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Research ArticlePortfolio Selection Based on Bayesian Theory

Daping Zhao 1 Yong Fang 2 Chaoliang Zhang3 and Zongrun Wang4

1School of Finance Capital University of Economics and Business Beijing 100070 China2Institute of Systems Science Academy ofMathematics and Systems Sciences Chinese Academy of Sciences Beijing 100190 China3Founder Fubon Fund Company Beijing 100037 China4Business School Central South University Changsha Hunan Province 410083 China

Correspondence should be addressed to Yong Fang yfangamssaccn

Received 17 June 2019 Accepted 17 September 2019 Published 20 October 2019

Academic Editor Vincenzo Vespri

Copyright copy 2019 Daping Zhao et al +is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

+e traditional portfolio selection model seriously overestimates its theoretic optimal return Aiming at this problem twoportfolio selection models are proposed to modify the parameters and enhance portfolio performance based on Bayesian theoryFirstly a Bayesian-GARCH(11) model is built Secondly Markov Chain is applied to curve the parametersrsquo state transfer and aBayesian Markov regime-Switching-GARCH(11) model is constructed Both the two models can handle the overestimationproblem and can obtain self-financing portfolios In the numerical experiments both the models are examined with data fromChina stock market and their performances are compared and analyzed +e results show that BMS-GARCH(11) model issuperior to the Bayesian-GARCH(11) model

1 Introduction

Portfolio optimization and diversification have been in-strumental in the development and understanding of fi-nancial markets and financial decision-making +e majorbreakthrough came in 1952 with the publication of HarryMarkowitz theory of portfolio selection +e theory is nowpopularly referred to as modern portfolio theory Accordingto this theory both the return and risk of a security can bequantified using the statistical measures of its expectedreturn and standard deviation +en the investors shouldconsider return and risk together and determine the allo-cation of funds among investment alternatives on the basisof their return-risk trade-off +e idea that sound financialdecision-making is a quantitative trade-off between returnand risk was revolutionary And it has had a major impacton academic research and the financial industry as a whole

However there are still some concerns with the theoryand one of the concerns is that the original mean-variancemodel depends on estimated parameters which are un-certain and may be mistaken Specifically return-risk op-timization can be very sensitive to changes in the inputsespecially when the return and risk estimates are not well

aligned or when the problem formulation uses multipleinteracting constraints As a result many practitionersconsider the output of risk-return optimization to be opa-que unstable andor unintuitive Besides estimation errorsin the forecasts significantly impact the resulting portfolioweights For example it is well-known that in practicalapplications equally weighted portfolios often outperformmean-variance portfolios [1ndash4] +e classical framework hasto be modified when used in practice in order to achievereliability stability and robustness with respect to modeland estimation errors [5 6] In other words the portfolioselection models face estimation risks +erefore how tominimize the estimation risk to portfolio draws much at-tention in academia

Recently more and more parameter uncertainties havebeen analyzed in Bayesian Framework+e Bayesianmethodhas advantages first of all it can fully use prior informationsecondly it explains the uncertainties of estimated risks andmodels thirdly it makes the computation in simulatingcomplicated variables easy In this way investors firstly setthe parameter prior then they use Bayesian rules to get theposterior As a result Bayesian methods are gradually usedin the portfolio selection analysis Avramov and Zhou [7]

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 4246903 11 pageshttpsdoiorg10115520194246903

reviewed the development of Bayesian portfolio in thefollowing respect (1) the returns are iid (2) the returns arepredicable (3) the mean follows regime switching and thevolatility is stochastic +ey mainly discussed the Bayesianportfolio in parameter uncertainties as well as with themodel uncertainties In Fabozzis review of ldquo60 Years ofPortfolio Optimization Practical Challenges and CurrentTrendsrdquo they also speak high of the application of Bayesianmethods in portfolio optimization [8] In this paper we takethe financial data with different lengths as the applicationbackground mainly focus on the uncertainties of parametersin portfolio selection and build two models based on Bayesandmodern portfolio theory a Bayesian-GARCH(11) modeland a Bayesian-Markov Regime Switching-GARCH(11)model Using the Bayesian optimization algorithm both themodels do not overestimate their theoretic optimal returnsand they can handle the overestimation problem

+e paper is organized as follows In Section 2 thebackground studies are briefly introduced including theliterature review and the relative theories and methods suchas Bayesian theory and Markov regime switching In Section3 a Bayesian-GARCH(11) model and a Bayesian-MarkovRegime Switching-GARCH(11) model are constructed InSection 4 the two models are taken into application and thenumerical results are illustrated and compared In Section 5the paper is concluded with summary comments andpointers to the future work

2 Background Studies

21 Literature Review It is well known that the parametersin portfolio selection may be uncertain or mistaken and thereturn-risk optimization can be very sensitive to changes inthe inputs In early studies most researchers assumed thatthe risky asset returns are iid and focused on the effect ofthe uncertainty of the mean and variance on the portfolioselection Williams [9] discussed how the parameter un-certainty affected the portfolio selection Bawa et al [10]firstly consider the uncertainty of the parameters of thereturn distribution in static portfolio selection Gennotte[11] analyzed portfolio selection when the expected returnwas uncertain Kandel and Stambaugh [12] showed thatwhen the returns were predictable investors who ignoredthe parameter uncertainties preferred risky stocks in a shortperiod Zenios et al [13] develop multiperiod dynamicmodels for fixed-income portfolio management under un-certainty using multistage stochastic programming withrecourse +e multiperiod models outperform classicalmodels based on portfolio immunization and single-periodmodels Brennan [14] studied the dynamic portfolio selec-tion with the assumptions that the expected risk premiumwas uncertain in discrete time He thought the assumptionsof financial models should be that all the invest opportunitiesdepended on a series of unobservable variables and theparameters were not certain Barberis [15] got similar resultswith the long-period investors Pastor and Stambaugh [16]discussed how the uncertainties of the mispricing affectedthe portfolio selection Pastor [17] treated the model un-certainty using the investorsrsquo opinions to the asset pricing as

prior information Bai et al [18 19] develop a bootstrap-corrected estimator to correct the overestimation in port-folio selection and further extend the theory to obtain self-financing portfolios Besides in order to improve theportfolio selection efficiency researchers tend to extend themodels with dynamic programming [13 20 21] withcontinuous time framework [22 23] using Markovian re-gime-switching methods [24] and so on

Currently most of the parameter uncertainties are an-alyzed in Bayesian Framework As we mentioned before theBayesian method has advantages first of all it can fully useprior information secondly it explains the uncertainties ofestimated risks and models thirdly it makes the compu-tation in simulating complicated variables easy In this wayinvestors firstly set the parameter prior then they useBayesian rules to get the posterior As a result Bayesianmethods are gradually used in the portfolio selectionanalysis One of the implications is that the Bayesian theorycan bemainly used to estimate the uncertain returns or otherfactors (such as a shrinkage estimator and so on) in portfolio[15 25ndash29] In addition for the parameter uncertainty ofportfolio selection in other words for the estimation riskevaluation or the estimation error correction researchersalso tend to choose Bayesian methods [4 30ndash34] Besidesthe Bayesian theory is also used in model analysis [35] riskmeasurement or return-risk trade off [36] portfolio opti-mization [37 38] and so on

22 Modern Portfolio eory According to Markowitzrsquosportfolio theory there are N assets Rt is the excess return ofasset N in time t Rt (R1t RNt)prime and they follow anormal distribution p(Rt | μΣ) N(μΣ) micro is a N times 1matrix and Σ is a N times N matrix +e weights are denoted asw (w1 wN) +e return of the portfolio is

Rp 1113944N

i1wiRit wprimeRt (1)

And the expected return and variance are

μp 1113944

N

i1wiμi wprimeμ

σ2p 1113944N

i11113944

N

j1wiwjcov Ri Rj1113872 1113873 wprime1113944 w

(2)

where μi is the return of asset i and cov(Ri Rj) is the co-variance of assets i and j

Suppose investors hold the portfolio for τ and theirtarget is to maximize the value at time T + τ where T is thetime when the portfolio is built +e mean-variance modelcan be expressed as the following optimization problem

minw

σ2p minw

wprime1113944T+τ

w

st wprimeμT+τ ge μlowast

wprime1 1

(3)

where μlowast is the permitted minimum return

2 Mathematical Problems in Engineering

23 Bayesian eory Bayes firstly presented Bayesian for-mula in his essay Later on Laplace put the new approachinto applications +en statisticians developed it as a sys-tematic statistical method the Bayesian theory With thenew theory each unknown variable can be seen as a sto-chastic variable and can be described with a probabilitydistribution+e distribution is the prior information beforesampling and hence it is called prior distribution or prior

Traditional statistical estimation depends on the givendistributions and data However prior is the base of theBayesian estimation Posterior is the key to Bayesian esti-mation For an unknown parameter θ its posterior π(θ | x)

is a conditional distribution of θ under sampler x and itcontains all the information that is available

24 GARCH Model Engle presented the ARCH (autore-gressive conditional heteroskedasticity) model in 1982+enthe method is popular in the volatility empirical testHowever in real stock market the conditional variancedepends not only on one or two previous variables but alsoon more alternating quantities which account for severallags and parameters +en Bollerslev presented the GARCHmodel

xt β0 + β1xtminus 1 + middot middot middot + βpxtminus p + ε

σ2t E μ2t1113872 1113873 α0 + α1μ2tminus 1 + middot middot middot + αqμ

2tminus q

+ β1σ2tminus 1 + middot middot middot + βpσ

2tminus p

(4)

where α0 gt 0 αi ge 0(i 1 q) and βj ge 0(j 1 p) Ingeneral the real market can be characterized well whenp q 1

25 MCMC Simulation MCMC sampling is proposed byMetropolis et al [36] and then Hastings [40] generalized itMCMC simulation depends on aMarkov chain and themainmethods are Gibbs sampler and MetropolisndashHastings Derinet al [41] presents Gibbs sampler in a statistics article whichis now the most popular sampler in MCMC According toGibbs sampler it is easy to figure out the total conditionalprobability density which makes the sampler not that dif-ficult However sometimes the Gibbs sampler cannot solvecomplicated Bayesian problems with combined probabilitydensity In these cases the MetropolisndashHastings algorithm isapplied Take a random variable θ as an example for itssymmetric proposal density function q(θ[j] | θlowast q(θlowastθ[j]))the acceptance probability with M-H algorithm is

minp θlowast | y( 1113857

p θ[j] | y1113872 1113873 1

⎧⎨

⎩

⎫⎬

⎭ (5)

In this way the acceptance probability is free fromcalculation It should be mentioned that the Bayesian pa-rameter estimation based on simulation depends on theastringency of Markov chain In this way the sample gen-erating from the simulation follows the posterior distribu-tion of expectation In the posterior simulation the aim is togenerate a fully ergodic Markov chain

3 Portfolio Selection Models Based on Bayes

31 Bayesian-GARCH(11) Model In this section the re-siduals of the return are assumed to follow t-distributionand the posterior is simulated by MetropolisndashHastings andGibbs sampler Suppose the return of asset follows theGARCH(11) model as belows

Rit μit + σit|tminus 1εit t 1 T (6)

σ2it|tminus 1 wi + αiμ2itminus 1 + βiσ

2itminus 1|tminus 2 (7)

Equations (6) and (7) show the process that the assetreturn and volatility follow Equation (6) is the meanequation and (7) is the volatility equation where Rit is thereturn of asset i at time t σ2it|tminus 1 is the variance (volatility) ofthe return of asset i between time t minus 1 and t and μitminus 1

Rit minus μit wi gt 0 αi ge 0 βi ge 0 For convenience assume μit isunchangeable If the variance is consistent then (6) turns toa linear regression For the parameters wi αi and βi theywill be estimated with Bayesian methods

32 Likelihood Function and Prior of Parameters

321 Likelihood Function With the assumptions of assetsand parameters above denote the parameter variable of themodel as θ (w α β υ μ) Suppose εit follows a t-distri-bution with υi degrees of freedom the likelihood functioncan be written as follows

L θ R1t

1113868111386811138681113868 R2t1113872 1113873 1113945N

i1L θi Rit

11138681113868111386811138681113872 1113873prop1113945

N1

i11113945

T

t1σ minus 1

it|tminus 1 1 +1υ

Rir minus μi1113872 11138732

σ2it|tminus 1⎛⎝ ⎞⎠⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

times 1113945

N2

j11113945

T

ts

σminus 1jt|tminus 1 1 +

1υ

Rjr minus μi1113872 11138732

σ2jt|tminus 1

⎛⎝ ⎞⎠

minus (υ+12)

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(8)where θi is a parameter variable of asset i and σ2it|tminus 1 is itsconditional variance +e initial conditional covariance σ2i0is a constant and εit follows a t-distribution with degreesof freedom υi Its conditional variance at time t isυi((υi minus 2)σ2it) υi gt 2

322 Prior of Parameters In many cases peoplersquos priors arevague and thus difficult to translate into an informative priorWe therefore want to reflect the uncertainty about the modelparameters without substantially influencing the posteriorparameter inference +e so-called noninformative priorsalso called vague or diffuse priors are employed to that endIn this section we suppose that there is a noninformationprior for the conditional variance parameters of asset i

π wi αi βi( 1113857prop 1IiθG (9)

where 1IiθGis an indicator function for constraints of pa-

rameters as follows

1IiθG

1 wi gt 0 αi gt 0 βi gt 0

0 others1113896 (10)

Mathematical Problems in Engineering 3

Geweke (1993) proposed that the prior of υi can be indexprior distribution and its density function is

pi υi( 1113857 λ exp minus υiλ( 1113857 (11)

Based on the distribution the mean is 1λ and λ can bedetermined by the degree of freedom of υi Several empiricalexperiments show that in the financial markets the degree offreedom is typically less than 20 Hence set the upper boundK as 20 According to Bauwens et al [42] if there is adiffusion prior for υi on [0infin) its posterior will not beappropriate So here we use the index prior as (11) shows+en assume the prior of μi follows normal distribution

μi sim N μi0 σ2i01113872 1113873 (12)

33 Parameter Estimation with Bayesian-GARCH(11)Assume there are N assets of which N1 assets are of Tperiod Denote the return of asset i as R1t (t 1 2 T)and all the T variables are iid following the same multi-variate normal distribution In addition the rest of the N2(N2 N minus N1) assets are of S periods and denote theirreturn as R1t (tgeT minus S + 1)

According to the prior assumptions the posterior of θi is

1113945

N

i1p θi Rit

11138681113868111386811138681113872 1113873prop1113945

N1

i11113945

T

t1σminus 1

it|tminus 1 1 +1υi

Rit minus μi1113872 11138732

σ2it|tminus 1⎛⎝ ⎞⎠

minus υi+12( )⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

times 1113945

N2

j11113945

T

ts


1υi

Rjt minus μi1113872 11138732

σ2jt|tminus 1

⎛⎝ ⎞⎠

minus υi+12( )⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

times 1113945

N

i1exp minus υiλ( 1113857

times 1113945

N

i11113945

N

i1exp

minus 12

μi minus μic1113872 11138731113874 1113875prime1113944

minus 1

ic

μi minus μic1113872 1113873⎛⎝ ⎞⎠

times 1113945N

i1IiθG

(13)

Obviously there is no analytic solution for the jointposterior density function +erefore it is acceptable to re-place tminus distribution with normal distribution before sam-pling Suppose the return of asset i follows tminus distributionthere is

Rit σ2it|tminus 111138681113868111386811138681113868 sim t μit σ

2it|tminus 1 υi1113872 1113873 (14)

It is equivalent to the normal distribution as follows

Rit σ2it|tminus 111138681113868111386811138681113868 ηt sim N μit

σ2it|tminus 1ηt

1113888 1113889 (15)

where ηt is called mixed variables following iid Gammadistribution is

ηt υi

1113868111386811138681113868 sim gammaυi

2υi

21113874 1113875 t 1 T (16)

+is substitution helps to make the calculation of theposterior easy After the replacement the parameter variableof asset i becomes θi (wi αi βi υi μi

prime ηtprime) and its logarithm

likelihood function turns to normal logarithm likelihoodfunction

lg L θ R1t

1113868111386811138681113868 R2t1113872 11138731113872 1113873 const minus12

1113944

T

tt0

lg σ2it|tminus 11113872 1113873 minus lg ηt( 1113857 +ηt Rit minus μi1113872 1113873

2

σ2it|tminus 1⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

(17)

According to the equation when i 1 N1 t0 1and when i 1 N2 t0 Y minus S + 1 Meantime theposterior of θi contains other mixed variables+e logarithmposterior is

lg L θi R1t

1113868111386811138681113868 R2t1113872 11138731113872 1113873 const minus12

1113944T

tt0


2

σ2it|tminus 1⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

minus12μi minus μic1113872 1113873prime1113944

minus 1

ic

μi minus μic1113872 1113873 +Tυi

2lgυi

2

minus T lg Γυi

21113874 11138751113874 1113875 +

υi

2minus 11113874 1113875 1113944

T

t1lg ηt( 1113857

minusυi

2t 11113944

T

t1ηt minus υtλ

(18)where wi gt 0 αi ge 0 βi ge 0

331 Conditional Posterior Distributions

(1) e Conditional Posterior Distribution of μi It can beproved that the full conditional posterior density of μi is anormal distribution

p μi θiG

1113868111386811138681113868 η υi R1t R2t1113872 1113873 N μlowastΣlowast( 1113857 (19)

(2) e Conditional Posterior Distribution of ηt It can beproved that the full conditional posterior density of ηt(t

1 T) is a gamma distribution

p η θiG

1113868111386811138681113868 μi υi Rit R2t1113872 1113873 gammaυi

2

Rit minus μi1113872 11138732

2σ2it|tminus 1+υi

2⎛⎝ ⎞⎠

(20)

(3)e Conditional Posterior Distribution of υi As there is nostandard form for the conditional posterior density of υi andthe degree of freedom (11) the kernel of the posteriordistribution is

p υi θiG

1113868111386811138681113868 μi η R1t R2t1113872 1113873propΓυi

21113874 1113875

minus T υi

21113874 1113875

Tυi2exp υiλ

lowast( 1113857

(21)where λlowast (12)1113936

Tt1(lg(ηt) minus ηt) minus λ


Here we put sampling with the Griddy Gibbs sampleralgorithm into use +e kernel of the conditional logarithmposterior distribution of υi is

lg p υi θminus υi

11138681113868111386811138681113868 R1t R2t1113874 11138751113874 1113875 const +Tυi

2log

υi

21113874 1113875 minus T lg Γ

υi

21113874 11138751113874 1113875

+υi

2minus 11113874 1113875 1113944

T

t1lg ηt( 1113857 minus

υi

21113944

T

t1ηt minus υiλ

(22)

Set the reasonable range of υi as (2 30) Divide υi equallyand denote the grid interval as (upsiloni1 upsiloniJ)Sample from the time m and the other parameters aredenoted as θ(mminus 1)

minus υi

(4) Conditional Posterior Destiny of θiG +e posterior dis-tribution kernel of θiG is

lg p θiG θminus iθG

11138681113868111386811138681113868 R1tR2t1113874 11138751113874 1113875 const minus12

1113944T

tt0

log σ2it|tminus 11113872 1113873 +ηt Rit minus μi1113872 1113873

2

σ2it|tminus 1⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

(23)

where wi gt 0 αi ge 0 and βi ge 0 Here we set a proposaldensity function for θiG and it follows tminus distribution

332 Sampling Algorithm of Bayesian-GARCH(11) Wedecompose the parameter variable θi (θiG υi μi

prime ηtprime) of

asset i where θiG (wi αi βi) +en we can use a coherentsampling strategy to sample from each conditional posteriordistribution

p θiG μi

1113868111386811138681113868 η υi R1t R2t1113872 1113873

p υi θiG

1113868111386811138681113868 μi η R1t R2t1113872 1113873

p η θiG

1113868111386811138681113868 μi υi R1t R2t1113872 1113873

p μi θiG

1113868111386811138681113868 η υi R1t R2t1113872 1113873

(24)

34 Portfolio Selection Based on BMS-GARCH(11) Manyresearches find that the returns are heteroscedastic but thereare still some other empirical experiments which show thatthe parameters are not fixed and unchangeable As for thereasons of changes it might be due to the potential transferin different mechanisms during data generating processesFor example business cycle fluctuations can be seen as anendogenous factor And in the regime-switching situationthe Markov regime-switching model is good at describingthe dynamics of the returns and variance

+e Markov regime-switching model is proposed byHamilton in 1989 providing great convenience in modelingthe volatility between different states +en several re-searchers try to introduce the method in the GARCHprocess For instance the regime-dependent parameter canbe integrated in the standard deviation [44]

Rt μt|tminus 1 +gSt

1113968σt|tminus 1εt (25)

where St is the state at time t Besides the regime-dependentparameters can be part of the variance equation [45] that is

σ2t|tminus 1 w + gSt1113872 1113873 + 1113944

p

i1αiu

2tminus i (26)

Models of (25) and (26) are both based on the dynamicsof the conditional variance since the status dependencymakes the likelihood function hard to deal with Accordingto Henneke et al [43] the conditional mean can be treated asa ARMA(11) process in Bayesian framework where thevolatilities will change in different statuses

341 Markov Chain Suppose that the volatility has threestates and denote i 1 2 3 as the low volatility normalvolatility and high volatility Denote πij as the transferprobability from state to state +e transfer probabilitymatrix is

Π

π11 π12 π13

π21 π22 π23

π31 π32 π33

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (27)

where the sum of each row is 1 And the property of Markovchain can be shown as follows

P St Stminus 1 Stminus 2 S11113868111386811138681113868 1113873 P St

1113868111386811138681113868 Stminus 111138721113872 1113873 (28)

In matrix (27) each row means the conditional prob-ability distribution of transferring from the realized state Stminus 1to state St where St1113864 1113865

T

t1 is a tridimensional Markov chainwith the transfer probability matrix Pi

Here we suppose all the parameters in GARCH(11) arestate-dependent and for convenience the calculation belowis for asset j ignoring the subscript and the conditionalvariance equation is

σ2t|tminus 1 w St( 1113857 + α St( 1113857u2tminus 1 + β St( 1113857σ2tminus 1|tminus 2 (29)

and in every stage there is

w St( 1113857 α St( 1113857 β St( 1113857( 1113857

w1 α1 β1( 1113857 St 1

w2 α2 β2( 1113857 St 2

w3 α3 β3( 1113857 St 3

⎧⎪⎪⎨

⎪⎪⎩(30)

As for the return there is Rt μSt+ σt|tminus 1εt t 1 t

If the given states are the same the conditional variances arethe same according to equations (29) and (7)

342 Prior Information Denote the parameter variables ofasset and Markov chain St1113864 1113865

T

t1 in BMS-GARCH(11) modelas

θ υ μprime ηprime θG1 θG2 θG3 π1 π2 π3 S( 1113857 (31)

where θGi (wi αi βi) i 1 2 3 πi (πi1 πi2 πi3) i

1 2 3 and S is the state path in all the periods S

(S1 S2 ST)For parameters μj η and υ they have the non-

information prior which means they are not affected by thestates of the BMS-GARCH(11) model +e following are forthe other parameters


(1) Prior Distribution of θGi For θGi (i 1 2 3) there is anormal distribution prior

θGi sim N μθGiΣθGi

1113872 1113873I θGi (32)

where I θGi is an indicator function(2) Prior Distribution of πi In a binomial environment

beta distribution can be a proper prior for probability pa-rameters and it is called Dirichlet distribution in multi-variate situation +erefore there is

πi sim Dirichlet ai1 ai2 ai3( 1113857 (33)

In order to get the prior parameters aij it is necessary toget the prior of every expected value of each transferprobability and figure out the parameters with the equations

343 Parameter Estimation with BMS-GARCH(11)BMS-GARCH(11) evolves with invisible status variableSt(t 1 T) Hence the discrete Markov Chain is calledinvisible Markov process as well However the parameterscan be estimated by Bayesian methods simulate the invisiblevariable and the parameters at the same time Here thedistribution of S is multivariable distribution

p(S | π) 1113945Tminus 1

t1p St+1 St

1113868111386811138681113868 π1113872 1113873

πn1111 π

n1212 middot middot middot πn32

32 πn3333

πn1111 π

n1212 1 minus π11 minus π12( 1113857

n13 middot middot middot πn3232 1 minus π31 minus π32( 1113857

n33

(34)where nij is the number of times the chain transferring fromstatus i to status j between time 1 and time T +e firstequation in (34) draws the property of a Markov Chain

According to the GARCH(11) model withtminus distribution the invisible Markov process πi andθGi(i 1 2 3) the joint log posterior distribution of pa-rameter variables in BMS-GARCH(11) is shown as follows

lg L θ R1t

1113868111386811138681113868 R2t1113872 11138731113872 1113873 const minus12

1113944

T

tt0

lg σ2t|tminus 11113872 1113873 minus lg ηt( 1113857 +ηt Rt minus μ( 1113857

2

σ2t|tminus 11113890 1113891

minus12μ minus μc1113872 1113873prime1113944

minus 1

c

μ minus μc1113872 1113873

minus12

1113944

3

i1θGi minus μθGi

1113872 1113873prime1113944minus 1

θGi

θGi minus μθGi1113872 1113873I S(t)i

+Tυ2lg

υ2

1113874 1113875 minus T lg Γυ2

1113874 11138751113874 1113875

+υ2

minus 11113874 1113875 1113944T


υ2

1113944T

t1ηt minus υλ

+ 11139443

i11113944

3

j1aij + nij minus 11113872 1113873lg πij1113872 1113873

(35)where wi gt 0 αi ge 0 βi ge 0 and i is a certain status +elikelihood function depends on the whole sequence of in-visible variable S

(1) e Conditional Posterior Distribution of πi Fortransfer probability πi(i 1 2 3) its conditional posteriordistribution of πi is as follows

lg p πi Rt

1113868111386811138681113868 θminus πi1113872 11138731113872 1113873 const + 1113944

3


(36)

where θminus πimeans the variable of all the other parameters

except πi Equation (36) can be seen as the log of kernel ofDirichlet distribution aij can be set with a prior and nij isthe times of St1113864 1113865

T

t1 shifting from status i to j

(2) e Conditional Posterior Distribution of S With thethree states the number of potential possible status paths is3T Hence it is impossible to get the whole variable at a timeBut part of the parameters can be sampled out In otherwords we can do sampling from the full conditional pos-terior density of St And the full conditional posteriordensity is

p St i Rt

1113868111386811138681113868 θminus S Sminus t1113872 1113873 (37)

where θminus S is the variable with S removed and Sminus t is the statuspath without the state at time t With Bayes theorem p(St

i | Rt θminus S Sminus t) becomes

p St i Rt

1113868111386811138681113868 θminus S Sminus t1113872 1113873 p St i Rt Sminus t θminus S

11138681113868111386811138681113872 1113873

p Rt Sminus t θminus S

11138681113868111386811138681113872 1113873

p Rt θminus S

1113868111386811138681113868 St St i1113872 1113873p St i Sminus t θminus S

11138681113868111386811138681113872 1113873


11138681113868111386811138681113872 1113873

(38)

where p(Rt | θminus S Sminus t St i) is the likelihood function withSt i according to equation (38) Considering Markovproperties there is

p St i Sminus t θminus S

1113868111386811138681113868 1113873 p St i Stminus 1 l St+1 k1113868111386811138681113868 θminus S11138721113872 1113873 πliπik

(39)

+e denominator is


11138681113868111386811138681113872 1113873 11139443

s1p St s Rt Sminus t θminus S

11138681113868111386811138681113872 1113873 (40)

Combine equation (38)ndash(40) then it is possible to figureout the conditional posterior distribution of St

p St i Rt

1113868111386811138681113868 θminus S Sminus t1113872 1113873 p Rt θminus S

1113868111386811138681113868 Sminus t St i1113872 1113873πliπik

11139363s1p Rt θminus S

1113868111386811138681113868 Sminus t St s1113872 1113873πliπik

i 1 2 3

(41)

In addition the proposal density of θGi is similar to thatin single statuses with GARCH(11) and tminus distribution


344 Sampling Algorithm of Parameter Estimation with theBMS-GARCH(11) Model +e sampling procedures are asfollows In the iteration

(1) Get π(m)i from the posterior density according to

equation (36) i 1 2 3(2) Get S(m) with equation (41)(3) Get η(m) by equation (20)(4) Get υ(m) according to equation (21)(5) Get μ(m)

j with equation (19)(6) Get θGi from the proposal distribution of

θjGi(i 1 2 3)

(7) Check that if each element is satisfying the constraintor continue sampling until all the parameter con-straints are satisfied

(8) Calculate the acceptable probability mentioned inSection 2 then decide whether or not θlowastGi(i 1 2 3)

can be accepted

With the new parameters from sampling update thevariable θ Repeat all the procedures for a certain time untilthe Markov Chain is convergent

According to the parameters and Bayes formula figureout the posterior distribution p(μ V | R1t R2t) +e forecastdistribution of return RT+1 at time T + 1 is

p RT+1 R1t

1113868111386811138681113868 R2t1113872 1113873prop 1113946 p RT+1 | μ V( 1113857p μ V R1t

1113868111386811138681113868 R2t1113872 1113873dμ dV

(42)

In most cases there is not an accurate analytic solutionto the formula above But fortunately we can find a nu-merical solution by simulation and construct a portfoliowith the returns from sampling Besides the models canobtain self-financing portfolios if we set wprime1 0 instead of 1in model (3) as well as removing the restriction of wi gt 0 inequation (10) and others In this way self-financing port-folios are constructed and themodels can be used to estimatethe parameters and optimize the portfolios

4 Numerical Examples

41 Data We select 30 stocks from China stock markets intotal 10 stocks are picked up from GEM board market andthey are from the very beginning of the market from Nov 62009 to Oct 21 2011 Besides 10 stocks are selected fromShanghai stock exchange and 10 from Shenzhen stock ex-change +ese 20 stocks have a longer history from Jan 72000 to Oct 21 2011 All the data are weekly Consideringthe interest market situation of China we choose monthlyrate of interest that is converted with the three monthsdeposit rate as risk-free rate because the interest rate doesnot follow the market principle yet In the following analysisthe sample is taken between Nov 6 2009 and Oct 21 2011

42 Parameter Estimation of Bayesian-GARCH(11)

421 Parameter Estimation We set the freedom degreeυ 5 which means λ 02 +e initial variance σ20 can be

treated as the variance of residuals We make 10000 itera-tions and choose the last 5000 as the posterior According tothe Bayesian methods and models stated above the un-certain parameters are estimated

422 Efficient Frontiers According to the mean-variancemodel and with the expected return and covariance matrixthe efficient frontiers are shown as follows (risk-averseparameter A 3) And the Sharpe ratios are shown inTable 1

In Figure 1 it can be seen that the efficient frontierwithout short-selling is below the frontier with short-sellingand the Sharpe ratio is smaller +is suggests short-sellingcan hedge most nonsystematic risks

43 Parameter Estimation of BMS-GARCH(11)

431 Parameter Estimation In this model we suppose thatwith Dirichlet distribution the prior parameteraij 1 (i j 1 2 3) which means the prior of the transferprobability follows uniform distribution +e prior mean ofthe parameters in conditional variance is set as θG1

(000020106)prime θG2 (020304)prime and θG3 (20601)prime+e parameter priors are selected with the following rule

prior mean reflects the different statuses (status 1 status 2and status 3) of asset We suppose the sum of αi and βi isfixed and its value fluctuates with statuses In the case ofhigh volatility investorsrsquo reaction to the unknown in-formation may be sharper than that in the low volatility caseand then α is greater than β For simplicity we suppose theprior covariance matrix of θGi(i minus 1 2 3) is a unit matrixWith M-H algorithm running 10000 iterations keep the last5000 as the posterior inference +en the modified meanand variance in BMS-GARCH(11) can be estimated underthe posterior

According to the estimation results it is easy to findthat the values of posterior mean of conditional varianceparameters are nearly the same with prior but it isonly satisfied with status 1 and status 2 In the case ofstatus 3 there is α3 + β3 gt 1 implying the status is notsteady

We stochastically pick up one stock with short historydata (Hanwei Electronics A3) and one stock with longhistory data (Dazhonggongyong A11) +eir state transitionmatrixes are shown in Table 2 and the state transferprobability is shown in Figures 2 and 3

From Table 2 and Figure 2 we can see that for stock A3(Hanwei Electronics) its status 3 is a transfer status com-pared with statuses 1 and 2 Besides stock A3 is newly publicon GEM board it is interesting that it gradually transfersfrom state 3 (high volatility) to a steady one state 1 (lowvolatility) Instead with stock A11 (Dazhonggongyong sinceits data length is longer than that of A3 here the status

Table 1 Sharpe ratios with Bayesian-GARCH(11)

Short-selling No short-sellingSharpe ratio 0411 0097


0 01 02 03 04 05 06 07 080

002

004

006

008

01

012

014

016

018

02

Short-selling allowed

(a)

004 005 006 007 008 009 01 0113

4

5

6

7

8

9

10

11times10ndash3

No short-selling

(b)

Figure 1 Efficient frontiers with (a) short-selling or (b) without short-selling with Bayesian-GARCH(11)

Table 2 +e state transition matrixes of stock A3 and A11

A3 A11

πj1 πj2 πj3 πj1 πj2 πj3

π1k 0854 0976 0013 0011 0142 0004π2k 0041 0029 0955 0016 0936 0023π3k 0103 0329 0245 0426 0116 0781

1

05

0

2009

11

6

2010

16

2010

36

2010

56

2010

76

2010

96

2010

11

6

2011

16

2011

36

2011

56

2011

76

2011

96

1

05

0

2009

11

6

2010

16

2010

36

2010

56

2010

76

2010

96

2010

11

6

2011

16

2011

36

2011

56

2011

76

2011

96

1

05

0

2009

11

6

2010

16

2010

36

2010

56

2010

76

2010

96

2010

11

6

2011

16

2011

36

2011

56

2011

76

2011

96

Figure 2 Status (statuses 1 2 and 3) transfer probability of stock A3


transfer sample period is from Jan 7 2000) when status 3appears it will soon transfer to status 1 or status 2 Butduring the year of 2008 status 3 shows consistency We

believe this is because during the financial crisis there aremore noises in stock market and the asset is trapped in astate of high volatility

1

05

0

2000

17

2001

17

2002

17

2003

17

2004

17

2005

17

2006

17

2007

17

2008

17

2009

17

2010

17

2011

17

1

05

0

2000

17

2001

17

2002

17

2003

17

2004

17

2005

17

2006

17

2007

17

2008

17

2009

17

2010

17

2011

17

1

05

020

001

7

2001

17

2002

17

2003

17

2004

17

2005

17

2006

17

2007

17

2008

17

2009

17

2010

17

2011

17


Table 3 Sharpe ratios of BMS-GARCH(11)


0 01 02 03 04 05 06 07 080

002

004

006

008

01

012

014

016

018

02


(a)

004 005 006 007 008 009 01 0113

4

5

6

7

8

9

10

11

12times10ndash3

No short-selling

(b)

Figure 4 Efficient frontiers with (a) short-selling or (b) without short-selling with BMS-GARCH(11)


432 Efficient Frontier According to te BMS-GARCH(11)model with the expected return and covariance in lastsection the portfolio weights can be calculated (the risk-averse parameter A 3)

Comparing the portfolio weights with BMS-GARCH(11)to those with Bayesian-GARCH(11) it can be seen that thereare obvious differences We believe that is because there is astate variable in BMS-GARCH(11) and the Markov chaincontains a lot of information hence making the results dif-ferent Table 3 shows the Sharpe ratios It can be seen that theSharpe ratio with BMS-GARCH(11) model is much greaterthan that with Bayesian-GARCH(11) and the Sharpe ratios inSection 42 It suggests that the state variables overcome theproblem of not considering the relationship between longseries and short series In other words the state variables carrya lot of useful information

Figure 4 is about the efficient frontier of portfoliosunder short-selling or without short-selling It is clearthat in the case of no short-selling the frontier is belowthat with short-selling and the corresponding Sharperatio is smaller as well +is also proves that short-sellingcan hedge some nonsystematic risk which improves theSharpe ratio Comparing Figure 4 with Figure 1 it can befound that the portfolio frontiers based on BMS-GARCH(11) are above the others implying the statevariable the Markov chain carries useful information andis helpful to the portfolio selection +erefore the port-folio selection model based on BMS-GARCH(11) per-forms better than Bayesian-GARCH(11)

5 Conclusions

+is paper introduces and constructs two models and op-timizes portfolios based on them With the Bayesian-GARCH(11) model the asset returns are assumed to followtminus distribution and the likelihood function can be figuredout +en with Gibbs sampler the parameters are estimatedand modified

BMS-GARCH(11) can be seen as introducing Markovstate variables to the Bayesian-GARCH(11) In this modelthe parameters are state-dependent and all the states con-struct a Markov chain +e state changes can be curves withstate transfer matrixes and the parameters can be estimatedwith the MCMC method At last we compare the twomodels with real stock market data Numerical results showthat portfolios with Bayesian-GARCH(11) perform poorerthan those with BMS-GARCH(11) implying that after in-troducing state variables the transferred states carry usefulinformation for the investment portfolio

People always want to make the optimal financial de-cision However many investors ignore the uncertainties ofthe parameters and models which lead to a suboptimalportfolio at last From this point of view these models maybe of some practical significance and enlightenment Be-sides in the future work we can try to take other informativepriors into consideration try to expand the models to themulti-stage situation or even try other frameworks insteadof mean-variance framework such as the utility functionsafety-first framework and so on

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Disclosure

An earlier version of this paper was presented as a pre-sentation in 20th Conference of the International Federationof Operational Research Societies

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is work was supported by the National Natural ScienceFoundation of China 71271201 71631008 and 71701138

References

[1] J D Jobson and R M Korkie ldquoPutting Markowitz theory toworkrdquo e Journal of Portfolio Management vol 7 no 4pp 70ndash74 1981

[2] P Jorion ldquoInternational portfolio diversification with esti-mation riskrdquo e Journal of Business vol 58 no 3pp 259ndash278 1985

[3] L Chiodi R Mansini and M G Speranza ldquoSemi-absolutedeviation rule for mutual funds portfolio selectionrdquo Annals ofOperations Research vol 124 no 1ndash4 pp 245ndash265 2003

[4] V DeMiguel L Garlappi and R Uppal ldquoOptimal versusnaive diversification how inefficient is the 1N portfoliostrategyrdquo Review of Financial Studies vol 22 no 5pp 1915ndash1953 2009

[5] F J Fabozzi D Huang and G Zhou ldquoRobust portfolioscontributions from operations research and financerdquo Annalsof Operations Research vol 176 no 1 pp 191ndash220 2010

[6] D Huang S Zhu F J Fabozzi and M Fukushima ldquoPortfolioselection under distributional uncertainty a relative robustCVaR approachrdquo European Journal of Operational Researchvol 203 no 1 pp 185ndash194 2010

[7] D Fukushima and G Zhou ldquoBayesian Portfolio AnalysisrdquoAnnual Review of Financial Economics vol 2 no 1 pp 25ndash472010

[8] P N Kolm R Tutuncu and F J Fabozzi ldquo60 Years ofportfolio optimization Practical challenges and currenttrendsrdquo European Journal of Operational Research vol 234no 2 pp 356ndash371 2014

[9] J T Williams ldquoCapital asset prices with heterogeneous be-liefsrdquo Journal of Financial Economics vol 5 no 2 pp 219ndash239 1977

[10] V S Bawa S J Brown and R W Klein Estimation Risk andOptimal Portfolio Choice North-Holland Publishing Com-pany Amsterdam Netherlands 1979

[11] G Gennotte ldquoOptimal portfolio choice under incompleteinformationrdquo e Journal of Finance vol 41 no 3pp 733ndash746 1986

[12] S Kandel and R F Stambaugh ldquoOn the predictability of stockreturns an asset-allocation perspectiverdquo e Journal of Fi-nance vol 51 no 2 pp 385ndash424 1996

[13] S A Zenios M R Holmer R McKendall and C Vassiadou-Zeniou ldquoDynamic models for fixed-income portfolio


management under uncertaintyrdquo Journal of Economic Dy-namics and Control vol 22 no 10 pp 1517ndash1541 1998

[14] M J Brennan ldquo+e role of learning in dynamic portfoliodecisionsrdquo Review of Finance vol 1 no 3 pp 295ndash306 1998

[15] N Barberis ldquoInvesting for the long run when returns arepredictablerdquo e Journal of Finance vol 55 no 1 pp 225ndash264 2000

[16] L Pastor and R F Stambaugh ldquoCosts of equity capital andmodel mispricingrdquo e Journal of Finance vol 54 no 1pp 67ndash121 1999

[17] L Pastor ldquoPortfolio selection and asset pricing modelsrdquo eJournal of Finance vol 55 no 1 pp 179ndash223 2000

[18] Z Bai H Liu and W K Wong ldquoOn the Markowitz mean-variance analysis of self-financing portfoliosrdquo Social ScienceElectronic Publishing vol 1 no 1 pp 35ndash42 2009

[19] Z Bai H Liu and W K Wong ldquoEnhancement of the ap-plicability of Markowitzrsquos portfolio optimization by utilizingrandom matrix theoryrdquo Mathematical Finance vol 19 no 4pp 639ndash667 2010

[20] S A Zenios Financial Optimization Cambridge UniversityPress Cambridge UK 2002

[21] N Topaloglou H Vladimirou and S A Zenios ldquoA dynamicstochastic programming model for international portfoliomanagementrdquo European Journal of Operational Researchvol 185 no 3 pp 1501ndash1524 2008

[22] F-Y Chen and S-L Liao ldquoModelling VaR for foreign-assetportfolios in continuous timerdquo Economic Modelling vol 26no 1 pp 234ndash240 2009

[23] H Yao Z Li and S Chen ldquoContinuous-time mean-varianceportfolio selection with only risky assetsrdquo Economic Model-ling vol 36 pp 244ndash251 2014

[24] R J Elliott T K Siu and A Badescu ldquoOn mean-varianceportfolio selection under a hidden Markovian regime-switching modelrdquo Economic Modelling vol 27 no 3pp 678ndash686 2010

[25] Z Wang ldquoA shrinkage theory of asset allocation using asset-pricing modelsrdquo SSRN Electronic Journal vol 18 no 2pp 673ndash705 2001

[26] L MacLean M E Foster andW T Ziemba ldquoEmpirical bayesestimation with dynamic portfolio modelsrdquo SSRN ElectronicJournal 2004

[27] T L Lai H Xing and Z Chen ldquoMean-variance portfoliooptimization whenmeans and covariances are unknownrdquoeAnnals of Applied Statistics vol 5 no 2A pp 798ndash823 2011

[28] E Vilkkumaa J Liesio and A Salo ldquoBayesian evaluation andselection strategies in portfolio decision analysisrdquo EuropeanJournal of Operational Research vol 233 no 3 pp 772ndash7832012

[29] T Bodnar S Mazur and Y Okhrin ldquoBayesian estimation ofthe global minimum variance portfoliordquo European Journal ofOperational Research vol 256 no 1 pp 292ndash307 2017

[30] R Kan and G Zhou ldquoOptimal Portfolio Choice with Pa-rameter Uncertaintyrdquo Journal of Financial and QuantitativeAnalysis vol 42 no 3 pp 621ndash656 2007

[31] D Niedermayer Portfolio Optimization under ParameterUncertainty Springer Boston MA USA 2008

[32] A Kourtis G Dotsis and R N Markellos ldquoParameter un-certainty in portfolio selection shrinking the inverse co-variance matrixrdquo Journal of Banking amp Finance vol 36 no 9pp 2522ndash2531 2012

[33] V DeMiguel A Martın-Utrera and F J Nogales ldquoParameteruncertainty in multiperiod portfolio optimization withtransaction costsrdquo Journal of Financial and QuantitativeAnalysis vol 50 no 6 pp 1443ndash1471 2015

[34] A A Santos A M Monteiro and R Pascoal Portfolio Choiceunder Parameter Uncertainty Bayesian Analysis and RobustOptimization Comparison GEMF-Faculdade de EconomiaUniversidade de Coimbra Coimbra Portugal 2014

[35] L Pastor and P Veronesi ldquoLearning in financial marketsrdquoAnnual Review of Financial Economics vol 1 no 1pp 361ndash381 2009

[36] J Li M Li D Wu Q Dai and H Song ldquoA Bayesian net-works-based risk identification approach for software processrisk the context of Chinese trustworthy softwarerdquo In-ternational Journal of Information Technology amp DecisionMaking vol 15 no 6 pp 1391ndash1412 2016

[37] W Bessler A Leonhardt and D Wolff ldquoAnalyzing hedgingstrategies for fixed income portfolios a Bayesian approach formodel selectionrdquo International Review of Financial Analysisvol 46 pp 239ndash256 2016

[38] J Liu Essays on Risk and Uncertainty in Financial DecisionMaking Bayesian Inference of Multi-Factor Affine TermStructure Models and Dynamic Optimal Portfolio Choices forRobust Preferences Boston University Boston MA USA2014

[39] N Metropolis and S Ulam ldquoA property of randomness of anarithmetical functionrdquo American Mathematical Monthlyvol 60 no 4 pp 252-253 1953

[40] W K Hastings ldquoMonte Carlo sampling methods usingMarkov chains and their applicationsrdquo Biometrika vol 57no 1 pp 97ndash109 1970

[41] H Derin H Elliott R Cristi and D Geman ldquoBayessmoothing algorithms for segmentation of binary imagesmodeled by markov random fieldsrdquo IEEE Transactions onPattern Analysis amp Machine Intelligence vol 6 no 6pp 707ndash720 1984

[42] L Bauwens M Lubrano and J F Richard Bayesian Inferencein Dynamic Econometric Models OUP Oxford Oxford UK2000

[43] J D Hamilton and R Susmel ldquoAutoregressive conditionalheteroskedasticity and changes in regimerdquo Journal ofEconometrics vol 64 no 1-2 pp 307ndash333 1994

[44] J Cai ldquoAMarkovmodel of switching-regime ARCHrdquo Journalof Business amp Economic Statistics vol 12 no 3 pp 309ndash3161994

[45] J S Henneke S T Rachev F J Fabozzi and M NikolovldquoMCMC-based estimation of Markov Switching ARMAndashC-GARCH modelsrdquo Applied Economics vol 43 no 3pp 259ndash271 2011


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Submit your manuscripts atwwwhindawicom

reviewed the development of Bayesian portfolio in thefollowing respect (1) the returns are iid (2) the returns arepredicable (3) the mean follows regime switching and thevolatility is stochastic +ey mainly discussed the Bayesianportfolio in parameter uncertainties as well as with themodel uncertainties In Fabozzis review of ldquo60 Years ofPortfolio Optimization Practical Challenges and CurrentTrendsrdquo they also speak high of the application of Bayesianmethods in portfolio optimization [8] In this paper we takethe financial data with different lengths as the applicationbackground mainly focus on the uncertainties of parametersin portfolio selection and build two models based on Bayesandmodern portfolio theory a Bayesian-GARCH(11) modeland a Bayesian-Markov Regime Switching-GARCH(11)model Using the Bayesian optimization algorithm both themodels do not overestimate their theoretic optimal returnsand they can handle the overestimation problem

+e paper is organized as follows In Section 2 thebackground studies are briefly introduced including theliterature review and the relative theories and methods suchas Bayesian theory and Markov regime switching In Section3 a Bayesian-GARCH(11) model and a Bayesian-MarkovRegime Switching-GARCH(11) model are constructed InSection 4 the two models are taken into application and thenumerical results are illustrated and compared In Section 5the paper is concluded with summary comments andpointers to the future work

2 Background Studies

21 Literature Review It is well known that the parametersin portfolio selection may be uncertain or mistaken and thereturn-risk optimization can be very sensitive to changes inthe inputs In early studies most researchers assumed thatthe risky asset returns are iid and focused on the effect ofthe uncertainty of the mean and variance on the portfolioselection Williams [9] discussed how the parameter un-certainty affected the portfolio selection Bawa et al [10]firstly consider the uncertainty of the parameters of thereturn distribution in static portfolio selection Gennotte[11] analyzed portfolio selection when the expected returnwas uncertain Kandel and Stambaugh [12] showed thatwhen the returns were predictable investors who ignoredthe parameter uncertainties preferred risky stocks in a shortperiod Zenios et al [13] develop multiperiod dynamicmodels for fixed-income portfolio management under un-certainty using multistage stochastic programming withrecourse +e multiperiod models outperform classicalmodels based on portfolio immunization and single-periodmodels Brennan [14] studied the dynamic portfolio selec-tion with the assumptions that the expected risk premiumwas uncertain in discrete time He thought the assumptionsof financial models should be that all the invest opportunitiesdepended on a series of unobservable variables and theparameters were not certain Barberis [15] got similar resultswith the long-period investors Pastor and Stambaugh [16]discussed how the uncertainties of the mispricing affectedthe portfolio selection Pastor [17] treated the model un-certainty using the investorsrsquo opinions to the asset pricing as

prior information Bai et al [18 19] develop a bootstrap-corrected estimator to correct the overestimation in port-folio selection and further extend the theory to obtain self-financing portfolios Besides in order to improve theportfolio selection efficiency researchers tend to extend themodels with dynamic programming [13 20 21] withcontinuous time framework [22 23] using Markovian re-gime-switching methods [24] and so on

Currently most of the parameter uncertainties are an-alyzed in Bayesian Framework As we mentioned before theBayesian method has advantages first of all it can fully useprior information secondly it explains the uncertainties ofestimated risks and models thirdly it makes the compu-tation in simulating complicated variables easy In this wayinvestors firstly set the parameter prior then they useBayesian rules to get the posterior As a result Bayesianmethods are gradually used in the portfolio selectionanalysis One of the implications is that the Bayesian theorycan bemainly used to estimate the uncertain returns or otherfactors (such as a shrinkage estimator and so on) in portfolio[15 25ndash29] In addition for the parameter uncertainty ofportfolio selection in other words for the estimation riskevaluation or the estimation error correction researchersalso tend to choose Bayesian methods [4 30ndash34] Besidesthe Bayesian theory is also used in model analysis [35] riskmeasurement or return-risk trade off [36] portfolio opti-mization [37 38] and so on

22 Modern Portfolio eory According to Markowitzrsquosportfolio theory there are N assets Rt is the excess return ofasset N in time t Rt (R1t RNt)prime and they follow anormal distribution p(Rt | μΣ) N(μΣ) micro is a N times 1matrix and Σ is a N times N matrix +e weights are denoted asw (w1 wN) +e return of the portfolio is

Rp 1113944N

i1wiRit wprimeRt (1)

And the expected return and variance are

μp 1113944

N

i1wiμi wprimeμ

σ2p 1113944N

i11113944

N

j1wiwjcov Ri Rj1113872 1113873 wprime1113944 w

(2)

where μi is the return of asset i and cov(Ri Rj) is the co-variance of assets i and j

Suppose investors hold the portfolio for τ and theirtarget is to maximize the value at time T + τ where T is thetime when the portfolio is built +e mean-variance modelcan be expressed as the following optimization problem

minw

σ2p minw

wprime1113944T+τ

w

st wprimeμT+τ ge μlowast

wprime1 1

(3)

where μlowast is the permitted minimum return








2tminus q


2tminus p

(4)




p θ[j] | y1113872 1113873 1

⎧⎨

⎩

⎫⎬

⎭ (5)











L θ R1t

1113868111386811138681113868 R2t1113872 1113873 1113945N

i1L θi Rit

11138681113868111386811138681113872 1113873prop1113945

N1

i11113945

T

t1σ minus 1

it|tminus 1 1 +1υ

Rir minus μi1113872 11138732

σ2it|tminus 1⎛⎝ ⎞⎠⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

times 1113945

N2

j11113945

T

ts


1υ

Rjr minus μi1113872 11138732

σ2jt|tminus 1

⎛⎝ ⎞⎠

minus (υ+12)

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦





rameters as follows

1IiθG


0 others1113896 (10)





μi sim N μi0 σ2i01113872 1113873 (12)



1113945

N

i1p θi Rit

11138681113868111386811138681113872 1113873prop1113945

N1

i11113945

T

t1σminus 1

it|tminus 1 1 +1υi

Rit minus μi1113872 11138732


minus υi+12( )⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

times 1113945

N2

j11113945

T

ts


1υi

Rjt minus μi1113872 11138732

σ2jt|tminus 1

⎛⎝ ⎞⎠

minus υi+12( )⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

times 1113945

N


times 1113945

N

i11113945

N

i1exp

minus 12


minus 1

ic

μi minus μic1113872 1113873⎛⎝ ⎞⎠

times 1113945N

i1IiθG

(13)



2it|tminus 1 υi1113872 1113873 (14)



σ2it|tminus 1ηt

1113888 1113889 (15)


ηt υi

1113868111386811138681113868 sim gammaυi

2υi

21113874 1113875 t 1 T (16)




lg L θ R1t

1113868111386811138681113868 R2t1113872 11138731113872 1113873 const minus12

1113944

T

tt0


2

σ2it|tminus 1⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

(17)


lg L θi R1t

1113868111386811138681113868 R2t1113872 11138731113872 1113873 const minus12

1113944T

tt0


2

σ2it|tminus 1⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦


minus 1

ic


2lgυi

2

minus T lg Γυi

21113874 11138751113874 1113875 +

υi

2minus 11113874 1113875 1113944

T

t1lg ηt( 1113857

minusυi

2t 11113944

T

t1ηt minus υtλ




p μi θiG




p η θiG


2

Rit minus μi1113872 11138732

2σ2it|tminus 1+υi

2⎛⎝ ⎞⎠

(20)


p υi θiG

1113868111386811138681113868 μi η R1t R2t1113872 1113873propΓυi

21113874 1113875

minus T υi

21113874 1113875

Tυi2exp υiλ

lowast( 1113857






11138681113868111386811138681113868 R1t R2t1113874 11138751113874 1113875 const +Tυi

2log

υi

21113874 1113875 minus T lg Γ

υi

21113874 11138751113874 1113875

+υi

2minus 11113874 1113875 1113944

T


υi

21113944

T

t1ηt minus υiλ

(22)


minus υi



11138681113868111386811138681113868 R1tR2t1113874 11138751113874 1113875 const minus12

1113944T

tt0


2

σ2it|tminus 1⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

(23)



prime ηtprime) of


p θiG μi

1113868111386811138681113868 η υi R1t R2t1113872 1113873

p υi θiG

1113868111386811138681113868 μi η R1t R2t1113872 1113873

p η θiG

1113868111386811138681113868 μi υi R1t R2t1113872 1113873

p μi θiG

1113868111386811138681113868 η υi R1t R2t1113872 1113873

(24)






σ2t|tminus 1 w + gSt1113872 1113873 + 1113944

p

i1αiu

2tminus i (26)



Π

π11 π12 π13

π21 π22 π23

π31 π32 π33

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (27)



1113868111386811138681113868 Stminus 111138721113872 1113873 (28)


T





w St( 1113857 α St( 1113857 β St( 1113857( 1113857

w1 α1 β1( 1113857 St 1

w2 α2 β2( 1113857 St 2

w3 α3 β3( 1113857 St 3

⎧⎪⎪⎨

⎪⎪⎩(30)




T










1113872 1113873I θGi (32)






p(S | π) 1113945Tminus 1

t1p St+1 St

1113868111386811138681113868 π1113872 1113873

πn1111 π


32 πn3333

πn1111 π

n1212 1 minus π11 minus π12( 1113857


n33



lg L θ R1t

1113868111386811138681113868 R2t1113872 11138731113872 1113873 const minus12

1113944

T

tt0


2

σ2t|tminus 11113890 1113891


minus 1

c

μ minus μc1113872 1113873

minus12

1113944

3

i1θGi minus μθGi

1113872 1113873prime1113944minus 1

θGi


+Tυ2lg

υ2

1113874 1113875 minus T lg Γυ2

1113874 11138751113874 1113875

+υ2

minus 11113874 1113875 1113944T


υ2

1113944T

t1ηt minus υλ

+ 11139443

i11113944

3




lg p πi Rt

1113868111386811138681113868 θminus πi1113872 11138731113872 1113873 const + 1113944

3


(36)



T



p St i Rt

1113868111386811138681113868 θminus S Sminus t1113872 1113873 (37)



p St i Rt


11138681113868111386811138681113872 1113873


11138681113868111386811138681113872 1113873

p Rt θminus S


11138681113868111386811138681113872 1113873


11138681113868111386811138681113872 1113873

(38)




(39)

+e denominator is


11138681113868111386811138681113872 1113873 11139443


11138681113868111386811138681113872 1113873 (40)


p St i Rt





i 1 2 3

(41)







θjGi(i 1 2 3)



can be accepted



p RT+1 R1t


1113868111386811138681113868 R2t1113872 1113873dμ dV

(42)



















0 01 02 03 04 05 06 07 080

002

004

006

008

01

012

014

016

018

02


(a)

004 005 006 007 008 009 01 0113

4

5

6

7

8

9

10

11times10ndash3

No short-selling

(b)



A3 A11


π1k 0854 0976 0013 0011 0142 0004π2k 0041 0029 0955 0016 0936 0023π3k 0103 0329 0245 0426 0116 0781

1

05

0

2009

11

6

2010

16

2010

36

2010

56

2010

76

2010

96

2010

11

6

2011

16

2011

36

2011

56

2011

76

2011

96

1

05

0

2009

11

6

2010

16

2010

36

2010

56

2010

76

2010

96

2010

11

6

2011

16

2011

36

2011

56

2011

76

2011

96

1

05

0

2009

11

6

2010

16

2010

36

2010

56

2010

76

2010

96

2010

11

6

2011

16

2011

36

2011

56

2011

76

2011

96





1

05

0

2000

17

2001

17

2002

17

2003

17

2004

17

2005

17

2006

17

2007

17

2008

17

2009

17

2010

17

2011

17

1

05

0

2000

17

2001

17

2002

17

2003

17

2004

17

2005

17

2006

17

2007

17

2008

17

2009

17

2010

17

2011

17

1

05

020

001

7

2001

17

2002

17

2003

17

2004

17

2005

17

2006

17

2007

17

2008

17

2009

17

2010

17

2011

17




0 01 02 03 04 05 06 07 080

002

004

006

008

01

012

014

016

018

02


(a)

004 005 006 007 008 009 01 0113

4

5

6

7

8

9

10

11

12times10ndash3

No short-selling

(b)






5 Conclusions




Data Availability


Disclosure




Acknowledgments


References
























































Journal of












Journal of







Volume 2018






Volume 2018














2tminus q


2tminus p

(4)




p θ[j] | y1113872 1113873 1

⎧⎨

⎩

⎫⎬

⎭ (5)











L θ R1t

1113868111386811138681113868 R2t1113872 1113873 1113945N

i1L θi Rit

11138681113868111386811138681113872 1113873prop1113945

N1

i11113945

T

t1σ minus 1

it|tminus 1 1 +1υ

Rir minus μi1113872 11138732

σ2it|tminus 1⎛⎝ ⎞⎠⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

times 1113945

N2

j11113945

T

ts


1υ

Rjr minus μi1113872 11138732

σ2jt|tminus 1

⎛⎝ ⎞⎠

minus (υ+12)

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦





rameters as follows

1IiθG


0 others1113896 (10)





μi sim N μi0 σ2i01113872 1113873 (12)



1113945

N

i1p θi Rit

11138681113868111386811138681113872 1113873prop1113945

N1

i11113945

T

t1σminus 1

it|tminus 1 1 +1υi

Rit minus μi1113872 11138732


minus υi+12( )⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

times 1113945

N2

j11113945

T

ts


1υi

Rjt minus μi1113872 11138732

σ2jt|tminus 1

⎛⎝ ⎞⎠

minus υi+12( )⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

times 1113945

N


times 1113945

N

i11113945

N

i1exp

minus 12


minus 1

ic

μi minus μic1113872 1113873⎛⎝ ⎞⎠

times 1113945N

i1IiθG

(13)



2it|tminus 1 υi1113872 1113873 (14)



σ2it|tminus 1ηt

1113888 1113889 (15)


ηt υi

1113868111386811138681113868 sim gammaυi

2υi

21113874 1113875 t 1 T (16)




lg L θ R1t

1113868111386811138681113868 R2t1113872 11138731113872 1113873 const minus12

1113944

T

tt0


2

σ2it|tminus 1⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

(17)


lg L θi R1t

1113868111386811138681113868 R2t1113872 11138731113872 1113873 const minus12

1113944T

tt0


2

σ2it|tminus 1⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦


minus 1

ic


2lgυi

2

minus T lg Γυi

21113874 11138751113874 1113875 +

υi

2minus 11113874 1113875 1113944

T

t1lg ηt( 1113857

minusυi

2t 11113944

T

t1ηt minus υtλ




p μi θiG




p η θiG


2

Rit minus μi1113872 11138732

2σ2it|tminus 1+υi

2⎛⎝ ⎞⎠

(20)


p υi θiG

1113868111386811138681113868 μi η R1t R2t1113872 1113873propΓυi

21113874 1113875

minus T υi

21113874 1113875

Tυi2exp υiλ

lowast( 1113857






11138681113868111386811138681113868 R1t R2t1113874 11138751113874 1113875 const +Tυi

2log

υi

21113874 1113875 minus T lg Γ

υi

21113874 11138751113874 1113875

+υi

2minus 11113874 1113875 1113944

T


υi

21113944

T

t1ηt minus υiλ

(22)


minus υi



11138681113868111386811138681113868 R1tR2t1113874 11138751113874 1113875 const minus12

1113944T

tt0


2

σ2it|tminus 1⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

(23)



prime ηtprime) of


p θiG μi

1113868111386811138681113868 η υi R1t R2t1113872 1113873

p υi θiG

1113868111386811138681113868 μi η R1t R2t1113872 1113873

p η θiG

1113868111386811138681113868 μi υi R1t R2t1113872 1113873

p μi θiG

1113868111386811138681113868 η υi R1t R2t1113872 1113873

(24)






σ2t|tminus 1 w + gSt1113872 1113873 + 1113944

p

i1αiu

2tminus i (26)



Π

π11 π12 π13

π21 π22 π23

π31 π32 π33

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (27)



1113868111386811138681113868 Stminus 111138721113872 1113873 (28)


T





w St( 1113857 α St( 1113857 β St( 1113857( 1113857

w1 α1 β1( 1113857 St 1

w2 α2 β2( 1113857 St 2

w3 α3 β3( 1113857 St 3

⎧⎪⎪⎨

⎪⎪⎩(30)




T










1113872 1113873I θGi (32)






p(S | π) 1113945Tminus 1

t1p St+1 St

1113868111386811138681113868 π1113872 1113873

πn1111 π


32 πn3333

πn1111 π

n1212 1 minus π11 minus π12( 1113857


n33



lg L θ R1t

1113868111386811138681113868 R2t1113872 11138731113872 1113873 const minus12

1113944

T

tt0


2

σ2t|tminus 11113890 1113891


minus 1

c

μ minus μc1113872 1113873

minus12

1113944

3

i1θGi minus μθGi

1113872 1113873prime1113944minus 1

θGi


+Tυ2lg

υ2

1113874 1113875 minus T lg Γυ2

1113874 11138751113874 1113875

+υ2

minus 11113874 1113875 1113944T


υ2

1113944T

t1ηt minus υλ

+ 11139443

i11113944

3




lg p πi Rt

1113868111386811138681113868 θminus πi1113872 11138731113872 1113873 const + 1113944

3


(36)



T



p St i Rt

1113868111386811138681113868 θminus S Sminus t1113872 1113873 (37)



p St i Rt


11138681113868111386811138681113872 1113873


11138681113868111386811138681113872 1113873

p Rt θminus S


11138681113868111386811138681113872 1113873


11138681113868111386811138681113872 1113873

(38)




(39)

+e denominator is


11138681113868111386811138681113872 1113873 11139443


11138681113868111386811138681113872 1113873 (40)


p St i Rt





i 1 2 3

(41)







θjGi(i 1 2 3)



can be accepted



p RT+1 R1t


1113868111386811138681113868 R2t1113872 1113873dμ dV

(42)



















0 01 02 03 04 05 06 07 080

002

004

006

008

01

012

014

016

018

02


(a)

004 005 006 007 008 009 01 0113

4

5

6

7

8

9

10

11times10ndash3

No short-selling

(b)



A3 A11


π1k 0854 0976 0013 0011 0142 0004π2k 0041 0029 0955 0016 0936 0023π3k 0103 0329 0245 0426 0116 0781

1

05

0

2009

11

6

2010

16

2010

36

2010

56

2010

76

2010

96

2010

11

6

2011

16

2011

36

2011

56

2011

76

2011

96

1

05

0

2009

11

6

2010

16

2010

36

2010

56

2010

76

2010

96

2010

11

6

2011

16

2011

36

2011

56

2011

76

2011

96

1

05

0

2009

11

6

2010

16

2010

36

2010

56

2010

76

2010

96

2010

11

6

2011

16

2011

36

2011

56

2011

76

2011

96





1

05

0

2000

17

2001

17

2002

17

2003

17

2004

17

2005

17

2006

17

2007

17

2008

17

2009

17

2010

17

2011

17

1

05

0

2000

17

2001

17

2002

17

2003

17

2004

17

2005

17

2006

17

2007

17

2008

17

2009

17

2010

17

2011

17

1

05

020

001

7

2001

17

2002

17

2003

17

2004

17

2005

17

2006

17

2007

17

2008

17

2009

17

2010

17

2011

17




0 01 02 03 04 05 06 07 080

002

004

006

008

01

012

014

016

018

02


(a)

004 005 006 007 008 009 01 0113

4

5

6

7

8

9

10

11

12times10ndash3

No short-selling

(b)






5 Conclusions




Data Availability


Disclosure




Acknowledgments


References
























































Journal of












Journal of







Volume 2018






Volume 2018











μi sim N μi0 σ2i01113872 1113873 (12)



1113945

N

i1p θi Rit

11138681113868111386811138681113872 1113873prop1113945

N1

i11113945

T

t1σminus 1

it|tminus 1 1 +1υi

Rit minus μi1113872 11138732


minus υi+12( )⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

times 1113945

N2

j11113945

T

ts


1υi

Rjt minus μi1113872 11138732

σ2jt|tminus 1

⎛⎝ ⎞⎠

minus υi+12( )⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

times 1113945

N


times 1113945

N

i11113945

N

i1exp

minus 12


minus 1

ic

μi minus μic1113872 1113873⎛⎝ ⎞⎠

times 1113945N

i1IiθG

(13)



2it|tminus 1 υi1113872 1113873 (14)



σ2it|tminus 1ηt

1113888 1113889 (15)


ηt υi

1113868111386811138681113868 sim gammaυi

2υi

21113874 1113875 t 1 T (16)




lg L θ R1t

1113868111386811138681113868 R2t1113872 11138731113872 1113873 const minus12

1113944

T

tt0


2

σ2it|tminus 1⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

(17)


lg L θi R1t

1113868111386811138681113868 R2t1113872 11138731113872 1113873 const minus12

1113944T

tt0


2

σ2it|tminus 1⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦


minus 1

ic


2lgυi

2

minus T lg Γυi

21113874 11138751113874 1113875 +

υi

2minus 11113874 1113875 1113944

T

t1lg ηt( 1113857

minusυi

2t 11113944

T

t1ηt minus υtλ




p μi θiG




p η θiG


2

Rit minus μi1113872 11138732

2σ2it|tminus 1+υi

2⎛⎝ ⎞⎠

(20)


p υi θiG

1113868111386811138681113868 μi η R1t R2t1113872 1113873propΓυi

21113874 1113875

minus T υi

21113874 1113875

Tυi2exp υiλ

lowast( 1113857






11138681113868111386811138681113868 R1t R2t1113874 11138751113874 1113875 const +Tυi

2log

υi

21113874 1113875 minus T lg Γ

υi

21113874 11138751113874 1113875

+υi

2minus 11113874 1113875 1113944

T


υi

21113944

T

t1ηt minus υiλ

(22)


minus υi



11138681113868111386811138681113868 R1tR2t1113874 11138751113874 1113875 const minus12

1113944T

tt0


2

σ2it|tminus 1⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

(23)



prime ηtprime) of


p θiG μi

1113868111386811138681113868 η υi R1t R2t1113872 1113873

p υi θiG

1113868111386811138681113868 μi η R1t R2t1113872 1113873

p η θiG

1113868111386811138681113868 μi υi R1t R2t1113872 1113873

p μi θiG

1113868111386811138681113868 η υi R1t R2t1113872 1113873

(24)






σ2t|tminus 1 w + gSt1113872 1113873 + 1113944

p

i1αiu

2tminus i (26)



Π

π11 π12 π13

π21 π22 π23

π31 π32 π33

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (27)



1113868111386811138681113868 Stminus 111138721113872 1113873 (28)


T





w St( 1113857 α St( 1113857 β St( 1113857( 1113857

w1 α1 β1( 1113857 St 1

w2 α2 β2( 1113857 St 2

w3 α3 β3( 1113857 St 3

⎧⎪⎪⎨

⎪⎪⎩(30)




T










1113872 1113873I θGi (32)






p(S | π) 1113945Tminus 1

t1p St+1 St

1113868111386811138681113868 π1113872 1113873

πn1111 π


32 πn3333

πn1111 π

n1212 1 minus π11 minus π12( 1113857


n33



lg L θ R1t

1113868111386811138681113868 R2t1113872 11138731113872 1113873 const minus12

1113944

T

tt0


2

σ2t|tminus 11113890 1113891


minus 1

c

μ minus μc1113872 1113873

minus12

1113944

3

i1θGi minus μθGi

1113872 1113873prime1113944minus 1

θGi


+Tυ2lg

υ2

1113874 1113875 minus T lg Γυ2

1113874 11138751113874 1113875

+υ2

minus 11113874 1113875 1113944T


υ2

1113944T

t1ηt minus υλ

+ 11139443

i11113944

3




lg p πi Rt

1113868111386811138681113868 θminus πi1113872 11138731113872 1113873 const + 1113944

3


(36)



T



p St i Rt

1113868111386811138681113868 θminus S Sminus t1113872 1113873 (37)



p St i Rt


11138681113868111386811138681113872 1113873


11138681113868111386811138681113872 1113873

p Rt θminus S


11138681113868111386811138681113872 1113873


11138681113868111386811138681113872 1113873

(38)




(39)

+e denominator is


11138681113868111386811138681113872 1113873 11139443


11138681113868111386811138681113872 1113873 (40)


p St i Rt





i 1 2 3

(41)







θjGi(i 1 2 3)



can be accepted



p RT+1 R1t


1113868111386811138681113868 R2t1113872 1113873dμ dV

(42)



















0 01 02 03 04 05 06 07 080

002

004

006

008

01

012

014

016

018

02


(a)

004 005 006 007 008 009 01 0113

4

5

6

7

8

9

10

11times10ndash3

No short-selling

(b)



A3 A11


π1k 0854 0976 0013 0011 0142 0004π2k 0041 0029 0955 0016 0936 0023π3k 0103 0329 0245 0426 0116 0781

1

05

0

2009

11

6

2010

16

2010

36

2010

56

2010

76

2010

96

2010

11

6

2011

16

2011

36

2011

56

2011

76

2011

96

1

05

0

2009

11

6

2010

16

2010

36

2010

56

2010

76

2010

96

2010

11

6

2011

16

2011

36

2011

56

2011

76

2011

96

1

05

0

2009

11

6

2010

16

2010

36

2010

56

2010

76

2010

96

2010

11

6

2011

16

2011

36

2011

56

2011

76

2011

96





1

05

0

2000

17

2001

17

2002

17

2003

17

2004

17

2005

17

2006

17

2007

17

2008

17

2009

17

2010

17

2011

17

1

05

0

2000

17

2001

17

2002

17

2003

17

2004

17

2005

17

2006

17

2007

17

2008

17

2009

17

2010

17

2011

17

1

05

020

001

7

2001

17

2002

17

2003

17

2004

17

2005

17

2006

17

2007

17

2008

17

2009

17

2010

17

2011

17




0 01 02 03 04 05 06 07 080

002

004

006

008

01

012

014

016

018

02


(a)

004 005 006 007 008 009 01 0113

4

5

6

7

8

9

10

11

12times10ndash3

No short-selling

(b)






5 Conclusions




Data Availability


Disclosure




Acknowledgments


References
























































Journal of












Journal of







Volume 2018






Volume 2018










11138681113868111386811138681113868 R1t R2t1113874 11138751113874 1113875 const +Tυi

2log

υi

21113874 1113875 minus T lg Γ

υi

21113874 11138751113874 1113875

+υi

2minus 11113874 1113875 1113944

T


υi

21113944

T

t1ηt minus υiλ

(22)


minus υi



11138681113868111386811138681113868 R1tR2t1113874 11138751113874 1113875 const minus12

1113944T

tt0


2

σ2it|tminus 1⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

(23)



prime ηtprime) of


p θiG μi

1113868111386811138681113868 η υi R1t R2t1113872 1113873

p υi θiG

1113868111386811138681113868 μi η R1t R2t1113872 1113873

p η θiG

1113868111386811138681113868 μi υi R1t R2t1113872 1113873

p μi θiG

1113868111386811138681113868 η υi R1t R2t1113872 1113873

(24)






σ2t|tminus 1 w + gSt1113872 1113873 + 1113944

p

i1αiu

2tminus i (26)



Π

π11 π12 π13

π21 π22 π23

π31 π32 π33

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (27)



1113868111386811138681113868 Stminus 111138721113872 1113873 (28)


T





w St( 1113857 α St( 1113857 β St( 1113857( 1113857

w1 α1 β1( 1113857 St 1

w2 α2 β2( 1113857 St 2

w3 α3 β3( 1113857 St 3

⎧⎪⎪⎨

⎪⎪⎩(30)




T










1113872 1113873I θGi (32)






p(S | π) 1113945Tminus 1

t1p St+1 St

1113868111386811138681113868 π1113872 1113873

πn1111 π


32 πn3333

πn1111 π

n1212 1 minus π11 minus π12( 1113857


n33



lg L θ R1t

1113868111386811138681113868 R2t1113872 11138731113872 1113873 const minus12

1113944

T

tt0


2

σ2t|tminus 11113890 1113891


minus 1

c

μ minus μc1113872 1113873

minus12

1113944

3

i1θGi minus μθGi

1113872 1113873prime1113944minus 1

θGi


+Tυ2lg

υ2

1113874 1113875 minus T lg Γυ2

1113874 11138751113874 1113875

+υ2

minus 11113874 1113875 1113944T


υ2

1113944T

t1ηt minus υλ

+ 11139443

i11113944

3




lg p πi Rt

1113868111386811138681113868 θminus πi1113872 11138731113872 1113873 const + 1113944

3


(36)



T



p St i Rt

1113868111386811138681113868 θminus S Sminus t1113872 1113873 (37)



p St i Rt


11138681113868111386811138681113872 1113873


11138681113868111386811138681113872 1113873

p Rt θminus S


11138681113868111386811138681113872 1113873


11138681113868111386811138681113872 1113873

(38)




(39)

+e denominator is


11138681113868111386811138681113872 1113873 11139443


11138681113868111386811138681113872 1113873 (40)


p St i Rt





i 1 2 3

(41)







θjGi(i 1 2 3)



can be accepted



p RT+1 R1t


1113868111386811138681113868 R2t1113872 1113873dμ dV

(42)



















0 01 02 03 04 05 06 07 080

002

004

006

008

01

012

014

016

018

02


(a)

004 005 006 007 008 009 01 0113

4

5

6

7

8

9

10

11times10ndash3

No short-selling

(b)



A3 A11


π1k 0854 0976 0013 0011 0142 0004π2k 0041 0029 0955 0016 0936 0023π3k 0103 0329 0245 0426 0116 0781

1

05

0

2009

11

6

2010

16

2010

36

2010

56

2010

76

2010

96

2010

11

6

2011

16

2011

36

2011

56

2011

76

2011

96

1

05

0

2009

11

6

2010

16

2010

36

2010

56

2010

76

2010

96

2010

11

6

2011

16

2011

36

2011

56

2011

76

2011

96

1

05

0

2009

11

6

2010

16

2010

36

2010

56

2010

76

2010

96

2010

11

6

2011

16

2011

36

2011

56

2011

76

2011

96





1

05

0

2000

17

2001

17

2002

17

2003

17

2004

17

2005

17

2006

17

2007

17

2008

17

2009

17

2010

17

2011

17

1

05

0

2000

17

2001

17

2002

17

2003

17

2004

17

2005

17

2006

17

2007

17

2008

17

2009

17

2010

17

2011

17

1

05

020

001

7

2001

17

2002

17

2003

17

2004

17

2005

17

2006

17

2007

17

2008

17

2009

17

2010

17

2011

17




0 01 02 03 04 05 06 07 080

002

004

006

008

01

012

014

016

018

02


(a)

004 005 006 007 008 009 01 0113

4

5

6

7

8

9

10

11

12times10ndash3

No short-selling

(b)






5 Conclusions




Data Availability


Disclosure




Acknowledgments


References
























































Journal of












Journal of







Volume 2018






Volume 2018










1113872 1113873I θGi (32)






p(S | π) 1113945Tminus 1

t1p St+1 St

1113868111386811138681113868 π1113872 1113873

πn1111 π


32 πn3333

πn1111 π

n1212 1 minus π11 minus π12( 1113857


n33



lg L θ R1t

1113868111386811138681113868 R2t1113872 11138731113872 1113873 const minus12

1113944

T

tt0


2

σ2t|tminus 11113890 1113891


minus 1

c

μ minus μc1113872 1113873

minus12

1113944

3

i1θGi minus μθGi

1113872 1113873prime1113944minus 1

θGi


+Tυ2lg

υ2

1113874 1113875 minus T lg Γυ2

1113874 11138751113874 1113875

+υ2

minus 11113874 1113875 1113944T


υ2

1113944T

t1ηt minus υλ

+ 11139443

i11113944

3




lg p πi Rt

1113868111386811138681113868 θminus πi1113872 11138731113872 1113873 const + 1113944

3


(36)



T



p St i Rt

1113868111386811138681113868 θminus S Sminus t1113872 1113873 (37)



p St i Rt


11138681113868111386811138681113872 1113873


11138681113868111386811138681113872 1113873

p Rt θminus S


11138681113868111386811138681113872 1113873


11138681113868111386811138681113872 1113873

(38)




(39)

+e denominator is


11138681113868111386811138681113872 1113873 11139443


11138681113868111386811138681113872 1113873 (40)


p St i Rt





i 1 2 3

(41)







θjGi(i 1 2 3)



can be accepted



p RT+1 R1t


1113868111386811138681113868 R2t1113872 1113873dμ dV

(42)



















0 01 02 03 04 05 06 07 080

002

004

006

008

01

012

014

016

018

02


(a)

004 005 006 007 008 009 01 0113

4

5

6

7

8

9

10

11times10ndash3

No short-selling

(b)



A3 A11


π1k 0854 0976 0013 0011 0142 0004π2k 0041 0029 0955 0016 0936 0023π3k 0103 0329 0245 0426 0116 0781

1

05

0

2009

11

6

2010

16

2010

36

2010

56

2010

76

2010

96

2010

11

6

2011

16

2011

36

2011

56

2011

76

2011

96

1

05

0

2009

11

6

2010

16

2010

36

2010

56

2010

76

2010

96

2010

11

6

2011

16

2011

36

2011

56

2011

76

2011

96

1

05

0

2009

11

6

2010

16

2010

36

2010

56

2010

76

2010

96

2010

11

6

2011

16

2011

36

2011

56

2011

76

2011

96





1

05

0

2000

17

2001

17

2002

17

2003

17

2004

17

2005

17

2006

17

2007

17

2008

17

2009

17

2010

17

2011

17

1

05

0

2000

17

2001

17

2002

17

2003

17

2004

17

2005

17

2006

17

2007

17

2008

17

2009

17

2010

17

2011

17

1

05

020

001

7

2001

17

2002

17

2003

17

2004

17

2005

17

2006

17

2007

17

2008

17

2009

17

2010

17

2011

17




0 01 02 03 04 05 06 07 080

002

004

006

008

01

012

014

016

018

02


(a)

004 005 006 007 008 009 01 0113

4

5

6

7

8

9

10

11

12times10ndash3

No short-selling

(b)






5 Conclusions




Data Availability


Disclosure




Acknowledgments


References
























































Journal of












Journal of







Volume 2018






Volume 2018












θjGi(i 1 2 3)



can be accepted



p RT+1 R1t


1113868111386811138681113868 R2t1113872 1113873dμ dV

(42)



















0 01 02 03 04 05 06 07 080

002

004

006

008

01

012

014

016

018

02


(a)

004 005 006 007 008 009 01 0113

4

5

6

7

8

9

10

11times10ndash3

No short-selling

(b)



A3 A11


π1k 0854 0976 0013 0011 0142 0004π2k 0041 0029 0955 0016 0936 0023π3k 0103 0329 0245 0426 0116 0781

1

05

0

2009

11

6

2010

16

2010

36

2010

56

2010

76

2010

96

2010

11

6

2011

16

2011

36

2011

56

2011

76

2011

96

1

05

0

2009

11

6

2010

16

2010

36

2010

56

2010

76

2010

96

2010

11

6

2011

16

2011

36

2011

56

2011

76

2011

96

1

05

0

2009

11

6

2010

16

2010

36

2010

56

2010

76

2010

96

2010

11

6

2011

16

2011

36

2011

56

2011

76

2011

96





1

05

0

2000

17

2001

17

2002

17

2003

17

2004

17

2005

17

2006

17

2007

17

2008

17

2009

17

2010

17

2011

17

1

05

0

2000

17

2001

17

2002

17

2003

17

2004

17

2005

17

2006

17

2007

17

2008

17

2009

17

2010

17

2011

17

1

05

020

001

7

2001

17

2002

17

2003

17

2004

17

2005

17

2006

17

2007

17

2008

17

2009

17

2010

17

2011

17




0 01 02 03 04 05 06 07 080

002

004

006

008

01

012

014

016

018

02


(a)

004 005 006 007 008 009 01 0113

4

5

6

7

8

9

10

11

12times10ndash3

No short-selling

(b)






5 Conclusions




Data Availability


Disclosure




Acknowledgments


References
























































Journal of












Journal of







Volume 2018






Volume 2018








0 01 02 03 04 05 06 07 080

002

004

006

008

01

012

014

016

018

02


(a)

004 005 006 007 008 009 01 0113

4

5

6

7

8

9

10

11times10ndash3

No short-selling

(b)



A3 A11


π1k 0854 0976 0013 0011 0142 0004π2k 0041 0029 0955 0016 0936 0023π3k 0103 0329 0245 0426 0116 0781

1

05

0

2009

11

6

2010

16

2010

36

2010

56

2010

76

2010

96

2010

11

6

2011

16

2011

36

2011

56

2011

76

2011

96

1

05

0

2009

11

6

2010

16

2010

36

2010

56

2010

76

2010

96

2010

11

6

2011

16

2011

36

2011

56

2011

76

2011

96

1

05

0

2009

11

6

2010

16

2010

36

2010

56

2010

76

2010

96

2010

11

6

2011

16

2011

36

2011

56

2011

76

2011

96





1

05

0

2000

17

2001

17

2002

17

2003

17

2004

17

2005

17

2006

17

2007

17

2008

17

2009

17

2010

17

2011

17

1

05

0

2000

17

2001

17

2002

17

2003

17

2004

17

2005

17

2006

17

2007

17

2008

17

2009

17

2010

17

2011

17

1

05

020

001

7

2001

17

2002

17

2003

17

2004

17

2005

17

2006

17

2007

17

2008

17

2009

17

2010

17

2011

17




0 01 02 03 04 05 06 07 080

002

004

006

008

01

012

014

016

018

02


(a)

004 005 006 007 008 009 01 0113

4

5

6

7

8

9

10

11

12times10ndash3

No short-selling

(b)






5 Conclusions




Data Availability


Disclosure




Acknowledgments


References
























































Journal of












Journal of







Volume 2018






Volume 2018










1

05

0

2000

17

2001

17

2002

17

2003

17

2004

17

2005

17

2006

17

2007

17

2008

17

2009

17

2010

17

2011

17

1

05

0

2000

17

2001

17

2002

17

2003

17

2004

17

2005

17

2006

17

2007

17

2008

17

2009

17

2010

17

2011

17

1

05

020

001

7

2001

17

2002

17

2003

17

2004

17

2005

17

2006

17

2007

17

2008

17

2009

17

2010

17

2011

17




0 01 02 03 04 05 06 07 080

002

004

006

008

01

012

014

016

018

02


(a)

004 005 006 007 008 009 01 0113

4

5

6

7

8

9

10

11

12times10ndash3

No short-selling

(b)






5 Conclusions




Data Availability


Disclosure




Acknowledgments


References
























































Journal of












Journal of







Volume 2018






Volume 2018











5 Conclusions




Data Availability


Disclosure




Acknowledgments


References
























































Journal of












Journal of







Volume 2018






Volume 2018

















































Journal of












Journal of







Volume 2018






Volume 2018








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