Faculdade de Economia da Universidade de Coimbra

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1 Faculdade de Ecoomia da Uiversidade de Coimbra Grupo de Estudos Moetários e Fiaceiros (GEMF) Av. Dias da Silva, COIMBRA, PORTUGAL gemf@fe.uc.pt PEDRO GODINHO Estimatig State-Depedet Volatility of Ivestmet Projects: A Simulatio Approach ESTUDOS DO GEMF N.º 0 05

2 Estimatig state-depedet volatility of ivestmet projects: A simulatio approach Pedro Godiho a a GEMF ad Faculty of Ecoomics of the Uiversity of Coimbra, Av. Dias da Silva, 65, Coimbra, Portugal. Phoe: pgodiho@fe.uc.pt Abstract Project volatility is a essetial parameter for real optios aalysis, ad it may also be useful for risk aalysis. May volatility estimatio procedures oly cosider the volatility i the first year of the project. Others cosider that differet years may have differet values of the project volatility. I this paper I show that volatility may chage ot oly with time but also with the state of the project. I cosider two possible defiitios for the project volatility, the log-variace ad the variace of the project value, ad I propose three procedures for estimatig state-depedet volatility: two-level simulatio, oe ad a half level simulatio ad a regressio procedure. Computatioal experimets show that the oe ad a half level simulatio procedure ad the regressio procedure lead to the most accurate estimatios of project volatility. Keywords: Fiace; Simulatio; Project volatility; Real optios; Ivestmet aalysis

3 . Itroductio Despite havig ejoyed some sigificat success i academe, the adoptio of real optios aalysis by firms has bee quite slow (e.g., Triatis, 005, Block, 007, Baker et al., 0). Sice real optios models have bee aroud for some decades, ad they allow practitioers to avoid some severe drawbacks of the classical discouted cash flows methodologies (see, e.g., Trigeorgis, 993), such slow rate of adoptio may seem surprisig. However, it may be explaied by the mathematical sophisticatio that is required to uderstad ad use most real optios models (Baker et al., 0). Mu (00, p. ) poits out that it is advatageous to use lattice approaches i maagemet discussios, sice they are ituitive ad easy to uderstad. Bradão et al. (005, p. 85) poit out that biomial decisio trees may have eve more ituitive appeal. The quality of results obtaied with lattice or decisio tree approaches depeds, amog other thigs, o a accurate modelig of uderlyig asset volatility. If there is oly oe sigificat source of project risk ad that source of risk is a traded asset, the such asset may be used as the sigle uderlyig asset of the project, ad market data may be used for estimatig its volatility (e.g., Smit, 997). However, most real life projects have several sources of risk, ad some of them are ot traded. If may risk sources are simultaeously modeled, the problem becomes multi-dimesioal, ad the use of lattices or decisio trees may become impractical. Copelad ad Atikarov (00) propose usig the project without optios as the uderlyig asset of the aalysis i a lattice-based approach. Such approach may lead to some sub-optimal decisios (if the optimal decisios would deped o the values of all state variables), but it allows the problem to remai sigle-dimesioal. This is, therefore, a useful approach, sice it provides a simple way of icorporatig risk ad maagerial flexibility i the aalysis, allowig a more accurate valuatio tha traditioal discouted cash flows methodologies. Copelad ad Atikarov s (00) approach requires a estimate of the project volatility, as is also the case with other real optios models. The authors also propose a procedure for estimatig project volatility based o a Mote Carlo model of the project, but this procedure has sice bee show to grossly over-estimate the true project volatility (Bradão et al., 0, Godiho, 006, Smith, 005). Other authors have also addressed the estimatio of project volatility, either usig Mote Carlo-based procedures (e.g., Bradão et al., 0, Godiho, 006, Haahtela, 0, Herath ad Park, 00, Pamploa et al., 03) or performig aalytical calculatios based o a pre-defied model of the project (e.g., Costa Lima ad Suslick, 006a & 006b, Davis, 998). Aalytical calculatio of project volatility provides accurate results,

4 but it is oly applicable whe the project fits the pre-defied uderlyig model (Godiho, 006). May real life projects have complex cash flow structures, which are ot ameable to the aalytical calculatio of volatility. The aalytical estimatio of volatility may also be impractical for the developmet of software tools aimed at supportig capital ivestmet decisios. Mote-Carlo based procedures, o the other had, are more flexible, providig ways of estimatig volatility wheever it is possible to build a Mote Carlo model of the project. Most procedures for volatility estimatio oly cosider the first year volatility. However, a complete real optios aalysis must take ito accout the operatioal flexibility i the remaiig years, ad the way it affects the first year decisios cocerig the project so, the volatility i the remaiig years is also ecessary. Davis (998) estimates the volatility i differet years. Sice the author uses a aalytical procedure, it ca oly be applied to projects fittig his pre-defied model. Bradão et al. (0) propose a procedure for estimatig volatility i ay year which assumes that the aalyst is able to write aalytical expressios for the expected value of the cash flows occurrig after, coditioed o iformatio available at time. Haahtela (0) proposes a procedure for estimatig the volatility i differet years, based o the residual sum of squares of a liear regressio. I this paper I show that volatility may chage ot oly with time but also with the state of the project (the state of the project beig defied as the vector composed by the radom variables o which the cash flows deped, like demad, prices of market-traded iputs ad outputs, stochastic fixed costs, etc.) This meas that, i each year after startig the project, the best estimate of the future project volatility depeds o time, ad also o the remaiig uderlyig stochastic variables. Failig to recogize the state depedece of volatility i real optios aalysis will lead to sub-optimal decisios, as well as a iaccurate valuatio of the project. I will address the estimatio of volatility for ay give project state, based o Mote Carlo models of the project. As far as I am cocered, there are o previous works o the estimatio of state-depedet project volatility based o a simulatio model of the project. The mai cotributios of this paper are: to show that project volatility may chage with the state of the project; to propose differet procedures for estimatig project volatility i a give state; to preset computatioal tests to compare the performace of the proposed procedures. This paper focuses o the calculatio of project volatility. Real optios aalysis is used as the mai motivatio for volatility estimatio, but I must also stress that accurate estimatio 3

5 of state-depedet volatility may also be useful for several other purposes. For example, accurate volatility estimatio may be useful for some kids of risk aalysis (e.g., calculatio of value-at-risk or coditioal value-at-risk), or eve for defiig a appropriate discout rate for projects udertake by some private compaies (particularly compaies with few owers with udiversified portfolios ad restrictios i the access to credit markets). So, the focus of this paper will be the estimatio of project volatility. The paper his structured as follows. After this itroductio, i Sectio I defie the cocepts of volatility that I use. I Sectio 3 I defie some simple projects for which I am able to aalytically estimate the volatility, ad I show that the volatility may chage sigificatly with the state of the project. Sectio presets the procedures that ca be used to estimate volatility i give project states, as well as some computatioal tests for comparig their performace. Fially, the coclusios are preseted i Sectio 5. The Appedix cotais a mathematical result cocerig oe of the volatility estimatio procedures.. Cocepts of project volatility Project volatility is a measure of the ucertaity over expected project returs from period to period. Volatility is usually defied as the variace, or stadard deviatio, of a give fuctio of the project value. I this paper I choose to use variace as the measure for volatility (of course, if the stadard deviatio is ecessary, I oly have to calculate the square root of the variace). Several real optios models assume that project value follows a geometric Browia motio, ad i this case the variace of the logarithm of project value is a coveiet measure of volatility. I refer to this measure as the log-variace, ad I preset it i Subsectio.. I spite of beig widely used, the log-variace has a very importat drawback: it ca oly be used if the project value does ot become egative. So, I cosider aother measure that ca be applied to all projects: the simple variace of project value. This measure has ot bee as widely used as the log-variace, but it may be very coveiet, for example to approximate the local project behavior by a Arithmetic Browia Motio (ABM). I refer to this measure simply as variace, ad I discuss it i Subsectio.... The log-variace I cosider a ivestmet project with a kow iitial ivestmet cash flow F0 ad a series of future ucertai cash flows F, t =,,T, ad a cotiuously compouded discout

6 rate r. I defie the expected value of the project at time ( V ) as the expected value of the cash flows that will occur after time, discouted to time. The value of the project is coditioed by the values, at time, of the radom variables o which the cash flows deped, like demad, prices of market-traded iputs ad outputs, stochastic fixed costs, etc. The vector with these radom variables will be termed the state of the project, ad deoted by may be writte as: V T = E t= + r( t ) ( F ) e t. The project value, () where E( ) deotes the expected value. I defie the et value of the project at time ( ) as the project value at time plus the curret cash flow. By oticig that F = E( ) write: T = V + F = E t t= r ( t ) ( F ) e F (), I may The et value at time 0 is the et preset value (NPV) of the project. I let k be a radom variable that represets the cotiuously compouded rate of retur o the project betwee time - ad time. The: k = V e (3) From expressio (3) it follows that I ca write k as: k l = = l( ) l( V ) () V Sice I wat to measure the ucertaity from time - to time, I must ackowledge that will be kow at time -. Therefore, the relevat variace, which I deote as lv ( ), must be coditioed o : ( ) = var ( ) = var ( l ( ) ) lv k, (5) where the secod equality takes ito accout that V is completely determied by. If I were to estimate a sigle value for the variace of the project from - to, like Bradão et al. (0) or Haahtela (0), I would just be calculatig the expected value of lv ( ) deote this expected volatility as e lv :. I will I fact, V is a fuctio of explicitly. The same will be doe for. I order to avoid otatioal clutter, I do ot represet this depedece. 5

7 ( ) ( ( )) var ( l ( ) ) lv = E lv = E. (6) e By applyig the law of total variace, it may be see that if I fail to coditio the variace of k o, I will obtai a upward boud o e lv. Fially, I otice that a questio may arise whether lv ( ) or e lv should be used whe buildig a project value model. Let me address real optios models. A real optios model usually iteds to determie the optimal strategy (or at least a early optimal strategy) for maagig the project, ad to calculate the project value i case that strategy is followed. If the state variables ca be observed, or if there is some observable iformatio that allows me to coditio the distributio of the state variables, the the use of e lv will result i a distortio of project behavior (sice, whe I reach time - ad observe the state the best estimate of its variace is lv ( ) ). To illustrate this, cosider the decisios made at time - i a project with a abadomet optio that allows sellig the project for a value assume that i the states for which model that uses lv istead of lv ( ) e VAb, I have lv ( ) < V < lv e V Ab. Additioally. If I build a project, the variace of project value that I cosider at time - will be larger tha the true variace wheever V < VAb. So, if the project reaches time - with V < VAb, I over-estimate the probability of the project value risig above V Ab agai, therefore choosig a abadomet threshold that will be smaller tha the true optimal threshold. Therefore, if I use variace e lv o a real optios model, I fail to maximize the project value. So, the coditioal variace should be used i real optios models. A similar reasoig will apply to other uses of stochastic project value models, ad lv ( ) accurate depictio of project behavior. will provide a more.. The variace of project value The use of log-variace assumes that the project dyamics follows (3). This kid of dyamics may be appropriate for assets that are traded i capital markets, ad whose values will ever become egative. However, it will usually be less applicable to ivestmet projects, whose value may become egative ad may ot have a logormal distributio, ad therefore it has bee criticized by several authors (e.g., Wag ad Dyer, 00). Whe the project value 6

8 may become egative, the expressio () for calculatig k will ot be correctly defied. Other measures of the project rate of retur, based o ratios, might be used istead (see, e.g, Costa Lima ad Suslick, 006b). However, such measures will also become icorrectly defied whe there is a strictly positive probability of the project value becomig egative. I order to avoid such drawbacks associated with measures of the relative chage i project value, I cosider a measure of the absolute chage i project value: the uaticipated chage i project value betwee time - ad time, that is, the differece betwee the et value at ad the preset value at -, compouded to. Such a measure may be particularly useful for some approximatios of the dyamics of project value, e.g., whe the local dyamics of project value is approximated by a Arithmetic Browia Motio (ABM). Arithmetic Browia motios have, i fact, bee used by several authors to model the dyamics of cash flows or project values (e.g., Alexader et al., 0, Bar-Ila, 000, Dixit et al., 999, Lahma, 03). I deote the uaticipated chage i project value betwee time - ad time by ad, i order to defie it, I assume that the discout rate betwee time - ad time is r : ( r ) = V + (7) For the same reaso that was preseted i the previous subsectio, the variace of should be coditioed to. Assumig that r is kow at time -, ad sice V is determied by v preferred to, I may defie this variace (deoted by v ( ) ( ) = var ( ) = var ( ) ) as: (8) The expected variace betwee time - ad time is deoted by ( ( )) ( var ( )) e e v ad defied as: v = E v = E (9) Followig the argumet preseted i the previous subsectio, v ( ) estimatig v ( ) e v. should be e v whe buildig project value models. Therefore, i this paper I will focus o. I ote that the proposed procedures may also be easily used to estimate 3. Examples 7

9 I this sectio I defie some example projects that have bee origially cosidered i other papers, to which I will later apply the volatility estimatio procedures. The cosidered projects are simple projects, for which I am able to aalytically calculate the volatility. Notice that the procedures that I propose are particularly useful for complex projects, for which it is impracticable to aalytically calculate the volatility. However, i order to test the procedures, it is coveiet to use them i projects for which I ca aalytically calculate the volatility, i order to be able to assess the estimatio error. For each of the projects, I will focus o a give period for the purpose of assessig the accuracy of the estimatio procedures, ad I will preset the aalytical expressio for calculatig the volatility i that period. Period = correspods to the begiig of the project, whe there is o iformatio about how the project is evolvig. Therefore, = will be of little iterest for assessig procedures aimig at estimatig state-depedet volatility. I period = there is already iformatio about how the project is evolvig, ad there are also ucertai cash flows remaiig i the projects. Period = will, therefore, provide a iterestig assessmet of the estimatio procedures, ad I will use it for projects ad. I the case of project 3, the aalytical calculatio of state-depedet variace i periods = ad =3 is quite cumbersome, so I will istead focus o period =. Based o the aalytic expressios derived for the variace of the project value, I show that the project variace may chage sigificatly with the state of the project. 3.. Project Project is very simple, based o a project origially cosidered by Godiho (006), ad desiged i such a way that its log-variace is costat (that is, idepedet of the project state). The project cosists of producig 00 uits of a market-traded commodity that has a curret price of oe. The cotiuously compouded rate of retur o the commodity is ormally distributed with a mea μ = 0%/year ad a stadard deviatio σ = 5%/year. The project's oly cost is the iitial ivestmet ad, sice I am oly iterested i estimatig volatility, the value of that cost is irrelevat. The 00 uits of the commodity will oly be available three years after startig the project, ad the rate of retur shortfall is ull for the commodity. The cotiuously compouded risk-adjusted discout rate for the project is the average aual commodity price icrease, r =.5%. I deote the commodity price at year by P. The oly state variable of this project is the commodity price, so [ ] P. I the assessmet of the volatility estimatio procedures, 8

10 I will focus o the volatility of this project for =, so I will oly aalyze the volatility from the ed of the first year to the ed of the secod year. I this project, the log-variace will be costat for all years ad for all values of the state variable it is always As for the project variace, it chages from period to period, ad it is a fuctio of the state variable P. For =: ( ) = 0 ( ) v P P e e (0) The calculatio of both the log-variace ad the variace of the project value, for =, ca be foud i Appedix. 3.. Projects ad 3 Projects ad 3 were origially aalyzed by Cobb ad Chares (00), ad also cosidered by Godiho (006). Both of them are ivestmet projects that produce cash flows for five years. I each year t (t =,..,5), the relevat sources of ucertaity are the uit cotributio margi Xt ad the aual demad Dt. N(μ,σ ) represets the ormal distributio with mea μ ad variace σ ; T(a,b,c) is the triagular distributio with miimum a, mode b, ad maximum c. The distributios for X t ad D t are: X ~N(50,0); X ~N(60,5); X 3~N(70,); X~N(80,8); X5~N(90,36); D~T(95,00,05); D~T(8.5,00,7.5); D3~T(70,00,30); D ~T(57.5,00,.5); D 5~T(5,00,55). The discout rate is %, the tax rate is 0%, ad the fixed expeses are $,50 i the first year ad rise by $50 each year (all these values are o-stochastic). The required iitial ivestmet is irrelevat for the estimatio of project volatility. Cobb ad Chares (00) examie several differet scearios for the correlatios betwee the radom variables Xt ad Dt. I this paper, I cosider oly two projects based o the geeral settig preseted above. I project, all radom variables X t ad D t are ucorrelated; i project 3, there is serial correlatio i the uit price, ad cosequetly i the uit cotributio margi, with a correlatio coefficiet of 0.6 betwee Xt ad Xt+ (t =,...,). Notice that the defiitio project is somewhat urealistic, sice all cash flows are completely idepedet: the probability of the prices beig very high i a give year is idepedet of the price level i the previous year. Project 3 is more realistic, sice it takes ito accout some depedece i the price level. Both projects will be useful for assessig the procedures for estimatig variace, sice it is possible to aalytically calculate their variace. For the purpose 9

11 of assessig these procedures, I will focus o the volatility of each of these projects i a give period. I will cosider period = for project, ad period = for project 3. For both these projects, there is a strictly positive (although very small) probability of the project value becomig egative. I such a case, the log-variace will be improperly defied. Therefore, I choose to cosider oly the variace for the effect of assessig the volatility estimatio procedures. I defie X t ad D t as the state variables, so [ X D ]. Sice all radom variables are idepedet for project, the variace will be idepedet of the project state. For =, the exact variace is ( ) v = 0, 5.65,. The calculatio of this variace ca be foud i Appedix. Sice the uit cotributio margi is auto-correlated i project 3, the volatility may be differet for differet states. For = I have: (, ) 67, ( 5.70 ) v X D = + X + () The derivatio of () ca be foud i Appedix Project volatility as a fuctio of the state of the project As show i (0), the secod year variace of the value of project is a fuctio of the state variable, that is, the commodity price. I order to show how sigificat may be the chages i the variace of project value, I simulated 000 values for the commodity price i the begiig of the secod year, usig the correspodig distributio fuctio ad, for each of them, I calculated the secod-year variace of the project value usig the aalytical expressio (0). The results I obtaied are show i Figure. 0

12 Figure : Secod year variace of the value of project, as a fuctio of the state variable (the commodity price). Variace ,5,5 Commodity price Figure shows that eve i this project, which has a costat log-variace, the variace of project value may chage sigificatly with the state variable. For large values of the state variable the variace may be almost seve times as high as for small values of this variable. Similarly, it would be easy to show that some projects (e.g., projects with a costat variace of project value) may have very sigificat differeces i the log-variace. Followig the discussio i the ed of Subsectio., if I were to estimate a sigle value for the secod year volatility, this value would be a average of the values plotted i e Figure, which I deoted previously as v. Sice abadomet optios are usually valuable for low project values, if I were to use the average secod year volatility to value a abadomet optio, I would be cosiderig a volatility value that is larger tha the real volatility i the rage of state variable values for which this optio is valuable. Therefore, I would over-estimate the value of the optio. O the other had, sice expasio optios are usually valuable for high project values, if I were to use the average secod year volatility to value a expasio optio, I would be cosiderig a volatility value that is smaller tha the real volatility i the rage of state variable values for which this optio is valuable, therefore uderestimatig the value of the optio. I both cases, the optios would be icorrectly valued. Figure shows a similar issue with the fourth year variace of the value of project 3. The figure depicts the variace as a fuctio of a state variable (the uit cotributio margi), for 000 simulated values of this state variable. I this case, the variace also chages sigificatly with the value of the cosidered state variable for large values of the uit

13 cotributio margi, the variace may be almost oe ad a half times as high as for small values of this variable. This would lead to the problems discussed before i the valuatio of real optios. Figure : Fourth year variace of the value of project 3, as a fuctio of the state variable (the uit cotributio margi) Variace Uit cotributio margi. Estimatig project volatility for a give state I will ow preset some procedures for calculatig the volatility for a give period () ad for a give project state ( ). If I kow that is the state at time -, it may seem that I must "simply" simulate the behavior of the project from - to, calculate defie the volatility as the variace of the simulated ad the 's (or the variace of the logarithm of, if I am iterested i the log-variace). The difficulty with this simple procedure lies i calculatig project at time,. By simulatig the project behavior from - to, I will have the state of the. Although is uiquely determied by, there is usually o simple way of calculatig give. As ca be see i (), the calculatio of requires defiig the expected values of all future cash flows as a fuctio of the state simple cases, it may be possible to determie a aalytical expressio for. I some very as a fuctio of (Bradão et al., 0, rely o the ability of defiig such a aalytical expressio), but i

14 may real projects that will be impractical. So, I propose procedures that do ot rely o the ability of beig able to defie ad aalytical expressio for as a fuctio of. The first two procedures, termed "two-level simulatio" ad "regressio procedure", are based o the procedures proposed by Godiho (006) for estimatig the first-year volatility. The first of these procedures, estimates by usig a ew simulatio, while the secod estimates it by usig a liear regressio. The third procedure is based o results preseted by Su et al. (0), ad it is termed "oe ad a half level simulatio". This procedure is also based o a two level simulatio, but it uses a direct estimator of the volatility, without resortig to the itermediate estimatio of. This estimator is able to achieve accurate results whe the umber of iteratios of the secod level simulatio is small (thus the term "oe ad a half level simulatio"). Oe drawback of such procedure is that it is ot able to estimate the log-variace just the variace... Two-level simulatio The uderlyig idea of the two-level simulatio procedure is estimatig by performig a ew simulatio. This meas that I have a first-level simulatio, i which I simulate the project behavior from - to. This allows me to geerate the project state at year,. I each iteratio of this first-level simulatio, I also perform a complete secod-level simulatio. This secod-level simulatio allows me to estimate the expected value of the future cash flows, thus providig me with a estimate of for the state defied by the firstlevel iteratio. The procedure ca be described i Algorithm. Algorithm Two-level simulatio procedure umber of iteratios of the first-level simulatio umber of iteratios of the secod-level simulatio For i = to Simulate the project behavior from - to For j = to Simulate the project behavior from util the last period Calculate ij T r( t ), t= X = F e F, F +,... beig the simulated cash flows 3

15 Next j Calculate i = ij ( i j= X X X is a estimate of for iteratio i) Next i The variace of { X i }, is a estimator for v ( ) (the variace of { Zi} { l( X ii )} = is a estimator for lv ( ) ) Such two-level simulatios are used i project risk maagemet models (see, e.g., La et al., 007). Godiho (006) proposes such a procedure for estimatig the ucoditioal first-period log-variace. The most importat problem with this procedure is its computatioal burde. Sice I follow each iteratio of the first-level simulatio with a complete simulatio of the secod level, I must perform a very large umber of iteratios of the secod level. Moreover, reducig the umber of iteratios of the secod-level simulatio may lead to a bias i the results. This ca be easily see if I cosider the variace measure v ( ) v ( ) as the variace of { X i } simulated cash flows, give the project state ( X ). I Algorithm I estimate. If I defie V as the variace of the year sum of discouted, V = var, with X = F e ad if I apply the law of total variace, I get : T r( t ) () t= ( ( ) ) ( ( ) ) ( ) ( ) var( X ) = var E X + E var X = v + E V (3) i i i X Therefore this procedure will provide a biased estimator for v ( ) whe is fiite ad some of the cash flows occurrig after time are stochastic. If is large, the bias will be egligible; however, if is small, the bias may become sigificat. This meas that I caot use a small umber of iteratios i the secod-level simulatio. I order to avoid otatioal clutter, I do ot explicitly represet the coditioig o.

16 .. The regressio procedure The mai idea of the regressio procedure was origially proposed by Godiho (006), ispired i part by the least squares Mote Carlo approach for America optio valuatio. The uderlyig procedure cosists of usig a liear regressio model to estimate. I order to estimate the model, a sigle level regressio is performed for all the project cash flows. With this procedure, istead of performig a two-level simulatio, I perform two sigle-level simulatios. I the first oe I assume that the project is i state at time -, ad simulate the behavior of the project up to the ed of its life, that is, util time T. The results of this simulatio are used to estimate a model that defies as a fuctio of. I order to estimate this model, I calculate, for each simulated path i, the year discouted sum of the cash flows, T t= r( t ) X = F e, ad regress the values of X i o fuctios of the state variables i of. Similarly to Logstaff ad Schwartz (00), the fuctios that I use are the Laguerre polyomials of the state variables ad of their cross products. This way, I get a model X i = f the. ( ), where f ( ) ca be used as a estimate of the coditioal value of give The secod simulatio uses this model to estimate the project volatility. The project behavior is simulated from - to, ad is estimated by X f ( ). v ( ) i = ca be estimated as the variace of { X i }, ad lv ( ) ca be estimated as the variace of l ( X i ). The procedure is described i Algorithm. { } Algorithm Regressio procedure umber of iteratios of the first-level simulatio umber of iteratios of the secod-level simulatio For i = to Simulate the project behavior from - to the fial period (T), calculate Next i i T r( t ) ad store X ad t= X = F e. 5

17 From the results of the first simulatio, estimate a model = ( ) the values For i = to Next i X i o fuctios of the state variables of (e.g., Laguerre polyomials). Simulate the project behavior from - to Calculate X f ( ) i = ( The variace of { X i }, is a estimator for v ( ) X i f X i is a estimate of for iteratio i) (the variace of { Zi} = { l ( X i )} is a estimator for lv ( ) by regressig ad their cross products ) This procedure is less computatioally demadig tha the two-level simulatio. However, the accuracy of results usually depeds o the choice of the fuctios of the state variables, ad it ca be difficult to determie which oes produce the best model..3. Oe ad a half level simulatio Oe ad a half level simulatio is a procedure based o the results preseted by Su et al. (0) that uses a two-level simulatio with a small umber of iteratios i the secod level i order to estimate v ( ). There are three mai differeces betwee this procedure ad the two-level simulatio procedure preseted above: Istead of estimatig v ( ) as the variace of { } Y, which would lead to a biased estimatio (as show before, based o (3)), a ANOVA estimator, demostrated to be ubiased uder mild coditios, is used; The umber of secod-level iteratios is ot defied as a parameter of the procedure, but istead it is calculated i a way that miimizes the variace estimator, give a predefied computatioal budget; The results i which this procedure is based do ot exted to the estimatio of log-volatility, so this procedure caot be used to estimate lv ( ) I order to defie the estimator for v ( ) i., let me ow defie to be the umber of iteratios of the first level, to be the umber of iteratios of the secod level ad X ij to be 6

18 the year discouted sum of the cash flows obtaied i the j-th secod level iteratio of the i-th first level iteratio, that is: ij T r( t ) () t= X = F e Defie also: X i j = = X (5) ij = X Xi i= (6) X i is the value of of the estimated values if as (see Su et al., 0, eq. 9): estimated i the i-th first level iteratio, ad X is the average. The estimator for v ( ), deoted by v ( ) ( ) ( ) ( ) ( ) i ij i i= i= j= ˆ, is calculated vˆ = X X X X (7) I order to determie the optimal umber of first- ad secod-level iteratios i oe ad a half level simulatio, Su et al. (0) assume the existece of a give computatioal budget C, ad divide this budget by the two simulatio levels i order to miimize the variace of the estimator (i the case cosidered i this paper, the variace of v ( ) ˆ ). The authors show that there is a fiite, costat, asymptotically optimal level of secod-level iteratios, **, idepedet of the budget C. So, as this computatioal budget grows to ifiity, the policy of settig the umber of secod-level iteratios to ** is at least as good as ay policy of settig the secod-level size as a fuctio of C. I light of some computatioal tests, Su et al. recommed the usage of ** secod-level iteratios regardless of the total computatioal budget. This way, the umber of first-level iteratios is defied by the computatioal budget ad by **. I this paper I follow the recommedatio of Su et al. (0), ad use a estimator of the asymptotically optimal level of secod-level iteratios, **. However, I ote that the expressio used by Su et al. to calculate ** is derived assumig that the computatioal cost of geeratig a first-level sceario (that is, the cost of geeratig the project state ) is ull. This implies that all the computatioal cost is due to the secod-level iteratios, allowig 7

19 the authors to simplify the expressios they use. I the applicatio of oe ad a half level simulatio to the estimatio of project volatility, the geeratio of the project state correspods to the calculatio of the state variables values ad of the cash for year, ad each secod-level iteratio will correspod to the geeratio of the state variables values ad of the cash flows from year + util the ed of the project. If I am estimatig the project volatility i a period ear the ed of the project life, the relative cost of the geeratig will be relevat, so I will ot follow the assumptio of Su et al., ad I will derive a more geeral expressio for calculatig **. I use the cost of performig a secod-level iteratio as the uit of measuremet for the computatioal cost, ad I assume that the cost of geeratig the project state is γ. So, if I have iteratios of the first level ad iteratios of the secod level, the computatioal cost is give by: C = + γ (8) For a give year project state,, I defie V as the variace of the year sum of discouted simulated cash flows, give the project state (see ()). I also defie τ as the differece betwee the value of over all possible states τ : ( ) = E give state ad the expectatio of (9) For a secod-level iteratio of a first-level iteratio that defied the state calculated, I defie ε as the differece betwee the year discouted sum of the cash flows ad the et value give state : T r( t ) F e (0) t= ε = Accordig to Su et al. (0), eq. 0, for give values of ad, the variace of the estimator v ( ) ˆ ca be defied as: 8

20 ( 3) ( ˆ ( )) ( τ ) ( ) ( + ) σ ε v ( ) var v E v ( + + ) ( ) = + + σ ε ( ) + + ( ) E ( ε ) ( ) E V 3 ( ) + + E + 3 ( τ ε ) E ( τε ) Similarly to Su et al. (0), I aim at fidig a asymptotical approximatio to the ( ) miimizatio of var vˆ ( ) (). Assumig a fixed computatioal budget C whose relatio with ad is defied by (8), I may equivaletly miimize ( ) ˆ ( ) ( ) ( γ ) ( ˆ ( )) h, = C var v = + var v () I order to obtai a asymptotical approximatio, I make C ted to ifiity. For a fixed (whose optimal value I wat to fid out), this is the same as makig ted to ifiity. So I defie ( ) = lim h (, ) h + γ E v E V E ( τ ) ( ) ( ) ( τ ε ) = + + For the relevat values of, that is, for >, the miimum of ( ) of the followig equatio (see Appedix for the derivatio of this result): ( τ ) ( ) ( ) + ( τ ) ( ) ( ) E v E v 3 E ( τ ) v ( ) E ( τ ε ) E ( V ) ( ) ( γ ) E ( V )( ) ( γ ) E ( V ) + + = 0 (3) h will be the root + () The left side of () is a quartic i ( ) ad the equatio ca be solved by several differet methods (for a example see Borwei ad Erdelyi, 995, p. ). Notice that, sice I did ot assume that the computatioal cost of geeratig a first-level sceario is egligible, I arrive at a more complex expressio tha the oe derived by Su et al. (0). I eed a iteger value for the umber of secod-level iteratios, ad the root of () will usually be o-iteger. However, defiig ** as the earest iteger that is greater tha oe will provide a good approximatio to the maximizatio of h ( ). 9

21 A importat issue cocers four quatities that are ecessary for defiig () ad are ot kow at the outset: ( ) E τ, v ( ) E τ ε ad E ( V ), ( ). I fact, oe of the quatities we eed is the same oe that this procedure aims to estimate. I follow the approach of Su et al. (0), ad use a pilot simulatio to obtai rough estimates of these quatities, which are the plugged ito () i order to defie the umber of iteratios that allows me to reach a accurate estimate of v ( ). This pilot simulatio uses a small portio of the computatioal budget (e.g., 0% of this budget), ad it will be a two-level simulatio with a arbitrarily chose small secod-level size. The first-level size,, will be defied by the portio of the computatioal budget devoted to this pilot simulatio ad by the secod-level size. Su et al. propose estimators for the above metioed quatities, based o this pilot simulatio. Usig the otatio that I am followig, ad cosiderig ij = X i i= j= ( ) Eˆ V vˆ Eˆ Eˆ ( ) X X ij, X i ad X as defied i ()-(6): (5) ( X i X ) ( X ij X i ) i= i= j= = (6) ( ) ( τ ε ) vˆ ( ) ( τ ) = ( X ij X i ) ( ) i= j= (7) ( ) + ( ) ( i ) = ( ) + ( ) ( )( ) 3 3 X X vˆ 3 ( ) i= (8) 6 E ˆ ( τ ε ) So, after the pilot simulatio, (5)-(8) are used to estimate the required quatities, ad () is used to calculate the umber of secod-level iteratios. A two-level simulatio is performed ad (7) is used to estimate v ( ) Algorithm 3.. The complete procedure is described i 0

22 Algorithm 3 Oe ad a half level simulatio procedure C computatioal budget for the simulatios, i terms of secod-level iteratios γ computatioal cost of geeratig a project state iteratios, i terms of secod-level α percetage of the computatioal budget to be used i the pilot simulatio umber of secod-level iteratios i the pilot simulatio Calculate = αc ( + γ ) For i = to Simulate the project behavior from - to For j = to Next j Simulate the project behavior from util the last period Calculate ij T r( t ), t= X = F e Use (5)-(8) to estimate ( ) Plug the estimates of ( ) E τ, v ( ) E τ, v ( ) F, F +,... beig the simulated cash flows E τ ε ad E ( V ), ( ) E τ ε ad ( ), ( ) calculate the oly root of that equatio that is larger tha (oe), * E V ito (), ad Let ** be the closest iteger to * ad calculate ( α ) ( γ ) ** = C ** + For i = to ** Next i Simulate the project behavior from - to For j = to ** Next j Simulate the project behavior from util the last period Calculate Use (7) to estimate v ( ) ij T r( t ), t= X = F e F, F +,... beig the simulated cash flows

23 .. Computatioal tests... Defiitio of the tests Computatioal tests were defied for comparig the performace of the procedures described i the previous subsectios. The procedures were implemeted i C computer laguage, usig Microsoft Visual Studio. The ALGLIB library ( was used for liear regressio, calculatio of Laguerre polyomials ad radom umber geeratio. The projects described i Sectio 3, were used i order to compare the three procedures for volatility estimatio. The same computatioal budget for the simulatios was allocated to each procedure ad oe thousad (000) iitial project states were simulated for the momet before the begiig of the volatility estimatio year (, with = i the case of projects ad, ad = i the case of project 3). The volatility was the estimated for each of these states, usig the defied computatioal budget ad each of the three procedures. The estimated volatility was the compared with the theoretically correct value. The defiitio of a computatioal budget for the simulatios that is commo for the three procedures allows us to compare the cases i which the simulatio represets the most relevat computatioal effort. This computatioal budget takes ito accout the umber of project cash flows that it is ecessary to simulate. I fact, the regressio procedure also implies performig a liear regressio that is more complex tha the auxiliary calculatios ecessary i the other procedures. However, i order to avoid makig further assumptios about the complexity of the auxiliary calculatios, or havig results depedet o the efficiecy of the implemetatio of auxiliary procedures, I chose to compare the results of the procedures takig ito accout the required umber of simulated cash flows. For each procedure, three differet computatioal budgets were cosidered: 0 000, ad secod-level iteratios. The computatioal budget is shared by first- ad secod-level iteratios. I order to defie the umber of iteratios of each level, it was assumed that i each project the simulatio effort was proportioal to the umber of years that it was ecessary to simulate (that is, the umber of cash flows it is ecessary to calculate). Let me exemplify with project. This project has 3 years, I wat to calculate the secod-year volatility, a first-level iteratio will simulate oe cash flow (the secod year cash flow) ad a secod-level iteratio will also simulate oe cash flow (the third year cash flow). This meas

24 that, for project, the cost of a first-level iteratio is equal to the cost of a secod-level iteratio, cosequetly γ =. I the case of the regressio ad the oe ad a half level simulatio procedures, the way the computatioal budget is split by the two simulatio levels is defied by the methods: i the regressio procedure, there is oe secod-level iteratio ad i the case of the oe ad a half level simulatio the umber of secod-level iteratios is defied by (), ad (8) ca the be used to calculate the umber of first-level iteratios. However, for the pure two-level simulatio procedure, I do ot have a a priori rule for the umber of iteratios i each level. I each applicatio of this procedure, I defied a ratio betwee the umber of first- ad secod level iteratios, α =. I cosidered two values for this ratio: α = ad α = 0. The former value represets a equal umber of first- ad secod-level iteratios, while the latter represets a icrease i the umber of first-level iteratios at the expese of the umber of secod-level oes. I the regressio procedure, Laguerre polyomials of the state variables ad their cross products were used as idepedet variables for the regressio, similarly to Logstaff ad Schwartz (00). I the case of project, the first five Laguerre polyomials of the state variable were used. I projects ad 3, the first five Laguerre polyomials of each state variable were used, alog with the five cross-products of these polyomials. I the oe ad a half level simulatio procedure, 0% of the computatioal budget was allocated to the pilot estimatio, similarly to Su et al. (0). The umber of secod-level iteratios used i this pilot estimatio was 5. I the case of project, both the log-variace ad the variace were estimated. I the case of projects ad 3, oly the variace was estimated, sice the project value may be egative.... Results ad aalysis Table presets the results of the tests. The accuracy of the estimatios is assessed by the Mea Absolute Error (MAE) ad by the Mea Absolute Percetage Error (MAPE). As expected, the mea estimatio error always decreases whe the computatioal budget icreases. The results also show clearly that both the oe ad a half level simulatio procedure ad the regressio procedure perform much better tha two-level simulatio. 3

25 Table Estimatio of the project volatility usig the proposed procedures, for the three example projects. Project Project Project 3 Log-variace Variace Variace Variace Budget Method MAE (x0 3 ) MAPE MAE MAPE MAE (x0-3 ) MAPE MAE (x0-3 ) MAPE level (α=).8.0%.99.37% % % -level (α=0). 6.38% % % % Regressio 0.6.7% % % % ½-level % % % -level (α=). 6.38% % % % -level (α=0) %.8.% % % Regressio % %.58.% % ½-level % % % -level (α=) % % % % -level (α=0) 0..0% % 5.3.6% % Regressio % % % % ½-level % % % Budget: computatioal budget for the simulatios; MAE: Mea Absolute Error; MAPE: Mea Absolute Percetage Error; -level: two-level simulatio procedure; Regressio: regressio procedure; ½ level: oe ad a half level simulatio procedure; α: ratio betwee the umber of first- ad secod-level iteratios; -: the procedure was ot applied (the oe ad a half level simulatio procedure caot be used to estimate the log-variace). A iterestig result cocers the compariso betwee the results of the two-level simulatio procedure for differet ratios betwee the umbers of first- ad secod-level iteratios (α). While a larger umber of first-level iteratios (α=0) performs better i the estimatios made for projects ad 3, it performs much worse i the case of project. Remember that i the cases of projects ad 3, the project volatility was estimated for the year immediately before the last year of the project. I these cases, the volatility of the values obtaied i the secod-level iteratios will be limited, so the bias due to this volatility (the bias due to the last term of the rightmost expressio i (3)) will be small ad it is better to use a larger umber of first-level iteratios. I the case of project, the volatility of the secod year is beig estimated for a 5-year project. Sice there are still three years of project operatio after the year of volatility estimatio, the values obtaied i the secod-level estimatio are subject to sigificat volatility. Whe the umber of secod-level iteratios is small, the bias due to this secod-level volatility is large ad, therefore, a large umber of secod-level iteratios leads to a better performace.

26 I Table, it ca also be see that the regressio procedure usually performs better tha the oe ad a half level simulatio procedure for the same computatioal budget. However, I would recommed some prudece i makig coclusios based o this result, for two reasos. First, the differece betwee the mea absolute errors is ot very large. Secod, the overhead that is ot icluded i the computatioal budget is larger for the regressio procedure tha for the oe ad a half level simulatio (due to the eed to perform a liear regressio). So, while we have idicatios that, whe allocated the same budget for simulatios, the regressio procedure performs best, we caot be sure that this is really the most efficiet method, i computatioal terms. 5. Coclusios ad future research This article addressed the estimatio of state-depedet project volatility resortig to Mote Carlo simulatio. I cosidered two possible defiitios of the cocept of project volatility ad I showed that, i a give year, the project volatility may chage accordig to the project state. I briefly argued that usig a sigle average volatility may itroduce biases i the valuatio of real optios. I preseted three procedures for estimatig state-depedet volatility: two-level simulatio, oe ad a half level simulatio ad a regressio procedure. I cosidered three simple projects ad defied idetical computatioal budgets for the procedures (based o the umber of cash flows it is ecessary to simulate), ad I compared the mea absolute errors of the estimates. The regressio procedure usually performs better tha the other two, but some care must be take i iterpretig the results, sice the computatioal cost of performig the regressios is ot cosidered i the computatioal budgets. The oe ad a half level simulatio procedure also shows very good results, but it ca oly be used to estimate the project variace, ot the log-variace. The lattice-based approach proposed by Copelad ad Atikarov (00) has bee a importat motivatio for developig procedures for the estimatio of project volatility, although, as I argued, there may be other reasos for watig to estimate project volatility. I order to use the Copelad ad Atikarov approach, project volatility must be defied as a fuctio of the project value, ot as a fuctio of other state variables. The procedures proposed i this article may be easily used to estimate the average volatility associated with a give project value: oe way to do this may be to cosider several samples of the state variables ad estimate both the project values ad the volatilities associated with those samples; the project 5

27 values ad volatilities thus estimated may the be used to build a fuctio that maps a project value to a associated average volatility. Defiig the best ways to sample the values of state variables (e.g., simulate values or cosider evely spaced values?) ad the best way to build the volatility fuctio (usig liear regressio?) are left for future research. The Copelad ad Atikarov approach assumes that the project value follows a geometric Browia motio with costat log-variace. This assumptio is usually too strog, ad iapplicable to projects whose value may become egative, but it allows the use of recombiig lattices. Whe volatility chages with time ad/or project state, lattices o loger recombie, so the approach must be modified. Some authors have already proposed some ehacemets to the Copelad ad Atikarov approach that do ot rely o recombiig lattices (e.g., the approaches based o biomial trees proposed by Bradão ad Dyer, 005 ad Bradão et al., 005). The availability of methods for estimatig time ad state-depedet project volatility opes the way to the developmet of ew approaches that do ot require the geometric Browia motio assumptio: for examples, approaches that locally approximate the behavior of project value by a arithmetic Browia motio, ad the resort to orecombiig lattices or biomial trees. The developmet of such approaches is aother promisig way of future research. Ackowledgemets This work was supported by EMSURE (Eergy ad Mobility for Sustaiable Regios) - CENTRO-07-0-FEDER Refereces Alexader, D. R., Mo, M., & Stet, A.F. (0). Arithmetic Browia motio ad real optios. Europea Joural of Operatioal Research, 9(), -. Bar Ila, A. (000). Ivestmet with a arithmetic process ad lags. Maagerial ad Decisio Ecoomics, (5), Baker, H.K., Dutta, S., & Saadi, S. (0). Maagemet Views o Real Optios i Capital Budgetig. Joural of Applied Fiace, (),

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