An Introduction to Omega

Transcription

1 An Intoduction to Omega Con Keating and William F. Shadwick These distibutions have the same mean and vaiance. Ae you indiffeent to thei isk-ewad chaacteistics? The Finance Development Cente

2 Fom Alpha to Omega Compehensive Pefomance Measues While eveyone knows that mean and vaiance cannot captue all of the isk and ewad featues in a financial etuns distibution, except in the case whee etuns ae nomally distibuted, pefomance measuement taditionally elies on tools which ae based on mean and vaiance. This has been a matte of pacticality as econometic attempts to incopoate highe moment effects suffe both fom complexity of added assumptions and appaently insupeable difficulties in thei calibation and application due to spase and noisy data. A measue, known as Omega, which employs all the infomation contained within the etuns seies was intoduced in a ecent pape i. It can be used to ank and evaluate potfolios unequivocally. All that is known about the isk and etun of a potfolio is contained within this measue. With tongue in cheek, it might be consideed a Shape atio, o the successo to Jensen s alpha. The appoach is based upon new insights and developments in mathematical techniques, which facilitate the analysis of (etuns) distibutions. In the simplest of tems, as is illustated in Diagam 1, it involves patitioning etuns into loss and gain above and below a etun theshold and then consideing the pobability weighted atio of etuns above and below the patitioning. I 2 F () I 1 = 7 Diagam 1 The cumulative distibution F fo Asset A, which has a mean etun of 5. The loss theshold is at =7. I 2 is the aea above the gaph of F and to the ight of 7. I 1 is the aea unde the gaph of F and to the left of 7. Omega fo Asset A at =7 is the atio of pobability weighted gains, I 2, to pobability weighted losses, I 1. The Finance Development Cente

3 By consideing this Omega atio at all values of the etuns theshold, we obtain a function which is chaacteistic of the paticula asset o potfolio. We illustate this in Diagam 2. () Diagam 2 Omega fo Asset A as a function of etuns fom =-2 to =15. Omega is stictly deceasing as a function of and takes the value 1 at Asset A s mean etun of 5. The evaluation statistic Omega has a pecise mathematical definition as: Ω() = b (1 F(x))dx a F(x)dx whee (a,b) is the inteval of etuns and F is the cumulative distibution of etuns. It is in othe wods the atio of the two aeas shown in Diagam 2 with a loss theshold set at the etun level. Fo any etun level, the numbe Ω() is the pobability weighted atio of gains to losses, elative to the theshold. The Omega function possesses many pleasing mathematical popeties ii that can be intuitively and diectly intepeted in financial tems. As is illustated above, Omega takes the value 1 when is the mean etun. An impotant featue of Omega is that it is not plagued by sampling uncetainty, unlike standad statistical estimatos as it is calculated diectly fom the obseved distibution and equies no estimates. This function is, in a igoous mathematical sense, equivalent to the etuns distibution itself, athe than simply being an appoximation to it. It theefoe omits none of the infomation in the distibution and is as statistically significant as the etuns seies itself. As a esult, Omega is ideally suited to the needs of financial pefomance measuement whee what is of inteest to the pactitione is the isk and ewad The Finance Development Cente

4 chaacteistics of the etuns seies. This is the combined effect of all of its moments, athe than the individual effects of any of them which is pecisely what Omega povides. Now to use Omega in a pactical setting, all that is needed is a simple decision ule that we pefe moe to less. No assumptions about isk pefeences o utility ae necessay though any may be accommodated. The Omega function may be thought of as the canonical isk-etun chaacteistic function of the asset o potfolio. In use, Omega will usually show makedly diffeent ankings of funds, potfolios o assets fom those deived using Shape atios, Alphas o Value at Risk, pecisely because of the additional infomation it employs. In the cases whee highe moments ae of little significance, it agees with taditional measues while avoiding the need to estimate means o vaiances. In those cases whee highe moments do matte and when they do thei effects can have a significant financial impact it povides the cucial coections to these simple appoximations. It also makes evident that at diffeent levels of etuns, o maket conditions, the best allocation among assets may change. In many espects, Omega can be thought of as a pay-off function, a fom of bet whee we ae consideing simultaneously both the odds of the hose and its obseved likelihood of winning. Omega povides, fo each etun level, a pobability adjusted atio of gains to losses, elative to that etun. This means that at a given etun level, using the simple ule of pefeing moe to less, an asset with a highe value of Omega is pefeable to one with a lowe value. We illustate the use of the Omega function in a choice between two assets. Diagam 3 which shows thei Omegas as functions of the etun level. () I () µ B = 3.4 Diagam 3 The Finance Development Cente

5 Notice that the mean etun of asset A (the point at which is equal to 1) is highe than that of Asset B. The point I, whee the two Omegas ae equal, is an indiffeence point between A and B. At etun levels below this point, using just the we pefe moe to less citeion, we pefe Asset B, while above we pefe Asset A. This phenomenon of changes of pefeence, cossings of the Omega functions, is commonplace and multiple cossings can occu fo the same pai of assets. The additional infomation built into Omega, can lead to changes in ational pefeences which cannot be pedicted using only mean and vaiance. We emak that in the example, Asset A is iskie than Asset B in the sense that it has a highe pobability of exteme losses and gains. This aspect of isk is encoded in the slope of the Omega function: the steepe it is, the less the possibility of exteme etuns. A global choice in this example involves an investment tade-off between the elative safety of Asset B compaed to Asset A and the educed potential this caies fo lage gains. We may use the Omega functions calculated ove a selected sequence of times to investigate the pesistence o skill in a manage s pefomance. Some peliminay wok suggests that thee is fa moe pesistence than academic studies have indicated peviously. The Omega function can also be used in potfolio constuction, whee makedly diffeent weights fom those deived unde the standad mean vaiance analysis of Makowitz ae obtained. In fact those efficient potfolios can be shown to be a limited special case appoximation within the moe geneal Omega famewok. Omega, when applied to benchmak elative potfolios, povides a famewok in which tuly meaningful tacking eo analysis can be caied out, a significant expansion of existing capabilities. All things consideed, Omega looks set to become a pimay tool fo anyone concened with asset allocation o pefomance evaluation. Paticulaly those concened with altenative investments, leveaged investment o deivatives stategies. The fist of a new geneation of tools adapted fo eal isk ewad evaluation. The Finance Development Cente

6 Ω Notes: Ω and Nomal distibutions Hee we conside the simplest application of all, to etuns distibutions which ae nomal. The appoach to anking such distibutions via the Shape atio involves an implicit choice to conside the possibility of a etun above the mean and a etun below the mean as equally isky. Fo two nomal distibutions with the same mean, the Shape atio favous the one with the lowe vaiance, as this minimizes the potential fo losses. Of couse it also minimizes the potential fo gains. Thus, the use of vaiance as a poxy fo isk consides the downside as moe significant than the upside, even in the case whee these ae equally likely. Hee we conside two assets, A and B, which both have a mean etun of 2 and have standad deviations of 3 and 6 espectively. Thei pobability densities ae shown in Diagam1. In tems of thei Shape atios, A is pefeable to B. If we wee to ank these assets in tems of thei potential fo gains howeve, the ankings would be evesed. Conside the anking which an investo who equies a etun of 3 o highe to avoid a shotfall might make. Fom this point of view a etun below 3 is a loss, while one above 3 is a gain. To assess the elative attactiveness of assets A and B such an investo must be concened with the elative likelihood of gain o loss. It is appaent fom Diagam 1 that this is geate fo asset B than fo asset A. A B = 3 Diagam 1. The distibutions fo assets A and B with a loss theshold of 3 This is due to the fact that the distibution fo B has substantially moe mass to the ight of 3 than the distibution A. Fo asset B, about 43% of the etuns ae above 3 while fo asset A the popotion dops to 37%. The atios of the likelihood of gain to loss ae 0.77 fo B and 0.59 fo A. Fom this point of view, ou ank ode is the evese of the anking by Shape atios. We ae not simply evesing the Shape atio bias howeve. If the investo s loss The Finance Development Cente

7 theshold wee placed at a etun of 1 athe than 3, the same pocess would lead to a pefeence fo A ove B. The atios of the likelihood of gain to loss with the loss theshold set at a etun of 1 ae 1.71 fo A and 1.31 fo B. Clealy, at any loss theshold above the mean the pefeence will be fo B ove A, while fo any loss theshold below the mean the pefeence will be evesed. With the loss theshold set at the mean, both assets poduce a atio of 1. Like this simple pocess, the use of Ω teats the potential fo gains and losses on an equal footing and povides ankings of A and B which depend on a loss theshold. The function Ω() compaes pobability weighted gains to losses elative to the etun level. As a esult, ankings will vay with. Diagam 2 shows the Ωs fo assets A and B. Fo any value of geate than the common mean of 2, the pobability weighted gains to losses ae highe fo asset B than fo asset A. Fo any value of less than the mean, the ankings ae evesed. The elative advantage of asset A to asset B declines smoothly as appoaches thei common mean of 2 and theeafte the elative advantage of B to A inceases steadily. Diagam 2. Omega fo assets A B as a function of etun level The Finance Development Cente

8 Ω Notes: The Omega of a Shape Optimal Potfolio The mean-vaiance appoach to pefomance measuement and potfolio optimization is based on an appoximation of nomality in etuns. In this note we show that even in the case of two assets with nomally distibuted etuns, a potfolio which maximizes the Shape atio will be sub-optimal ove a significant ange of etuns. This is a manifestation of the inheent bias in egading losses and gains as equally isky. As Omega ankings change with the level of etuns, an Omega optimal potfolio s composition will vay ove diffeent anges of etuns. This exta flexibility can be vey impotant as we show hee. A potfolio composition which is independent of etuns level and is optimal on the downside, as the Shape optimal potfolio is, must be sacificing consideable upside potential. We conside two assets, A and B which have independent, nomally distibuted etuns with means and standad deviations of 6 and 4 and 7 and 3 espectively. We let a denote the weight of asset A and 1 a the weight of asset B in the potfolio. 6a + 7(1 a) The Shape atio fo the potfolio is then SR(a) =, which has its 16a (1 a) maximum value at about a = The Shape optimal potfolio has nomally distibuted etuns with a mean and standad deviation of 6.7 and 2.4. The potfolio distibution is shown in Diagam 1, with the distibutions fo assets A and B. S A B Diagam 1. The distibutions fo asset A, asset B and the Shape optimal potfolio. In Diagam 2 we show the Omegas fo asset A, asset B and the Shape optimal potfolio fo etuns below the level of 5. The Shape optimal potfolio clealy dominates both asset A and asset B ove this ange. Diagam 3 shows that fo etuns levels above about 5.4, the Shape optimal potfolio The Finance Development Cente

9 has a lowe value of Omega than a potfolio consisting only of asset A. Diagam 4 shows that the Shape optimal potfolio also has a lowe value of Omega than asset B fo etuns above 7.8. Fo etuns above 5.4 and below 10 we obtain a highe value of Omega by holding asset A. Fo etuns above 10, holding only asset A continues to be pefeable to the Shape optimal potfolio but holding only asset B is pefeable to both these options, as one sees in Diagam 5. It is appaent that in this example the Shape optimal potfolio is sacificing a consideable amount of the available upside. Fully 69% of the etuns fom the Shape optimal potfolio and fom asset A ae above 5.4. Ove 55% of the etuns fom asset B ae above this level. Ω S Diagam 2. The Omegas fo asset A, asset B and the Shape optimal potfolio as functions of the etun level. The cossing in the Omegas fo asset A and the Shape optimal potfolio identifies the point at which a potfolio consisting of 100% asset A has bette isk-ewad chaacteistics than the Shape optimal potfolio. Holding asset A, togethe with a put option with a stike at the etun level of this cossing is an obvious stategy fo a isk avese investo. Stategies fo any isk pefeence may be obtained by optimising Omega ove the appopiate ange of etuns, as is illustated in Diagam 6. The Finance Development Cente

10 Ω S Diagam 3. Above 5.4 the Shape optimal potfolio has a lowe value of Omega than asset A. Ω S Diagam 4. The Shape optimal potfolio has a lowe value of Omega than eithe asset A o asset B fo etuns above 7.8. The Finance Development Cente

11 Ω S Diagam 5. The Omega optimal potfolio is 100% asset B fo etuns above 10. Ω S Ω Opt Diagam 6. The Omega optimal potfolio (black) fo etuns between 4.4 and 5.4 is 80% asset A, 20% asset B. The Finance Development Cente

12 Ω Notes: Does negative skew and highe than nomal kutosis mean moe isk? This is the distibution fomed by combining 3 nomal distibutions with means of 0, 78.5 and 76 espectively and standad deviations of 11.2, 20.8 and Thei weights in the combination ae 62%, 7% and 31%. The mean and standad deviation of the esulting distibution ae 18 and 46. It has skew of -.05 and kutosis of 3.17 about 6% highe than a nomal distibution s kutosis of 3. These ae both widely egaded as signs of highe than nomal isk. We show the distibution (A) and a nomal distibution with the same mean and vaiance (B). A B In spite of the indications fom skew and kutosis, it is the nomal distibution which has the heavie tails on both the up and downsides. A 2-σ gain is almost 1.8 times as likely fom distibution A as fom the nomal howeve a 3-σ gain is only 0.85 times as likely. At the 4 -σ level the gain is 35 times moe likely fom the nomal. The downside is moe alaming. The pobability of a 1-σ loss is about 1.4 times highe than fo the nomal howeve a 2-σ loss is only 0.7 times as likely. The nomal is almost 80 times as likely to poduce a 3-σ loss and ove 100,000 times moe likely to poduce a 4-σ loss. Both the lage loss and lage gain egimes ae poduced by moments of ode 5 and highe which dominate the effects of skew and kutosis. It is not possible to estimate moments of such odes fom eal financial data. The Omegas fo these two distibutions, captue this infomation completely, with no need to compute moments of any ode. The cossings indicate a change of pefeence. The Finance Development Cente

13 This is the leftmost cossing of the Omegas. The combination has less downside and a highe value of Omega. This is the ightmost cossing of the Omegas. The nomal has moe upside. The Finance Development Cente

14 Ω Notes: How many moments do you need to descibe tail behaviou? This is a distibution fomed by combining thee nomal distibutions whose means ae 5, 0 and 5 with standad deviations of 0.5, 6.5 and 0.5 espectively. The espective weights ae 25%,50% and 25%. It is shown below with a nomal distibution with the same mean (0) and vaiance (5.8) A B The distibution A with a nomal B of the same mean and vaiance. The kutosis of this distibution is 2.65 o about 88% of the nomal value of 3. Although lowe kutosis is often egaded as indicating lowe isk, this distibution has heavie tails than a nomal with the same mean and vaiance. The distibution is symmetic so the odd moments ae, like those of the nomal, all zeo. The 6 th moment is identical with that of the nomal to within 2 pats in 1,000 and the eighth moment only diffes fom that of the nomal by 24%. It is only with the tenth moment that a moe substantial deviation fom the nomal appeas. The tenth moment is 55% geate fo the potfolio than fo the nomal. The dominant effects poducing the heavy tail behaviou theefoe come fom moments of 8, 10 and highe. These simply cannot be estimated fom eal data. The Finance Development Cente

15 The Omegas fo distibutions A and B aound thei common mean of 0. Cossings in Omegas indicate a change in pefeence. The lage loss egime and the left-most pefeence change (2σ is 11.6). Distibution A has a lowe Omega to the left of 12.2 as it the highe catastophic loss potential. Distibution A has almost 3 times the likelihood of a 4-σ loss o gain than the nomal with the same mean and vaiance. i Con Keating and William F. Shadwick, A Univesal Pefomance Measue The Finance Development Cente ii Ana Cascon, Con Keating and William F. Shadwick, The Mathematics of the Omega Measue The Finance Development Cente An Intoduction to Omega The Finance Development Cente, Febuay 2002 The Finance Development Cente