Capital Budgeting: The Valuation of Unusual, Irregular, or Extraordinary Cash Flows

Transcription

1 Capital Budgeting: The Valuation of Unusual, Irregular, or Extraordinary Cash Flows ichael C Ehrhardt and Phillip R Daves any projects have cash flows that are caused by the project but are not part of the project s normal operating cash flows We describe an appropriate technique for valuing such cash flows and we reconcile the conflicting recommendations currently found in the literature Although managers must still use their judgment when valuing such projects, we provide guidelines and a framework within which managers can systematically articulate and quantify their judgment [JEL:G1,G31] nany projects have ancillary cash flows that are caused by the project but are not part of the project s normal operating cash flows We define these unusual, irregular, or extraordinary cash flows as non-operating cash flows Examples include decommissioning costs for a power plant, reclamation costs for a strip mine, expected litigation costs for illnesses contracted at a water amusement park, or the residual value of an asset at the completion of a lease These cash flows usually have three distinguishing characteristics: 1) they are not part of the project s normal operating cash flows; ) they can be quite large; and 3) their risk characteristics are different from those of the normal operating cash flows Finance theory and textbooks give consistent recommendations for valuing a project s normal operating cash flows If the risk of the project is similar to the risks of the firm s other projects, then the value of the project is the present value of the cash flows discounted at the firm s cost of capital If the project s risk is different from that of the firm s other projects, ichael C Ehrhardt is a Professor of Finance at the University of Tennessee, Knoxville,TN Phillip R Daves is an Associate Professor of Finance at the University of Tennessee, Knoxville, TN We appreciate the suggestions from two anonymous referees In addition, we thank Gene Brigham and participants in the University of Tennessee Finance Department Seminar Series for helpful comments Any errors are our responsibility then the discount rate should be adjusted For example, many companies use divisional discount rates when divisions differ in risk, and some companies even adjust the discount rate to reflect the risk of individual projects For certain types of projects, such as leasing analysis, different discount rates are used sometimes for different cash flows within a single project In all cases, the principle remains the same: calculate the present value of the cash flows using a risk-adjusted discount rate However, there has been considerable debate in academic journals and textbooks concerning the appropriate technique for evaluating unusual, irregular, or extraordinary cash flows, especially when the cash flows are negative In fact, the literature provides conflicting recommendations to managers The objectives of this paper are to describe an appropriate risk-adjustment technique for valuing unusual negative cash flows, reconcile the conflicting recommendations in the literature, and provide guidelines for managers to use when implementing the valuation technique 1 The remainder of this paper is organized as follows Section I illustrates the problem with an example and provides a brief literature review Section II describes the risk-adjustment technique and 1 The technique can be applied to positive cash flows as well as negative cash flows Because the finance literature has conflicting recommendations for the valuation of negative cash flows, we focus upon costs rather than revenues 106

2 EHRHARDT AND DAVES CAPITAL BUDGETING reconciles conflicts in the finance literature Section III provides guidelines for managers in implementing the technique Section IV is a brief summary I Negative Cash Flows and the Impact of Risk on Value The following example illustrates the source of conflict in the finance literature Consider two projects that are identical except with respect to a terminal negative cash flow In particular, both projects last one year, and both have identical risky operating cash flows of $1,03 occurring at the end of the year These operating cash flows have risks similar to the firm s other cash flows, and should be discounted at the firm s 10% cost of capital The present value of the expected operating cash flows is $930 (see Exhibit 1 for all calculations and results) For project Lo-Risk, there is also a riskless -$1,000 cash flow that will occur when the project ends in one year Since the magnitude of this non-operating cost is known with certainty, it should be discounted at the risk free rate If the risk free rate is 5%, the present value of this cost is approximately -$95 Thus, the Net Present Value (NPV) of project Lo-Risk is -$ If only NPV is considered, this project should be rejected Project Hi-Risk is identical to project Lo-Risk, except with respect to the non-operating terminal cost For project Hi-Risk, the expected value of the terminal cash flow is still -$1,000, but the actual cost could be either higher or lower Since the risky operating cash flows of the project were discounted at a rate of 10%, which is greater than the risk free rate, it might seem reasonable to discount the risky non-operating cost at a rate greater than the risk free rate Suppose that a manager discounts this expected risky cost using a risk-adjusted discount rate equal to the firm s cost of capital, 10% In this case, the present value of the terminal cost is approximately -$909 The NPV of project Hi-Risk is approximately positive $1, and the manager would accept the project As this example illustrates, discounting a risky negative expected cash flow with a risk-adjusted discount rate greater than the risk free rate implies that the high-risk project is more valuable than the low-risk project This is apparently inconsistent with the premise that investors are risk averse any previous authors have recognized this fact, but have produced conflicting recommendations Beedles (1978) focuses on projects with cash flows of alternating signs, and concludes that the risk adjusted discount rate approach is inappropriate for such projects; his conclusion is dependent on his assumption that the same discount rate is applied to all cash flows Pettway and Celec (1975) allow discount 107 rates to vary for positive and negative cash flows They implicitly assume that all uncertainty in cash flows harms the utility of investors, and so, advocate a disutility approach, in which they penalize negative cash flows Their net result is an effective risk adjusted discount rate that is less than the risk free rate Weston and Copeland (199) also recommend decreasing the discount rate for a negative cash flow The recommendations of these authors are correct, given their implicit assumptions about risk In our discussion, we explicitly articulate the conditions under which their conclusions are correct, and we describe the appropriate risk adjustment for other underlying assumptions Some authors claim that the same risk-adjustment technique must be applied to all cash flows to preclude arbitrage, even if it leads to a higher discount rate and higher value for a project with a risky negative cash flow ost of these arguments are based on the assumption that a particular, positive, risky, expected cash flow should be discounted at a rate higher than the risk-free rate, and that a negative, risky cash flow is simply the positive cash flow multiplied by negative one To preclude arbitrage, both cash flows must be discounted at the same rate (eg, see iles and Choi, 1979, and Ariel, 1998) Given their assumptions, the authors are correct We extend their analysis to the case in which the negative, risky cash flow can have more complicated risk attributes Some authors have explicitly stated that the appropriate risk-adjustment can lead to either a higher or a lower discount rate, depending on the nature of the risk (see Celec and Pettway, 1979, Lewellen, 1977 & 1979, Booth, 198 &1983, Berry and Dyson, 1980, and Dyson and Berry, 1983) In particular, Booth (198) reaches this conclusion through an elegant assessment of the issue within the context of statepreference theory Booth (198) and Berry and Dyson (1980) employ the Capital Asset Pricing odel (CAP) to reach this conclusion We build upon their work and explicitly express the appropriate project beta in terms of the parameters of the underlying cash flows The previous literature is somewhat fragmented, with different authors employing different sets of assumptions, focusing on different aspects, and frequently reaching different conclusions In the next section, we reconcile these differences, synthesize the different sets of assumptions, and explicitly describe the appropriate riskadjustment technique Because the existing literature emphasizes theory rather than implementation issues, we also provide an illustrative application of the correct risk-adjustment technique Although managers always will be required to use their judgment, we provide a framework within which they can systematically translate their judgment into quantitative results

3 108 FINANCIAL PRACTICE AND EDUCATION FALL / WINTER 000 Exhibit 1 Impact of Risk on Net Present Value Lo-Risk Hi-Risk Operating Cash Flow $1,03 $1,03 Discount Rate for Operating Cash Flow 10% 10% Present Value of Operating Cash Flow $930 $1,03/110 $930 $1,03/110 Terminal Cash Flow -$1,000 -$1,000 Discount Rate for Terminal Cash Flow 5% 10% Present Value of Terminal Cash Flow -$95 $1,000/105 -$909 $1,000/11 Net Present Value -$ $930 - $95 $1 $930 - $909 II A Risk-Adjustment Technique for Non-Operating Costs The crux of the debate is the proper adjustment for risk As we show in this section, the same riskadjustment technique should be applied to positive and negative cash flows However, the proper application of this technique sometimes results in increasing the discount rate above the risk free rate or decreasing the discount rate below the risk free rate, depending on the specific type of risk that is inherent in the cash flow Furthermore, the application of this technique can lead to different discount rates for positive and negative cash flows that are otherwise similar The following two sub-sections define the appropriate risk-adjustment technique and reconcile the conflicting recommendations in the literature A The Appropriate Risk-Adjustment Technique We develop our results within the context of CAP, although they are easily extended to any linear multifactor model of returns such as the Arbitrage Pricing odel or the Fama-French three-factor model For expositional clarity we assume that the firm is financed solely with equity, but our conclusions are applicable to the case of a levered firm CAP states that the appropriate adjustment for risk depends on the project s beta, b p Specifically, the appropriate risk-adjusted discount rate, k, is defined as: k k RF + β p RP (1) where k RF denotes the risk free rate and RP is the arket Risk Premium The key to valuing a negative cash flow is to estimate the appropriate beta However, before discussing the case of a negative cash flow, it is helpful to examine some of the intuition underlying the application of CAP to capital budgeting First, the beta of a project is a function of the standard deviation of the project s expected return (s p ), the correlation between the project s expected return and the market s expected return (r p, ), and the standard deviation of the market s expected return (s ): b () p s r p p, s Second, notice that all variables in Equation are based on the project s return, and not on its cash flows Finally, the project s beta depends on the magnitude and direction of the project s risk The magnitude of risk is defined by s p and the direction of the risk is defined by r p, When the correlation is positive, beta is also positive, and the discount rate is larger than the risk free rate When the correlation is negative, the project beta is also negative, and the discount rate is less than the risk free rate We define a normal project as one with an initial negative cash flow and subsequent positive cash flows Consider two normal projects with cash flows that are identical with respect to their standard deviations and expected values, but whose returns have correlations with the market that are opposite in sign to each other; ie, one project has a positive beta and the other project has a negative beta In this case, the value of the project with the negative beta is greater than the value of the project with the positive beta, even though the expected values and standard deviations of the projects cash flows are the same athematically, this is because a negative beta produces a smaller discount rate Intuitively, this is because a negative beta project serves to hedge an investor s other investments, which typically have positive betas In other words, the project with the negative beta does well when the investor s other

4 EHRHARDT AND DAVES CAPITAL BUDGETING projects are doing poorly Therefore, a negative beta, normal project is more valuable than a positive beta project with the same expected cash flow and volatility Now consider a non-normal project that consists of a single, risky, non-operating cash flow with expected value An important step in valuing the cash flow is to use Equation 1 to estimate the risk-adjusted discount rate based on the project s beta Since this is a non-operating cash flow, its risk is likely to be different from the risk of the firm s other cash flows, and therefore the project s beta will be different than the firm s beta Thus, the project beta must be estimated from the characteristics of the cash flow rather than the characteristics of the firm The Appendix contains the derivations for all of the following equations; only results are shown here in the text First, the project s beta can be expressed in terms of the present value of the cash flow (V), the correlation between the cash flow and the return on the market (r, ), and the standard deviation of the cash flow (s ): ρ, βp V (3) Standard deviations are always non-negative, so Equation 3 shows that the project s beta and the correlation of the cash flow with the market return are of the same sign, if and only if V is positive Suppose, however, that V is negative If the cash flow has a positive correlation with the market, Equation 3 shows that the project s beta must be negative In other words, the project beta and the correlation of the cash flow with the market have opposite signs if V is negative As we discuss in section IIB, this negative relationship between beta and cash flow correlation for projects with negative cash flows is the source of the debate in the finance literature The project beta also can be expressed without explicit reference to V, the value of the project: ρ (1 + k, RF βp ( (4) ) ( ρ, RP) Equation 4 provides the key to the correct procedure for the valuation of a non-operating cash flow 3 First, Similarly, the Appendix shows that r, r p, when V is positive, but r, -r p, when V is negative 3 Equation 4 provides the correct procedure for the valuation of any cash flow whether operating or non-operating, positive or negative Since most operating cash flows are assumed to be similar in risk to the rest of the firm s cash flows, the application of Equation 4 would result in the same project beta as the company s beta Therefore, it is correct and easier to directly use the firm s beta when finding a discount rate for operating cash flows ) 109 estimate the project beta for the non-operating cash flow using equation 4 Second, use CAP to estimate the risk-adjusted discount rate Third, discount the expected non-operating cash flow at the risk-adjusted discount rate Section IV describes how to implement this technique within the context of a company s existing capital budgeting processes B Reconciliation of Conflicting Recommendations Some researchers have stated that the same procedure should be used to adjust the discount rates for positive and negative cash flows Others have stated that a negative cash flow should be discounted at a lower discount rate to avoid rewarding a project for being risky As we show in this section, both of these apparently contradictory statements can be true To see this, consider two projects P and N whose expected cash flows have identical absolute values, standard deviations, and correlations with the market Suppose project P has a positive expected cash flow > 0 and project N has negative expected cash flow < 0 Equation 4 shows that unless is equal to zero, the two projects will have different betas In particular, if the correlation between the cash flows and the market is positive, then Project N will have a negative project beta and Project P will have a positive project beta 4 This means that using the same procedure to adjust for risk will result in different discount rates for projects with negative versus positive expected cash flows, even if the risk parameters of the projects are identical 5 Another implication of this result is that an increase in risk (ie, an increase in the standard deviation of cash flow) will affect the betas of the positive and negative expected cash flow projects differently If the correlation is positive and the standard deviation of cash flow is increased, then b P will increase, while b N will decrease (become more negative) Translated to discount rates, higher cash flow risk results in a larger discount rate for P and a smaller discount rate for N This result is consistent with intuition An increase 4 There is a singularity in Equation 4 for extremely risky cash flows when s s r, RP If and r are both positive then if s increases enough, it will drive the value of P to zero and b P to infinity This induces a sign change from positive to negative for b P and V at the singularity Further increases in s will then increase b P from and make the value of P a greater negative See Gallagher and Zumwalt (1991) for a discussion of this issue For expository simplicity, we will focus on the situation with normal levels of cash flow risk that generate finite betas and for which positive expected value cash flows have a positive present value, and negative expected value cash flows have a negative present value 5 Note that in general b P will not be equal to b N, nor will it be equal to -b N

5 110 FINANCIAL PRACTICE AND EDUCATION FALL / WINTER 000 in risk reduces the value of Project P, which has a positive expected cash flow An increase in risk also reduces the value of Project N, which has a negative expected cash flow, from a negative value to an even more negative value This reduction in value is due to the positive correlation between the cash flows and the market Both projects have cash flows that are higher (ie, less negative for Project N) when the market is doing well, and lower when the market is doing poorly Therefore, an increase in risk appropriately penalizes the values of both projects Notice that the reverse occurs when the correlation is negative In this case, Project P has a negative project beta, and Project N has a positive project beta An increase in risk causes the discount rate for the positive cash flow project to decrease and the discount rate for the negative cash flow project to increase This causes the values of both projects to increase (ie, become less negative for Project N) This is also consistent with intuition; both projects have better cash flows when the market is doing poorly, which makes Project P more valuable to investors and Project N less harmful to investors In other words, the projects act as hedges Equations 4 and 1 show that a project with a riskless cash flow will have a beta of zero, and hence should be discounted at the risk free rate The discussion above also shows that all risky, negative, cash flows that are positively correlated with the market have negative betas and should therefore be discounted at rates that are lower than the risk free rate, with the discount rate decreasing as the risk of the cash flow increases Risky negative cash flows that are negatively correlated with the market have positive betas and should be discounted at rates that are higher than the risk free rate, with the discount rate increasing as the risk of the cash flow increases The risk-adjustment technique we have described is also consistent with the absence of arbitrage opportunities For example, consider Project P which has a single positive expected cash flow Project B is the same as Project P except that its cash flow is multiplied by negative one To prevent arbitrage opportunities, the value of Project B must equal the negative of the value of Project P, since the cash flow from the sum of the two projects is zero with certainty This will occur only if the betas of both projects are identical An inspection of Equation 4 reveals that this is indeed the case The expected cash flow () for Projects P and B are of opposite signs, but so are the correlations (r, ) of Projects P and B Substituting into Equation 4 shows that the betas of the two projects are identical Therefore, their values sum to zero, and there is no opportunity for arbitrage The risk-adjustment technique is also consistent with the standard practice of discounting all of the net operating cash flows of a project at the same discount rate, where net operating cash flows are defined as operating revenues minus operating costs Suppose per unit operating revenues and per unit operating costs are constants, but that the quantity sold is risky This implies that revenues and costs are scalar multiples of the quantity sold It also implies that the standard deviation of revenues and the standard deviation of costs are scalar multiples of the standard deviation of the quantity sold Notice also that the correlation for revenues is the same as the correlation for quantity sold, while the correlation for costs is the negative of the correlation for quantity sold The application of Equation 4 shows that the project beta for the net operating cash flows is the same as the project beta for just the revenues or the project beta for just the costs Therefore, discounting the net cash flows (ie, finding the present value of the difference between the revenues and costs) results in the same present value as finding the difference between the present value of the revenues and the present value of the costs In summary, the conflicting recommendations in the literature for finding the present value of a negative cash flow are due to different implicit assumptions about the correlation between the cash flow and the market As Equation 4 shows, application of the same risk-adjustment procedure to projects with a negative non-operating cash flow and to projects with a positive non-operating cash flow can lead to either different or identical discount rates, depending on the correlation of the cash flow with the market III anagerial Guidelines for Application of the Risk-Adjustment Technique We illustrate the application of the procedure using an example Consider a project that has operating cash flows that are similar in risk to the firm s other cash flows; ie, these operating cash flows are positively correlated with the performance of the economy As a part of the project, the company must expand its labor force However, many of these workers will be released at the end of the year as the project winds down Due to recently passed state laws, the company will incur substantial costs to re-train and aid in the relocation of these workers These costs have an expected value of -$1,000,000 However, these costs are very uncertain If the economy is doing well, the costs will be much lower, since it will be easier to find other employment for these workers However, if the economy is doing poorly, the costs will be much higher, since it will be more difficult to relocate the workers Notice that if this terminal cost were an operating cost with the same

6 EHRHARDT AND DAVES CAPITAL BUDGETING risk as other cash flows associated with the project, then it would be appropriate to discount this cash flow along with the other project cash flows at the project s cost of capital However this is a non-operating cost, and has a risk profile that is different from the risk of the project s operating cash flows, so it is inappropriate to discount it at the project s cost of capital As in all capital budgeting decisions, managerial judgment plays a key role in the valuation of this nonoperating cash flow However, Equation 4 provides a systematic way to translate managerial judgment into an appropriate discount rate for this cash flow The keys to applying Equation 4 are the estimation of the cost s standard deviation and correlation with the market A good starting point for the standard deviation of the expected cost is the standard deviation of the firm s other cash flows, based on past values of the firm s free cash flows This firm s free cash flows have averaged $00 million per year for the last eight years, with a standard deviation of $40 million The ratio of the standard deviation to the average, called the coefficient of variation, is $40/$00 0% If the nonoperating cost is similar in risk to the firm s other cash flows, then a first estimate of the standard deviation of the cost can be found by multiplying the absolute value of the expected cost by the firm s coefficient of variation of 0% This yields an estimate of the standard deviation of cost equal to $00,000; ie, 00,000 00(1,000,000) From here, simulation and/ or managerial judgment should be used to improve this estimate Suppose that simulation indicates that this cost is roughly twice as risky as the firm s other cash flows 6 Then, the final estimate of the cost s standard deviation is $400,000 Similarly, historical data can be used to provide a good starting point for the estimate of the correlation between the expected cash flow and the market return The firm s stock has a beta equal to 10 and a standard deviation of returns equal to 36%, based on past returns The standard deviation of the market return has averaged approximately 18%, depending on the particular stock index and historical period Rearranging Equation, we can find the correlation 6 When using simulation to estimate the standard deviation of the cash flow, it is important to convert the standard deviation to an annual basis if an annualized discount rate is being used, which is normally the case For example, suppose the project lasts five years, has an expected terminal cash flow of -$0 million, and simulation gives the standard deviation of this terminal cash flow of $10 million The appropriate annualized standard deviation is found by dividing the standard deviation of the terminal cash flow by the number of years; in this example, the annualized standard deviation is $ million (10/ 5) This adjustment is due to the underlying assumption about the resolution of risk that allows the use of an annual discount rate, compounded by the number of years of the project; see Copeland and Weston (1983, Chapter 9) for details between the firm s returns and the stock market: ( ) 111 βf ρf, 05 (a) f 036 As shown in equation A4 of the appendix, the absolute value of the correlation between the firm s expected cash flows and the market return is the same as the absolute value of the correlation between the firm s stock return and the market return Since equation a shows that 05 is an estimate of the correlation between the firm s stock return and the market return, then 05 is also a reasonable estimate of the absolute value of the correlation between the firm s expected operating cash flows and the market return However, we need the correlation between the firm s expected non-operating cash flow and the market return At this point, simulation and/or managerial judgment should be used to determine the sign of the correlation and to modify the estimate of the correlation for the non-operating cash flow associated with this particular project Since the cash flow will be larger (less negative) when the economy is doing well, managers believe that the expected negative cash flow of this project has a positive correlation with the market return However, they believe that it is much less correlated with the market than are the firm s other cash flows Therefore, the managers decide to use a correlation of 0, rather than the correlation of 05 Suppose the risk free rate is 5% and the market risk premium is 6% Using Equation 4, we find the beta of the non-operating cash flow: ρ, (1 + krf) βp ( ) ( ρ, RP) (400,000) (0)(105) 046 [(018)( 1,000,000)] [(400,000)(0)(006)] Using Equation 1, we find the discount rate: (4a) k krf +βp RP 005+ ( 046)(006) 00 (1a) Notice that this discount rate is less than the company s cost of capital of 11% (based on the company s beta of 10), and is even less than the risk free rate This is because the project beta is negative, arising from the negative expected value of the cash flow and the positive correlation of the cash flow with the market Notice also that an increase in risk, as measured by the cash flow s standard deviation, would make the beta in Equation 4a even lower Discounting the expected cost of $1,000,000 at the % discount rate results in a present value of -$978,474-1,000,000/10

7 11 FINANCIAL PRACTICE AND EDUCATION FALL / WINTER 000 In this example, the expected cost is relatively small in comparison to the firm s other cash flows, and is not large enough to pose a threat to the firm s solvency If the expected cost is large enough to materially affect the firm s likelihood of bankruptcy, then the expected bankruptcy cost should be added to the expected cost of the project For such bet-the-company projects, it might be better to use real option techniques rather than discounted cash flow techniques Finally, what if investors are also concerned with more than just systematic risk, as defined by beta? Although finance theory provides no formal models relating nonsystematic risk and required rates of return, there are some heuristic guidelines for managers For example, consider a project with a negative expected cash flow The first step is to estimate the risk-adjusted discount rate as describe in the previous sections The second step is to further adjust the discount rate to account for the impact of non-systematic risk In particular, does the project increase or decrease the standard deviation of the firm s total cash flow? In other words, is the nonsystematic portion of the project s cash flow correlated with the firm s other cash flows? If the project increases the standard deviation of the firm s cash flows, then the previously calculated discount rate should be decreased to penalize the project If the project decreases the standard deviation of the firm s cash flows, then the discount rate should be increased to reward the project for acting to hedge the firm s other cash flows IV Summary any projects have costs that are caused by a project but that are not part of the project s normal operating cash flows The key to finding the present value of such a non-operating cost is to explicitly estimate the standard deviation of the cost and the correlation between the cost and the market return As we show in this paper, these estimates can be used to calculate the appropriate discount rate to be used when finding the present value of the cost We show how to calculate the beta for a project that consists of a single cash flow If the value of the project is negative, which is typically the case for a project consisting of a single non-operating cost, then the project s beta and the correlation of the cash flow with the market return have opposite signs This observation reconciles previous conflicts in the literature In particular, we show how application of the same risk-adjustment procedure to positive and negative cash flows can lead to either different betas for positive and negative cash flows, or the same betas, depending on the correlation of the cash flow with the market When cash flows are positively correlated with the market, the addition of risk to a negative nonoperating cash flow means that the discount rate should be reduced below the risk free rate when finding the present value of the cost In other words, the addition of risk does reduce the value of such a project When cash flows are negatively correlated with the market, the addition of risk to a negative non-operating cash flow means that the discount rate should be increased above the risk free rate when finding the present value of the cost In this case the addition of risk increases the value of such a projectn Appendix We begin the analysis with a project that lasts one year and that has a single cash flow at the end of the year; later in the appendix we show that these results are also valid for a multi-year project Let R p denote the expected return on a project and R denote the expected market return Let s p denote the standard deviation of the expected project return, s denote the standard deviation of the expected market return, and r p, denote the correlation between the expected project return and the expected market return Notice that all returns and standard deviations are measured with units based on a year basis The beta of a project, denoted by b p, is defined as: COV[R p, R ] β p p ρp, Now consider a project consisting of a single expected cash flow,, that occurs in one year Let s denote the standard deviation of the cash flow and let r, denote the correlation between the cash flow and the market return V is defined as the present value of, when discounted at the required rate of return Given equilibrium in the financial markets, the required rate of return is equal to the expected return, R p Note that the cost of the project may be different than V, in which case the project has either a positive or negative net present value Solving the single-year valuation equation for R p yields R p 1 (A) V (A1)

8 EHRHARDT AND DAVES CAPITAL BUDGETING 113 and the standard deviation of the project s return is p STD 1 V abs(v) The correlation of the project s return with the market, r p,, is (A3) COV 1, R COV(R p, R ) V sign(v)cov[, R ] ρp, sign(v) ρ, p abs(v) (A4) Equation A4 shows that the correlation between the project s cash flow and the market return (r p, ) is numerically equal to the correlation between the project s return and the market return (r, ) multiplied by the sign of the value of the project (V) In other words, the absolute values of the cash flow correlation and the return correlation are the same If the value of the project is positive (ie, the cash flow is sufficiently positive), then the return correlation and the cash flow correlation are identical to one another If the value of the project is negative, then the return correlation is the negative of the cash flow correlation The beta of the cash flow is defined as: COV[, R ] β while the beta of the project is: β p COV 1, R V ρ, 1 V COV, [ R ] Note that b is different from a conventional beta in the sense that a b of 10 is only interpreted as average risk if the project s value is also 10 Otherwise, a b equal to the project s value, V, indicates that the project is of average risk Let k denote the required rate of return on a project, k RF denote the risk free rate, and RP denote the market risk premium CAP defines the required rate of return on a project as: β V (A5) (A6) k k RF + β P RP (A7) The value of the project is: V 1+ ( k + RP) RF β P (A8) Substituting the definition of the project beta from Equation A6 and the cash flow beta from Equation A5 into A8 and rearranging yields the well known risk neutral valuation result: V ρ, ( 1+ k ) RF RP Substituting Equation A9 and Equation A5 into Equation A6 yields an expression of the project beta in terms of the characteristics of the cash flow and the parameters of CAP: (A9)

9 114 FINANCIAL PRACTICE AND EDUCATION FALL / WINTER 000 ρ, (1 + k RF ) βp (A10) ( ) ( ρ, RP) Up until this point, we have considered a project that lasts a single year Now suppose the project is one that lasts for more than one year but still has a single cash flow occurring at the end of the last year In this paper, we consider only projects for which it is appropriate to use a constant discount rate each year In other words, we assume that the value of an N-year project can be expressed as: V [ 1 + ( k + RP) ] RF β P N (A11) Equation A11 is the typical valuation equation shown in all textbooks and the one that is used by most financial managers Copeland and Weston (1983) provide a detailed discussion of the assumptions underlying the use of this equation which include a constant risk-free rate, a constant market risk premium, and a constant beta, all of which imply a constant discount rate Since the discount rate is constant, it must be the same in all previous years as it is in the last year Hence, we can derive equation A10 for the last year and it will give us the beta to be used in estimating the appropriate discount rate for the entire life of the project References Ariel, Robert A, 1998, Risk Adjusted Discount Rates and the Present Value of Risky Costs, The Financial Review 33 (No 1, February), Beedles, William L, 1978, Evaluating Negative Benefits, Journal of Financial and Quantitative Analysis 13 (No 1, arch), Berry, RH and RG Dyson, 1980, On the Negative Risk Premium for Risk Adjusted Discount Rates, Journal of Business Finance and Accounting 7 (No 3, ay), Booth, Laurence D, 198, Correct Procedures for the Evaluation of Risky Cash Outflows, Journal of Financial and Quantitative Analysis 17 (No, June), Booth, Laurence D, 1983, On the Negative Risk Premium for Risk Adjusted Discount Rates: A Comment and Extension, Journal of Business Finance and Accounting 10 (No 1, January), Celec, Stephen E and Richard H Pettway, 1979, Some Observations on Risk-Adjusted Discount Rates: A Comment, Journal of Finance 34 (No 4, September), Dyson, RG and RH Berry, 1983, On the Negative Risk Premium for Risk Adjusted Discount Rates: A Reply, Journal of Business Finance and Accounting 10 (No 1, January), Gallagher, Timothy J and J Kenton Zumwalt, 1991, Risk- Adjusted Discount Rates Revisited, The Financial Review 6 (No 1, February), Lewellen, Wilbur G, 1977, Some Observations on Risk- Adjusted Discount Rates, Journal of Finance 3 (No 4, September), Lewellen, Wilbur G, 1979, Reply to Pettway and Celec, Journal of Finance 34 (No 4, September), iles, James and Dosoung Choi, 1979, Comment: Evaluating Negative Benefits, Journal of Financial and Quantitative Analysis 14 (No 5, December), Pettway, Richard H and Stephen E Celec, 1975, Thrust and Parry, Financial anagement 4 (No 4, Winter), 7-11 Weston, J Fred and Thomas E Copeland, 199, anagerial Finance, 9 th ed, New York, NY, Dryden Press Copeland, Thomas E and J Fred Weston, 1983, Financial Theory and Corporate Policy, nd ed, Reading, A, Addison- Wesley Publishing Company