Caital Budgeting: The Valuation of Unusual, Irregular, or Extraordinary Cash Flows ichael C. Ehrhardt Philli R. Daves Finance Deartment, SC 424 University of Tennessee Knoxville, TN 37996-0540 423-974-1717 mehrhard@utk.edu February 3, 1999 The authors would like to thank Gene Brigham and articiants in the University of Tennessee Finance Deartment Seminar Series for helful comments. Any errors are the resonsibility of the authors.
Caital Budgeting: The Valuation of Unusual, Irregular, or Extraordinary Cash Flows Abstract any rojects have cash flows that are caused by the roject but that are not art of the roject s normal oerating cash flows. This aer develos an aroriate technique for valuing such cash flows, reconciles the conflicting recommendations currently found in the literature, and rovides guidelines for managers to use when imlementing the valuation technique. I. Introduction any rojects have ancillary cash flows that are caused by the roject but that are not art of the roject s normal oerating cash flows. We define these unusual, irregular, or extraordinary cash flow as non-oerating cash flows. Examles include decommissioning costs for a ower lant, reclamation costs for a stri mine, exected litigation costs for illnesses contracted at a water amusement ark, or the residual value of an asset at the comletion of a lease. These cash flows usually have three distinguishing characteristics: (1) they are not art of the roject s normal oerating cash flows; (2) they can be quite large; and (3) their risk characteristics are different from those of the normal oerating cash flows. Finance theory and textbooks give consistent recommendations for valuing a roject s normal oerating cash flows. If the risk of the roject is similar to the risks of the firm s other rojects, then the value of the roject is the resent value of the cash flows discounted at the firm s cost of caital. If the roject s risk is different from that of the firm s other rojects, then the discount rate should be adjusted. For examle, many comanies use divisional discount rates when divisions differ in risk, and some comanies even adjust the discount rate to reflect the risk of individual rojects. For certain tyes of rojects, such as leasing analysis, different discount rates are used some times for different cash flows within a single roject. In all cases, the rincile remains the same: calculate the resent value of the cash flows using a risk-adjusted discount rate. However, there has been considerable debate in academic journals and textbooks concerning the aroriate technique for evaluating unusual, irregular, or extraordinary cash 1
flows, esecially when the cash flows are negative. In fact, the literature rovides conflicting recommendations to managers. The objectives of this aer are to develo an aroriate riskadjustment technique for valuing unusual negative cash flows, reconcile the conflicting recommendations in the literature, and rovide guidelines for managers to use when imlementing the valuation technique. 1 The remainder of this aer is organized as follows. Section II illustrates the roblem with an examle and rovides a brief literature review. Section III develos the risk-adjustment technique and reconciles conflicts in the finance literature. Section IV rovides guidelines for managers in imlementing the technique. Section V is a brief summary. II. Negative Cash Flows and the Imact of Risk on Value The following examle illustrates the source of conflict in the finance literature. Consider two rojects that are identical excet with resect to a terminal negative cash flow. In articular, both rojects last one year, and both have identical risky oerating cash flows that are similar in risk to the firm s other cash flows. The resent value of the exected oerating cash flows, when discounted at the firm s 10% cost of caital, is $930 thousand. For roject Lo-Risk, there is also a riskless $1 million negative cash flow that will occur when the roject ends in one year. Since the magnitude of this non-oerating cost is known with certainty, it should be discounted at the risk free rate. If the risk free rate is 5%, the resent value of this cost is aroximately negative $952 thousand. Thus, the Net Present Value (NPV) of roject Lo-Risk is negative $22 thousand. If only NPV is considered, this roject should be rejected. Project Hi-Risk is identical to roject Lo-Risk, excet with resect to the non-oerating terminal cost. For roject Hi-Risk, the exected value of the terminal cost is still a negative $1 million, but the actual cost could be either higher or lower. Since the risky oerating cash flows of the roject were discounted at a rate of 10%, which is greater than the risk free rate, it might seem reasonable to discount the risky non-oerating cost at a rate greater than the risk free rate. Suose that a manager discounts this exected risky cost with a risk-adjusted discount rate equal to the firm s cost of caital, 10%. In this case, the resent value of the terminal cost is negative $909 thousand. The NPV of roject Hi-Risk is ositive $21 thousand, and the roject 1 The technique can be alied to ositive cash flows as well as negative cash flows. Because the finance literature has conflicting recommendations for the valuation of negative cash flows, we focus uon costs rather than revenues. 2
should be acceted. As this examle illustrates, discounting a risky negative exected cash flow with a riskadjusted discount rate greater than the risk free rate imlies that the high-risk roject is more valuable than the low-risk roject. This is aarently inconsistent with the remise that investors are risk averse, and so some authors recommend reducing rather than increasing the discount rate for the risky negative cash flow in order to enalize the riskier roject. See Pettway and Celec (1975), Beedles (1978), Berry and Dyson (1980), and Weston and Coeland (1992) for examles of this recommendation. However, some authors claim that the same risk-adjustment technique must be alied to all cash flows to reclude arbitrage, even if it leads to a higher discount rate and higher value for a roject with a risky negative cash flow; see Lewellen (1977), iles and Choi (1979), and Ariel (1998). Along these same lines, some authors have exlicitly stated that the aroriate risk-adjustment can lead to either a higher or a lower discount rate, deending on the nature of the risk; see Celec and Pettway (1979), Lewellen (1979), Booth (1982, 1983), Berry and Dyson (1980), and Dyson and Berry (1983). In articular, Booth (1982) reaches this conclusion through an elegant assessment of the issue within the context of state-reference theory, and Booth (1982) and Berry and Dyson (1980) emloy the Caital Asset Pricing odel (CAP). In summary, different recommendations have been roosed for adjusting the discount rate for a negative non-oerating cash flow: (1) increase the discount rate to reflect the additional risk; (2) decrease the discount rate to enalize the roject for additional risk; and (3) establish a searate risk-adjusted discount rate for each non-oerating cash flow. Furthermore, the literature has emhasized theory rather than imlementation issues. As a result, financial managers have been left with comelling arguments that lead to contradictory recommendations for determining the aroriate discount rate for non-oerating cash flows. Financial managers also lack clear guidelines for a risk-adjustment technique that can be imlemented within the context of their existing caital budgeting rocesses. III. A Risk-Adjustment Technique for Non-Oerating Costs The crux of the debate revolves around the roer adjustment for risk. As we show in this section, the same risk-adjustment technique should be alied to ositive and negative cash flows. However, the roer alication of this technique sometimes results in increasing the 3
discount rate above the risk free rate and sometimes results in decreasing the discount rate below the risk free rate, deending on the secific tye of risk that is inherent in the cash flow. Furthermore, the alication of this technique can lead to different discount rates for ositive and negative cash flows that are otherwise similar. The following two sub-sections define the aroriate risk-adjustment technique and reconcile the conflicting recommendations in the literature. III.A. The Aroriate Risk-Adjustment Technique Our results are derived within the context the CAP, although they are easily extended to any linear multi-factor model of returns, such as Ross s Arbitrage Pricing odel or the Fama- French three-factor model. For exositional clarity we assume that the firm is financed solely with equity, but our conclusions are alicable to the case of a levered firm. The CAP states that the aroriate adjustment for risk deends on the roject s beta,. Secifically, the aroriate risk-adjusted discount rate, k, is defined as: k k + RP (1) where k RF denotes the risk free rate and RP is the arket Risk Premium. RF The key to valuing a negative cash flow is to estimate the aroriate beta. However, before discussing the case of a negative cash flow, it is helful to examine some of the intuition underlying the alication of CAP to caital budgeting. First, the beta of a roject is a function of the standard deviation of the roject s exected return,, the correlation between the roject s exected return and the market s exected return,,, and the standard deviation of the market s exected return, :,. (2) Second, notice that all variables in Equation (2) are based on the roject s return, and not on its cash flows. Finally, the roject s beta deends on the magnitude and direction of the roject s risk. The magnitude of risk is defined by and the direction of the risk is defined by,. When the correlation is ositive, beta is also ositive, and the discount rate is larger than the risk free rate. When the correlation is negative, the roject beta is also negative, and the discount rate is less than the risk free rate. 4
We define a normal roject as one with an initial negative cash flow and subsequent ositive cash flows. Consider two normal rojects with cash flows that are identical with resect to their standard deviations and exected values, but whose returns have correlations with the market that are oosite in sign to each other; i.e., one roject has a ositive beta and the other roject has a negative beta. In this case, the value of the roject with the negative beta is greater than the value of the roject with the ositive beta, even though the exected values and standard deviations of the rojects cash flows are the same. athematically, this is because a negative beta roduces a smaller discount rate. Intuitively, this is because a negative beta roject serves to hedge an investor s other investments, which tyically have ositive betas. In other words, the roject with the negative beta does well when the investor s other rojects are doing oorly. Therefore, a negative beta normal roject is more valuable than a ositive beta roject with the same exected cash flow and volatility. Now consider a non-normal roject that consists of a single risky non-oerating cash flow with exected value. An imortant ste in valuing the cash flow is to use Equation (1) to estimate the risk-adjusted discount rate based on the roject s beta. Since this is a nonoerating cash flow, its risk is likely to be different from the risk of the firm s other cash flows and therefore the roject s beta will be different than the firm s beta. Thus, the roject beta must be estimated from the characteristics of the cash flow rather than the characteristics of the firm. The Aendix contains the derivations for all of the following equations; only results are shown here in the text. First, the roject s beta can be exressed in terms of the resent value of the cash flow, V, the correlation between the cash flow and the return on the market,,, and the standard deviation of the cash flow, :,. (3) V Standard deviations are always non-negative, so Equation (3) shows that the roject s beta and the correlation of the cash flow with the market return are of the same sign if and only if V is ositive. Suose, however, that V is negative. If the cash flow has a ositive correlation with the market, Equation (3) shows that the roject s beta must be negative. In other words, the roject beta is of the oosite sign as the correlation of cash flow with the market if V is 5
negative. 2 As we discuss in section III.B., this inverse relationshi between beta and cash flow correlation for rojects with negative cash flows is the source of the debate in the finance literature. The roject beta also can be exressed without exlicit reference to the resent value of the cash flow:, (1 + k RF ). (4) ( ) ( RP) Equation (4) rovides the key to the correct rocedure for the valuation of a nonoerating cash flow. 3 First, estimate the roject beta for the non-oerating cash flow using equation (4). Second, use the CAP to estimate the risk-adjusted discount rate. Third, discount the exected non-oerating cash flow at the risk-adjusted discount rate. Section IV describes how to imlement this technique within the context of a comany s existing caital budgeting rocesses., III.A. Reconciliation of Conflicting Recommendations Some researchers have stated that the same rocedure should be used to adjust the discount rates for ositive and negative cash flows. Others have stated that a negative cash flow should be discounted at a lower discount rate to avoid rewarding a roject for being risky. As we show in this section, both of these aarently contradictory statements can be true. To see this, consider two rojects P and N whose exected cash flows have identical absolute values, standard deviations, and correlations with the market. Suose roject P has ositive exected cash flow > 0 and roject N has negative exected cash flow < 0. Equation (4) shows that unless is equal to zero, the two rojects will have different betas. In articular, if the correlation between the cash flows and the market is ositive, then Project N 2 Similarly, the Aendix shows that,, when V is ositive, but, -, when V is negative. 3 Equation (4) rovides the correct rocedure for the valuation of any cash flow whether oerating or nonoerating, ositive or negative. However, since most oerating cash flows are assumed to be similar in risk to the rest of the firm s cash flows, the alication of Equation (4) would result in the same roject beta as the comany s beta. Therefore, it is correct and easier to directly use the firm s beta when finding a discount rate for oerating cash flows. 6
will have a negative roject beta and Project P will have a ositive roject beta. 4 This means that using the same rocedure to adjust for risk will result in different discount rates for rojects with negative versus ositive exected cash flows, even if the risk arameters of the rojects are identical. 5 Another imlication of this result is that an increase in risk (i.e. an increase in the standard deviation of cash flow) will affect the betas of the ositive and negative exected cash flow rojects differently. If the correlation is ositive and the standard deviation of cash flow is increased, then P will increase, while 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 in risk reduces the value of Project P, which has a ositive exected cash flow. An increase in risk also reduces the value of Project N, which has a negative exected cash flow, from a negative value to an even more negative value. This reduction in value is due to the ositive correlation between the cash flows and the market. Both rojects have cash flows that are higher (i.e., less negative for Project N) when the market is doing well, and both rojects have cash flows that are lower when the market is doing oorly. Therefore, an increase in risk aroriately enalizes the values of both rojects. Notice that the reverse occurs when the correlation is negative. In this case, Project P has a negative roject beta, and Project N has a ositive roject beta. An increase in risk causes the discount rate for the ositive cash flow roject to decrease and the discount rate for the negative cash flow roject to increase. This causes the values of both rojects to increase (i.e., become less negative for Project N). This is also consistent with intuition: both rojects have better cash flows when the market is doing oorly, which makes the Project P more valuable to investors and Project N less harmful to investors. In other words, the rojects act as hedges. 4 There is a singularity in Equation (4) for extremely risky cash flows when, RP. If and are both ositive then if increases enough, it will drive the value of P to zero and P to infinity. This induces a sign change from ositive to negative for P and V at the singularity. Further increases in will then increase P from and make the value of P even more negative. See Gallagher and Zumwalt (1991) for a discussion of this issue. For exository simlicity, we will focus on the situation with normal levels of cash flow risk that generate finite betas and for which ositive exected value cash flows have a ositive resent value, and negative exected value cash flows have a negative resent value. 5 Note that in general P will not be equal to N, nor will it be equal to - N. 7
Equations (4) and (1) show that a roject with a riskless cash flow will have a beta of zero, and hence should be discounted at the risk free rate, as exected. The discussion above also shows that all risky negative cash flows that are ositively 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 ositive 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 oortunities. For examle, consider Project P, which has a single ositive exected cash flow. Project B is the same as Project P, excet that its cash flow is multilied by negative one. To revent arbitrage oortunities, 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 rojects is zero with certainty. This will occur only if the betas of both rojects are identical. An insection of Equation (4) reveals that this is indeed the case. The exected cash flow,, for Projects P and B are of oosite signs, but so are the correlations,,, of Projects P and B. Substituting into Equation (4) shows that the betas of the two rojects are identical. Therefore, their values sum to zero, and there is no oortunity for arbitrage. The risk-adjustment technique is also consistent with the standard ractice of discounting all of the net oerating cash flows of a roject at the same discount rate, where net oerating cash flows are defined as oerating revenues minus oerating costs. Suose er unit oerating revenues and er unit oerating costs are constants, but that the quantity sold is risky. This imlies that revenues and costs are scalar multiles of the quantity sold. It also imlies that the standard deviation of revenues and the standard deviation of costs are scalar multiles 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 alication of Equation (4) shows that the roject beta for the net oerating cash flows is the same as the roject beta for just the revenues or the roject beta for just the costs. Therefore, discounting the net cash flows (i.e., finding the resent value of the difference between the revenues and costs) results in the same resent value as finding the difference between the resent value of the revenues and the resent value of the costs. 8
In summary, the conflicting recommendations in the literature for finding the resent value of a negative cash flow are due to different imlicit assumtions about the correlation between the cash flow and the market. As Equation (4) shows, alication of the same riskadjustment rocedure to rojects with a negative non-oerating cash flow and to rojects with a ositive non-oerating cash flow can lead to either different or identical discount rates, deending on the correlation of the cash flow with the market. IV. anagerial Guidelines for Alication of the Risk-Adjustment Technique We illustrate the alication of the rocedure using an examle. Consider a roject that has in addition to its oerating cash flow a terminal non-oerating cost in one year that is exected to be -$10 million. If this were an oerating cost with the same risk as other cash flows associated with the roject, then it would be aroriate to discount this cash flow along with the other roject cash flows at the roject s cost of caital. However this is a non-oerating cost, and has risk that is different from the risk of the roject cash flows, so it is inaroriate to discount it at the roject s cost of caital. Equation (4) must be used to find the aroriate discount rate for this cash flow. The keys to alying Equation (4) are the estimation of the cost s standard deviation and correlation with the market. A good starting oint for the standard deviation of the exected cost is the standard deviation of the firm s other cash flows, based on ast values of the firm s free cash flows. This firm s free cash flows have averaged $200 million er year for the last 8 years, with a standard deviation of $40 million. The ratio of the standard deviation to the average, called the coefficient of variation, is $40/$200 20%. If the non-oerating cost is similar in risk to the risk of the firm s other cash flows, then a first estimate of the standard deviation of the cost can be found by multilying the exected cost by the firm s coefficient of variation of 20%. This yields an estimate of the standard deviation of cost equal to $2 million; i.e., 2 0.20(10). From here, simulation and/or managerial judgment should be used to imrove this estimate. Suose that simulation indicates that this cost is roughly twice as risky as the firm s other cash flows. Therefore, the final estimate of the cost s standard deviation is $4 million. Similarly, historical data can be used to rovide a good starting oint for the estimate of the correlation. The firm s stock has a beta equal to 1.0 and a standard deviation of returns equal to 36%, based on ast returns. The standard deviation of the market return has averaged 9
aroximately 18%, deending on the articular stock index and historical eriod. Rearranging Equation (2), we can find the correlation between the firm s returns and the stock market: ( 1.0) 0.18 0.36 f f, f 0.5. (2a) As shown in the aendix, the absolute value of the correlation between the firm s cash flows and the market return is the same as the absolute value of the correlation between the firm s return and the market return. Therefore, 0.5 is a reasonable estimate of the absolute value of the correlation between the firm s oerating cash flows and the market return. At this oint, 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-oerating cash flow associated with this articular roject. In this examle, managers believe that the negative cash flow has a ositive correlation with the market, but 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 0.2, rather than the correlation of 0.5. Suose the risk free rate is 5% and the market risk remium is 6%. Using Equation (4), the beta of the non-oerating cash flow is: (, ) ( (1 + k, RF ) (4)(0.2)(1.05) 0.46. RP) [(0.18)( 10)] [(4)(0.2)(0.06)] Using Equation (1), the discount rate is: k k RF + RP 0.05 + ( 0.46)(0.06) 0.022. (1a) Notice that this discount rate is less than the comany s cost of caital of 11% (based on the comany s beta of 1.0), and is even less than the risk free rate. This is because the roject beta is negative, arising from the negative exected value of the cash flow and the ositive 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 exected cost of $10 million at the 2.2% discount rate results in a resent value of -$9.78 million (-9.78-1/1.022). In this examle, the exected cost is relatively small in comarison to the firm s other cash flows, and is not large enough to ose a threat to the firm s solvency. If the exected cost is large enough to materially affect the firm s likelihood of bankrutcy, then the exected bankrutcy cost should be added to the exected cost of the roject. For such bet-the-comany (4a) 10
rojects, it might even be better to use real otion 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 rovides no formal models relating non-systematic risk and required rates of return, there are some heuristic guidelines for managers. For examle, consider a roject with a negative exected cash flow. The first ste is to estimate the riskadjusted discount rate as describe in the revious sections. The second ste is to further adjust the discount rate to account for the imact of non-systematic risk. In articular, does the roject increase or decrease the standard deviation of the firm s total cash flow? In other words, is the non-systematic ortion of the roject s cash flow correlated with the firm s other cash flows? If the roject increases the standard deviation of the firm s cash flows, then the reviously calculated discount rate should be decreased to enalize the roject. If the roject decreases the standard deviation of the firm s cash flows, then the discount rate should be increased to reward the roject for acting to hedge the firm s other cash flows. IV. Summary any rojects have costs that are caused by a roject but that are not art of the roject s normal oerating cash flows. The key to finding the resent value of such a non-oerating cost is to exlicitly estimate the standard deviation of the cost and the correlation between the cost and the market return. As we show in this aer, these estimates can be used to calculate the aroriate discount rate to be used when finding the resent value of the cost. For a roject that consists of a single cash flow, we show how to calculate the beta for the roject. If the value of the roject is negative, which is tyically the case for a roject consisting of a single non-oerating cost, there is an inverse relationshi between the roject s beta and the correlation of the cash flow with the market return. This inverse relationshi reconciles revious conflicts in the literature. In articular, we show how alication of the same risk-adjustment rocedure to ositive and negative cash flows can lead to either different betas for ositive and negative cash flows, or the same betas, deending on the correlation of the cash flow with the market. When cash flows are ositively correlated with the market, the addition of risk to a nonoerating cost means that the discount rate should be reduced below the risk free rate when finding the resent value of the cost. In other words, the addition of risk does in fact reduce the 11
value of such a roject. When cash flows are negatively correlated with the market, the addition of risk to a non-oerating cost means that the discount rate should be increased above the risk free rate when finding the resent value of the cost. In this case the addition of risk increases the value of such a roject. 12
References Ariel, R., 1998, Risk Adjusted Discount Rates and the Present Value of Risky Costs, The Financial Review (Vol. 33), 17-30. Beedles, W.L., 1978, Evaluating Negative Benefits, Journal of Financial and Quantitative Analysis (Vol. 13, arch), 173-175. Berry, R.H., and R.G. Dyson, 1980, On the Negative Risk Premium for Risk Adjusted Discount Rates, Journal of Business Finance and Accounting, (Vol. 7, No. 3), 427-436. Booth, L.D., 1982, Correct Procedures for the Evaluation of Risky Cash Outflows, Journal of Financial and Quantitative Analysis, (Vol. XVII, No. 2, June), 287-300. Booth, L.D., 1983, On the Negative Risk Premium for Risk Adjusted Discount Rates: A Comment and Extension, Journal of Business Finance and Accounting, (Vol. 10, No. 1), 147-155. Celic, S.E., and R.H. Pettway, 1979, Some Observations on Risk-Adjusted Discount Rates: A Comment, Journal of Finance, (Vol. XXXIV, No. 4, Setember), 1061-1063. Dyson, R.G., and R.H. Berry, 1983, On the Negative Risk Premium for Risk Adjusted Discount Rates: A Rely, Journal of Business Finance and Accounting, (Vol. 10, No. 1), 157-159. Gallagher, T.J., and J.K. Zumwalt, 1991, Risk-Adjusted Discount Rates Revisited, The Financial Review, (Vol. 26, No. 1, February), 105-114. Lewellen, W.G., 1977, Some Observations on Risk-Adjusted Discount Rates, Journal of Finance, (Vol. XXXII, No. 4, Setember), 1331-1337. Lewellen, W.G., 1979, Rely to Pettway and Celec, Journal of Finance, (Vol. XXXIV, No. 4, Setember), 1065-1066. iles, J., and D. Choi, 1979, Comment: Evaluating Negative Benefits, Journal of Financial and Quantitative Analysis, (Vol. XIV, No. 5, December), 1095-1099. Pettway, R. and S.E. Celic, Thrust and Parry, Financial anagement, (Vol. 4, Winter), 7-11. Weston, J.F., and T. E. Coeland, 1992, anagerial Finance, Ninth Edition. (Dryden Press, New York: NY). 13
Aendix Let R denote the return on a roject and R denote the market return. Let denote the standard deviation of a roject s return, denote the standard deviation of the market return, and, denote the correlation between the roject s return and the market return. The beta of a roject, denoted by, is defined as: COV [ R, R ],. (A1) 2 Now consider a roject consisting of a single cash flow,. Let denote the standard deviation of the cash flow and let, denote the correlation between the cash flow and the market return. If V is the resent value of, then the return to this roject is R V (A2) and the standard deviation of the roject s return is STD V (A3) abs(v ) The correlation of the roject s return with the market,,, is COV, R cov( R, R ) sign( V ) COV [, R ] V, sign( V ) abs( V ) and hence the correlation between the cash flow and the market return is numerically equal to the correlation between the roject s return and the market return multilied by the sign of V. The beta of the cash flow is defined as: COV [, R ],, (A4) 2 14
while the beta of the roject is: 1 COV, R COV [, R ] V V 2 2 V. (A5) Note that is different from a conventional beta in the sense that a of 1.0 is only interreted as average risk if the roject s value is also 1.0. Otherwise, a equal to the roject s value, V, indicates that the roject is of average risk. Let k denote the required rate of return on a roject, k RF denote the risk free rate, and RP denote the market risk remium. The CAP defines the required rate of return on a roject as: The value of the roject is: k k + RP. (A6) RF P V 1+ RP (A7) ( k + ). RF P Substituting the definition of the roject beta from Equation (A5) and the cash flow beta from Equation (A4) into (A7) and rearranging yields the well known risk neutral valuation result: V, ( 1+ k ) RF RP. (A8) Substituting Equation (A8) and Equation (A4) into Equation (A5) yields an exression of the roject beta in terms of the characteristics of the cash flow and the CAP:, (1 + k RF ). (A9) ( ) ( RP), 15