Using Omega Measure for Performance Assessment of a Real Options Portfolio

Transcription

1 Usig Oega Measure for Perforace Assesset of a Real Optios Portfolio Javier Gutiérrez Castro * Tara K. Nada Baidya * Ferado A. Lucea Aiube * * Departaeto de Egeharia Idustrial (DEI) Potifícia Uiversidade Católica do Rio de Jaeiro (PUC-Rio) Rua Marquês de São Vicete 225, sala L960 - C.E.P Gávea. Rio de Jaeiro, Brasil. Tel / Fax javiergc@aluo.puc-rio.br baidya@id.puc-rio.br aiube@id.puc-rio.br Petrobras, Petróleo Brasileiro S.A. Av. Chile 65, sala 402 C.E.P Cetro. Rio de Jaeiro, Brasil. Tel / Fax: aiube@petrobras.co.br Abstract The portfolio copositio of assets is a classic thee i fiace literature. Ivestors wat to get highest retur, iiizig as possible the ivolved risk. I a portfolio coposed by real assets, such as ivestet projects, it ust be decided which of the carry o. So, it is always a challege to easure the ivolved risk. Furtherore, i a portfolio of real assets it s possible to itroduce ad odel for each idividual project future ivestet decisios, like the right tie to ivest, akig a expasio, reductio of operatios, abadoet, etc. Bearig i id the possibility of exercisig these optios, the odel becoes ore realistic. For this type of portfolios, i order to do a adequate assesset of their risk-retur perforace, it is proposed to use the uiversal easure called Oega, developed by Keatig ad Shadwick (2002), which reflects all the properties statistics of the distributio of gais ad losses, icorporatig all oets of the distributio of returs, ad ot oly the ea ad variace as is doe i the classical approach of portfolio copositio i Markowitz (1952). It s deostrated that usig Oega easure for perforace assesset ad portfolio copositio, is ore advatageous tha usig the classic Mea-Variace approach. The ai objective of our research is to apply the Oega easure i portfolios cotaiig Real Optios i their Ivestet Projects. Keywords: Portfolio, Real Optios, Risk, Retur, Oega Measure. 1

2 1. Itroductio I fiacial literature, it is well kow the fact that ivestors always wat to get highest returs o their ivestets, iiizig as possible the ivolved risk. Markowitz (1952) desiged the foudatios of the portfolio copositio theory of ivestets. Accordig to his theory, ivestors ca deterie all optial portfolios, relatig to risk ad retur, ad to for a efficiet frotier. The efficiet frotier ca be described as the best possible set of portfolios, that is, all portfolios have the iiu level of risk for a give level of retur. Ivestors would focus o the selectio of a portfolio o the efficiet frotier ad they would igore others cosidered iferior. Although the classical theory of Markowitz (1952) is cosidered easy for ipleetatio ad efficiet i portfolio copositio of assets, coplicatios appear whe it is take ito accout ucertaity i the value of variables. I this case, a deteriistic approach does ot result loger valid, ad it s ecessary to do a probabilistic odelig of variables ad eploy siulatio ethods. Whe the portfolio is fored by real assets, the proble is ore coplex, because of there is t a history of returs ad that does t let the calculatio of a expected value or correlatios aog other assets. For exaple, a portfolio coposed of ivestet projects, ad we have to decide which ivestets perfor i order to get at least a desired retur by aagers. Moreover, i the portfolio of real assets is possible to itroduce ad odel future ivestet decisios for each idividual project, like the right tie to ivest, akig a expasio, reductio of operatios, abadoet, etc.. Bearig i id the possibility of exercisig these optios (called real optios for havig its applicatio i real assets), the odelig becoes ore realistic, thus iprovig the attractiveess of projects. Alog with the copositio of the portfolio, it is ecessary to assess the risk. I literature there is a great aout of risk easures; the ost popular risk easures are stadard deviatio ad Value at Risk (VaR). There is also a easure called Expected Shortfall (ES) which is ore iforative tha VaR, because it evaluates the expected ea loss i a cofidece level. Accordigly, VaR respods to the questio: "What is the iiu loss icurred by the portfolio i α% worst scearios?". I tur, ES respods the questio: "What is the ea loss icurred by the portfolio i α% worst scearios?". Recetly, several authors have proposed easures of risk-retur (kow also as perforace easures) ore cosistet with the expected distributio of gais observed i practice, that is, ot oral distributios. Aog the the easure Oega (Ω), itroduced by Keatig ad Shadwick (2002), seeks to iclude every oet fro the retur distributio whe assessig the risk of a asset. This paper explores the characteristics of Oega easure, which is a relatively ew ad there are few studies i portfolio optiizatio usig Oega. Siilarly, it will be showed its applicatio i optiizatio for copositio a real asset portfolio, particularly ivestet projects with optios, assessig their levels of risk ad retur, settig targets to be achieved. The proposed ethodology is iteded to be easy to ipleet to ay type of idustry. Optiizatio techiques ad Mote Carlo siulatio are ai tools i its applicatio. 2. Theoretical Fraework 2.1. The Portfolio Copositio Model of Markowitz Matheatically the risk ca be treated as a rado variable, ad the first two oets of 2

3 the probability distributio (ea ad variace) are idicators that defie the degree of risk exposure. It is relatively siple to study the risk of a asset uder this approach. But i a portfolio with ay assets, the coplexity of easurig the risk is great due to the fact the probability distributio of the portfolio retur ay differ sigificatly fro the probability distributio of idividual assets. For exaple, cosider a uiverse of stocks. If r j is the retur fro stock j (rado variable) ad x j is the aout, i cash, to ivest i stock j. Expected retur fro this portfolio is give by: r(x j j 1,..., x ) E rjx j E r x j1 j1 (1) where E [.] represets the expected value of the rado variable. Furtherore, the stadard deviatio of retur is: σ(x1,..., x ) E rjx j - E rjx j1 j1 2 Markowitz used the variace of returs, as a risk easure. It s desired to get a portfolio with iiu risk, that is, with iiu variace subject to costraits of capital ad iiu retur. Thus the odel ca be writte as the followig optiizatio progra: iiize sujeito a i1 j1 j1 R j ij i j σ x x x j j ρm 0 (2) (3) j1 x j M 0 0 x j u j, j 1,..., where M 0 is the capital available iitially, R j = E[r j ], σ ij = E[(r i R j )(r j - R j )] is the covariace betwee assets i ad j, r i or r j represet idividual returs fro asset distributio i or j, ρ is a paraeter that represets the iiu retur rate required by a ivestor, ad u j is the axiu aout of oey that could be ivested i j Real Optios Copaies ad fiacial istitutios, over ay years, have used traditioal ethods for project evaluatio i order to review their ivestet decisios. Net Preset Value (NPV) ad Iteral Rate of Retur (IRR) are classic ethods of evaluatio. They argue that projects with NPV positive or IRR higher tha the discout rate, these would be, i priciple, ivestets to be ade. But a few years ago, those ethods are beig severely questioed. The ai reaso is that they do ot deal properly with three iportat characteristics i ivestet decisios: 1) Irreversibility: that is, the fact that ivestet is a suk cost, so ivestor is uable to recover it totally i case of regret, 2) Ucertaity about future profits fro the ivestet, 3) Possibility of postpoeet of ivestet, which ca brig beefits for ew iforatio about the eviroet. So, give these features, copay could have flexibility to chage 3

4 the origial ters of the project to save possible losses or geerate additioal earigs depedig o the future sceario. Real optios theory ca evaluate i a way ore realistic flexibility i ivestet. It has a aalogy with fiacial optios: a call o a asset (preset value of future earigs o ivestet) gives right but ot obligatio to buy it i the future, at a exercise price (iitial cost of ivestet), i a aturity tie (axiu tie that project ca be postpoed). Optio exercise (do ivestet) is irreversible, but fir has opportuity to preserve the value of its optio (postpoe ivestet) util arket coditios becoe ore favorable. Real optios icrease the copay value, due to flexibility that projects would have to adapt to future coditios, i respose to arket chages. Thus, it applies the followig relatioship: Project Value = Project Value without optio + Optio Value (Calculated by NPV) There are differet types of Real Optios, betwee the: A) Siple Optios - Abadoet Optio: If future coditios becoe ufavorable to the project, you ca leave the busiess ad sell the assets to a pre-established salvage value. - Deferral Optio: It is the optio that deteries the optial tie for ivest, i order to geerate highest profits. - Cotractio Optio: This optio gives the right to reduce a portio of ivestet capacity. Future value of assets is reduced, but it is received a oey etry i the cotractio year. - Expasio Optio: If project resulted be better tha expected, this optio gives right to expad the origial project capacity. Thus, the uderlyig asset value icreases, but we eed to do prior a additioal ivestet. B) Copoud Optios Copoud optios are a cobiatio of siple optios that ca be perfored siultaeously or i sequetial order. For exaple, a fir that ivests i Research ad Developet ay eed soe iitial tie to obtai results of preliiary tests before doig the ivestet; the it ay decide to expad, cotract or abado the project, accordig to iforatio obtaied by waitig. I ost cases, there is always ore tha oe optio to exercise. C) Switchig Optios Switchig optios provide to the holder possibility of chage betwee differet types of resources, assets or techology. Also allow iitiate ad teriate operatios, or eter ad exit fro a particular activity. This high flexibility over the project adds value to it, i case that value of soe alterative becoes ore profitable i the future. D) Learig Optios Optios described previously, it is cosidered that as tie passes, ucertaities regardig asset value, ad possibility of exercisig the optio or ot, will be revealig. However, i ay situatios ucertaity does ot resolve by itself. Efforts ad ivestets are ecessary to obtai ore iforatio about project coditios ad reduce ucertaity. For exaple, i petroleu idustry, whe it decides ivest ore i geological research ad discover the exact agitude of reserves, or testig the arket before scale sales, for a few cases. 4

5 2.3. Risk Measures Below, a package of easures for traditioal risk obtaied fro a series of gais or returs laugh, i = (1.2,..., ), ad '' the total uber of observatios of retur o active: 2 (ri r) a) Stadard Deviatio: i1 DP (4) i 0;(ri r) b) Seivariace: i1 S V (5) i 0;(ri ri ) c) Dowside Risk: i1 DR (6) where r i is the iiu required gai ri r d) Mea Absolute deviatio: i1 DA (7) i 0; (ri r) i1 e) Mea Absolute Seideviatio: SDA (8) i 0; (ri ri ) f) Mea Absolute Dowside Risk: i1 DRA (9) g) Value at Risk VaR: VaR ethodology, developed by JP Morga Bak (1996), started i order to quatify, systeatically ad siple, potetial losses due to exposure to arket risk, that is coe fro the volatility of arket prices (exchage rate, iterest rate, stock arket, etc.). Sice the, this ethodology has bee widely used i aagig risk practice as total risk easure i project portfolios, beig etioed i various regulatory practices of iteratioal fiacial syste. VaR attepts to suarize i a sigle uber the axiu expected loss i a certai period, whit a specific statistical cofidece level. He evaluate the rado variable represetig gai (or loss).so, VaR (95%) idicates that there are 5 chaces i 100 that ijury is greater tha that idicated by VaR, i a give period. It becoes a uber of easy readig ad uderstadig that depeds o the ter (N) ad the degree (1-α)% of desired cofidece. VaR = V ca be read as: "We are (1 - α)% certai that we will ot lose ore tha V oetary uits i followig N days". Statistically this stateet is equivalet to: Prob [r j <VaR] = α% (10) VaR calculatio is quite siple, sice you kow i detail the retur distributio r j, because VaR is, by defiitio, soe quatile associated with a extree percetile fro portfolio distributio - usually α =1% or α=5%

6 h) Expected Shortfall Expected Shortfall (ES) is a easure which idicates the expected ea loss exceedig VaR, that is, it quatifies "how" great is the loss (risk), o average, that we are exposed i a specific portfolio, thus providig iforatio about the tail of NPV distributio (this statistic is also kow as VaR coditioal, VaR i the tail). You ca thik ES as "how heavy" is the gai distributio tail i a portfolio. So, while VaR respods the questio "What is the iiu loss icurred by portfolio i α% worst scearios?", ES respods the questio "What is the ea loss icurred by portfolio i α% worst scearios?". Matheatically, you ca set ES as a coditioal expectatio of portfolio losses higher tha VaR. ES = E [r j / r j < VaR] (11) 2.4. Portfolio Perforace Assesset (Risk Retur) The iportace of usig odels for assessig perforace of ivestets started with the priciple of diversificatio proposed by Markowitz (1952) ad his ea-variace, which argues that ivestors would prefer higher returs to the sae risk level. Thus, etrics required for portfolio selectio were based o the expected retur ad the stadard deviatio (risk) of returs. Later, a relatio betwee risk ad retur was statistically foralized by Treyor (1965), Sharpe (1966) ad Jese (1968). They assue that returs are orally distributed ad ivestors have a quadratic utility fuctio. Establishet of alteratives to calculate gai ad risk is ot a trivial task, ad there are various easures to assess portfolio perforace. Risk ca be established as ay of the easures listed above, that is, stadard deviatio, iiu expected loss or other easure that icludes higher oets fro the distributio. At this poit, Figure 1 extracted fro Keatig ad Shadwick (2002) is quite illustrative, as it idicates iportace of higher oets for ivestet evaluatio. Both distributios have the sae ea (10) ad stadard deviatio (152), but they differ i skewess, kurtosis ad i all higher oets. However, accordig to soe traditioal perforace idicators, like ea ad stadard deviatio, both would be equivalet. Figure 1 - Distributios with equal ea (10) ad variace (152) Best kow ratios for evaluatig portfolio perforace are Sharpe (SR), Treyor (TR), Jese (JR) ad Sortio (SoR). Idicators developed by Jese ad Treyor evaluate perforace takig ito accout the ivestet portfolio perforace i relatio to arket 6

7 perforace. Sharpe ratio evaluates the ivestet perforace takig ito accout oly the portfolio behavior, while Sortio ratio uses Dowside Risk cocept to assess risks. Oega easure (Ω), proposed by Keatig ad Shadwick (2002), is cosidered ore cosistet because it ca satisfactorily deal with gai distributios observed i practice (heavy tails ad extree values). a) Sharpe Ratio: Aog statistics for perforace evaluatio, Sharpe Ratio (SR) is the best-kow. Extreely acclaied betwee acadeics ad fiacial arket practitioers, SR has bee widely used i evaluatio of ivestet fuds. Forulated by Willia Sharpe (1966), SR is based o portfolio selectio theory, poitig poits o the capital arket lie, which represet optiu portfolios. SR is defied i equatio (17), where r f is the risk free iterest rate, ad E[R p ] e p represet respectively expected retur ad volatility of portfolio. ERp rf SR (12) σp Mea ad variace theory of Markowitz deterie optiu copositio of portfolio o a risk-retur space. It is easy to show that portfolios with highest SR are exactly optiu portfolios, cosiderig orality i retur distributio. b) Sortio Ratio Accordig Duarte (2000), Sortio ratio (SoR) differs fro Sharpe ratio, by tacklig the risk cocept called Dowside Risk, which cosiders i calculatio of variace just fiacial losses. Sortio perceived that stadard deviatio easures oly the risk of failig to achieve a ea. However, the ost iportat would be to capture the risk of ot achievig the gai i relatio to the goal. Therefore, the Sortio ratio depeds explicitly o the iiu acceptable retur (MAR), for purposes of copariso betwee the aalyzed portfolio or asset ad that iiu. Therefore, SoR is give by the followig equatio: E[Rp ] r SoR σ DR MAR 2 i(0; R p i rmar ) where, i1 σ (14) DR 3. A ew perforace easure: Oega Idex (Ω) Due to criticiss cocerig to ea-variace approach proposed by Markowitz (1952), which is based o the assuptio of oral distributio of gais, Keatig ad Shadwick (2002) preset a uiversal perforace easure called Oega, which reflects all statistic properties of gais distributio, icorporatig all its oets, ot oly ea ad variace. Most perforace idicators cosider two ai siplificatios: - Mea ad variace copletely describe the retur distributio. - Risk-retur characteristics of a portfolio ca be described oly with ea of returs. These siplificatios are valid if it is assued a oral distributio of returs, but it is (13) 7

8 geerally accepted the epirical fact that returs o ivestets do ot have a oral distributio. Thus, besides ea ad variace, higher order oets would be required to better distributio descriptio. Oega easure (Ω) icorporates all oets of the distributio. It provides a coplete descriptio of the risk-retur characteristics, so that results i a easure ituitively attractive ad easily coputable. Istead of estiatig soe idividual oets, Oega easures total ipact, which is certaily of iterest of decisio-akers. I order to defie Oega fuctio (Ω), priarily, it s ecessary to defie exogeously the liit retur (L). This divides probability distributio of returs i two areas: area of earigs, ad area of losses. This liit varies by idividual ad by type of ivestet. Figure 2 illustrates the distributio of returs of a asset, which was established a liit L = 1,4. 0,25 L=1,4 0,2 0,15 0,1 0,05 0-0,2 0 0,3 0,7 0,9 1,4 1,7 1,9 2,4 2,9 Prob 0,03 0,06 0,13 0,18 0,22 0,15 0,1 0,07 0,04 0,02 Figure 2 - Distributio of returs with liit L=1,4 Usig the procedure described by Keatig ad Shadwick (2002), it is estiated Oega easure (Ω), through the cuulative distributio fuctio, show i Figure 3. Gais (g i ) ad losses (l i ) ay occur with soe probability i gai areas (r i >L) or loss areas (r i <L). 1 0,9 g3=0,5 g4=0,5 0,8 g2=0,2 0,7 0,6 l 5=0,6 g1=0,3 0,5 l 4=0,2 0,4 0,3 l 3=0,4 0,2 l 2=0,3 0,1 l 1=0,2 0-0,2 0 0,3 0,7 0,9 1,4 1,7 1,9 2,4 2,9 Figure 3 - By reducig itervals betwee returs to refie estiates of gais ad losses relatig to L =1,4 Accordig to Figure 3, weighted total gai would be calculated: 8

9 r>=l g i = r i+1-r i [1-F(r)] g*[1-f(r)] 1,4 0,3 0,23 0,069 1,7 0,2 0,13 0,026 1,9 0,5 0,06 0,03 2,4 0,5 0,02 0,01 2,9 Weighted Total Gai = g i*f(r i) = 0,135 ad, weighted total loss would be: r<l l i = r i+1-r i F(r) l*f(r) -0,2 0,2 0,03 0, ,3 0,09 0,027 0,3 0,4 0,22 0,088 0,7 0,2 0,4 0,08 0,9 0,5 0,62 0,31 1,4 Weighted Total Loss = l i *F(r i)= 0,511 So, Ω = 0,135 / 0511 = 0,2642 Whe probability distributio ceases to be discreet, it is said, a cotiuous desity fuctio, Figure 3 i the liit, whe itervals becoe icreasigly saller, becoes Figure 4. Figure 4 Liit areas whe profit ad loss uits ted to zero. Cosiderig a cotiuous desity fuctio, is defied: (a,b) = lower ad upper liits, respectively, the rage of returs distributio. Most of the tie, a = - e b=. I 2 (L) = weighted ea gais above a level L (upper area of Figure 4). I 1 (L) = weighted ea losses below a level L (lower area of Figure 4). Perforace easure Oega (Ω), i cotiue way, is defied by the followig expressio: 9

10 I Ω(L) I Where: F = Cuulative distributio fuctio of gais L = Miiu required level of gais a = Miiu retur b = Maxiu retur 2 1 b 1 F(x) dx L L F(x) a (15) Fuctio Ω (L) copares returs fro differet assets ad rak the i relatio to agitude of their Oegas. A Ω (L) = 1, idicate that weighted gais equal to weighted losses. It s always desirable Ω (L)> 1. Kazei, Scheeweis ad Gupta (2003) preset Oega easure i a ore ituitive way, deostratig that equatio (15) ca be writte as a divisio of two expected values. I equatio (16), uerator is the expected value of gai excess (x-l) coditioal to positive results (where 'x' is soe retur fro the distributio), ad deoiator is the expected value of losses (L-x) coditioal to egative results. Thus, these authors develop a aalogy with the optios theory ad write the uerator like the axiu expected value betwee (x-l) ad zero, siilar to payoff of a call without discout. Deoiator, therefore, would be the payoff of a put without discout. 4. Optiizatio with Oega Measure 4.1. Optiizatio Progra I order to optiize the portfolio with Oega, Ick ad Nowak (2006) follows the progra: Where: Ω(L) b b 1 F(x) dx (x - L)f(x)dx -rf L L e Eax(x L;0) Call(L) L L -rf F(x) (L - x)f(x)dx e Eax(L- x; 0) Put(L) a a ES (L) E ax(l- Rp ;0) P sujeto a : EC (L) E ax(rp - L;0) P ECP(L) ax Ω(L) P ES (L) i Rp i i j1 w R j ij w j j1 w 0 w (16) (16) = Expected Shortfall Risk for portfolio P = Excess Chace for portfolio P = Portfolio Retur o period i = Part of portfolio ivested i asset j 4.2. Applicatio Exaple It s optiized by Oega a portfolio coposed of four assets, ad the we copared with the optiizatio applyig Markowitz odel, with the purpose of better uderstadig the advatages betwee oe ad aother ethodology. j P 1 j 1 10

11 I Table 1, it s showed four assets of the portfolio, ad ai statistics. For each asset, it is beig cosidered 500 observatio periods, which it geerates its retur distributio. Table 1 Statistic Properties fro Returs Historical Data of Four Assets Mea Variace Skewess Kurtosis Miiu Value Maxiu Value JB Test Asset X 0,15 0,25-0,90 3,42-1,69 0,84 71,63 Asset Y 0,20 1,44 1,96 8,38-1,00 7,41 925,06 Asset Z 0,25 1,00 1,90 8,17-0,80 6,39 856,89 Asset W 0,05 0,16-1,42 5,51-2,02 0,53 299,31 Four assets showed i Table 1 are very far fro oral distributio, which is idicated by Jarque-Bera test. Observe that all values are uch greater tha 5.99 which is the value of chi-square with two degrees of freedo ad probability of 95%, which is take as a bechark for testig orality hypothesis. Graphs of these distributios are show i Figure 5. 1,200 Asset W 1,000 0,800 Asset Z 0,600 Asset Y 0,400 0,200 Asset X E2 E3 E4 0, Figure 5 Retur Probability Distributios of Four Assets Assets X ad W have egative skewess ad a slight excess of kurtosis, i cotrast, assets Y ad Z have positive skewess ad a cosiderable excess of kurtosis ad they have greater dispersio of their returs (high variace.) Moreover, assets are correlated, which is show i Table 2. Observe that assets X ad Z, which have the sae skewess sig, show a positive correlatio betwee the ad egatively with the rest of assets. A siilar situatio happes aog assets Y ad Z. Table 2 Correlatio Coefficiets Matrix X Y Z W X 1-0,42-0,46 0,44 Y -0,42 1 0,46-0,45 Z -0,46 0,46 1-0,48 W 0,44-0,45-0,48 1 Results fro optiizatio of portfolio copositio usig Markowitz ad usig Oega 11

12 easure with differet levels of liit returs L, are show i Table 3. Optiizatio through Mea-Variace ethodology (Markowitz) does ot take ito accout other distributio oets, ad hece, copositio percetages vary sigificatly i alost all assets, copared to optiizatio by Oega. Table 3 Portfolio Copositio Markowitz Oega (L=0%) Methodology Oega (L=3%) Oega (L=15%) w1 = %Asset X 27,12% 35,10% 44,54% 47,90% w2 = % Asset Y 9,46% 10,10% 10,84% 2,74% w3 = % Asset Z 15,34% 28,51% 28,95% 49,37% w4 = % Asset W 48,08% 26,30% 15,66% 0,00% I Markowitz, the goal is to achieve the iiu variace, so, asset W was chose i greater proportio due to which presets the sallest variace i relatio to others. Whe it s axiized the Oega easure, assets X ad Z gai greater represetatio i the portfolio, because i these assets is ore likely to obtai returs above the liit value L. Figure 6 shows the distributio shape of optial portfolios, depedig o the ethodology used. 2,0 2,0 1,8 1,6 1,4 1,2 (a) E(Rp) = 12,20% Var(P) = 5,06% Oega(L=0%)=3,71 1,8 1,6 1,4 1,2 (b) E(Rp) = 15,74% Var(P) = 8,17% Oega(L=0%)=5,21 1,0 1,0 0,8 0,8 0,6 0,6 0,4 0,4 0,2 0,2 0,0-1,0-0,5 0,0 0,5 1,0 1,5 2,0 0,0-1,0-0,5 0,0 0,5 1,0 1,5 2,0 2,0 2,0 1,8 1,6 1,4 1,2 (c) E(Rp) = 16,88% Var(P) = 9,28% Oega(L=3%)=3,83 1,8 1,6 1,4 1,2 (d) E(Rp) = 20,09% Var(P) = 20,13% Oega(L=15%)=1,38 1,0 1,0 0,8 0,8 0,6 0,6 0,4 0,4 0,2 0,2 0,0-1,00-0,75-0,50-0,25 0,00 0,25 0,50 0,75 1,00 1,25 1,50 1,75 2,00 Figure 6 Returs probability distributio of the optiized portfolio 'P'. (a) Mea-Variace (Markowitz) Optiizatio, (b) (c) ad (d) Optiizatio by Oega easure with differet levels of L. 0,0-1,00-0,75-0,50-0,25 0,00 0,25 0,50 0,75 1,00 1,25 1,50 1,75 2,00 Table 4 presets ai statistics of the optiized portfolios usig both ethodologies (Markowitz ad Oega). 12

13 Table 4 Mai Statistics of Optiized Portfolios Markowitz Oega (L=0%) Methodology Oega (L=3%) Oega (L=15%) Mea 12,20% 15,74% 16,88% 20,09% Variace 5,06% 8,16% 9,28% 20,13% Skewess -0,34 1,24 1,12 1,77 kurtosis 3,92 7,37 7,15 9,07 JB Test 27,06 524,85 462, ,53 Miiu Value -75,15% -60,47% -68,03% -68,75% Maxiu Value 101,18% 198,08% 208,43% 324,49% EC 16,70% 19,47% 18,80% 18,41% ES 4,50% 3,74% 4,91% 13,32% Oega 3,71* 5,21 3,83 1,38 * Oega is calculated with L=0 It s showed i Table 4, through Jarque Bera test, that all distributios of optiized portfolios are t orality distributed. It also is appreciated that Markowitz ethodology always fids the portfolio with the sallest variace, but its idex Oega (L = 0) copared with the Oega idex fro the optiized portfolio with this easure (with the sae liit retur, L = 0) is cosiderably saller (3,71 vs. 5,21). Ratio of weighted earigs vs. weighted losses is cosidered ore sigificat whe portfolio copositio is optiized by Oega. No-oral distributios are resposible by differeces i results betwee a ethodology ad aother, ad Oega optiizatio is better for to deal with o-oral distributios. Graphically, i Figure 7 are copared o the sae scale (ES vs. EC) the efficiet frotiers of Mea Variace optiizatio ad Oega optiizatio, with L = 0. 0,4 0,37 0,34 Max Ω(0)= 5,21 EC 0,31 0,28 0,25 Maxiu Oega Poit, Ω(0)= 4,69 0,22 0,19 Miiu Variace Poit, Ω(0)= 3,71 0,16 0,13 0 0,02 0,04 0,06 0,08 0,1 0,12 0,14 0,16 0,18 ES Mea-Variace Oega(L=0%) Figure 7 Efficiet frotiers i the scale ES vs. EC It is observed i Figure 7 that the efficiet frotier calculated by Oega easure is superior to the frotier calculated by Markowitz (Mea-Variace), especially i poits with lower expected shortfall (ES). The axiu Oega idex fro Markowitz frotier is less tha 13

14 the axiu Oega idex of the other frotier (oega optiizatio), i the sae way, the Oega idex of iiu variace is uch lower. Thus, it is showed that for a expected shortfall (ES) level, through Oega easure optiizatio i foud higher excess chace (EC) values. By akig the graph of both frotiers o the scale Variace Mea, you get the graph show i Figure 8. 0,3 0,25 Miiu Variace= 0,0506 Sharpe R. = 0,5427 Ω(0) = 3,71 E[Rp] 0,2 0,15 Variace = 0,0817 Sharpe R. = 0,5508 Maxiu Ω(0)=5,21. 0,1 0,05 Miiu Variace=0,0584 Sharpe R. = 0,4381 Ω(0) = 3, ,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Variace Mea-Variace Oega(L=0%) Figure 8 Efficiet frotiers i the scale Variace vs. Portfolio Mea Give the assuptio of orality i Markowitz ethodology, its efficiet frotier is slightly higher (Figure 8) tha efficiet frotier obtaied by axiizig Oega (L = 0). If assets had a oral distributio, both frotiers would be equal, providig sae results, but this differece appears because the frotier calculated by Oega, takes ito accout the shape of the portfolio returs distributio, which is ot oral. I the frotier calculated by Markowitz, is assued oral distributio for portfolio returs, so, it is always foud the sallest variace. Nevertheless, both frotiers do ot deviate uch aog the, ad as the variace icreases the curves will becoe equal. Furtherore, for purposes of easurig perforace usig the Sharpe classic easure, the Markowitz frotier at the poit of iiu variace, calculated a value for this idex slightly lower tha the Sharpe ratio obtaied at the poit of axiu Oega, showed o the efficiet frotier calculated by this easure. But it is kow that Sharpe ratio assue the orality of returs, therefore, that is ot the ost appropriate i other situatios, as i our exaple Aalysis of results ad coparisos aog both ethodologies Through the developed exaple i previous sectio, it is observed: Whe there is orality i the returs distributio of assets, ad therefore orality i portfolio returs, optiizatio by Markowitz provides the sae results as optiizatio through axiizig the Oega easure. Whe there is o orality, should be take ito accout all the distributio oets 14

15 (skewess, kurtosis, heavy tails, extree values, etc.), ad i this way, to easure the real ipact o expected gais ad losses. All this ca be icorporated through the Oega easure. The Figure 7 allows appreciate the differece betwee a efficiet frotier optiized by Oega, ad aother usig Markowitz. Usig Oega is foud a better frotier, which takes ito accout the actual forat of the portfolio returs distributio ad ot just the first two distributio oets. I Table 4 is appreciated that if the liit retur (L) icreases, the Oega easure decreases, because as L oves to the right, the EC area will becoe saller ad ES area will becoe bigger. Thus, Oega idex value teds to decrease. Doig L=0, it will provide the best Oega easure, which eas that o losses are accepted. Efficiet frotiers fro Oega with three liit retur levels are illustrated i Figure 9. Observe that the frotier with better perforace is obtaied whe L = 0. 0,55 0,5 0,45 0,4 EC 0,35 0,3 0,25 L=0 L=0,03 L = 0,15 0,2 0,15 0,1 0,05 0 0,03 0,06 0,09 0,12 0,15 0,18 0,21 0,24 0,27 0,3 0,33 ES Figure 9 Efficiet frotiers for three liit retur levels (L) 5. Optiizatio for a Real Optios Portfolio Next, it is described a ethodology that allows optiizig portfolio copositio for real assets, particularly projects with ivestet optios. This optiizatio atteds objectives for risk levels ad desired returs, usig the Oega easure (Ω). The proposed ethodology cosists of three steps: Step 1: Portfolio coforatio of real assets becoes ore coplicated tha i the case of tradable assets o the arket. This is because there is o usually a history of siilar projects i which oe ight assue a certai behavior of their retur ad/or volatility. Ad eve, if there is a siilar project, future cotext certaily will be differet, doig uique each ivestet. I order to estiate a returs distributio, Mote Carlo siulatio techique is the ost appropriate. It ca odel several future coditios for arket variables that affect the project. Bradão et al. (2005) estiate the arket value of a project, through Mote Carlo 15

16 siulatio, based o the MAD assuptio (Market Asset Disclaier). This assuptio cosiders that the project value without optios would be its arket value, ad hece, the project would be cosidered as a egotiable asset. It is proposed to use this approach to value projects fro the portfolio, which it s previously required to defie variables of the project that will be cosidered stochastic, for exaple, the product price ad/or the productio levels. Stochastic odelig with possible correlatios betwee variables is highly recoeded. Step 2: Secod step is to isert optios to the projects. This akes the value of the icrease. The odel cosiders possibility to exercise those optios whe arket coditios are favorable. Modelig optios by Mote Carlo siulatio is very useful i that sese, doig parity with treatet give to Aerica optios. I those, exercise ca be doe withi a tie util the expiratio. Threshold curve cocept will be very useful for idetifyig the right tie for exercisig the optio. Whe project value reaches a value equal to soe o the curve, it idicates that at that oet should be ade the ivestet. If arket coditios are ufavorable, it ca be cosidered optio to leave, or akig a teporary cessatio of operatios. Favorable or ufavorable coditios ca also lead to decide expasio or reductio i operatios, prior kowledge of the axiu productio capacity ad iiu capacity of operatio. Step 3: Oce odeled projects as if they were egotiable assets, ow the task cosist i forig a portfolio with several goals, as desirables risk ad retur levels. It is uderstood by risk the Expected Shortfall (ES), which is the deoiator of Oega easure (Ω), ad retur is the Excess Chace (EC), which is the uerator of Oega easure (Ω). Both Oega copoets ca be expressed i oetary uits, or i retur percetages. As i Markowitz odel, a goal could be to iiizig the risk for a give retur level. Traslatig these cocepts to Oega easure (prior stipulatio of the liit retur (L)), it is set up a level for EC, ad o its efficiet frotier, we fid the lowest ES. There are possibilities for aager choose a goal that he/she would like to achieve, ad the ethodology will allow hi/her to kow decisios about portfolio copositio ad optios to be cosidered. It is very iportat a high flexibility, both i odelig of variables such as results to be obtaied. Fially, we desire to create basis for a project assesset, i which it s possible to cobie i dyaic way, goals for risk, retur ad perforace levels, takig advatage of ew acadeic researches. This ethodology ca be easily adapted to ay type of idustry. 6. Coclusios ad Fial Cosideratios Nowadays, copaies face up to high degree of ucertaity regardig future perforace of their ivestets. With arket idicators chagig costatly ad vertigious eergeces of ew eterprises, forecast based o the past is icreasigly difficult to be justified, especially whe they iclude arket variables, o which we have o cotrol. Therefore, portfolio copositio of ivestet projects ust lear to deal with this dyais, ad be prepared to chage its structure i a flexible way, depedig o chages 16

17 that ay occur i the eviroet. Mote Carlo siulatio, ais to facilitate odelig of uerous scearios for variables. For exaple, you ca adopt stochastic odelig for sale price of products, productio costs, productio levels, etc. Siilarly, iclusio of real optios i projects is doe easily by siulatio. Over project life is odeled various scearios ad ca be idetified appropriate oets to exercise real optios. Doig a appropriate aalysis for risk, returs ad perforace i portfolio of real assets, it is very crucial i akig decisios. Flexibility i techiques ad/or odels is favorable i order to iprove ability of the copay for reactio. Refereces [1] BERA, A. K.; JARQUE, C. M. (1980). "Efficiet tests for orality, hooscedasticity ad serial idepedece of regressio residuals". Ecooics Letters 6 (3): [2] BRANDÃO, L.; DYER, J.; WARREN, J. Usig Bioial Decisio Trees to Solve Real-Optio Valuatio Probles. Decisio Aalysis, v.2,.2, Jue 2005, p [3] BREALEY, R.A.; MYERS, S.C. "Priciples of Corporate Fiace", McGraw-Hill, Ic., 6.ed., 2000, 1093 pp. [4] CASCON, A.; SHADWICK, W. New Statistical Tools Fro Oega Fuctios. The Fiace Developet Cetre. Lodo, [5] CASCON, A., KEATING, C., SHADWICK, W. The Oega Fuctio. The Fiace Developet Cetre. Lodo, [6] COPELAND, T.; ANTIKAROV, V. Real Optios A Practitioer s Guide. New York: Texere LLC Publishig, 2001, 372 pp. [7] DIXIT, A.; PINDYCK, R. Ivestet uder Ucertaity. Priceto Uiversity Press, New Jersey, [8] DUARTE, Jr., A.M. Model Risk ad Risk Maageet. Derivatives Quarterly, v.3, 1997, p [9] DUARTE Jr., A. M. Aálise de perforace de ivestietos. Uibaco Global Risk Maageet, Available i: < /ANAPERFO.pdf>. Accessed i: 1 Aug [10] ICK, M.; NOWAK, E. Oega based Portfolio Optiizatio a siulatio study o Private Equity ivestets. Workig Paper Uiversity of Lugao, Switzerlad, [11] INUI, K.; KIJIMA, M. O the sigificace of expected shortfall as a coheret risk easure. Joural of Bakig & Fiace, v.29, 2005, p [12] JENSEN, M. The Perforace of Mutual Fuds i the Period The Joural of Fiace, v.23,.2, May 1968, pp [13] J.P. Morga. RiskMetrics. Techical Docuet, New York, [14] KAZEMI, H.; SCHNEEWEIS, T.; GUPTA R. Oega as a Perforace Measure. Workig Paper CISDM. Uiversity of Massachusetts, Iseberg School of Maageet,

18 [15] KEATING, C.; SHADWICK, W. A Uiversal Perforace Measure. Joural of Perforace Measureet, Sprig 2002, p [16] KONNO, H.; YAMAZAKI, H. Mea-Absolute Deviatio Portfolio Optiizatio Model ad its Applicatio to Tokyo Stock Market. Maageet Sciece, v.37 (5), 1991, p [17] LEWIS, A.L. Seivariace ad the Perforace of Portfolios with Optios. Fiacial Aalysts Joural, v.46 (4), 1990, p [18] LONGSTAFF, F.A.; SCHUWARTZ, E. Valuig Aerica Optios by Siulatio: a Siple Least-Squares Approach. The Review of Fiacial Studies, v.14,.1, 2001, p [19] MARKOWITZ, H. Portfolio Selectio. The Joural of Fiace, v.7,.1, Mar. 1952, p [20] SHARPE, W. Mutual Fud Perforace. Joural of Busiess, v.39,.1, 1966, p [21] TREYNOR, J. How to rate aageet of ivestet fuds. Harvard Busiess Review, v.43,.1, Jauary-February 1965, p [22] TRIGEORGIS, L. Real Optios i Capital Ivestet: Models, Strategies, ad Applicatios. Praeger, Lodo, [23] YAMAI, Y.; YOSHIBA, T. Value-at-risk versus expected shortfall: A practical perspective. Joural of Bakig & Fiace, v.29, 2005, pp