -2000-250-8007001000600300-772.68

25. 13.5.3 Simulation Analysis Table 13.4a Variables for simulation VariablesRange 1. Market size (units) 2,500,000 – 3,000,000 2. Selling price ($/unit) 40 – 60 3. Market growth 0 – 5% 4. Market share 10 – 15% 5. Total investment required ($) 8,000,000 – 10,000,000 6. Useful life of facilities (years)5-9 7. Residue value of investment ($) 1,000,000 – 2,000,000 8. Operating cost ($) 30 – 45 9. Fixed cost ($) 400,000 – 500,000 10. Tax rate40% 11. Discount rate12% Notes: a. Random numbers from Wonnacott and Wonnacott (1977) are used to determine the value of the variable for simulation. b. Useful life of facilities is an integer. Random number Year 10-19 5 20-39 6 40-59 7 60-79 8 80-99 9 00 10

26. 13.5.3 Simulation Analysis Table 13.4b Simulation Variables1234 VMARK 1(39)2,695,000(47)2,735,000(67)2,835,000(12)2,580,000 PRICE 2(73)$54.6(93)$58.6(59)51.8(78)55.6 GROW 3(72)3.6%(21)1.05%(63)0.0315(03)0.0015 SMARK 4(75)13.75%(95)14.75%(78)0.139(04)0.102 TOINV 5(37)8,740,000(97)9,940,000(87)9,740,000(61)9,220,000 KUSE 6(02)5 years(68)8 years(47)7 years(23)6 years RES 7(87)1,870,000(41)1,410,000(56)1,560,000(15)1,150,000 VAR 8(98)$44.7(91)$43.65(22)33.3(58)38.7 FIX 9(10)$410,000(80)$480,000(19)419,000(93)493000 TAX 1040%0.4 DIS 1112%0.12 NPV$197,847.561$7,929,874.287$12,146,989.579$1,169,846.55

27. 13.5.3 Simulation Analysis Table 13.4b Simulation (Cont’d) VARIABLES5678 VMARK 1(78)2,890,000(89)2,945,000(26)2,630,000(60)2,800,000 PRICE 2(61)$52.2(18)43.6(47)49.4(88)$57.6 GROW 3(42)0.021(83)0.0415(94)0.047(17)0.0085 SMARK 4(77)0.1385(08)0.104(06)0.103(36)0.118 TOINV 5(65)9,300,000(90)9,800,000(72)9,440,000(77)9,540,000 KUSE 6(71)8 years(05)5 years(40)7 years(43)7 years RES 7(20)1,200,000(89)1,890,000(62)1,620,000(28)1,280,000 VAR 8(17)$32.55(18)$32.7(47)37.05(31)$34.65 FIX 9(48)448,000(08)408,000(68)468,000(06)406,000 TAX 100.400.4 DIS 110.12 NPV$15,306,245.293$-1,513,820.475$11,327,171.67$839,650.211

28. 13.5.3 Simulation Analysis Table 13.4b Simulation (Cont’d) VARIABLES910 VMARK 1(68)2,840,000(23)2,615,000 PRICE 2(39)$47.8(47)$49.4 GROW 3(71)0.0355(25)0.0125 SMARK 4(22)0.111(79)0.1395 TOINV 5(76)9,520,000(08)8,160,000 KUSE 6(81)9 years(15)5 years RES 7(88)1,880,000(71)1,710,000 VAR 8(94)44.1(58)$38.7 FIX 9(76)476,000(56)456,000 TAX 100.4 DIS 110.12 NPV$-6,021,018.052$563,687.461 NPV=4,194,647.207 Notes. 1. Definitions variables can be found in Table 13.4a. 2.NPV calculator procedure can be found in Table 13.4c. Random number 01-19 20-39 40-59 60-79 80-99 00 Useful life 5 6 7 8 9 10

29. 13.5.3 Simulation Analysis [Sales Volume]t = [(Market Size)  (1 + Market Growth Rate)t]  (Share of Market), EBITt = [Sales Volume]t  [Selling Price - Operating Cost] - [Fixed Cost]; [Cash Flow]t = [EBIT]t  [1 - Tax Rate];

30. 13.5.3 Simulation Analysis Table 13.4c Cash Flow Simulations Period1234 1$2,034,382.335$3,368,605.531$4,260,506.327$2,376,645.064 22,116,476.0993,406,999.8894,402,631.3772,380,653.731 32,201,525.2393,445,797.3884,549,233.3652,384,668.412 42,289,636.1473,485,002.2614,700,453.3162,388,689.114 52,380,919.0493,524,618.7854,856,436.6952,392,715.848 63,564,651.2825,017,333.5512,396,748.622 73,605,104.1205,183,298.658 83,645,981.714

31. 13.5.3 Simulation Analysis Table 13.4c Cash Flow Simulations (Cont’d) Period5678 1$4,549,425.961$1,841,398.655$1,820,837,760$4,344,679.668 24,650,608,7071,927,975.8991,919,614.7354,383,680.045 34,753,916.2892,018,146.0992,023,034.2284,423,011.926 44,859,393.3312,112,058.3622,131,314,4364,462,678.127 54,967,085.3912,209,867.9842,244,683.8154,502,681.491 65,077,038.9852,363,381.5544,543,024.884 75,189,301.6032,487,658.0874,583,711.195 85,303,921.737

32. 13.5.3 Simulation Analysis Table 13.4c Cash Flow Simulations (Cont’d) Notes. 1. NPVs are listed in Table 13.4b Period910 1$439,076.864$2,097,642.448 2464,802.8932,127,282.979 3491,442.1962,157,294,016 4519,027.1942,187,680.191 5547,591.4592,218,446.194 6577,169.756 7607,798.082 8639,513.714 9672,355.251

33. 13.6Inflationary effects in the capital-budgeting procedure (13.15) where k = A real rate of return in the absence of inflation (i) plus an inflation premium (η) plus a risk adjustment to a riskless rate of return (ρ).

34. 13.6Inflationary effects in the capital-budgeting procedure (13.16a) (13.16b) (13.17)

35. 13.6Inflationary effects in the capital-budgeting procedure (13.18) (13.19)

36. 13.6Inflationary effects in the capital-budgeting procedure (13.20) where R t = Expected growth in cash flow, O t = Outflow for variable operating expense, θ j = The percentage change in O t induced by inflation in period j. F t = Expected fixed cash charge, dep t = Fixed noncash charge, τ= Marginal corporate tax rate, i= Real risk-free rate, η= Inflation rate, ρ= Risk premium associated with uncertainty of nominal cash flow.

37. 13.6Inflationary effects in the capital-budgeting procedure (13.21) (13.22) (13.23)

38. 13.6Inflationary effects in the capital-budgeting procedure (13.24) (13.24’) (13.25)

39. 13.6Inflationary effects in the capital-budgeting procedure (13.26) (13.27) G j = a + bL j + cX τ A j Z j - dr j Z j - eDX j + u j, (13.28)

40. 13.6Inflationary effects in the capital-budgeting procedure TABLE 13.5 From Kim, M. K. L., “Inflationary effects in the capital-investment process: An empirical examination,” Journal of Finance 34 (September 1979): Table 1. Reprinted by permission. ConditionLX/TA1 + rD/X L > L ’ L = L ’ L < L ’ ++++++ +0-+0- -0+-0+ ------

41. 13.6Inflationary effects in the capital-budgeting procedure TABLE 13.6 Regression results of Eq. (10.28) (Figures in parentheses are t values) From Kim, M. K. L., “Inflationary effects in the capital-investment process: An empirical examination.” Journal of Finance 34 (September 1979): Table 4. Reprinted by permission. Periodabcde 1965-76 1965-70 1971-76.0884.0882.1044.0103 (14.74).0131 (9.93).0097 (5.90).0957 (4.88) -.0079 (0.37).1046 (4.05) -1.0007 (2.35).0670 (0.15) -.8642 (1.87) -.0545 (4.42) -.0297 (1.46) -.0870 (5.63)

42. 13.7Multi-period capital budgeting ( 13.29) where V 1 = Random value of the firm at the end of the time period; V M1 = Random value of the market value of all firms at the end of the time period; L 0 = Market-determined price of risk; r f = Riskless rate-of-return available to all investors.

43. 13.7Multi-period capital budgeting (13.30) E[V p1 ] - L 0 E[Cov (V p1, V M1 )] - V p0 > 0. (13.31)

44. 13.7Multi-period capital budgeting (13.32) where η = Elasticity of expectations of future earnings stream; b = Firm-specific constant measuring sensitivity of the disturbance term to unanticipated changes in the economic index; = Variance of the market asset’s rate-of-return; σ Im = ; represents the unanticipated changes in some general economic index; R mT = Market return; Q t = Cash-flow multiplier for period t.

45. 13.7Multi-period capital budgeting E[R i ] = r f + L[cov (Cov (R i, R m ))] (13.33) where R i = Rate-of-return on the risky asset i; r f = Riskless rate-of-return available to all market participants; L = Price of risk in the market, (E(R m ) - r f )/ [variance of the returns on the market]; R M = Return on a market portfolio of risky securities.

46. 13.7Multi-period capital budgeting (13.34) (13.35)

47. 13.8Summary and concluding remarks The preceding discussion outlined three alternative capital- budgeting procedures, each useful when cash flows are not known exactly but only within certain specifications. Depending on the correlation of the cash flows and the number of possible outcomes, they will all yield meaningful results. Simulation was introduced as a tool to deal with those situations in which the most uncertainty existed. Also discussed were various means of forecasting cash flows, most notably the product life-cycle approach. The emphasis in the later portion of the chapter was on the effects inflation has on the ability to forecast cash flows and on the appropriate discount-rate selection. We stress the importance of this factor as its nonrecognition can lead to disastrous results. In addition, we touched upon more theoretical issues by attempting to apply the CAPM, which created problems, but the basic approach is still applicable.

48. 13.8Summary and concluding remarks Lastly we investigated some of the more generalized mean- variance pricing frameworks and found that they were essentially equivalent. The application of these approaches is much the same as that of the CAPM, and further supports the increasing use of this technique in dealing with a very large, if not the largest problem area in applied finance theory, capital budgeting under uncertainty. The concepts, theory, and methods discussed in Chapters 12 and 13 are essentially based upon a myopic view of capital budgeting and decision making. This weakness can be improved by using Pinches’ (1982) recommendations. Concern should be given to how capital budgeting actually interfaces with the firm’s strategic positioning decision: to deal with risk effectively; to improve the control phase; and to take advantage of related findings from other disciplines. A broader examination of the capital-budgeting process, along with many effective business/academic interchanges, can go a long way toward improving the capital-budgeting process.

49. Appendix 13A. The use of the time-state preference and the option-pricing models in capital budgeting under uncertainty (13.A.1) where V st = current value (price) of a dollar for state s and time t, Z st = present value of cash flow for state s and time t, PV= present value of project.

50. Appendix 13A. The use of the time-state preference and the option-pricing models in capital budgeting under uncertainty TABLE 13.A.1 Expected cash flows for Project, PV = $1000(0.1672) + $800(0.2912) + $500(0.5398) + $500(0.1693) + $400(0.2915) + $200(0.5333) + $300(0.1686) + $200(0.2903) + $100(0.5313) = $1,140. State of EconomyYear 1Year 2Year 3 Boom Normal Recession $1000 $800 $500 $400 $200 $300 $200 $100

51. Appendix 13A. The use of the time-state preference and the option-pricing models in capital budgeting under uncertainty V j = e -i N [d 2 (M j )], (13.A.2) (13.A.3) where i= Interest rate, = Instantaneous variance of the rate-of-return on the market portfolio, N()= Probability of d 2 (M j ) obtained from a normal distribution.

52. Appendix 13A. The use of the time-state preference and the option-pricing models in capital budgeting under uncertainty (13.A.4) where r Mj = Rate-of-return on the market portfolio for the period if M 0  M j.

53. Appendix 13A. The use of the time-state preference and the option-pricing models in capital budgeting under uncertainty ΔV j = V j - V j+1. (13.A.5) V j = e -1 (N[d 2 (M j )] - N [d 2 M j+1 ]), (13.A.6) v j = e -i (N[D 2 (r Mj )] - N[d 2 (r M,(j+1) )]. (13.A.7) V j = v j + V j+1. (13.A.8)

54. Appendix 13A. The use of the time-state preference and the option-pricing models in capital budgeting under uncertainty