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 1

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. 2

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. (13.2) Where C t = Certainty-equivalent cash flow at period t, I = Riskless interest rate, N = Life of the project. (13.3) and (13.2’) 13.3Certainty equivalent method 3

(13.4) (13.4’) (13.2’’) If R f = 10%, E(R m ) = 15%, = $300, I 0 =$2,000, B j = 0.80, =0.02, then Cov (, R m ) = (080)(0.02)(2,000) = 32, and from Eq. (13.4′) we have CAPM Vi : market value of firm i Multiply V and rearrange the equation above, we can have Equation (13.4) 4

13.4The relationship of the risk-adjustment discount rate method to the certainty equivalent method (13.5) and (13.6) (13.7) and (13.8) (13.9) and (13.10) Under the Arrow (1971) and Pratt (1964) risk-aversion framework, α can be derived as Where E( ) = Actuarial value of risk; X is asset; and U′ (X) and U′(X) are second and first derivatives with respect to utility function U(x) and 5

13.4The relationship of the risk-adjustment discount rate method to the certainty equivalent method  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.  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. 6

The Statistical Distribution Method (13.11) and (13.12a) (13.12b) (13.12’) and (13.13) (13.14) 13.5Three other related stochastic approaches to capital budgeting Cov(C t, C τ ) = ρ τt σ τ σ t, If the cash flows are mutually independent, then Eq. (13.12b’) reduces to (13.13) Hillier (1969) combines the assumption of mutual independence and perfect correlation in developing a model to deal with mixed situations. 7

Derivation of Equation (13.14)  Hillier (1969) combines the assumption of mutual independence (Y t ) and perfect correlation in developing a model to deal with mixed situations. (13.14)  The derivation of Equation (13.14) is as follows  Assume the net cash flow at time t, X t, is related to the sources as follows 8

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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 Production Cash Outflow Production Cash Outflow Production Cash Outflow Production Outflow – salvage Value Marketing Marketing Marketing Marketing Marketing

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(5|4,3,2,1) (6) Cash Flow (7) Joint Probability (8) ,

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)

The Statistical Distribution Method Table 13.2b NPV and joint probability Discount Rate = 4 percent NPV = $2, Variance = $20,359, Standard Deviation = $4, PVANPVProbability 4, , , , , , , , , , , , , , , , , , , , , PVANPVProbability 12, , , , , , , , , , , , , , , , , , , , , , , , , , , , ,

The Decision-Tree Method Fig Decision tree. Numbers in parentheses are probabilities; numbers without parentheses are NPV 13.5Three other related stochastic approaches to capital budgeting Decision Variables For the First Stage Net Present Value* Standard Deviation * Coefficient Of Variation Regional $ $ National International Expressed in thousands of dollars. NPV = 0.7(1,248.84) + 0.2(814.73) + 0.1(-14.54) = $1,

The Decision-Tree Method Table 13.3 Expected cash flows for various branches of the decision tree NPV Regional Distribution throughout: High Medium Low National Distribution throughout: High , Medium Low , International Distribution Throughout: High , Medium Low ,

The Decision-Tree Method Table 13.3 Expected cash flows for various branches of the decision tree (Cont’d) NPV High National – High Inter , High National – Medium Inter , High National – Low Inter Medium National – High Inter , Medium National – Medium Inter Medium National – Low Inter Low National – High Inter Low National – Medium Inter Low National – Low Inter ,

The Decision-Tree Method Table 13.3 Expected cash flows for various branches of the decision tree (Cont’d) NPV High Region – High National , High Region – Medium National High Region – Low National Medium Region – High National Medium Region – Medium National Medium Region – Low National Low Region – High National Low Region – Medium National Low Region – Low National

Simulation Analysis Table 13.4a Variables for simulation VariablesRange 1. Market size (units) 2,500,000 – 3,000, Selling price ($/unit) 40 – Market growth 0 – 5% 4. Market share 10 – 15% 5. Total investment required ($) 8,000,000 – 10,000, Useful life of facilities (years) Residue value of investment ($) 1,000,000 – 2,000, Operating cost ($) 30 – Fixed cost ($) 400,000 – 500, 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 Three other related stochastic approaches to capital budgeting 18

Simulation Analysis Table 13.4b Simulation Variables12345 VMARK 1(39)2,695,000(47)2,735,000(67)2,835,000(12)2,580,000(78)2,890,000 PRICE 2(73)$54.6(93)$58.6(59)51.8(78)55.6(61)$52.2 GROW 3(72)3.6%(21)1.05%(63)0.0315(03)0.0015(42)0.021 SMARK 4(75)13.75%(95)14.75%(78)0.139(04)0.102(77) TOINV 5(37)8,740,000(97)9,940,000(87)9,740,000(61)9,220,000(65)9,300,000 KUSE 6(02)5 years(68)8 years(47)7 years(23)6 years(71)8 years RES 7(87)1,870,000(41)1,410,000(56)1,560,000(15)1,150,000(20)1,200,000 VAR 8(98)$44.7(91)$43.65(22)33.3(58)38.7(17)$32.55 FIX 9(10)$410,000(80)$480,000(19)419,000(93)493000(48)448,000 TAX 1040% DIS 1112%0.12 NPV$197, $7,929, $12,146, $1,169,846.55$15,306, Number in parentheses is the generated random number.

Simulation Analysis Table 13.4b Simulation (Cont’d) VARIABLES VMARK 1(89)2,945,000(26)2,630,000(60)2,800,000(68)2,840,000(23)2,615,000 PRICE 2(18)43.6(47)49.4(88)$57.6(39)$47.8(47)$49.4 GROW 3(83)0.0415(94)0.047(17)0.0085(71)0.0355(25) SMARK 4(08)0.104(06)0.103(36)0.118(22)0.111(79) TOINV 5(90)9,800,000(72)9,440,000(77)9,540,000(76)9,520,000(08)8,160,000 KUSE 6(05)5 years(40)7 years(43)7 years(81)9 years(15)5 years RES 7(89)1,890,000(62)1,620,000(28)1,280,000(88)1,880,000(71)1,710,000 VAR 8(18)$32.7(47)37.05(31)$34.65(94)44.1(58)$38.7 FIX 9(08)408,000(68)468,000(06)406,000(76)476,000(56)456,000 TAX DIS NPV$-1,513, $11,327,171.67$839, $-6,021, $563, NPV=4,194, Notes. 1. Definitions variables can be found in Table 13.4a. 2. NPV calculator procedure can be found in Table 13.4c. Random number Useful life

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]; 21

Simulation Analysis Table 13.4c Cash Flow Simulations Period $2,034, $3,368, $4,260, $2,376, $4,549, ,116, ,406, ,402, ,380, ,650,608,707 32,201, ,445, ,549, ,384, ,753, ,289, ,485, ,700, ,388, ,859, ,380, ,524, ,856, ,392, ,967, ,564, ,017, ,396, ,077, ,605, ,183, ,189, ,645, ,303,

Simulation Analysis Table 13.4c Cash Flow Simulations (Cont’d) Notes. 1. NPVs are listed in Table 13.4b Period $1,841, $1,820,837,7 60 $4,344, $439, $2,097, ,927, ,919, ,383, , ,127, ,018, ,023, ,423, , ,157,294, ,112, ,131,314,43 6 4,462, , ,187, ,209, ,244, ,502, , ,218, ,363, ,543, , ,487, ,583, , ,

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). (13.16a), (13.16b), and (13.17) (13.18), and (13.19) (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. 24

13.6Inflationary effects in the capital-budgeting procedure Define PV= present value of a one-period project, X= Net cash flow received at the end of the period, I= net investment outlay at time 0, r=risk-adjusted noninflation-adjusted required rate of return, τ=tax rate applicable to the firm, and p’= change in the price level expected to occur over the coming period. (13.21), and (13.22) (13.23) (13.24) 25

13.6Inflationary effects in the capital-budgeting procedure (13.24’) (13.25) (13.26), and (13.27) 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 ’

13.6Inflationary effects in the capital-budgeting procedure TABLE 13.6 Regression results of Eq. (13.28) (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 (14.74).0131 (9.93).0097 (5.90).0957 (4.88) (0.37).1046 (4.05) (2.35).0670 (0.15) (1.87) (4.42) (1.46) (5.63) G j = a + bL j + cX τ A j Z j - dr j Z j - eDX j + u j, (13.28) 27

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. (13.30) (13.31) E[V p1 ] - L 0 E[Cov (V p1, V M1 )] - V p0 > 0. 28

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. 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. (13.34), and (13.35) 29

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. 30

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. 31

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. 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 32

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), and(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. (13.A.4) where r Mj = Rate-of-return on the market portfolio for the period if M 0  M j. ΔV j = V j - V j+1, V j = e -1 (N[d 2 (M j )] - N [d 2 M j+1 ]) (13.A.5), and (13.A.6) v j = e -i (N[D 2 (r Mj )] - N[d 2 (r M,(j+1) )], V j = v j + V j+1. (13.A.7), and(13.A.8) 33

Appendix 13A. The use of the time-state preference and the option-pricing models in capital budgeting under uncertainty 34