- Email: [email protected]
Volatility estimation for stochastic project value models
European Journal of Operational Research 220 (2012) 642–648
Contents lists available at SciVerse ScienceDirect
European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Stochastics and Statistics
Volatility estimation for stochastic project value models Luiz E. Brandão a, James S. Dyer b, Warren J. Hahn c,⇑ a
Pontifícia Universidade Católica do Rio de Janeiro, Rua Marquês de São Vicente 225, Gávea, Rio de Janeiro, 22450-900, RJ, Brazil McCombs School of Business, University of Texas at Austin, 1 University Station, B6000, Austin, TX 78712-1178, United States c Graziadio School of Business and Management, Pepperdine University, 24255 Pacific Coast Highway, Malibu, CA 90263, United States b
a r t i c l e
i n f o
Article history: Received 20 January 2011 Accepted 31 January 2012 Available online 6 February 2012 Keywords: Volatility Real options Simulation Investment decisions
a b s t r a c t One of the key parameters in modeling capital budgeting decisions for investments with embedded options is the project volatility. Most often, however, there is no market or historical data available to provide an accurate estimate for this parameter. A common approach to estimating the project volatility in such instances is to use a Monte Carlo simulation where one or more sources of uncertainty are consolidated into a single stochastic process for the project cash flows, from which the volatility parameter can be determined. Nonetheless, the simulation estimation method originally suggested for this purpose systematically overstates the project volatility, which can result in incorrect option values and non-optimal investment decisions. Examples that illustrate this issue numerically have appeared in several recent papers, along with revised estimation methods that address this problem. In this article, we extend that work by showing analytically the source of the overestimation bias and the adjustment necessary to remove it. We then generalize this development for the cases of levered cash flows and non-constant volatility. In each case, we use an example problem to show how a revised estimation methodology can be applied. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction Many capital budgeting decisions involve projects with uncertain cash flows that have significant managerial flexibilities, or options. In these cases, real options pricing models are the method of choice to support decision making, as traditional methods such as the discounted cash flow (DCF) do not capture the value of the options that may be embedded in the project. A critical part of this approach, however, is the estimation of the volatility for the stochastic project value model, since this parameter effectively specifies the degree of uncertainty confronting the decision-maker. The relationship between volatility and real option value is well known (Hull, 2006). Trigeorgis (1990) shows that a 50% increase in volatility may result in a 40% increase in the value of a real option, while Keswani and Shackleton (2006) find variations of up to 210% in project real option values when volatility increases from 10% to 30%, depending on the types of options under consideration. These studies demonstrate the importance of the correct estimation of the volatility parameter for a real options model. In the case of financial options, the determination of the volatility for the stochastic process model of the underlying asset is relatively straightforward, since volatility can be estimated from market or historical data. For commodities such as oil or electricity, ⇑ Corresponding author. Tel.: +1 310 506 8542; fax: +1 310 506 4126. E-mail address: [email protected] (W.J. Hahn). 0377-2217/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2012.01.059
the volatility can be obtained from futures market data (Bøckman et al., 2008). However, for non-market traded assets such as real projects, this data is usually unavailable. Some authors adopt ad hoc rules to exogenously determine the project volatility. One common assumption is that the volatilities of the underlying risky asset and that of the project are the same (Paddock et al., 1988; Trigeorgis, 1996). Dixit and Pindyck (1994) adopt the volatility of the stock market as a proxy for the project volatility. On the other hand, Copeland and Antikarov (2003), hereafter referred to as C&A, show that the volatility of the underlying asset and that of the project are not the same, as the project volatility is also influenced by the fixed costs that arise from the operational and financial leverage of the project (Tufano, 1998). Given the sensitivity of estimates of the value of real options projects to changes in volatility values, errors in volatility estimation can lead to suboptimal investment decisions. C&A present an approach to valuing real options based on the notion that the present value of the project without options is the best unbiased estimator of the ‘‘market’’ value of the project, an assumption they term the Marketed Asset Disclaimer (MAD). Thus, under this assumption, the project without options serves as the underlying asset in the option valuation model. If the changes in the value of the project without options are then assumed to vary over time according to a Geometric Brownian Motion (GBM) stochastic process, which is a commonly used model for asset values, then the project options can be valued with
L. E. Brandão et al. / European Journal of Operational Research 220 (2012) 642–648
traditional option pricing methods, such as the binomial lattice approach of Cox et al. (1979). The C&A approach utilizes Monte Carlo simulation to estimate the volatility of the GBM process when the underlying asset is the project value without options. The procedure begins with the project pro forma worksheet using expected values for project uncertainties, which is used to calculate the discounted net present value V0 of the project in time period 0. Next, key project uncertainties are entered as simulation input variables in the project cash flow pro-forma spreadsheet, so that each iteration in a simulation of the worksheet provides a new randomly sampled set of future cash flows from which the project value at the end of the ~ 1 is computed. A sample of the random variable c ~ first period V can then be calculated in each iteration as:
~ V c~ ¼ ln 1 V0
!
0
1 Pn ~ lðt1Þ F e t A; ¼ ln @P t¼1 h i n E F~t elt
ð1Þ
t¼1
~ is the project return between time zero and time 1 and F~i where c are the stochastic project cash flows in each period, i = 1, 2, . . . , n. ~, denoted as s, can be obThe estimate of the standard deviation of c tained from the simulation results. The project volatility r is then defined as the annualized percentage standard deviation of the reffi where Dt is the turns, and is estimated from the relationship psffiffiffi Dt length of the time period in years used in the cash flow pro forma ~ 1 and V0 is one year, then worksheet. If the time period between V r = s. Given the estimate of the volatility associated with the GBM process, a binomial lattice or tree can be used to approximate the evolution of the project value over time in a manner that is consistent with the project cash flows using, for example, the approach of Cox et al. (1979). This type of lattice can then be used to value real options associated with the project as discussed by C&A and by Brandao et al. (2005b). Several researchers have subsequently utilized this approach to volatility estimation in their work. Herath and Park (2002) analyzed the case of compound options with several uncorrelated underlying variables and apply the approach to an example multi-stage R&D investment problem. Cobb and Charnes (2004) extend this concept further by investigating the case of correlated inputs. Unfortunately, however, there is an issue with the C&A approach. As noted by Smith (2005), the volatility estimates derived with Eq. (1) are upwardly biased, which causes the project value model to overstate the variance of project cash flows. Brandao et al. (2005a) used an alternative approach to volatility estimation that isolates the uncertainty resolved within each period in the simulation model and eliminates this bias. This work also demonstrated how the volatility, rather than being constant throughout the life of the project, may change from period to period. The main focus of that paper, however, was on the implementation of the corrected stochastic project value model in a binomial decision tree format, which can accommodate the non-constant volatility, rather than on the volatility estimation procedure. Godinho (2006) develops a similar modification to remove this bias as well as an enhanced technique that uses regression to reduce the complexity of the volatility estimation process. However, only the first period volatility estimate is evaluated in the primary example in that work, and therefore the case of non-constant volatility is not explored. Furthermore, although Brandao et al. (2005a) and Godinho (2006) show examples of the volatility estimation bias under the C&A method and provide an intuitive solution to this problem, neither contains a rigorous analytical explanation for the bias or for the method for addressing it. In this paper, we show the source of the upward bias in the volatility estimation simulation methodology of C&A and provide a
643
derivation of an unbiased estimation approach, demonstrating its application, both analytically and empirically, to a range of problems. Using a similar approach, Davis (1998) develops a closed form model for volatility estimation and shows that the volatilities of the project and the underlying uncertainty are linked by means of a positive elasticity term. Costa Lima and Suslick (2006) extend the work of Davis (1998) to two uncertain underlying assets and apply the results to the case of a gold mine project. Both of these models assume that the uncertainties of the underlying asset follow a Geometric Brownian diffusion process and that there are no fixed costs, which limits their possible application to practical problems. In contrast, our approach is not restricted to any particular form of diffusion process, and can be adapted to situations where there are fixed costs, as well as cases where the volatility associated with the stochastic project value may change from time period to time period. This paper is organized as follows. In the next section we present an overview of the problem of real option valuation and the determination of the project volatility for a simple project. In Section 3 we evaluate the approach to estimating the project volatility proposed by C&A, comparing the results from that approach to the analytically derived volatility for a simple project. We then discuss an unbiased estimate of project volatility in Section 4, and extend this approach to the case with leveraged cash flows in Section 5. Section 6 illustrates the application of this approach to an example problem with non-constant volatility, and in Section 7 we conclude with a brief discussion.
2. Volatility of the project value Consider a project V subject to a single source of uncertainty S(t), such as the revenue from selling a product with stochastic price, and only variable costs, C(t) = cS(t), where c < 1 is a constant and the project revenues follow a GBM diffusion process as shown in Eq. (2):
dS ¼ aSdt þ rS Sdz;
ð2Þ
where a is a drift rate (mean return or growth rate), rS is the pffiffiffiffi ffi process volatility (standard deviation of returns), and dz ¼ e dt is the standard Wiener process. Let F(t) be the project cash flows, so that F(t) = S(t) cS(t) = (1 c)S(t), and define k ¼ ð1 cÞ. Since F(t) is a linear function of S(t), by Itô’s Lemma it can be shown that the process for F is also a GBM with the same parameters as S(t) (Eq. (3):
dF ¼ aFdt þ rS Fdz;
ð3Þ
where F ¼ kS. This implies that for the sample project we have described, the volatility of the underlying uncertainty S(t) and the volatility of the cash flows are the same. The stochastic model for the project value can be determined as follows. If l is the project discount rate, then at any time t = s we R1 have V s ¼ t¼s E½FðtÞelðtsÞ dt. Since E[F(s)] = F0eas, then the relationship between V(s) and F(s) is VðsÞ ¼ lFðsaÞ , l > a. Applying Itô to the cash flows process F, we obtain dV ¼ a lF a dt þ rS lF a dz. Therefore, the stochastic process for project value V can be written as:
dV ¼ aVdt þ rS Vdz:
ð4Þ
Eq. (4) shows that the volatility of the project is the same as the volatility of the cash flows, which is also equal to the volatility of the underlying uncertainty in this case, and is independent of k. These results provide the basis for evaluating volatility estimation methods in the following two sections, where we generalize this model.
644
L. E. Brandão et al. / European Journal of Operational Research 220 (2012) 642–648
3. Volatility estimation with simulation Consider a single period project where the revenues S(t) follow a GBM as in Eq. (2). The volatility of S(t) is the annualized standard deviation of the returns, which we define as G = ln(S) and dG ¼ a 12 r2S dt þ rS dz. This stochastic process is an arithmetic Brownian motion (ABM) with a volatility of rS, which is the same ~ ¼ a, as the volatility of S. The expected return for the asset S is E½a ~ ¼ ln ~ ~¼G and we denote the returns in stochastic form as a S. As shown in Eq. (1), C&A define the project volatility as the stan~ ~ ¼ ln VV 1 , V0 is the dard deviation of the project returns c , where c 0 ~ 1 ¼ Pn expected present value of the project at time 0 and V t¼1 F~t elðt1Þ is the stochastic project value at time 1. The standard deviation can be determined with a Monte Carlo simulation. Since we know from Eq. (4) that the volatility of the cash flows and of the project are the same, it must also be that ~Þ ¼ Varða ~ Þ. To check this, we determine the expression for c ~ Varðc and derive the variance and standard deviation analytically. The process for the cash flows is defined by Eq. (3). The stochastic cash flow in t = 1 is F~1 ¼ F 0 ea~ and the expected value of this cash flow at time 0 is E0 ½F~1 ¼ F 0 ea . Accordingly, the stochastic project ~ 1 for this project is V ~ 1 ¼ F 0 ea~ , and the expected value of value V the project is V 0 ¼ E0 ½F 1 el ¼ F 0 eal and the project returns can ~ ¼ lnðea~aþl Þ. The right hand side of this therefore be written as c ~ a þ l, and since the constants do not affect equation is simply a ~Þ ¼ Varða ~ Þ, which the variance of the uncertain variable, Varðc shows that the C&A simulation approach is correct for a single period project. However, we next assume that the project has two periods, as shown in Fig. 1. In this project, the deterministic and stochastic expressions for cash flows for period 1 are respectively F1 = F0ea and F~1 ¼ F 0 ea~ 0 . Similarly, for period 2, we have F2 = F0e2a and F~2 ¼ F 0 ea~ 0 þa~1 where a~ 0 and a~ 1 are the i.i.d. stochastic returns for the asset for the first (0–1) and second (1–2) periods, respectively. C&A model the stochastic cash flow F~2 as a function of the uncertainty in both periods 1 and 2. Using this approach, the expected value of the project in t = 0 will be:
V 0 ¼ F 0 eal ð1 þ eal Þ:
ð5Þ
The stochastic value of the project in period 1 can be expressed as the sum of the stochastic cash flows in period 1 plus the discounted ~ 1 ¼ F~1 þ F~2 el , which in terms of stochastic cash flows of period 2, V the period zero cash flows is:
~ 1 ¼ F 0 ea~0 þ ea~0 þa~1 l : V
ð6Þ h
~ ~ F 0 ðea0 þea0 þ~a1 l Þ F eal ð1þeal Þ
~ ¼ ln Using Eqs. (5) and (6) we obtain c 0 ~ can then be expressed as: iance of the returns c
i , and the var-
~ 0 þ ln 1 þ ea~1 l ; ~Þ ¼ Var a Var ðc
ð7Þ
which shows that with this method we obtain an upwardly biased ~ 0 and estimate of the true variance for the project value, since a ~Þ > Varða ~ 0 Þ. This result ln½1 þ ea~ 1 l > 0 are independent and Varðc is also consistent with the empirical observations made by Smith (2005). It is easy to see that this problem would be compounded for a project with three or more periods, so one should expect the
0
1
2
F0
F1
F2
V0
V1 Fig. 1. Two period project.
error in the estimate of the volatility of a project to increase as a function of t when this approach is used. To understand why this error occurs, note that the simulation process models possible realizations of future cash flows at a given ~ t is taken with respect to the relevant downtime t, so that Et ½V stream (future) uncertainty that may affect the value of Vt. Thus, in the previous two period example, each iteration of the simulation of project value at time t = 1 provides a new realization of ~ 1 ¼ F~1 þ F~2 el . However, the first step of the discrete process V can only model the uncertainty in the first year of the project, as information concerning actual realized cash flow in period 2 is not yet available. Therefore, the best unbiased estimate of this future cash flow is its expected value at time t = 1. 4. An unbiased estimate of project volatility To demonstrate how to correct this bias, we now provide an analytical development of the intuitive approach proposed by Brandao et al. (2005a). Their approach states that the volatility of the project from period 0 to period 1 depends only on the outcome of uncertainty at period 1, as the uncertainty in future periods is yet to be resolved. At time t = 1 then, the best unbiased estimator of the cash flows F~2 ; . . . ; F~n are their expected values conditional ~ is then computed as: on the outcome for F~1 . The random variable c
V~ c~ ¼ ln 1 V0
!
1 Pn h i F~1 þ t¼2 E F~t elðt1Þ A: ¼ ln @ Pn h~ i lt t¼1 E F t e 0
ð8Þ
In general, the numerator in Eq. (8) can be the value in any period t, in which case the denominator is the value in the corresponding period t 1. We will refer to this specification as the Generalized Conditional Expectations (GCE) approach, and test it using the same two period project defined in Section 3. To do this we use the project values between periods 0 and 1, but limit the simulation of uncertainty to this period only. The intuition for this can be seen by considering the analogy to the method of determining the volatility for a stock with price Pt from historical price series. In this case, the volatility is computed as the , where the market price Pt in standard deviation of returns, ln PPtþ1 t any period t is a function of the expected future returns for the stock based on current information available at time t. Therefore, in the above expression, the value Pt+1 does not include any information about the actual realized price in the next period, t + 2, or any other future period. Similarly, the period 1 uncertainty will generate stochastic values, F~1 ¼ F 0 ea~ , at the end of that period, but since no further uncertainty can yet be resolved, all subsequent cash flows will be conditional expected values such that F~2 ¼ F~1 ea , and so on, since the best unbiased estimate of F2 is the conditional expectation of F2 given the realized value of F1. The project value at time zero can be written as in Eq. (5). However, by isolating the uncertainty in period 1, we now have a different expression for the value in period 1:
~ 1 ¼ F 0 ea~ 0 ð1 þ eal Þ: V
ð9Þ
~ ¼ ln Substituting Eq. (9) into Eq. (8), we obtain c
h~ i V1 V0
~ 0 a þ l. ¼a
Since the addition of a constant to a random variable does not change the variance, the variance of the returns is then ~Þ ¼ Varða ~ 0 Þ. It can easily been shown that this method can be Varðc extended to projects with any number of periods. From Eq. (8), let Pn
a~ Pn
~ F e 0þ F ea0 þðalÞðt1Þ F 0 ea~ 1þ eðalÞðt1Þ t¼2 P ~ ¼ ln c~ ¼ ln 0 Pnt¼2 0 ðalÞt . Then c n ðalÞ ðalÞðt1Þ F e t¼1 0
F0 e
1þ
t¼2
e
~ 0 ða lÞ as be~¼a and from further simplification we arrive at c fore. This result shows that the GCE approach is unbiased in this
L. E. Brandão et al. / European Journal of Operational Research 220 (2012) 642–648
example, since the variance of the returns depends only on the var~ 0 of the project. iance of the drift rate a 5. Including a leverage effect We now consider the operating leverage effect that arises from a fixed cost x such that the project cash flows F are F ¼ kS x; 0 < k < 1, and x is a positive constant. Applying Itô’s Lemma to Eq. (2) yields dF ¼ akSdt þ rS kSdz. Since kS = F + x, we can substitute to obtain dF ¼ aðF þ xÞdt þ rS ðF þ xÞdz, which shows that the volatility of the cash flows F increases by a constant equivalent to xrS due to the operating leverage when there are fixed costs. Again, we can see that the volatility term is independent of k, and that the effect of the fixed cost x decreases as F increases. As before, the volatility of the project value V can be determined from the process for the present value of the expected cash flows F. As we are analyzing the particular discrete time period, we divide the cash flows in two: the immediate cash flows of the first period and the present value of all remaining cash flows1:
Z
1
E½FðtÞelðt1Þ dt Z 1 ¼ ððF 0 þ xÞ xÞ þ ðF 0 þ xÞeat x elðt1Þ dt;
V ¼Fþ
t¼1
or
t¼1
V¼
ðF 0
þ xÞðl a þ ea Þ
la
x x: l
ð10Þ
Applying Itô’s Lemma to the process for project cash flows, we obtain the laþea la
a
dV ¼ llaþe a ðF þ xÞadtþ ðF þ xÞrS dz. Using Eq. (10), we obtain: dV ¼ aðV þ x þ x l Þdtþ
process
for
the
project
value,
rS ðV þ x þ xlÞdz. Therefore, if X = ln V is the process for the returns of the project, the dynamics of the project returns are:
"
2 #
V 1 2 V V dX ¼ a r dz; dt þ rS V 2 V V
ð11Þ
where V ¼ V þ x þ x l. This implies that in the presence of fixed costs, the volatility of the project value is extended by the term VV . While the volatility of the project value does not depend on the proportion k of the variable cost, it is a function of the fixed cost x and increases by an amount proportional to its discounted present value, xV=l ð1 þ lÞ. If this proportion changes throughout the full project life, the project volatility will not be constant, and in general, the estimation process may require the calculation of the volatility of the remaining project value in each period. 6. Application We apply the models developed in the previous sections to an example to demonstrate how this approach corrects the upward bias in the estimation of the project volatility, and also how it can be used to adjust for non-constant volatility. Assume that there is a five period project, as shown in Fig. 2, subject to a single source of revenue uncertainty approximated by a GBM stochastic process, with a growth rate a = 6% and a volatility rS = 25%, and variable costs (1 k) equal to 30% of revenues. The discount rate is l = 10%. As there are no fixed costs, the project volatility given by dV ¼ aVdt þ rS Vdz must be the same as the volatility of the revenues, or 25%, as shown in Section 4. Monte Carlo simulation is commonly used for the purpose of risk analysis (Titman and Martin, 1 Note that F + x = kS, represents the project cash flows considering no fixed costs. Then E½F s ¼ E½kSs x ¼ kS0 eas x and E½F s ¼ ðF 0 þ xÞeas x.
645
2008), and in this case can be used to determine project volatility. For simulation purposes, the GBM process can be approximated in discrete time in Excel by setting the revenue estimate in period t + 1 equal to the revenue pffiffiffiffiffiffi estimate in period t times expðv Dtþ rS NORMINVðRANDð0; 1Þ DtÞ where v ¼ a r2S =2 and Dt = 1, since the time periods are one year.2 Using simulation, the C&A approach provides a value of rS = 36.1%, which, as we have seen, overestimates the true project volatility. To implement the GCE approach, we resort to Eq. (8) where we model the uncertainty for the Period 1 revenues and use conditional expected values for Periods 2–5. A Monte Carlo simulation with 10,000 iterations using this approach provides, as expected, a value of rS = 25%. We next consider the case where the project also has fixed cost of $3,000 in each year, as illustrated in Fig. 3. According to Eq. (11), in the presence of fixed costs the project volatility will be factored by VþxVþx=l, where the term x/l represents the perpetuity value of the fixed costs. Since the project has a life of only five years, the present value of the fixed costs during the five years at 10% is $11,372. The project volatility predicted analytically is r = (19,990 + 14,372)/19,990 25% = 42.97%, treating the five-year present value of the fixed costs as though it represents the value of a perpetuity. Applying the equation derived for perpetual cash flows to the finite life case implies that there will be an approximation error. Nonetheless we will see that this error is quite small, as illustrated in Table 1. The simulation results for the five year project provide a value of 42.5% for the GCE approach, which is within an error margin of approximately 1% of the corresponding analytic value. On the other hand, the C&A approach provides a simulation result of 66.8%. Since Eq. (11) was derived assuming perpetual cash flow streams, increasing the number of periods decreases the discrepancy between the simulation using the GCE approach and the analytic results based on the perpetuity assumption, as can be seen in Table 1. Note that since the impact of the fixed costs decreases as the revenues increase with time, the project volatility for longer project lifetimes will also decrease. The discussion in this example has been about the volatility estimate for the first period (Time 0 to Time 1). If the assumption that the changes in the value of the project follow a GBM distribution over time is true, then the volatility would be constant and the estimated volatility for the first period would apply to all subsequent periods. However, as discussed earlier, for some applications this assumption may not be appropriate, in which case the volatility must be specific to each time period. To illustrate a case in which the volatility may change in each period, we change the stochastic process for revenue from a GBM to a mean-reverting process, and also add a stochastic jump to the variable cost process (Fig. 4). The mean-reverting process for revenue is a simple one factor process (Ornstein–Uhlenbeck) of Rt Þdt þ rR dzt , where Rt is the log of revenue, the form dRt ¼ gðR g is the mean reversion coefficient, R is the log of the long term equilibrium revenue, rR is the revenue process volatility and dz is a standard Weiner process. To model variable costs, we use a simple discrete jump process for the variable cost ratio, which has an initial value of 30% of revenue. We assume that this ratio will remain constant until a one-time technology breakthrough reduces it to 15%, and that there is a 50% probability of the breakthrough occurring in each year. These modifications might apply to a real example where revenue depends on a price that reaches
2 Note that the Excel function used to approximate the GBM process in each period can be simplified if a commercial Monte Carlo simulation add-in such as @Risk or Crystal Ball is used. For example, the Excel function using @Risk would simply be pffiffiffiffiffiffi expðRISKNORMALv Dt; rS Dt Þ.
646
L. E. Brandão et al. / European Journal of Operational Research 220 (2012) 642–648
0
1
2
3
4
5
Revenue 10,000 Variable Costs 30% Investment (20,000)
$ 1000
10,600 (3,180)
11,236 (3,371)
11,910 (3,573)
12,625 (3,787)
13,382 (4,015)
7,420
7,865
8,337
8,837
9,368
Free Cash Flow
(20,000)
PV0 = Invest. = NPV0 =
31,362 (20,000) 11,362
Revenue Parameters
µ =
10%
Growth (α) = σ=
6.0% 25%
Fig. 2. Example project with variable costs only.
$ 1000
0
Revenue 10,000 Variable Costs 30% Fixed Costs Investment (20,000) Free Cash Flow
Invest. = NPV0 =
1
2
3
4
5
10,600 11,236 (3,180) (3,371) (3,000) (3,000)
11,910 (3,573) (3,000)
12,625 (3,787) (3,000)
13,382 (4,015) (3,000)
5,337
5,837
6,368
(20,000)
4,420
V0 19,990 (20,000) (10)
V1 21,989
Revenue Parameters
4,865
µ =
10%
Growth (α) = σ=
6.0% 25%
Fig. 3. Example project with variable and fixed costs.
Table 1 Comparison of analytic and simulation results for GCE approach. Number of periods
Vol. Analytic (%)
Vol. GCE (%)
Vol. C&A (%)
5 10 20 50 100 200
43.0 38.7 35.0 31.5 30.5 30.3
42.5 39.0 35.2 31.5 30.5 30.3
66.8 82.7 95.8 108.2 111.7 112.9
Infinite
30.3
30.3
–
(reverts to) some equilibrium level and variable costs are affected by changes in technology. In such an example, one might expect the volatility to change from period to period because, unlike the case with a GBM, there are two processes that have variances that do not change in constant proportion with time. Fortunately, a multi-period analysis can be easily performed by applying Eq. (8) to each time interval t and t + 1 and by defining the project volatility in each particular ~t where year t as the standard deviation of the returns c ~ V tþ1 c~t ¼ ln V t and Vt is the project value in year t. Thus, we estimate
Fig. 4. Example project with mean-reverting revenues and modified variable costs.
L. E. Brandão et al. / European Journal of Operational Research 220 (2012) 642–648 Table 2 Yearly volatility for revised example.
647
Table 3 Regression equations for conditional expectations.
Period
Vol. (C&A) (%)
Vol. (GCE) (%)
R2 jR1 ¼ 1452:56128 þ 1:21312 R1 þ 0:00004 ðR1 Þ2
1 2 3 4 5
27.5 27.8 28.2 28.7 30.4
15.6 18.0 20.9 24.7 30.4
R3 jR2 ¼ 2642:30177 þ 0:81290 R2 þ 0:00001 ðR2 Þ2 R4 jR3 ¼ 4391:84083 þ 0:50303 R3 þ 0:00000 ðR3 Þ2 R5 jR4 ¼ 3394:18464 þ 0:69232 R4 þ 0:00000 ðR4 Þ2 VC2 jVC1 ¼ 305:46346 þ 0:99277 VC1 þ 0:00012 ðVC1 Þ2 VC3 jVC2 ¼ 555:93824 þ 0:63474 VC2 þ 0:00001 ðVC2 Þ2 VC4 jVC3 ¼ 659:55541 þ 0:56747 VC3 þ 0:00002 ðVC3 Þ2
the volatility five times, once in each period, with the results shown in Table 2, along with the volatility estimated by the C&A approach. In an example such as this one, it becomes difficult, if not impossible, to calculate the volatility analytically for comparison and benchmarking. However, we can use the same simulation run in which we estimate the volatility for the binomial tree to numerically generate the distribution of cash flows in each period. Then, a binomial tree using the Cox et al. (1979) formulae can be built with the different volatility estimates from Table 2 used in each period. This binomial tree also produces probability distributions for cash flows in each period (obtained by multiplying all of the endpoints by their respective cash flow values). By comparing the distributions, we can evaluate how well the binomial tree with the estimated period-by-period volatilities approximates the actual distribution of cash flows, as modeled by simulation. We conducted this analysis for the example above, and focused on the cash flows in the last period. For all three methods – simulation and binomial trees using volatilities estimated using both the C&A method and the GCE approach, the mean value for Period 5 cash flow was $6,486. However the standard deviation of this cash flow, determined to be $2,466 by simulation, was $3,989 (62% higher) in the binomial tree with the volatility estimates from the C&A approach. The standard deviation of the Period 5 cash flow from the binomial tree using the GCE approach was also slightly high, but was a much closer approximation to the simulation standard deviation, at $2,932. Additionally, we have plotted the three distributions in Fig. 5, which shows the improved fit of the binomial tree with GCE estimates of period-by-period volatility, which had a root mean square error of 2.25%, compared to a 10.55% for the binomial tree with the C&A volatility estimates. To implement the GCE approach, it is obviously necessary to calculate the conditional expectation for each relevant uncertainty in the project. This is relatively straightforward in cases where the uncertainties can be modeled with stochastic processes. For example, for the revenue uncertainty in this example, which has a sto Rt Þdt þ rR dzt , we can chastic process of the form dRt ¼ gðR calculate the conditional expectation in the relevant downstream periods simply by omitting the second term (variance term). However, in general, we may not have stochastic processes for all pro-
Cumulative Probability
1.0 0.8 0.6 Simulated
0.4
GCE C&A
0.2
VC5 jVC4 ¼ 621:70545 þ 0:56831 VC4 þ 0:00001 ðVC4 Þ2
ject uncertainties, or otherwise may not be able to calculate the conditional expectations analytically. Godinho (2006) discusses this problem and proposes using two different numerical methods based on simulation and regression to determine the conditional expectation of the present value of the project based on the information about the uncertainties. In both of the proposed methods, a regression equation of the following form is obtained for the conditional expectation:
E½VjX 1 . . . X n ¼ a0 þ a1 X 1 þ a2 ðX 1 Þ2 þ þ ak1 X n þ ak ðX n Þ2 ;
ð12Þ
based on simulated data for uncertainties X1 to Xn. Unfortunately, this approach requires regression equations with 2n+1 terms for each conditional expectation to be estimated, which may require large numbers of iterations for an accurate estimate, especially if the numbers for the present value and the uncertainties are of different orders of magnitude. We propose instead a simpler two-step method that easily fits within the framework of our GCE method, and illustrate it by reworking the above example. In the first step, we simulate both stochastic processes in the proforma shown in Fig. 4 fully from beginning to period 5. We then use the simulated data to perform regressions of Rt on Rt1 and (Rt1)2 and of VCt on VCt1 and (VCt1)2 for t = 2 to 5, where Rt and VCt are the revenue and variable cost, respectively in period t. The fitted regression equations are shown below in Table 3. Using these numerically estimated expressions for the conditional expectations in place of the analytical expressions in the GCE approach, we essentially obtain the same volatility estimates as shown in Table 2. Note that this procedure involves only regression of each uncertain variable on its own previous value, rather than regressions that potentially involve several variables, and that the subsequent calculation of cash flows and present values in the proforma is unchanged from our originally proposed method. Perhaps as a result, this approach appears to be much more efficient; we were able to obtain volatility estimates that were within one decimal place of the values in Table 2 with regressions using as few as 1000 simulated paths. This example illustrates the importance of using a volatility estimation approach that isolates the variability within each period and includes the flexibility to model changes in the magnitude of that variability across time. Even though the expected values of these distributions were identical using the three approaches that we compared, estimates of option value based on these binomial trees would be more accurate using volatility estimates derived using the GCE approach, due to an improved approximation of the variance of the distributions of cash flows. 7. Conclusions
0.0 0
2000
4000
6000
8000
10000 12000 14000 16000
Year 5 Cash Flow Fig. 5. Simulated vs. binomial tree model distributions for year 5 cash flows.
Volatility is a critical input variable for stochastic process models in real option valuation; however, as has been shown here and elsewhere, values derived via existing simulation-based estimation
648
L. E. Brandão et al. / European Journal of Operational Research 220 (2012) 642–648
methods are biased high. In this paper we have analytically verified the source of this bias and confirmed that an approach that appropriately models the temporal resolution of uncertainty and updating of conditional expectations in each discrete time increment is unbiased when the cash flows can be closely approximated by a GBM stochastic process. Furthermore we have explicitly shown how this approach can be extended to cases with leveraged cash flows and non-constant volatility. The latter is an important consideration when the assumption that the project value varies according to a GBM stochastic process is not a reasonable one. This would be the case, for example, when there is a leveraging effect on a stochastic variable or when multiple stochastic variables are to be combined into a single underlying uncertainty in an option valuation model. Accordingly, by comparing the estimated values from simulation with the GCE approach to the exact analytically derived values for an example with five time periods and two stochastic variables, we have verified that this approach provides accurate volatility estimates in each period. These results provide additional theoretical support for the application of the GCE approach to practical problems. For example, Brandao and Saraiva (2008) use the GCE approach to determine the volatility of the project value for a toll road concession project where there is a government provided minimum revenue guarantee that has option-like characteristics. Caporal and Brandao (2008) analyze the case of a small hydroelectric power generation plant under uncertainty where there is flexibility to choose the optimal power purchase agreement, and use the GCE model to estimate the volatility for the GBM process for the plant’s project value. Freitas and Brandao (2010) also adopt the same approach to project volatility estimation to analyze a college level MBA distance learning project in a fast growing market, and to show how option pricing methods can be used to determine both the project value and the optimal investment strategy. In addition, the new approach to calculating the conditional expectation for each relevant uncertainty in the project illustrated in Section 6 should enhance the use of the GCE methodology in a wider range of real world applications. The GCE approach therefore not only resolves a critical error in the estimation of project volatility using Monte Carlo simulation methods, but it also provides for more accurate valuation in a wide range of real option applications.
References Bøckman, T., Fleten, S.-E., Juliussen, E., Langhammer, H.J., Revdal, I., 2008. Investment timing and optimal capacity choice for small hydropower projects. European Journal of Operational Research 190 (1), 255–267. Brandao, L.E., Saraiva, Eduardo, 2008. The option value of government guarantees in infrastructure projects. Construction Management & Economics 26, 1171–1180. Brandao, L.E., Dyer, J.S., Hahn, W.J., 2005a. Response to comments on Brandao et al. (2005). Decision Analysis 2 (2), 103–109. Brandao, L.E., Dyer, J.S., Hahn, W.J., 2005b. Using binomial decision trees to solve real option valuation problems. Decision Analysis 2 (2), 69–88. Caporal, A., Brandao, L.E., 2008. Valuation of a power plant with the real options approach. Brazilian Business Review 5 (2), 103–120. Cobb, B.R., Charnes, J.M., 2004. Real options volatility estimation with correlated inputs. The Engineering Economist: A Journal Devoted to the Problems of Capital Investment 49 (2), 119–137. Copeland, T., Antikarov, V., 2003. Real Options: A Practitioner’s Guide. Texere, New York. Costa Lima, G.A., Suslick, S.B., 2006. Estimating the volatility of mining projects considering price and operating cost uncertainties. Resources Policy 31 (2), 86– 94. Cox, J.C., Ross, S.A., Rubinstein, M., 1979. Option pricing: A simplified approach. Journal of Financial Economics 7 (3), 229–263. Davis, G.A., 1998. Estimating volatility and dividend yield when valuing real options to invest or abandon. The Quarterly Review of Economics and Finance 38 (3), 725–754, Part 2. Dixit, A.K., Pindyck, R.S., 1994. Investment Under Uncertainty. Princeton University Press, Princeton. Freitas, A., Brandao, L.E., 2010. Real options valuation of e-learning projects. International Journal on E-Learning 9, 363–393. Godinho, P., 2006. Monte Carlo estimation of project volatility for real option analysis. Journal of Applied Finance 16 (1), 15–30. Herath, H.S.B., Park, C.S., 2002. Multi-stage capital investment opportunities as compound real options. The Engineering Economist: A Journal Devoted to the Problems of Capital Investment 47 (1), 1–27. Hull, J., 2006. Options, Futures and Other Derivatives, 6th ed. Prentice Hall, Upper Saddle River, NJ. Keswani, A., Shackleton, M.B., 2006. How real option disinvestment flexibility augments project NPV. European Journal of Operational Research 168 (1), 240– 252. Paddock, J.L., Siegel, D.R., Smith, J.L., 1988. Option valuation of claims on real assets: The case of offshore petroleum leases. Quarterly Journal of Economics 103 (3), 479–508. Smith, J.E., 2005. Alternative approaches for solving real-options problems: (Comment on Brandao et al. 2005). Decision Analysis 2 (2), 89–102. Titman, S., Martin, J.H., 2008. Valuation: The Art and the Science of Corporate Investment Decisions. Prentice Hall, Boston. Trigeorgis, L., 1990. A real options application in natural resource investments. Advances in Futures and Options Research 4, 153–164. Trigeorgis, L., 1996. Real Options, Managerial Flexibility and Strategy in Resources Allocation. MIT Press, Cambridge, MA. Tufano, P., 1998. The determinants of stock price exposure: Financial engineering and the gold mining industry. The Journal of Finance 53 (3), 1015–1052.
Contents lists available at SciVerse ScienceDirect
European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Stochastics and Statistics
Volatility estimation for stochastic project value models Luiz E. Brandão a, James S. Dyer b, Warren J. Hahn c,⇑ a
Pontifícia Universidade Católica do Rio de Janeiro, Rua Marquês de São Vicente 225, Gávea, Rio de Janeiro, 22450-900, RJ, Brazil McCombs School of Business, University of Texas at Austin, 1 University Station, B6000, Austin, TX 78712-1178, United States c Graziadio School of Business and Management, Pepperdine University, 24255 Pacific Coast Highway, Malibu, CA 90263, United States b
a r t i c l e
i n f o
Article history: Received 20 January 2011 Accepted 31 January 2012 Available online 6 February 2012 Keywords: Volatility Real options Simulation Investment decisions
a b s t r a c t One of the key parameters in modeling capital budgeting decisions for investments with embedded options is the project volatility. Most often, however, there is no market or historical data available to provide an accurate estimate for this parameter. A common approach to estimating the project volatility in such instances is to use a Monte Carlo simulation where one or more sources of uncertainty are consolidated into a single stochastic process for the project cash flows, from which the volatility parameter can be determined. Nonetheless, the simulation estimation method originally suggested for this purpose systematically overstates the project volatility, which can result in incorrect option values and non-optimal investment decisions. Examples that illustrate this issue numerically have appeared in several recent papers, along with revised estimation methods that address this problem. In this article, we extend that work by showing analytically the source of the overestimation bias and the adjustment necessary to remove it. We then generalize this development for the cases of levered cash flows and non-constant volatility. In each case, we use an example problem to show how a revised estimation methodology can be applied. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction Many capital budgeting decisions involve projects with uncertain cash flows that have significant managerial flexibilities, or options. In these cases, real options pricing models are the method of choice to support decision making, as traditional methods such as the discounted cash flow (DCF) do not capture the value of the options that may be embedded in the project. A critical part of this approach, however, is the estimation of the volatility for the stochastic project value model, since this parameter effectively specifies the degree of uncertainty confronting the decision-maker. The relationship between volatility and real option value is well known (Hull, 2006). Trigeorgis (1990) shows that a 50% increase in volatility may result in a 40% increase in the value of a real option, while Keswani and Shackleton (2006) find variations of up to 210% in project real option values when volatility increases from 10% to 30%, depending on the types of options under consideration. These studies demonstrate the importance of the correct estimation of the volatility parameter for a real options model. In the case of financial options, the determination of the volatility for the stochastic process model of the underlying asset is relatively straightforward, since volatility can be estimated from market or historical data. For commodities such as oil or electricity, ⇑ Corresponding author. Tel.: +1 310 506 8542; fax: +1 310 506 4126. E-mail address: [email protected] (W.J. Hahn). 0377-2217/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2012.01.059
the volatility can be obtained from futures market data (Bøckman et al., 2008). However, for non-market traded assets such as real projects, this data is usually unavailable. Some authors adopt ad hoc rules to exogenously determine the project volatility. One common assumption is that the volatilities of the underlying risky asset and that of the project are the same (Paddock et al., 1988; Trigeorgis, 1996). Dixit and Pindyck (1994) adopt the volatility of the stock market as a proxy for the project volatility. On the other hand, Copeland and Antikarov (2003), hereafter referred to as C&A, show that the volatility of the underlying asset and that of the project are not the same, as the project volatility is also influenced by the fixed costs that arise from the operational and financial leverage of the project (Tufano, 1998). Given the sensitivity of estimates of the value of real options projects to changes in volatility values, errors in volatility estimation can lead to suboptimal investment decisions. C&A present an approach to valuing real options based on the notion that the present value of the project without options is the best unbiased estimator of the ‘‘market’’ value of the project, an assumption they term the Marketed Asset Disclaimer (MAD). Thus, under this assumption, the project without options serves as the underlying asset in the option valuation model. If the changes in the value of the project without options are then assumed to vary over time according to a Geometric Brownian Motion (GBM) stochastic process, which is a commonly used model for asset values, then the project options can be valued with
L. E. Brandão et al. / European Journal of Operational Research 220 (2012) 642–648
traditional option pricing methods, such as the binomial lattice approach of Cox et al. (1979). The C&A approach utilizes Monte Carlo simulation to estimate the volatility of the GBM process when the underlying asset is the project value without options. The procedure begins with the project pro forma worksheet using expected values for project uncertainties, which is used to calculate the discounted net present value V0 of the project in time period 0. Next, key project uncertainties are entered as simulation input variables in the project cash flow pro-forma spreadsheet, so that each iteration in a simulation of the worksheet provides a new randomly sampled set of future cash flows from which the project value at the end of the ~ 1 is computed. A sample of the random variable c ~ first period V can then be calculated in each iteration as:
~ V c~ ¼ ln 1 V0
!
0
1 Pn ~ lðt1Þ F e t A; ¼ ln @P t¼1 h i n E F~t elt
ð1Þ
t¼1
~ is the project return between time zero and time 1 and F~i where c are the stochastic project cash flows in each period, i = 1, 2, . . . , n. ~, denoted as s, can be obThe estimate of the standard deviation of c tained from the simulation results. The project volatility r is then defined as the annualized percentage standard deviation of the reffi where Dt is the turns, and is estimated from the relationship psffiffiffi Dt length of the time period in years used in the cash flow pro forma ~ 1 and V0 is one year, then worksheet. If the time period between V r = s. Given the estimate of the volatility associated with the GBM process, a binomial lattice or tree can be used to approximate the evolution of the project value over time in a manner that is consistent with the project cash flows using, for example, the approach of Cox et al. (1979). This type of lattice can then be used to value real options associated with the project as discussed by C&A and by Brandao et al. (2005b). Several researchers have subsequently utilized this approach to volatility estimation in their work. Herath and Park (2002) analyzed the case of compound options with several uncorrelated underlying variables and apply the approach to an example multi-stage R&D investment problem. Cobb and Charnes (2004) extend this concept further by investigating the case of correlated inputs. Unfortunately, however, there is an issue with the C&A approach. As noted by Smith (2005), the volatility estimates derived with Eq. (1) are upwardly biased, which causes the project value model to overstate the variance of project cash flows. Brandao et al. (2005a) used an alternative approach to volatility estimation that isolates the uncertainty resolved within each period in the simulation model and eliminates this bias. This work also demonstrated how the volatility, rather than being constant throughout the life of the project, may change from period to period. The main focus of that paper, however, was on the implementation of the corrected stochastic project value model in a binomial decision tree format, which can accommodate the non-constant volatility, rather than on the volatility estimation procedure. Godinho (2006) develops a similar modification to remove this bias as well as an enhanced technique that uses regression to reduce the complexity of the volatility estimation process. However, only the first period volatility estimate is evaluated in the primary example in that work, and therefore the case of non-constant volatility is not explored. Furthermore, although Brandao et al. (2005a) and Godinho (2006) show examples of the volatility estimation bias under the C&A method and provide an intuitive solution to this problem, neither contains a rigorous analytical explanation for the bias or for the method for addressing it. In this paper, we show the source of the upward bias in the volatility estimation simulation methodology of C&A and provide a
643
derivation of an unbiased estimation approach, demonstrating its application, both analytically and empirically, to a range of problems. Using a similar approach, Davis (1998) develops a closed form model for volatility estimation and shows that the volatilities of the project and the underlying uncertainty are linked by means of a positive elasticity term. Costa Lima and Suslick (2006) extend the work of Davis (1998) to two uncertain underlying assets and apply the results to the case of a gold mine project. Both of these models assume that the uncertainties of the underlying asset follow a Geometric Brownian diffusion process and that there are no fixed costs, which limits their possible application to practical problems. In contrast, our approach is not restricted to any particular form of diffusion process, and can be adapted to situations where there are fixed costs, as well as cases where the volatility associated with the stochastic project value may change from time period to time period. This paper is organized as follows. In the next section we present an overview of the problem of real option valuation and the determination of the project volatility for a simple project. In Section 3 we evaluate the approach to estimating the project volatility proposed by C&A, comparing the results from that approach to the analytically derived volatility for a simple project. We then discuss an unbiased estimate of project volatility in Section 4, and extend this approach to the case with leveraged cash flows in Section 5. Section 6 illustrates the application of this approach to an example problem with non-constant volatility, and in Section 7 we conclude with a brief discussion.
2. Volatility of the project value Consider a project V subject to a single source of uncertainty S(t), such as the revenue from selling a product with stochastic price, and only variable costs, C(t) = cS(t), where c < 1 is a constant and the project revenues follow a GBM diffusion process as shown in Eq. (2):
dS ¼ aSdt þ rS Sdz;
ð2Þ
where a is a drift rate (mean return or growth rate), rS is the pffiffiffiffi ffi process volatility (standard deviation of returns), and dz ¼ e dt is the standard Wiener process. Let F(t) be the project cash flows, so that F(t) = S(t) cS(t) = (1 c)S(t), and define k ¼ ð1 cÞ. Since F(t) is a linear function of S(t), by Itô’s Lemma it can be shown that the process for F is also a GBM with the same parameters as S(t) (Eq. (3):
dF ¼ aFdt þ rS Fdz;
ð3Þ
where F ¼ kS. This implies that for the sample project we have described, the volatility of the underlying uncertainty S(t) and the volatility of the cash flows are the same. The stochastic model for the project value can be determined as follows. If l is the project discount rate, then at any time t = s we R1 have V s ¼ t¼s E½FðtÞelðtsÞ dt. Since E[F(s)] = F0eas, then the relationship between V(s) and F(s) is VðsÞ ¼ lFðsaÞ , l > a. Applying Itô to the cash flows process F, we obtain dV ¼ a lF a dt þ rS lF a dz. Therefore, the stochastic process for project value V can be written as:
dV ¼ aVdt þ rS Vdz:
ð4Þ
Eq. (4) shows that the volatility of the project is the same as the volatility of the cash flows, which is also equal to the volatility of the underlying uncertainty in this case, and is independent of k. These results provide the basis for evaluating volatility estimation methods in the following two sections, where we generalize this model.
644
L. E. Brandão et al. / European Journal of Operational Research 220 (2012) 642–648
3. Volatility estimation with simulation Consider a single period project where the revenues S(t) follow a GBM as in Eq. (2). The volatility of S(t) is the annualized standard deviation of the returns, which we define as G = ln(S) and dG ¼ a 12 r2S dt þ rS dz. This stochastic process is an arithmetic Brownian motion (ABM) with a volatility of rS, which is the same ~ ¼ a, as the volatility of S. The expected return for the asset S is E½a ~ ¼ ln ~ ~¼G and we denote the returns in stochastic form as a S. As shown in Eq. (1), C&A define the project volatility as the stan~ ~ ¼ ln VV 1 , V0 is the dard deviation of the project returns c , where c 0 ~ 1 ¼ Pn expected present value of the project at time 0 and V t¼1 F~t elðt1Þ is the stochastic project value at time 1. The standard deviation can be determined with a Monte Carlo simulation. Since we know from Eq. (4) that the volatility of the cash flows and of the project are the same, it must also be that ~Þ ¼ Varða ~ Þ. To check this, we determine the expression for c ~ Varðc and derive the variance and standard deviation analytically. The process for the cash flows is defined by Eq. (3). The stochastic cash flow in t = 1 is F~1 ¼ F 0 ea~ and the expected value of this cash flow at time 0 is E0 ½F~1 ¼ F 0 ea . Accordingly, the stochastic project ~ 1 for this project is V ~ 1 ¼ F 0 ea~ , and the expected value of value V the project is V 0 ¼ E0 ½F 1 el ¼ F 0 eal and the project returns can ~ ¼ lnðea~aþl Þ. The right hand side of this therefore be written as c ~ a þ l, and since the constants do not affect equation is simply a ~Þ ¼ Varða ~ Þ, which the variance of the uncertain variable, Varðc shows that the C&A simulation approach is correct for a single period project. However, we next assume that the project has two periods, as shown in Fig. 1. In this project, the deterministic and stochastic expressions for cash flows for period 1 are respectively F1 = F0ea and F~1 ¼ F 0 ea~ 0 . Similarly, for period 2, we have F2 = F0e2a and F~2 ¼ F 0 ea~ 0 þa~1 where a~ 0 and a~ 1 are the i.i.d. stochastic returns for the asset for the first (0–1) and second (1–2) periods, respectively. C&A model the stochastic cash flow F~2 as a function of the uncertainty in both periods 1 and 2. Using this approach, the expected value of the project in t = 0 will be:
V 0 ¼ F 0 eal ð1 þ eal Þ:
ð5Þ
The stochastic value of the project in period 1 can be expressed as the sum of the stochastic cash flows in period 1 plus the discounted ~ 1 ¼ F~1 þ F~2 el , which in terms of stochastic cash flows of period 2, V the period zero cash flows is:
~ 1 ¼ F 0 ea~0 þ ea~0 þa~1 l : V
ð6Þ h
~ ~ F 0 ðea0 þea0 þ~a1 l Þ F eal ð1þeal Þ
~ ¼ ln Using Eqs. (5) and (6) we obtain c 0 ~ can then be expressed as: iance of the returns c
i , and the var-
~ 0 þ ln 1 þ ea~1 l ; ~Þ ¼ Var a Var ðc
ð7Þ
which shows that with this method we obtain an upwardly biased ~ 0 and estimate of the true variance for the project value, since a ~Þ > Varða ~ 0 Þ. This result ln½1 þ ea~ 1 l > 0 are independent and Varðc is also consistent with the empirical observations made by Smith (2005). It is easy to see that this problem would be compounded for a project with three or more periods, so one should expect the
0
1
2
F0
F1
F2
V0
V1 Fig. 1. Two period project.
error in the estimate of the volatility of a project to increase as a function of t when this approach is used. To understand why this error occurs, note that the simulation process models possible realizations of future cash flows at a given ~ t is taken with respect to the relevant downtime t, so that Et ½V stream (future) uncertainty that may affect the value of Vt. Thus, in the previous two period example, each iteration of the simulation of project value at time t = 1 provides a new realization of ~ 1 ¼ F~1 þ F~2 el . However, the first step of the discrete process V can only model the uncertainty in the first year of the project, as information concerning actual realized cash flow in period 2 is not yet available. Therefore, the best unbiased estimate of this future cash flow is its expected value at time t = 1. 4. An unbiased estimate of project volatility To demonstrate how to correct this bias, we now provide an analytical development of the intuitive approach proposed by Brandao et al. (2005a). Their approach states that the volatility of the project from period 0 to period 1 depends only on the outcome of uncertainty at period 1, as the uncertainty in future periods is yet to be resolved. At time t = 1 then, the best unbiased estimator of the cash flows F~2 ; . . . ; F~n are their expected values conditional ~ is then computed as: on the outcome for F~1 . The random variable c
V~ c~ ¼ ln 1 V0
!
1 Pn h i F~1 þ t¼2 E F~t elðt1Þ A: ¼ ln @ Pn h~ i lt t¼1 E F t e 0
ð8Þ
In general, the numerator in Eq. (8) can be the value in any period t, in which case the denominator is the value in the corresponding period t 1. We will refer to this specification as the Generalized Conditional Expectations (GCE) approach, and test it using the same two period project defined in Section 3. To do this we use the project values between periods 0 and 1, but limit the simulation of uncertainty to this period only. The intuition for this can be seen by considering the analogy to the method of determining the volatility for a stock with price Pt from historical price series. In this case, the volatility is computed as the , where the market price Pt in standard deviation of returns, ln PPtþ1 t any period t is a function of the expected future returns for the stock based on current information available at time t. Therefore, in the above expression, the value Pt+1 does not include any information about the actual realized price in the next period, t + 2, or any other future period. Similarly, the period 1 uncertainty will generate stochastic values, F~1 ¼ F 0 ea~ , at the end of that period, but since no further uncertainty can yet be resolved, all subsequent cash flows will be conditional expected values such that F~2 ¼ F~1 ea , and so on, since the best unbiased estimate of F2 is the conditional expectation of F2 given the realized value of F1. The project value at time zero can be written as in Eq. (5). However, by isolating the uncertainty in period 1, we now have a different expression for the value in period 1:
~ 1 ¼ F 0 ea~ 0 ð1 þ eal Þ: V
ð9Þ
~ ¼ ln Substituting Eq. (9) into Eq. (8), we obtain c
h~ i V1 V0
~ 0 a þ l. ¼a
Since the addition of a constant to a random variable does not change the variance, the variance of the returns is then ~Þ ¼ Varða ~ 0 Þ. It can easily been shown that this method can be Varðc extended to projects with any number of periods. From Eq. (8), let Pn
a~ Pn
~ F e 0þ F ea0 þðalÞðt1Þ F 0 ea~ 1þ eðalÞðt1Þ t¼2 P ~ ¼ ln c~ ¼ ln 0 Pnt¼2 0 ðalÞt . Then c n ðalÞ ðalÞðt1Þ F e t¼1 0
F0 e
1þ
t¼2
e
~ 0 ða lÞ as be~¼a and from further simplification we arrive at c fore. This result shows that the GCE approach is unbiased in this
L. E. Brandão et al. / European Journal of Operational Research 220 (2012) 642–648
example, since the variance of the returns depends only on the var~ 0 of the project. iance of the drift rate a 5. Including a leverage effect We now consider the operating leverage effect that arises from a fixed cost x such that the project cash flows F are F ¼ kS x; 0 < k < 1, and x is a positive constant. Applying Itô’s Lemma to Eq. (2) yields dF ¼ akSdt þ rS kSdz. Since kS = F + x, we can substitute to obtain dF ¼ aðF þ xÞdt þ rS ðF þ xÞdz, which shows that the volatility of the cash flows F increases by a constant equivalent to xrS due to the operating leverage when there are fixed costs. Again, we can see that the volatility term is independent of k, and that the effect of the fixed cost x decreases as F increases. As before, the volatility of the project value V can be determined from the process for the present value of the expected cash flows F. As we are analyzing the particular discrete time period, we divide the cash flows in two: the immediate cash flows of the first period and the present value of all remaining cash flows1:
Z
1
E½FðtÞelðt1Þ dt Z 1 ¼ ððF 0 þ xÞ xÞ þ ðF 0 þ xÞeat x elðt1Þ dt;
V ¼Fþ
t¼1
or
t¼1
V¼
ðF 0
þ xÞðl a þ ea Þ
la
x x: l
ð10Þ
Applying Itô’s Lemma to the process for project cash flows, we obtain the laþea la
a
dV ¼ llaþe a ðF þ xÞadtþ ðF þ xÞrS dz. Using Eq. (10), we obtain: dV ¼ aðV þ x þ x l Þdtþ
process
for
the
project
value,
rS ðV þ x þ xlÞdz. Therefore, if X = ln V is the process for the returns of the project, the dynamics of the project returns are:
"
2 #
V 1 2 V V dX ¼ a r dz; dt þ rS V 2 V V
ð11Þ
where V ¼ V þ x þ x l. This implies that in the presence of fixed costs, the volatility of the project value is extended by the term VV . While the volatility of the project value does not depend on the proportion k of the variable cost, it is a function of the fixed cost x and increases by an amount proportional to its discounted present value, xV=l ð1 þ lÞ. If this proportion changes throughout the full project life, the project volatility will not be constant, and in general, the estimation process may require the calculation of the volatility of the remaining project value in each period. 6. Application We apply the models developed in the previous sections to an example to demonstrate how this approach corrects the upward bias in the estimation of the project volatility, and also how it can be used to adjust for non-constant volatility. Assume that there is a five period project, as shown in Fig. 2, subject to a single source of revenue uncertainty approximated by a GBM stochastic process, with a growth rate a = 6% and a volatility rS = 25%, and variable costs (1 k) equal to 30% of revenues. The discount rate is l = 10%. As there are no fixed costs, the project volatility given by dV ¼ aVdt þ rS Vdz must be the same as the volatility of the revenues, or 25%, as shown in Section 4. Monte Carlo simulation is commonly used for the purpose of risk analysis (Titman and Martin, 1 Note that F + x = kS, represents the project cash flows considering no fixed costs. Then E½F s ¼ E½kSs x ¼ kS0 eas x and E½F s ¼ ðF 0 þ xÞeas x.
645
2008), and in this case can be used to determine project volatility. For simulation purposes, the GBM process can be approximated in discrete time in Excel by setting the revenue estimate in period t + 1 equal to the revenue pffiffiffiffiffiffi estimate in period t times expðv Dtþ rS NORMINVðRANDð0; 1Þ DtÞ where v ¼ a r2S =2 and Dt = 1, since the time periods are one year.2 Using simulation, the C&A approach provides a value of rS = 36.1%, which, as we have seen, overestimates the true project volatility. To implement the GCE approach, we resort to Eq. (8) where we model the uncertainty for the Period 1 revenues and use conditional expected values for Periods 2–5. A Monte Carlo simulation with 10,000 iterations using this approach provides, as expected, a value of rS = 25%. We next consider the case where the project also has fixed cost of $3,000 in each year, as illustrated in Fig. 3. According to Eq. (11), in the presence of fixed costs the project volatility will be factored by VþxVþx=l, where the term x/l represents the perpetuity value of the fixed costs. Since the project has a life of only five years, the present value of the fixed costs during the five years at 10% is $11,372. The project volatility predicted analytically is r = (19,990 + 14,372)/19,990 25% = 42.97%, treating the five-year present value of the fixed costs as though it represents the value of a perpetuity. Applying the equation derived for perpetual cash flows to the finite life case implies that there will be an approximation error. Nonetheless we will see that this error is quite small, as illustrated in Table 1. The simulation results for the five year project provide a value of 42.5% for the GCE approach, which is within an error margin of approximately 1% of the corresponding analytic value. On the other hand, the C&A approach provides a simulation result of 66.8%. Since Eq. (11) was derived assuming perpetual cash flow streams, increasing the number of periods decreases the discrepancy between the simulation using the GCE approach and the analytic results based on the perpetuity assumption, as can be seen in Table 1. Note that since the impact of the fixed costs decreases as the revenues increase with time, the project volatility for longer project lifetimes will also decrease. The discussion in this example has been about the volatility estimate for the first period (Time 0 to Time 1). If the assumption that the changes in the value of the project follow a GBM distribution over time is true, then the volatility would be constant and the estimated volatility for the first period would apply to all subsequent periods. However, as discussed earlier, for some applications this assumption may not be appropriate, in which case the volatility must be specific to each time period. To illustrate a case in which the volatility may change in each period, we change the stochastic process for revenue from a GBM to a mean-reverting process, and also add a stochastic jump to the variable cost process (Fig. 4). The mean-reverting process for revenue is a simple one factor process (Ornstein–Uhlenbeck) of Rt Þdt þ rR dzt , where Rt is the log of revenue, the form dRt ¼ gðR g is the mean reversion coefficient, R is the log of the long term equilibrium revenue, rR is the revenue process volatility and dz is a standard Weiner process. To model variable costs, we use a simple discrete jump process for the variable cost ratio, which has an initial value of 30% of revenue. We assume that this ratio will remain constant until a one-time technology breakthrough reduces it to 15%, and that there is a 50% probability of the breakthrough occurring in each year. These modifications might apply to a real example where revenue depends on a price that reaches
2 Note that the Excel function used to approximate the GBM process in each period can be simplified if a commercial Monte Carlo simulation add-in such as @Risk or Crystal Ball is used. For example, the Excel function using @Risk would simply be pffiffiffiffiffiffi expðRISKNORMALv Dt; rS Dt Þ.
646
L. E. Brandão et al. / European Journal of Operational Research 220 (2012) 642–648
0
1
2
3
4
5
Revenue 10,000 Variable Costs 30% Investment (20,000)
$ 1000
10,600 (3,180)
11,236 (3,371)
11,910 (3,573)
12,625 (3,787)
13,382 (4,015)
7,420
7,865
8,337
8,837
9,368
Free Cash Flow
(20,000)
PV0 = Invest. = NPV0 =
31,362 (20,000) 11,362
Revenue Parameters
µ =
10%
Growth (α) = σ=
6.0% 25%
Fig. 2. Example project with variable costs only.
$ 1000
0
Revenue 10,000 Variable Costs 30% Fixed Costs Investment (20,000) Free Cash Flow
Invest. = NPV0 =
1
2
3
4
5
10,600 11,236 (3,180) (3,371) (3,000) (3,000)
11,910 (3,573) (3,000)
12,625 (3,787) (3,000)
13,382 (4,015) (3,000)
5,337
5,837
6,368
(20,000)
4,420
V0 19,990 (20,000) (10)
V1 21,989
Revenue Parameters
4,865
µ =
10%
Growth (α) = σ=
6.0% 25%
Fig. 3. Example project with variable and fixed costs.
Table 1 Comparison of analytic and simulation results for GCE approach. Number of periods
Vol. Analytic (%)
Vol. GCE (%)
Vol. C&A (%)
5 10 20 50 100 200
43.0 38.7 35.0 31.5 30.5 30.3
42.5 39.0 35.2 31.5 30.5 30.3
66.8 82.7 95.8 108.2 111.7 112.9
Infinite
30.3
30.3
–
(reverts to) some equilibrium level and variable costs are affected by changes in technology. In such an example, one might expect the volatility to change from period to period because, unlike the case with a GBM, there are two processes that have variances that do not change in constant proportion with time. Fortunately, a multi-period analysis can be easily performed by applying Eq. (8) to each time interval t and t + 1 and by defining the project volatility in each particular ~t where year t as the standard deviation of the returns c ~ V tþ1 c~t ¼ ln V t and Vt is the project value in year t. Thus, we estimate
Fig. 4. Example project with mean-reverting revenues and modified variable costs.
L. E. Brandão et al. / European Journal of Operational Research 220 (2012) 642–648 Table 2 Yearly volatility for revised example.
647
Table 3 Regression equations for conditional expectations.
Period
Vol. (C&A) (%)
Vol. (GCE) (%)
R2 jR1 ¼ 1452:56128 þ 1:21312 R1 þ 0:00004 ðR1 Þ2
1 2 3 4 5
27.5 27.8 28.2 28.7 30.4
15.6 18.0 20.9 24.7 30.4
R3 jR2 ¼ 2642:30177 þ 0:81290 R2 þ 0:00001 ðR2 Þ2 R4 jR3 ¼ 4391:84083 þ 0:50303 R3 þ 0:00000 ðR3 Þ2 R5 jR4 ¼ 3394:18464 þ 0:69232 R4 þ 0:00000 ðR4 Þ2 VC2 jVC1 ¼ 305:46346 þ 0:99277 VC1 þ 0:00012 ðVC1 Þ2 VC3 jVC2 ¼ 555:93824 þ 0:63474 VC2 þ 0:00001 ðVC2 Þ2 VC4 jVC3 ¼ 659:55541 þ 0:56747 VC3 þ 0:00002 ðVC3 Þ2
the volatility five times, once in each period, with the results shown in Table 2, along with the volatility estimated by the C&A approach. In an example such as this one, it becomes difficult, if not impossible, to calculate the volatility analytically for comparison and benchmarking. However, we can use the same simulation run in which we estimate the volatility for the binomial tree to numerically generate the distribution of cash flows in each period. Then, a binomial tree using the Cox et al. (1979) formulae can be built with the different volatility estimates from Table 2 used in each period. This binomial tree also produces probability distributions for cash flows in each period (obtained by multiplying all of the endpoints by their respective cash flow values). By comparing the distributions, we can evaluate how well the binomial tree with the estimated period-by-period volatilities approximates the actual distribution of cash flows, as modeled by simulation. We conducted this analysis for the example above, and focused on the cash flows in the last period. For all three methods – simulation and binomial trees using volatilities estimated using both the C&A method and the GCE approach, the mean value for Period 5 cash flow was $6,486. However the standard deviation of this cash flow, determined to be $2,466 by simulation, was $3,989 (62% higher) in the binomial tree with the volatility estimates from the C&A approach. The standard deviation of the Period 5 cash flow from the binomial tree using the GCE approach was also slightly high, but was a much closer approximation to the simulation standard deviation, at $2,932. Additionally, we have plotted the three distributions in Fig. 5, which shows the improved fit of the binomial tree with GCE estimates of period-by-period volatility, which had a root mean square error of 2.25%, compared to a 10.55% for the binomial tree with the C&A volatility estimates. To implement the GCE approach, it is obviously necessary to calculate the conditional expectation for each relevant uncertainty in the project. This is relatively straightforward in cases where the uncertainties can be modeled with stochastic processes. For example, for the revenue uncertainty in this example, which has a sto Rt Þdt þ rR dzt , we can chastic process of the form dRt ¼ gðR calculate the conditional expectation in the relevant downstream periods simply by omitting the second term (variance term). However, in general, we may not have stochastic processes for all pro-
Cumulative Probability
1.0 0.8 0.6 Simulated
0.4
GCE C&A
0.2
VC5 jVC4 ¼ 621:70545 þ 0:56831 VC4 þ 0:00001 ðVC4 Þ2
ject uncertainties, or otherwise may not be able to calculate the conditional expectations analytically. Godinho (2006) discusses this problem and proposes using two different numerical methods based on simulation and regression to determine the conditional expectation of the present value of the project based on the information about the uncertainties. In both of the proposed methods, a regression equation of the following form is obtained for the conditional expectation:
E½VjX 1 . . . X n ¼ a0 þ a1 X 1 þ a2 ðX 1 Þ2 þ þ ak1 X n þ ak ðX n Þ2 ;
ð12Þ
based on simulated data for uncertainties X1 to Xn. Unfortunately, this approach requires regression equations with 2n+1 terms for each conditional expectation to be estimated, which may require large numbers of iterations for an accurate estimate, especially if the numbers for the present value and the uncertainties are of different orders of magnitude. We propose instead a simpler two-step method that easily fits within the framework of our GCE method, and illustrate it by reworking the above example. In the first step, we simulate both stochastic processes in the proforma shown in Fig. 4 fully from beginning to period 5. We then use the simulated data to perform regressions of Rt on Rt1 and (Rt1)2 and of VCt on VCt1 and (VCt1)2 for t = 2 to 5, where Rt and VCt are the revenue and variable cost, respectively in period t. The fitted regression equations are shown below in Table 3. Using these numerically estimated expressions for the conditional expectations in place of the analytical expressions in the GCE approach, we essentially obtain the same volatility estimates as shown in Table 2. Note that this procedure involves only regression of each uncertain variable on its own previous value, rather than regressions that potentially involve several variables, and that the subsequent calculation of cash flows and present values in the proforma is unchanged from our originally proposed method. Perhaps as a result, this approach appears to be much more efficient; we were able to obtain volatility estimates that were within one decimal place of the values in Table 2 with regressions using as few as 1000 simulated paths. This example illustrates the importance of using a volatility estimation approach that isolates the variability within each period and includes the flexibility to model changes in the magnitude of that variability across time. Even though the expected values of these distributions were identical using the three approaches that we compared, estimates of option value based on these binomial trees would be more accurate using volatility estimates derived using the GCE approach, due to an improved approximation of the variance of the distributions of cash flows. 7. Conclusions
0.0 0
2000
4000
6000
8000
10000 12000 14000 16000
Year 5 Cash Flow Fig. 5. Simulated vs. binomial tree model distributions for year 5 cash flows.
Volatility is a critical input variable for stochastic process models in real option valuation; however, as has been shown here and elsewhere, values derived via existing simulation-based estimation
648
L. E. Brandão et al. / European Journal of Operational Research 220 (2012) 642–648
methods are biased high. In this paper we have analytically verified the source of this bias and confirmed that an approach that appropriately models the temporal resolution of uncertainty and updating of conditional expectations in each discrete time increment is unbiased when the cash flows can be closely approximated by a GBM stochastic process. Furthermore we have explicitly shown how this approach can be extended to cases with leveraged cash flows and non-constant volatility. The latter is an important consideration when the assumption that the project value varies according to a GBM stochastic process is not a reasonable one. This would be the case, for example, when there is a leveraging effect on a stochastic variable or when multiple stochastic variables are to be combined into a single underlying uncertainty in an option valuation model. Accordingly, by comparing the estimated values from simulation with the GCE approach to the exact analytically derived values for an example with five time periods and two stochastic variables, we have verified that this approach provides accurate volatility estimates in each period. These results provide additional theoretical support for the application of the GCE approach to practical problems. For example, Brandao and Saraiva (2008) use the GCE approach to determine the volatility of the project value for a toll road concession project where there is a government provided minimum revenue guarantee that has option-like characteristics. Caporal and Brandao (2008) analyze the case of a small hydroelectric power generation plant under uncertainty where there is flexibility to choose the optimal power purchase agreement, and use the GCE model to estimate the volatility for the GBM process for the plant’s project value. Freitas and Brandao (2010) also adopt the same approach to project volatility estimation to analyze a college level MBA distance learning project in a fast growing market, and to show how option pricing methods can be used to determine both the project value and the optimal investment strategy. In addition, the new approach to calculating the conditional expectation for each relevant uncertainty in the project illustrated in Section 6 should enhance the use of the GCE methodology in a wider range of real world applications. The GCE approach therefore not only resolves a critical error in the estimation of project volatility using Monte Carlo simulation methods, but it also provides for more accurate valuation in a wide range of real option applications.
References Bøckman, T., Fleten, S.-E., Juliussen, E., Langhammer, H.J., Revdal, I., 2008. Investment timing and optimal capacity choice for small hydropower projects. European Journal of Operational Research 190 (1), 255–267. Brandao, L.E., Saraiva, Eduardo, 2008. The option value of government guarantees in infrastructure projects. Construction Management & Economics 26, 1171–1180. Brandao, L.E., Dyer, J.S., Hahn, W.J., 2005a. Response to comments on Brandao et al. (2005). Decision Analysis 2 (2), 103–109. Brandao, L.E., Dyer, J.S., Hahn, W.J., 2005b. Using binomial decision trees to solve real option valuation problems. Decision Analysis 2 (2), 69–88. Caporal, A., Brandao, L.E., 2008. Valuation of a power plant with the real options approach. Brazilian Business Review 5 (2), 103–120. Cobb, B.R., Charnes, J.M., 2004. Real options volatility estimation with correlated inputs. The Engineering Economist: A Journal Devoted to the Problems of Capital Investment 49 (2), 119–137. Copeland, T., Antikarov, V., 2003. Real Options: A Practitioner’s Guide. Texere, New York. Costa Lima, G.A., Suslick, S.B., 2006. Estimating the volatility of mining projects considering price and operating cost uncertainties. Resources Policy 31 (2), 86– 94. Cox, J.C., Ross, S.A., Rubinstein, M., 1979. Option pricing: A simplified approach. Journal of Financial Economics 7 (3), 229–263. Davis, G.A., 1998. Estimating volatility and dividend yield when valuing real options to invest or abandon. The Quarterly Review of Economics and Finance 38 (3), 725–754, Part 2. Dixit, A.K., Pindyck, R.S., 1994. Investment Under Uncertainty. Princeton University Press, Princeton. Freitas, A., Brandao, L.E., 2010. Real options valuation of e-learning projects. International Journal on E-Learning 9, 363–393. Godinho, P., 2006. Monte Carlo estimation of project volatility for real option analysis. Journal of Applied Finance 16 (1), 15–30. Herath, H.S.B., Park, C.S., 2002. Multi-stage capital investment opportunities as compound real options. The Engineering Economist: A Journal Devoted to the Problems of Capital Investment 47 (1), 1–27. Hull, J., 2006. Options, Futures and Other Derivatives, 6th ed. Prentice Hall, Upper Saddle River, NJ. Keswani, A., Shackleton, M.B., 2006. How real option disinvestment flexibility augments project NPV. European Journal of Operational Research 168 (1), 240– 252. Paddock, J.L., Siegel, D.R., Smith, J.L., 1988. Option valuation of claims on real assets: The case of offshore petroleum leases. Quarterly Journal of Economics 103 (3), 479–508. Smith, J.E., 2005. Alternative approaches for solving real-options problems: (Comment on Brandao et al. 2005). Decision Analysis 2 (2), 89–102. Titman, S., Martin, J.H., 2008. Valuation: The Art and the Science of Corporate Investment Decisions. Prentice Hall, Boston. Trigeorgis, L., 1990. A real options application in natural resource investments. Advances in Futures and Options Research 4, 153–164. Trigeorgis, L., 1996. Real Options, Managerial Flexibility and Strategy in Resources Allocation. MIT Press, Cambridge, MA. Tufano, P., 1998. The determinants of stock price exposure: Financial engineering and the gold mining industry. The Journal of Finance 53 (3), 1015–1052.