PORTFOLIO OPTIMIZATION WHEN ASSET RETURNS HAVE THE GAUSSIAN MIXTURE DISTRIBUTION

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1 PORTFOLIO OPTIMIZATION WHEN ASSET RETURNS HAVE THE GAUSSIAN MIXTURE DISTRIBUTION IAN BUCKLEY, GUSTAVO COMEZAÑA, BEN DJERROUD, AND LUIS SECO Abstact. Potfolios of assets whose etuns have the Gaussian mixtue distibution ae optimized in the static setting to find potfolio weights and efficient fonties using the pobability of outpefoming a taget etun and Hodges modified Shape atio objective functions. The sensitivities of optimal potfolio weights to the pobability of the maket being in the distessed egime ae shown to give valuable diagnostic infomation. A two-stage optimization pocedue is pesented in which the high-dimensional non-linea optimization poblem can be decomposed into a elated quadatic pogamming poblem, coupled to a lowe-dimensional non-linea poblem. Contents 1. Intoduction 2 2. Evidence of covaiance egimes 3 3. Gaussian mixtue distibution Definitions and identities Visualizing the GM distibution 5 4. Potfolio optimization Pobability of shotfall as a isk measue Hodges atio Lowe patial moments Optimization poblems fo diffeent objectives Investment oppotunity set Numeical examples Altenative algoithm fo solving the non-linea poblem Conclusions 17 Acknowledgment 17 Appendix A. Gaussian mixtue distibution 18 A.1. Definitions 18 A.2. Moments 18 A.3. Linea combinations of andom vaiables with the GM distibution 20 Refeences 20 Date: Tuesday, Febuay 18, 2003 at 20:32 and, in evised fom,... Key wods and phases. Mixtue of nomals distibution, Gaussian mixtue distibution, potfolio optimization, maket distess, hedge fund potfolio, Shape atio, efficient fontie, Hodges modified Shape atio, exponential utility, coelation switching, egime switching, asset allocation, commodity tading adviso, pobability of shotfall, distess sensitivities. This wok was completed with the suppot of... Scholaship. 1

2 2 I. BUCKLEY, G. COMEZAÑA, B. DJERROUD, AND L. SECO 1. Intoduction In this aticle Makowitz mean-vaiance potfolio theoy [12], the foundation fo single-peiod investment theoy, is genealized to descibe potfolios of assets whose etuns ae descibed by the (finite) Gaussian mixtue (GM) (altenatively mixtue of nomals) distibution. Whilst the assets in the univese could be of the conventional vaiety, such as equities o bonds, ou pimay goal is to develop a famewok which lends itself to the management of potfolios of hedge funds o even fo optimally combining the ecommendations fom a goup of commodity tading advisos (CTAs). That is to say, we seek an appoach suitable fo finding an optimal fund of funds. Because of the infinite vaiety of hedge fund and CTA stategies and the speed at which a given fund s composition can be changed, the altenative assets that we descibe ae not expected to behave like conventional assets such as individual equities, bonds o even long-only funds such as index tacke funds and exchange taded funds (ETFs). Indeed we need to be pepaed fo thei pices to be less pedictable, moe volatile and to have moe exotic distibutions. The new appoach is ideal fo an industial setting, poviding consideable additional flexibility ove and above a standad Makowitz appoach, with only a modest incease in complexity. The assumption that asset etuns have the multivaiate Gaussian distibution is a easonable fist appoximation to eality and gives ise to tactable theoies. Many theoies foming the foundations of mathematical finance adopt this conjectue, including Black-Scholes-Meton option picing theoy, Makowitz potfolio theoy, and the CAPM and APT equity picing models. Howeve, it is well-known that fo assets, both in the conventional sense of equities and bonds, but also in a boade sense, fo example in the fom of county o secto-based equity o bond indices, and even moe so fo altenative investments such as hedge funds and CTAs, the situation is moe complex. The pupose of the genealization descibed in this pape is to addess two well-known limitations with the assumption that asset etuns obey the multivaiate Gaussian distibution with constant paametes ove time: The skewed (asymmetic aound the mean) and leptokutotic (moe kutotic o fat-tailed than a Gaussian distibution) natue of maginal pobability density functions (pdfs) The asymmetic coelation (o coelation beakdown) phenomenon, which descibes the tendency fo the coelations between asset etuns to be dependent on the pevailing diection of the maket. Typically coelations ae lage in a bea maket than a bull maket. The fist point efes to the univaiate distibutions fo etuns that ae obseved if assets ae consideed one at a time. The second point descibes effects that can only be obseved when the etuns to multiple assets ae investigated togethe. One way to captue the dependence stuctue of multiple andom vaiables in a isk management setting is by using copulas [22], [7], [9], [13], [18]. Copulas descibe that pat of the shape of the pdf that cannot be descibed by the maginal distibutions. A (finite) Gaussian mixtue distibution, as descibed in this pape, can be used to appoximate a geneal multivaiate distibution, as equivalently expessed eithe as a pdf o decomposed into a seies of univaiate maginal distibutions and a copula.

3 PORTFOLIO OPTIMIZATION IN A GAUSSIAN MIXTURE ENVIRONMENT 3 In fact, stictly the tem asymmetic coelation descibes a poposed explanation fo a phenomenon, athe than the undelying, fundamental phenomenon itself. The latte is simply that the iso-pobability contous of the multivaiate pdf fo asset etuns ae less symmetic than the ellipsoidal contous of the multivaiate nomal distibution. Apat fom those distibutions with pdfs with ellipsoidal iso-pobability sufaces, such as the Gaussian and multivaiate t, fo all othe multivaiate distibutions the coelation matix does not povide an adequate summay of the dependence stuctue of the constituent isks. Consideing multiple egimes within which the odinay constant paamete Gaussian assumptions pevail successfully gives ise to a model that eflects eality bette than a standad Gaussian model, without having to depat too adically fom it. Howeve, it is cetainly not the only way to constuct a model with non-ellipsoidal iso-pobability contous. Indeed altenative assumptions to achieve the same ends may ende the concept of coelation edundant and statements about its evolution ove time, meaningless. Apat fom the distibutional assumptions, the second fundamental diffeence between the novel appoach descibed in this pape (the GM appoach) and the Makowitz mean-vaiance appoach, is that we adopt a diffeent objective function. This is essential in ode to get diffeent optimal potfolio weights. When we efe to the GM appoach, the use of an altenative objective will be implicit. A dawback of using a moe exotic objective than the vaiance is that the esultant optimization poblem is not a linea-quadatic pogam (LQP), as it is when vaiance is the chosen isk measue. Howeve, the nonlinea optimization poblem with multiple local extema that eplaces it can be solved eliably and quickly with commonly available outines, at least with a modeate numbe of assets (i.e. less than ten). The plan fo the pape is as follows. Evidence of covaiance egimes ove time is given in Section 2, including a summay of existing esults fom the liteatue fo conventional assets and oiginal esults fo altenative investments. We ae intoduced to the GM distibution in Section 3 in which we find definitions of GM distibuted andom vaiables, estimation issues, and some identities concening moments and linea combinations of GM vaiables. In Section 4 we take the pobability of shotfall objective, assume GM distibuted asset etuns and conside a theoy based on these ingedients. Key theoetical and numeical esults ae descibed. Section 5 contains ou conclusions. 2. Evidence of covaiance egimes Duing the aftemath of the 1987 maket cash, a deficiency in the isk models based on the multivaiate nomal distibution eceived inceased attention: a simultaneous downwad movement in all the makets of the wold was a moe fequent occuence than the models pedicted when calibated using asset etuns obseved duing tanquil peiods. Divesifying amongst diffeent assets o makets was less effective at educing isk than many paticipants had hitheto believed. Impotant investigations of the contagion phenomenon include [8], [4], [20]. In common with thei conventional asset countepats, altenative assets exhibit the coelation beakdown phenomenon. As evidence we pesent Figue 1, which shows the coelation matices between hedge fund etuns in tanquil and distessed egimes. Duing tanquil peiods coelations ae small, wheeas duing peiods of maket distess, the asset etuns become highly coelated, with off-diagonal coelation values close to one.

4 4 I. BUCKLEY, G. COMEZAÑA, B. DJERROUD, AND L. SECO Figue 1. Ba chats to show typical values of coelations between the etuns of a goup of eight hedge funds duing tanquil (left) and distessed (ight) peiods. We note in passing that because finding potfolio weights in a mean-vaiance setting is tantamount to inveting the covaiance matix, then eos in the optimal potfolio weights will be most sensitive to eos in the covaiance matix when the latte is closest to being singula (having a deteminant close to zeo). The covaiance matix will be moe singula in the distessed egime than in the tanquil egime. 3. Gaussian mixtue distibution The Gaussian mixtue distibution is selected fom the ange of paametic altenatives to the nomal distibution fo its tactability: calculations using it often closely esemble those using the nomal distibution. Whilst thee ae many univaiate paametic pobability distibutions, (e.g. hypebolic, t, genealized beta, α-stable) the list fo multivaiate distibutions is shote (e.g. t, α-stable). The GM distibution has been used befoe in the field of finance, mostly in its univaiate guise fo the estimation of Value at Risk (VaR). [16] develop a model fo estimating VaR in which the use is fee to choose any pobability distibution fo the daily changes in each maket vaiable and employ the univaiate mixtue of nomals distibution as an example. In the same field, [28] assumes pobability distibutions fo each of the paametes descibing the mixtue of nomals and uses a Bayesian updating scheme; and [26] uses a quasi-bayesian maximum likelihood estimation pocedue. The cuent RiskMetics TM methodology uses GM with a mixtue of two nomal distibutions. Moe ecently GM models have been used [19] to model futues makets and fo potfolio isk management and by [10] fo cedit isk. [27] develops an efficient analytical Monte Calo method fo geneating changes in asset pices using a multivaiate mixtue of nomal distibutions with abitay covaiance matix. [25] descibe computational tools fo the calculation of VaR and othe moe sophisticated isk measues such as shotfall, Max-VaR,

5 PORTFOLIO OPTIMIZATION IN A GAUSSIAN MIXTURE ENVIRONMENT 5 conditional VaR and conditional isk measues that aim to take account of the heteoskedastic stuctue of time seies. This pape descibes the static, single peiod setting only, in which distibutions of andom vaiables ae sufficient to specify the model. In this setting the key idea is that we build an exotic distibution by mixing simple ones, namely copies of the nomal distibution. Ou only assumption is that ove an inteval of time, the etuns to the assets in the univese ae descibed by the multivaiate GM distibution. It is unnecessay to make futhe assumptions about the natue of the asset pice evolution duing this inteval. Howeve, we do maintain an inteest in the dynamic case, i.e. the multiple peiod discete o continuous time setting, because we wish to motivate the use of the GM distibution and we pefe to constuct static models that extend natually to the dynamic case. Thee is a gowing body of wok in which exotic (asset etun) stochastic pocesses have been constucted by mixing simple ones. Pocesses fo asset etuns may be constucted fom unconditional o conditional distibutions. As an example of the latte case, by mixing autoegessive pocesses such as ARCH and GARCH, pocesses can be constucted that can account fo both the heteoscedastic and leptokutic natue of financial time seies. See [23] and [11]. GM distibutions can aise natually as the level of cetain stochastic pocesses at a point in time, conditional on the level at an ealie time e.g. Makov (egime) switching models and jump pocesses. Regime switching models descibe pocesses in which paametes of a continuous diffusion pocess may change discontinuously accoding to the ealized stochastic path though an associated Makov chain. We conclude that GM distibutions ae bette motivated and less contived than fist impessions might suggest. Recent applications of egime switching asset etun pocesses include: in the field of Meton-style option picing theoy, [17], and in potfolio management [5] (CAPM ) and [2], [1] (intenational divesification). Mixtue distibutions have the appeal that by adding togethe a sufficient numbe of component distibutions, any multivaiate distibution may be appoximated to abitay accuacy. With an infinite numbe of contibutions, any distibution can be econstucted exactly. As fa as estimation is concened, a disadvantage of using the GM distibution is that the log-likelihood function does not have a global minimum. A esolution to this poblem exploed in [14] is to use a modified log likelihood function. Because of the use of the GM distibution and othe mixtue distibutions in image pocessing, clusteing, and unsupevised leaning a host of estimation techniques have been developed to addess this poblem [6]. When using the GM distibution to model asset etuns, [27] employs the EM algoithm Definitions and identities. In Appendix A the definition of the GM distibution and vaious identities involving it ae pesented Visualizing the GM distibution. In Figue 2, in fou contou plots of pdfs, we see how two bivaiate Gaussian distibutions (top ow) can be added to yield a GM distibution (bottom, left). Note the potential fo highly non-elliptic iso-pobability contous when the GM distibution is used. Fo compaison a bivaiate nomal with the same sample means, vaiances and covaiances is included (bottom, ight). If the GM distibution given wee used to descibe the etuns to two assets, the lobe pointing down and to the left of the figue would descibe the

6 6 I. BUCKLEY, G. COMEZAÑA, B. DJERROUD, AND L. SECO Figue 2. Contou plots of pobability density functions. The top ow contains two bivaiate Gaussian distibutions - potentially fo the tanquil (left) and distessed (ight) egimes. The bottom ollustates the composite Gaussian mixtue distibution obtained by mixing the two distibutions fom the top ow (left) and a bivaiate nomal distibution with the same means and vaiancecovaiance matix as the composite (ight). popensity of the two assets to decline shaply togethe (the asymmetic coelation phenomenon). The Gaussian distibution, with its elliptic contous, is clealy unable to captue this featue. These examples illustate the two asset case. With thee assets the contous fo the Gaussian distibution ae thee-dimensional ellipsoids (athe than ellipses in two dimensions), and fo the GM distibution the contous ae complicated lobed sufaces embedded in a thee-dimensional space. 4. Potfolio optimization In ou famewok, the n Gaussian contibutions to the GM distibution ae associated with n egimes. In all the numeical examples, we take n = 2, and call the two egimes the tanquil and distessed egimes Pobability of shotfall as a isk measue. When vaiance is used as the optimization objective, the efficient fontie is independent of the distibution fo

7 PORTFOLIO OPTIMIZATION IN A GAUSSIAN MIXTURE ENVIRONMENT 7 the asset etuns. Theefoe, to achieve non-tivial esults in the GM setting, we adopt altenative, non-quadatic objective functions. One with the advantage of being intuitive fo pactitiones is the pobability of shotfall below a taget etun (PoS). When etuns have a nomal distibution, minimizing the PoS is equivalent to maximizing the out-pefomance Shape atio (atio of diffeence between ealized etun and taget etun in the numeato to volatility in the denominato). Theefoe adopting the GM appoach will not epesent a significant depatue fom cuent pactice fo many investos. Note that the PoS is not the same as the VaR: the fome is the pobability beyond a given point on the distibution (e.g. a taget etun of 18.2%), the latte is the point on the distibution such that the pobability of being beyond that point is equal to a given value (e.g. 1% o 5%). In the GM setting of this pape the pobability of shotfall objective is the pobability that a univaiate mixtue of nomals andom vaiable, with egime means µ i = µ i.θ and egime vaiances σ 2 i = θ.v i.θ, exceeds the taget etun k. Fom the expession fo the univaiate mixtue of nomals CDF Eqn. A.1.2, above, it can be shown that: Poposition The pobability that the potfolio etun falls shot of the taget k is: ( ) µ (4.1.1) F k (θ) = Φ i.θ k θ.v i.θ 4.2. Hodges atio. To addess the paadoxes inheent in using the Shape atio [24] as a measue fo anking the desiability of payoff distibutions, Hodges [15] intoduced a measue of isk fo pefomance anking based on the exponential utility function (4.2.1) U(w) = e λw, which is intuitive and is a genealization of the Shape atio, educing to it fo nomally distibuted etuns. We efe to Hodges modified Shape atio as the Hodges atio (HR). Because the HR uses a utility function with constant absolute isk avesion, the composition of the optimal potfolio is independent of the coefficient of isk toleance and so without loss of geneality this can be taken to be one. It educes to the Shape atio fo nomally distibuted etuns and yet can be applied to any etun distibution. It is compatible with stochastic dominance: an investment oppotunity that outpefoms anothe in evey state of the wold necessaily has a highe HR. This is one of the popeties of a coheent isk measue [3]. Madan and McPhail [21] point out an appaent limitation with the HR, namely that it sometimes pevesely consides distibutions with lage negative skew to be desiable investments, because the appoach effectively shots the isky asset. It tuns out that this poblem is simply esolved by scaling the utility function by the sign of the weight in the isky assets, ξ. In this pape, fo all sensible paamete values ξ > 0, so this efinement was unnecessay Lowe patial moments. Patial moments of a andom vaiable ae its moments ove a sub-inteval of the domain of the pdf. In paticula, the lowe and uppe nth-patial moments (LPM, UPM) ae the nth ode moments fo the andom vaiable when it takes nonzeo values below and above a given value, espectively.

8 8 I. BUCKLEY, G. COMEZAÑA, B. DJERROUD, AND L. SECO Necessaily, as a statistic it summaises the popeties of the distibution only to one side of the paamete. The LPM is a good candidate fo a isk measue because it only consides those states in which etun on an asset is below a pe-specified taget ate. To eal individuals a downside isk measue such as this coesponds moe closely to thei concept of isk than the ubiquitous vaiance isk measue. We pesent some useful identities: (4.3.1) (4.3.2) (4.3.3) (4.3.4) (4.3.5) a a a (a x) φ(x) dx = a Φ(a) + φ(a) = a Φ(x) dx (a x) 2 φ(x) dx = ( 1 + a 2) Φ(a) + a φ(a) a x σ φ µ,σ (x) dx = a µ σ Φ µ,σ (a) + σ φ µ,σ (a) ( 1 a σ 2 (a x) 2 φ µ,σ (x) dx = (a µ) φ µ,σ (a) ) (a µ)2 σ 2 Φ µ,σ (a) 4.4. Optimization poblems fo diffeent objectives. In each case µ T is the taget potfolio expected etun. Makowitz: (4.4.1) min θ θ.v S.θ s.t. θ.1 = 1 µ S.θ µ T whee µ S = n µ i and V S ae the sample mean and sample vaiancecovaiance matix, espectively. Note that the latte is not equal to the weighted sum of the egime vaiances. Instead it is calculated using the identity in Equation A.2.5 o equivalently A.2.6. Moments of GM distibutions ae the sums of the component distibution moments. Howeve, fo cental moments, such as the vaiance, the expessions ae moe complicated. Shape Ratio: (4.4.2) µ max S.θ k θ V S s.t. θ.1 = 1 µ S.θ µ T

9 PORTFOLIO OPTIMIZATION IN A GAUSSIAN MIXTURE ENVIRONMENT 9 (4.4.3) (4.4.4) (4.4.5) (4.4.6) Pobability of shotfall: Hodges atio: max θ,ξ max θ ( ) µ Φ i.θ k θ.v i.θ s.t. θ.1 = 1 µ i.θ µ nx T i. exp λ ξ µ i.θ 1 2 λ2 ξ 2.V s.t. θ.1 = 1 µ i.θ µ T whee ξ is the weight in the isky assets and λ is the paamete fo the exponential utility function. The emainde, 1 ξ is invested in a isk-fee bank account, taken to have zeo etun fo simplicity. The above optimization poblem is clealy equivalent to a simila one in which ξ = 1 and the budget constaint is emoved. Howeve, in the numeical examples, both the total weight in isky assets vaiable, ξ, and the budget constaint, ae etained so that the nomalised weights of the individual assets within the isky asset potion can be compaed with the esults fom othe objective functions. E.g., see 6. Note that the exponential tem in the objective can be deived by obseving that if the andom vaiable X has mean µ and vaiance V then ( ( E[U(ξX)] = exp λ ξ µ 1 )) 2 λ2 ξ 2 V whee the utility function U(x) = e λ x. Lowe patial moment, ode n = 1: ( ) k µi max Φ µi,σ θ σ i (k) + σ i φ µi,σ i (k) i s.t. θ.1 = 1 µ i µ T whee µ i = µ i.θ and σ i = θ.v i.θ Lowe patial moment, ode n = 2: ( ) max (k µ i ) φ µi,σ i (k) (k µ i) 2 Φ θ σ 2 µi,σ i (k) i s.t. θ.1 = 1 µ i µ T

10 10 I. BUCKLEY, G. COMEZAÑA, B. DJERROUD, AND L. SECO Figue 3. Scatte plot of potfolios in the investment oppotunity set, ovelaid on the contou plot of the pobability of shotfall objective function. The axes ae the out-pefomance Shape atios in the tanquil (x) and distessed (y) egimes. The GM appoach optimal potfolio is maked in the top ighthand cone, on the efficient fontie. Typically, the potfolios that ae optimal with espect to the mean-vaiance objectives in the tanquil and distessed egimes will be sub-optimal with espect to the pobability of shotfall objective used in the GM appoach. This is a thee asset example Investment oppotunity set. Figue 3 shows the investment oppotunity set (IOS) in the 3-asset case, ovelaid on the contou plot fo the PoS objective function, in tanquil and distessed egime out-pefomance Shape atio space. The investment oppotunity set is the set of all attainable points in a given space that may be eached by constucting potfolios of the assets in a given univese. Typically vaiance and etun ae taken as the dimensions of a space to exploe, but any statistics may be used. The GM appoach optimal potfolio is maked in the top ighthand cone, on the efficient fontie. Compae this with Figue 4, which shows the IOSs fo the tanquil and distessed egimes supeimposed onto the same plot. The axes ae the potfolio means and vaiances in the two egimes. Clealy the potfolio that maximizes the PoS objective, will be suboptimal with espect to the mean-vaiance efficient fontie in each of the component egimes Numeical examples. We pesent a thee asset example with five objectives: Vaiance Pobability of shotfall Hodges atio Lowe patial fist moment

11 PORTFOLIO OPTIMIZATION IN A GAUSSIAN MIXTURE ENVIRONMENT 11 Figue 4. Investment oppotunity sets fo the tanquil and distessed egimes supeimposed onto the same plot. The axes ae the potfolio mean and vaiance. Typically the GM appoach optimal potfolio will be sub-optimal with espect to both the tanquil and distessed mean-vaiance objectives. This is a thee asset example. Lowe patial second moment All esults ae pesented gaphically. Paamete values ae: (4.6.1) (4.6.2) (4.6.3) µ T = σ T = ρ T = µ D = σ D = ρ D = Figue 5 shows 2-dimensional pojections of the GM pdf as contou plots. The contous indicate iso-pobability cuves. The pictues ae slices in the sense that the suppessed dimension has asset weight zeo. Figue 6 shows the optimal potfolio weights along the fontie. All heights on the plot ae optimal objectives fo a given taget expected etun. Figue 7 shows the objective values fo all five objectives based on optimal potfolio weights obtained by using a given, single objective. Fo this example, all optimal potfolio weights ae simila and the five plots coespond closely. Figue 8 shows the dependence of the optimal potfolio weights on the mixing paamete w. Note that pue tanquil (distessed) egime is on the ight (left) of the figue. Theefoe assets with positive (negative) slope on this diagam, i.e., those with a tendency to have small (big) positions in the distessed egime, ae good fo insuing against (speculating on) the maket enteing a distessed state.

12 12 I. BUCKLEY, G. COMEZAÑA, B. DJERROUD, AND L. SECO Figue 5. Contou plots of 2d pojections of 3d asset etun pobability density function Figue 9 shows the dependence of the ate of change of asset weight with espect to mixing paamete along the efficient fontie, i.e., fo diffeent values of desied expected etun Altenative algoithm fo solving the non-linea poblem. The PoS objective gives ise to the non-linea optimization poblem (Equation 4.4.6) of dimension equal to the numbe of assets in the univese, m. This can be solved using standad non-linea pogam (NLP) optimizes, but such tools ae unable to exploit the close esemblance between the non-linea poblem and a linea-quadatic pogam (LQP). An altenative appoach is to ecognize that the optimal potfolio weights, fo a given taget etun µ T, will be embedded in the 2(n 1)-dimensional solution hype-suface of a elated LQP, whee n is the numbe of egimes. The latte is that of minimizing a linea sum of the egime vaiances subject to the linea constaint that a linea sum of the egime expected etuns be less than the etun taget. An NLP is still equied to find the optimal solution, but the dimensionality of the poblem is educed fom m to 2(n 1). Typically the fome exceeds the latte. E.g., if thee ae thity assets in the poblem and two egimes. We define some notation: Linea and quadatic functions of the potfolio weight vecto, θ and the Shape atio, fo each egime i: (4.7.1) (4.7.2) (4.7.3) L i (θ) = µ i.θ Q i (θ) = θ T V i.θ x i (θ) = L i(θ) k Qi (θ)

13 PORTFOLIO OPTIMIZATION IN A GAUSSIAN MIXTURE ENVIRONMENT 13 i. Vaiance ii. Shape Ratio iii. Pobability iv. Hodges MSR v. LPM1 vi. LPM2 Figue 6. Optimal asset weights along the efficient fontie Objective functions, one non-linea, one quadatic in θ: (4.7.4) f α (θ) = α i Φ(x i (θ)) (4.7.5) g β (θ) = whee α i, β i 0 i. β i Q i (θ)) The non-linea and linea-quadatic pogammes ae: Definition NLP(1) (4.7.6) whee α.1 = 1. max f α (θ) θ s.t. θ.1 = 1 α i L i (θ) p Solutions to NLP(1) will be denoted by θ α,p.

14 14 I. BUCKLEY, G. COMEZAÑA, B. DJERROUD, AND L. SECO Obj i. Vaiance Obj ii. Shape Ratio Obj iii. Pobability Obj iv. Hodges MSR Obj v. LPM1 Obj vi. LPM2 Figue 7. Efficient fonties fo diffeent objectives Definition LQP(2) (4.7.7) min g β (θ) θ s.t. θ.1 = 1 whee β.1 = 1 L i (θ) q i Solutions to LQP(2) will be denoted by θ β,q. The set of solutions fo all possible paamete values ae denoted fo the NLP and LQP espectively as: (4.7.8) Θ NLP = {θ α,p} α R n,α.1=1,p R Θ LQP = {θ β,q} β R n,β.1=1,q R n,q.1=1 Poposition All solutions to NLP(1) ae solutions of LQP(2). That is to say, thee always exist paamete values of β, q in LQP(2), such that the solution θ β,q coesponds to the solution θ α,p to NLP(1), fo all paamete values α, p. (4.7.9) Θ NLP Θ LQP Poof. The poof is by contadiction. The stategy is to conside a solution of NLP(1) that will necessaily satisfy the constaints of LQP(2), and hypothesize the existence of anothe potfolio, which also satisfies the constaints of LQP(2), but

15 PORTFOLIO OPTIMIZATION IN A GAUSSIAN MIXTURE ENVIRONMENT 15 i. Vaiance ii. Shape Ratio l l iii. Pobability iv. Hodges MSR l l v. LPM1 vi. LPM2 l l Figue 8. Dependence of optimal weights on the mixing paamete which gives a smalle objective fo LQP(2) (i.e., is moe optimal) and show that this leads to a contadiction. Given values fo the paametes fo NLP(1), choose specific values fo the LQP(2) paametes: (4.7.10) (4.7.11) (4.7.12) (4.7.13) q i = L i (θ α,p) β i = 1 f α θ N Q i α,p = 1 f α x i θ N x i Q i α,p = 1 2N α iφ(x i ) x i θ Q i α,p whee N is a nomalisation facto to ensue that β.1 = 1. Fo the LQP objective to be convex, we equie that β i 0, and this holds because f α (x) is inceasing in 1 x, whilst Q is deceasing in Q. Due to the choice of the chosen LQP paametes, θ α,p satisfies the constaints of the LQP poblem constaints. What is less obvious is whethe θ α,p is the optimal solution to the poblem.

16 16 I. BUCKLEY, G. COMEZAÑA, B. DJERROUD, AND L. SECO DS i. Vaiance DS ii. Shape Ratio DS iii. Pobability DS iv. LPM1 DS v. LPM2 Figue 9. Distess sensitivities along the efficient fontie If θ α,p is not the solution to LQP(2), then thee exists a θ = θ α,p +ɛ that gives a smalle value fo the LQP (constained) objective, g β (θ ), with paamete values β(α, p), (objective paametes) and q(α, p) (constaint paametes), than θ α,p. (4.7.14) g β (θ ) < g β (θ α,p ) Taylo s expansion on NLP(1): (4.7.15) f α (θ ) f α (θ α,p ) + = f α (θ α,p ) + n,m i,j=1 n,m i,j=1 ( fα L i + f ) α Q i ɛ j L i θ j Q i θ j θ α,p f α Q i Q i θ j θ α,p ɛ j The tem involving L i vanishes because θ and θ α,p both satisfy the linea constaints of LQP(2) by assumption and design, espectively. Theefoe, (4.7.16) m j=1 L i θ j θ α,p ɛ j = 0.

17 PORTFOLIO OPTIMIZATION IN A GAUSSIAN MIXTURE ENVIRONMENT 17 Similaly Taylo s expansion on LQP(2): (4.7.17) g β (θ ) g β (θ α,p ) + = g β (θ α,p ) + = g β (θ α,p ) 1 N n,m i,j=1 n,m i,j=1 g β Q i Q i θ j θ α,p ɛ j β i Q i θ j θ α,p ɛ j n,m i,j=1 f α Q i Q i θ j θ α,p ɛ j suggesting that n,m f α Q i θ i,j=1 Q i θ j ɛ j > 0 and theefoe that α,p (4.7.18) f α (θ ) > f α (θ α,p ) which contadicts the optimality of θ α,p to NLP(1), which was ou initial assumption. 5. Conclusions We ae cetain that the GM appoach, namely the assumption of a multivaiate finite Gaussian mixtue distibution fo asset etuns used in conjunction with a pobability of shotfall objective, will be useful fo a whole ange of potfolio management applications. It is only slightly hade to implement than standad Makowitz, with many featues in common (e.g. we etain the use of covaiance matices). Howeve, it is moe flexible because of its ability to handle non-elliptic etun distibutions. Because it is intuitive, the technique is unlikely to face esistance fom pactitiones aleady familia with mean-vaiance appoaches. With two egimes, the objective function does not possess moe than two maxima, so ou numeical examples have been obust and quick to solve. We have compaed the GM and mean-vaiance appoaches. The obvious questions to ask ae whethe the two appoach give diffeent optimal weights fom one anothe, and if so, whethe holding the GM weights will give impoved pefomance using measues pefeed by pactitiones. The esponse to both questions is in the affimative. The optimal weights ae significantly diffeent between the appoaches. The optimal weights fo the GM appoach will by definition bette seve an investo seeking to minimize the pobability of shotfall in an envionment with multiple egimes. Because the GM distibution is a bette model fo eality than the Gaussian distibution, we believe that the GM appoach will do a bette job fo managing potfolios in the eal wold. Acknowledgment. IB would like to thank M. Gey Salkin and Pof. Nicos Chistofides of the CQF, Impeial College fo funding.

18 18 I. BUCKLEY, G. COMEZAÑA, B. DJERROUD, AND L. SECO A.1. Definitions. Appendix A. Gaussian mixtue distibution Definition A.1.1. A (scala) andom vaiable Z has the univaiate GM distibution if its pobability density function f Z (z) is of the fom (A.1.1) f Z (z) = f Xi (z) = φ( z µ i ) σ i whee the andom vaiables X i ae nomally distibuted with nomal pobability density functions φ Xi (x) = φ( z µ i σ i ), φ(z) is the standad nomal pobability density function and the weights sum to one. The andom vaiables X i have means µ i and vaiances σi 2. The finite sum is ove the desied numbe of nomal components to combine, n. Remak A.1.2. The cumulative distibution function is tivially: (A.1.2) F Z (z) = Φ( z µ i ) σ i whee Φ(z) is the standad nomal cumulative density function. We will made use of this obsevation in Section 4.1 when we define the PoS objective. Similaly, Definition A.1.3. A vecto andom vaiable Z has the multivaiate GM distibution if its pobability density function f Z (z ) is of the fom (A.1.3) f Z (z ) = f X (i)(z ) = φ µ (i),v (i)(z ) whee the (vecto) andom vaiables X (i) ae multivaiate nomally distibuted with pobability density functions φ µ (i),v (i)(z ), and the weights sum to one. The vecto andom vaiable X (i) has mean µ (i) and vaiance-covaiance matix V (i). E.g. if we take the ath and bth components of X (i), thei covaiance is the element (a, b) of V (i) ; i.e. Cov(X a (i), X (i) b ) = V (i) ab. Also E[X(i) a ] = µ (i) a. Remak A.1.4. Note that by definition Cov(X a (i), X (j) b ) = 0 fo i j. Remak A.1.5. In the numeical expeiments descibed late, the mixtue distibution contains two nomal components, descibing asset etuns unde tanquil and distessed conditions. We shall efe to the weights as egime weights. A.2. Moments. The mean of a andom vaiable with the mixtue distibution is simply expessed as a linea combination of the means of the component nomal distibutions. Poposition A.2.1. The expectation of a function f of a andom vaiable with the GM distibution can be expessed in tems of the expectations of functions of the component nomally distibuted vaiables: (A.2.1) E[f(Z )] = E[f(X (i) )]

19 PORTFOLIO OPTIMIZATION IN A GAUSSIAN MIXTURE ENVIRONMENT 19 Remak A.2.2. In paticula, (A.2.2) E[Z ] = µ (i) N.B. This is a potential souce of confusion given that it is not tue in geneal that Z = n X (i). The vaiance depends not only on the vaiances of the components, but also on the diffeences between the means of the components. Poposition A.2.3. The vaiance of a andom vaiable with the (univaiate) GM distibution can be expessed in tems of the expectations and vaiances of the component nomally distibuted vaiables: (A.2.3) Va[Z a ] = = Va[X (i) a ] + (σ (i) a ) 2 + n,n i,j<i n,n i,j<i w j (E[X a (i) ] E[X a (j) ]) 2 w j (µ (i) a µ (j) a ) 2 Remak A.2.4. If we pemit the vaiance and expectation opeatos to thead ove the components of the vecto aguments f(x ) a := f(x a ), this can be witten in altenative vecto fom as (A.2.4) Va[Z ] = = Va[X (i) ] + σ (i)2 + n,n i,j<i n,n i,j<i w j (E[X (i) ] E[X (j) ]) 2 w j (µ (i) µ (j) ) 2 The vaiance esult, above is a special case of the following esult fo the covaiance: Poposition A.2.5. The covaiance between two elements of a vecto andom vaiable with the (multivaiate) GM distibution can be expessed in tems of the expectations of functions of the component nomally distibuted vaiables: (A.2.5) Cov[Z a, Z b ] = = n,n Cov[X a (i), X (i) b ]+ i,j<i w j (E[X (i) a V (i) ab n,n + i,j<i ] E[X (j) a w j (µ (i) a ])(E[X (i) b ] E[X(j) b ]) µ (j) a )(µ (i) b µ (j) b ) Remak A.2.6. In matix notation, whee W ab := Cov[Z a, Z b ], n,n (A.2.6) W = V (i) + w j (µ (i) µ (j) ).(µ (i) µ (j) ) whee indicates matix tanspose. i,j<i

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