TPLVM: Portfolio Construction by Student's t -process Latent Variable Model
aa r X i v : . [ q -f i n . P M ] J a n T PLV M: P
ORTFOLIO C ONSTRUCTION B Y S TUDENT ’ S t - PROC ESS L ATENT V AR IABLE M ODEL
A P
REPRINT
Yusuke Uchiyama
MAZIN Inc.,3-29-14 Nishi-Asakusa, Tito city, Tokyo 111-0035 Japan [email protected]
Kei Nakagawa
Innovation LabNomura Asset Management Co., Ltd.1-11-1 Nihonbashi, Chuo-ku, Tokyo, 103-8260, Japan [email protected]
February 18, 2020 A BSTRACT
Optimal asset allocation is a key topic in modern finance theory. To realize the optimal asset al-location on investor’s risk aversion, various portfolio construction methods have been proposed.Recently, the applications of machine learning are rapidly growing in the area of finance. In this ar-ticle, we propose the Student’s t -process latent variable model (TPLVM) to describe non-Gaussianfluctuations of financial timeseries by lower dimensional latent variables. Subsequently, we applythe TPLVM to minimum-variance portfolio as an alternative of existing nonlinear factor models. Totest the performance of the proposed portfolio, we construct minimum-variance portfolios of globalstock market indices based on the TPLVM or Gaussian process latent variable model. By compar-ing these portfolios, we confirm the proposed portfolio outperforms that of the existing Gaussianprocess latent variable model. K eywords Student’ t -process · Latent variable model · Factor model · Portfolio theory · Global stock markets
Estimation of covariance matrix of timeseries plays a dominant role in applications of modern financial theory. Theoptimization of mean-variance portfolio, which is one of the pioneering works of the modern finance theory [1], isbased on the covariance matrix of the multi-dimensional timeseries of return of assets. Since the return of assetsare modelled by non-stationary stochastic processes, the covariance matrix should be estimated as a time-dependentsymmetric matrix. In practice, we often estimate the covariance matrix by empirical time averaging, because of thelack of complete information of the corresponding probabilistic space. It is however pointed out that time averagingoften causes serious estimation error of the covariance matrix in the case of larger assets [2, 3]. To overcome thisproblem, several inference methods are proposed from the point of view of the random matrix theory [4, 5].With the aid of recently growing machine learning techniques, we can improve the accuracy of the estimation of thecovariance matrix [6, 7]. Furthermore, the applications of the machine learning techniques have been spreading inboth theoretical and practical financial problems [8, 9]. The prediction of the future price is implemented by the deepneural networks of various modeling [10, 11]. The Gaussian process is used as a model of dynamics of the covariancematrix of multi-dimensional timeseries. In the literature of option pricing theory, the model of the volatility of a riskyasset is given by the Gaussian process [12]. In particular, the application of the machine learning techniques for theportfolio optimization has attracted the interest of both academia and industry [13, 14].In the field of mathematical finance, stochastic volatility models have been utilized in estimating dynamic covariancematrix of the return of assets. One of the most popular conditional volatility models is the generalized autoregressiveconditional heteroscedasticity (GARCH) model [15], which describes the volatility clustering of the return of assets.To introduce a time-varying correlation structure to these conditional volatility models, the dynamic conditional cor-
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18, 2020relation (DCC) GARCH model has been proposed [16]. The parameters of the GARCH and DCC GARCH can beestimated by the method of maximum likelihood.On the other hand, in the literature of the machine learning, some kinds of latent variable models can be utilized toinfer the dynamics of the covariance matrix. Recently, the Gaussian process latent variable model (GPLVM) has beenemployed to the problem of the portfolio optimization, where latent variables are introduced as factors of return of theassets. Namely, this model can be interpreted as a latent variable factor model [17].Despite these practical applications, we should reconsider the assumption and validation of the use of the GPLVMfor finance because the GPLVM assumes that observed data follows the Gaussian distribution. In the most case offinancial problems, the return of assets is regarded as an observed variable. It is well known that the fluctuations of thereturn of assets follow non-Gaussian distributions[18]. To describe such fluctuations, some fat-tailed distributions havebeen presented and applied to the financial timeseries. Thus, the GPLVM should be extended to fat-tailed distributionswhen we use it for the financial problems.In this paper, we propose Student’s- t process latent variable model (TPLVM) as an extension of the GPLVM. Thismodel is developed based on the Student’s t -distribution, which is a symmetric fat-tailed distribution. Since theStudent’s t -distribution converges to the Gaussian distribution with the limit of a parameter, degree of freedom, theTPLVM includes the GPLVM as a special case. To use the TPLVM in practice, as well as the GPLVM, we derive itspredictive distribution as closed form and an estimator of hyper parameters by the variational inference in Bayesiansense.The reminder of this paper is organized as follows. Chapter 2 gives a brief introduction the GPLVM including theGaussian process with the concept of kernel functions. In Chap. 3, we introduce the formula of TPLVM, whichconsists of the kernel functions, predictive distribution and variational inference for estimating hyper parameters. As apreliminary preparation of finance, we explain the basis of factor model and portfolio optimization in Chap. 4. Chapter5 implements portfolio optimization, where we compare the performance of the GPLVM and TPLVM. Chapter 6 isdedicated to conclusions and future works. The Gaussian process, a kind of stochastic processes, is a non-parametric method of machine learning method [19, 20].This has been firstly introduced to describe random dynamics such as a fluctuating pollen on water surface known asBrownian motion [21]. Without loss of generality, the argument of the Gaussian process can be extended from one-dimensional time to multi-dimensional feature space. In this chapter, we provide a short review of the Gaussian processfor multi-dimensional features as the preliminary preparation of the proposed model.For a sequence of input features { x , x , · · · , x n } , a stochastic process f ( · ) is the Gaussian process when the sequenceof random variables { f ( x ) , f ( x ) , · · · , f ( x n ) } is sampled from the multivariate Gaussian distribution. In general, theform of the multivariate Gaussian distribution is determined by the mean vector and covariance matrix. Likewise, theGaussian process are specified by the mean and covariance function of input features. Thus, the Gaussian process isregarded as the representation of the infinite dimensional Gaussian distribution.The mean and covariance functions are defined as follows: m ( x ) = E [ f ( x )] , (1) k ( x, x ′ ) = E [( f ( x ) − m ( x ))( f ( x ′ ) − m ( x ′ ))] , (2)where the operator E [ · ] denotes expectation, m ( · ) and k ( · , · ) are respective mean and covariance functions. The meanvector and covariance matrix of the Gaussian process for given dataset are represented by m i = m ( x i ) , (3) K i,j = k ( x i , x j ) . (4)On these setting, the stochastic process f ( · ) is sampled from N ( m ( · ) , K ( · , · )) . In this situation, the stochastic process f ( · ) is the Gaussian process expressed as f ∼GP ( m, K ) . The covariance function satisfies to be symmetric andpositive definite, and thus is also called as kernel function. In the literature of the Gaussian process, the covariancematrix is often called as kernel matrix. The mathematical characteristics of the kernel functions are explained in [22].Given an additional dataset D ∗ = { x ′ , x ′ , · · · , x ′ n ′ } , the corresponding outputs { y ′ , y ′ , · · · , y ′ n ′ } can be predicted bythe conditional Gaussian process with prior dataset D = { ( x , y ) , ( x , y ) , · · · , ( x n , y n ) } . With notations that X = PREPRINT - F
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18, 2020 [ x , x , · · · , x n ] T , X ∗ = [ x ′ , x ′ , · · · , x n ′ ] T and Y = [ y , y , cdots, y n ] , the predictive distribution of the conditionalGaussian process is also given by the Gaussian process GP ( f ∗ , K ∗ ) , where f ∗ = m X + K X ∗ ,X K − X,X Y, (5) K ∗ = K X ∗ ,X ∗ − K X ∗ ,X K − X,X K X,X ∗ . (6)In Eqs. (5) and (6), it is seen that the covariance functions propagate the information about D to D ∗ . Hence, thecovariance functions play the dominant role in the use of the Gaussian process. In the literature of big data analysis, it is often expected that observed variables can be explained by lower dimensionallatent variables. For this purpose, various methods of dimension reduction have been developed. One of the mostpopular methods is the principal component analysis (PCA), which extracts latent variables by the singular value de-composition. To extend the PCA for nonlinear and random data, the Gaussian process latent variable model (GPLVM)has been proposed [23]. The GPLVM expresses the nonlinearity of both observed and latent variables by the covari-ance function. The randomness is assumed to be originate from the Gaussian distribution.To describe an observed variable y ∈ R D , we introduce a latent variable x ∈ R Q with Q < D , and a nonliner map f : R Q → R D with a Q -dimensional noise ǫ ∼N (0 , σ I ) as y = f ( x ) + ε. (7)For this latent variable model, we assume that the nonlinear map f ( · ) is sampled from the Gaussian process as f ∼N (0 , K ) . This model is known as the GPLVM. For the sake of brevity, we introduce notations for the set oflatent and observed variables as X = [ x , x , · · · , x N ] T and Y = [ y , y , · · · , y N ] T . Assume that the columns ofthe observed matrix Y ∈ R N × D are samples from the independently identical distributed Gaussian distributions whichhave the covariance functions with respect to the latent variable matrix X ∈ R N × Q , the probability density function ofthe GPLVM is introduced as follows: p ( Y | X ) = 1(2 π ) ND/ | K X,X | D/ exp (cid:18) − Y T K − X,X Y (cid:19) . (8)In the GPLVM, hyperparameters of the covariance functions and latent variables are inferred by several existingmethods such as gradient methods, variational inference and Markov Chain Monte Carlo methods. t -process latent variable model t -process The Gaussian process has diverse applications in the fields of computer science, robotics and others. However, itseems not to be applicable to problems in finance because the fluctuations of the financial data follow non-Gaussiandistributions with fat-tails. It is thus necessary to extend the methods of the Gaussian process non-Gaussian stochasticprocesses with fat-tails.For this purpose, the Student’s t -process was proposed as a generalization of the Gaussian process [24]. This stochasticprocess follows the Student’s t -distribution, of which tails show power-law behaviours. As with the Gaussian process,the Student’s t -process is specified by the mean and covariance functions. Given the mean and covariance functions,the probability density function of the Student’s t -process is defined as T ( m, K, ν ) = Γ (cid:0) ν + N (cid:1) [( ν − π ] N Γ (cid:0) ν (cid:1) | K | (cid:20) ν − y − m ) T K − ( y − m ) (cid:21) − ν + N , (9)where Γ( · ) is the multivariate gamma function and the positive real parameter ν is degrees of freedom. In this setting,the stochastic process f ( · ) is the Student’s t -process expressed as f ∼T P ( m, K ; ν ) . Note that the Student’s t -processconverges to the Gaussian process at the limit of ν →∞ .The conditional distribution of the Student’s t -process can be also derived analytically and given as the conditionalStudent’s t -distribution. Namely, we can update the mean and covariance functions and the degrees of freedom fromthe conditional distribution. Through cumbersome calculations, the renewal formulas of the mean and covariance3 PREPRINT - F
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18, 2020functions and the degrees of freedom are derived as follows: m ∗ = m + K X ∗ ,X K − X,X Y, (10) K ∗ = ν − β − ν − N − h K X ∗ ,X ∗ − K X ∗ ,X K − X,X K X,X ∗ i , (11) β = ( Y − m X ) T K − X,X ( Y − m X ) , (12) ν ∗ = ν + N. (13)It is seen that the renewal formula of the covariance function explicitly depends on the number of observed variables,which property does not appear in the case of the Gaussian process. Hence, the Student’s t -process is regarded toutilize prior information more effectively than the Gaussian process. t -process latent variable model To extend the GPLVM to stochastic processes following non-Gaussian distributions, we propose the Student’s- t pro-cess latent variable model (TPLVM). Suppose an observed variable y ∈ R D is explained by a low dimensional latentvariable x ∈ R Q ( Q < D ) by a nonlinear map f : R D → R Q , f ∼T P ( m, K ; ν ) , the TPLVM is introduced as follows: p ( Y | X ) = Γ (cid:0) ν + D (cid:1) [( ν − π ] D Γ (cid:0) ν (cid:1) | K X,X | (cid:20) ν − Y − m X ) T K − X,X ( Y − m X ) (cid:21) − ν + D . (14)The nonlinear dependency of the latent matrix X ∈ R N × Q is given through the covariance matrix. It is expected thatthe TPLVM provides a robust estimation especially for observed data with large fluctuations because the Student’s t -distribution can capture large deviated data from the Gaussian distribution in its sampling.As with the GPLVM, the latent variable and hyperparameters of the TPLVM can be estimated from its likelihood. Thelogarithmic likelihood of the TPLVM is given as log p ( Y | X ) = log Γ (cid:18) ν + D (cid:19) − D ν − π ] − log Γ (cid:16) ν (cid:17) −
12 log | K X,X |− ν + D (cid:20) ν − Y − m X ) T K − X,X ( Y − m X ) (cid:21) , (15)By means of existing optimization methods, we can estimate the latent variables and hyperparameters of the covariancefunction and the degrees of freedom. However, it is known that the optimization of the covariance function with respectto the latent variables often induces numerical instability because of its complexity. Hence, we should carefully selectthe initial values of optimization procedures and repeat with diverse seeds of the initial values to refuse dropping inlocal minima. To overcome the shortcomings of the method of maximum-likelihood, we utilize the method of variational infer-ence [25]. Instead of optimizing the logarithmic likelihood in Eq. (15), we consider that of posterior p ( X | Y ) = p ( Y | X ) p ( X ) /p ( Y ) in the Bayesian sense. In solving the optimization problem with respect to the posterior, wetry to approximate p ( X | Y ) by q ( X ) . As a measure of the difference between two probability density functions, weintroduce the Kullback-Leibler (KL) divergence as follows: KL[ q ( X ) || p ( X | Y )] = Z log q ( X ) p ( X | Y ) q ( X )d X. (16)With the use of the Bayes theorem, the KL divergence is alternatively represented as KL[ q ( X ) || p ( X | Y )] = − Z log p ( Y | X ) p ( X ) q ( X ) q ( X )d X + log p ( Y ) . (17)Since the second term in the right hand side in Eq. (17) does not depend on q ( · ) , we just have to maximize the firstterm in the right hand side, which is known as the evidence lower bound (ELBO), to minimize the KL divergence. TheELBO provides the lower bound of the evidence log p ( Y ) because the KL divergence is non-negative. Therefore, thisprocedure realizes the sufficient fitting of the observed data at the same time. Indeed, the maximization of the ELBOserves the best explanation of the reduced dimension Q of the latent variable.4 PREPRINT - F
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Arbitrage pricing theory [26] assumes that the D -days expected return of an asset r n ∈ R N is explained by the factormodel as r n = α n + F β n + ǫ, (18)where α n ∈ R D is an excess return, β n ∈ R Q is weight coefficients, F ∈ R D × Q is a factor matrix, and ǫ ∈ R D is an errorterm with zero mean and a finite covariance. The factor model manifests that the return of the asset is originated fromthe returns of Q -factors. In fact, without the excess return α n , the expected return of the factor model is derived asfollows: E [ r n ] = E [ F ] β n . (19)The special case of this formula with only one factor is known as the model of the capital asset pricing model, whichis a cornerstone of the modern finance theory [27].The weight coefficients β n in the factor model in Eq. (18) can be interpreted as latent variables which explain thereturn of the asset. Based on this idea, we introduce a nonlinear factor model as r n = f ( β n ) . (20)This model is regarded as a latent variable counterpart of nonlinear factor model [10]. Here, we employ the Student’s t -process as the model of nonlinear mapping f : R Q → R D . In other words, the nonlinear factor model in Eq. (20)is given by the TPLVM. The nonlinear correlation of the latent variable factors depends on the specific form of thecovariance function of the TPLVM, and the predicted return of the asset can be inferred by the predicted distribution.Furthermore, the nonlinear factor model can be interpreted as a dimension reduction model when Q < D . Hence wecan expect to obtain the essential lower dimensional variable which explains the dynamics of the return of the asset.
Markowitz established the modern portfolio theory on the mean-variance portfolio. In this theory, a portfolio consistsof multi assets classes such as stock, bond, currency and commodity with their optimal allocations based on bothindividual and entangled risk of assets.The mean-variance portfolio is designed by the constrained quadratic programming problem with respect to the objec-tive function as w T Kw − λ ( E [ r ] − µ ) , (21)where w ∈ R D is the weight coefficients of the portfolio, K ∈ R D × D is the covariance matrix of the returns, λ is aLagrangian multiplier, r is the return of the portfolio and µ is the expected return of the portfolio. In practical use,the return of the portfolio is quite hard to be estimated, whereby, without the constraint condition of the expectedreturn, the mean-variance portfolio is often replaced by the minimum-variance portfolio with empirically estimatedcovariance matrix. In this section, we test the performance of the minimum-variance portfolio with the TPLVM by comparing with thatwith the GPLVM. Before proceeding, we explain the experimental dataset of our performance test.As the experimental data, we use the following global stock market indices: S&P 500 (US), S&P/TSX 60 (Canada),FTSE 100 (UK), CAC 40 (France), DAX (Germany), IBEX 35 (Spain), FTSE MIB (Italy), AEX (the Netherlands),OMX 30 (Sweden), SMI (Switzerland), Nikkei 225 (Japan), HKHSI (Hong Kong), ASX 200 (Australia), KOSPI(Korea), OBX (Norway), MSCI (Singapore). These stock indices are sampled every month between Jun 1998 to Jun2019 from the Bloomberg’s data platform. The statistics of the return of the stock indices are shown in Table 1. In thistable, mean (Mean), standard deviation (Std.), the ratio of mean and standard deviation (R/R), skewness (Skew) andkurtosis (Kurtosis) of returns of the stock indices are presented.With the use of the historical returns of the stock indices, we construct the minimum-variance portfolios based onthe GPLVM (
Port G ) and TPLVM ( Port t ). The covariance matrix of each portfolio is estimated by the covariancefunction with 120 past samples. As the kernel function, we utilize the exponential kernel defined as k Exp ( x, x ′ ) = θ exp ( − θ − || x − x ′ || ) (22)5 PREPRINT - F
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18, 2020Table 1: Statistics of global market indicesUS Canada UK France Germany Spain Italy NetherlandsMean [%] 6.00 5.41 2.39 4.08 6.87 3.20 1.35 2.96Std. [%] 14.93 14.92 13.62 18.12 21.13 20.66 21.71 19.13R/R 0.40 0.36 0.18 0.23 0.33 0.15 0.06 0.15Skew -0.66 -0.92 -0.55 -0.38 -0.50 -0.17 0.03 -0.74Kurtosis 5.23 7.36 4.53 4.52 6.12 4.96 4.80 5.88Sweden Switzerland Japan HongKong Australia Korea Norway SingaporeMean [%] 6.32 2.80 3.35 7.27 4.70 12.98 10.72 5.05Std. [%] 19.51 14.68 19.24 23.46 12.40 28.80 21.49 21.71R/R 0.32 0.19 0.17 0.31 0.38 0.45 0.50 0.23Skew -0.19 -0.73 -0.54 0.28 -0.69 1.39 -0.93 -0.26Kurtosis 5.29 6.11 4.75 5.78 4.54 11.63 6.84 6.81Table 2: Performance of
Port G and Port t Port G Port t DifferenceAnterior half (Jun 2008 - Jun 2013)Return -4.89% -2.63% 2.25%Risk 19.57% 18.33% -1.24%R/R -0.25 -0.14 0.11Posterior half (Jul 2013 - Jun 2019)Return 6.08% 6.30% 0.22%Risk 11.16% 10.56% -0.60%R/R 0.54 0.60 0.05Whole period (Jun 2008 - Jun 2019)Return 0.64% 1.87% 1.23%Risk 15.92% 14.93% -0.99%R/R 0.04 0.12 0.09with θ l ( l = 1 , being hyper parameters. For the sake of brevity, the dimension of the latent variables are fixed Q = 1 . Under these conditions, we compare the performance of the Port G and Port t by its annualized return(Return), annualized risk as the standard deviation of return (Risk), risk/return (R/R) as return divided by risk. Return = 12 T T X t =1 R Pt (23) Risk = r T − × ( R Pt − µ P ) (24) R / R = Return / RISK (25)Here, R Pt indicates GPLVM or TPLVM portfolio return at time t , and µ P = (1 /T ) P Tt =1 R Pt denotes the averagereturn of the GPLVM or TPLVM portfolio.Table 2 shows the performances of the portfolios by comparing annual return, risk and return-risk ratio. The sampleperiod is separated into anterior half period (Jun 2008 - Jun 2013) and posterior half period (Jul 2013 - Jun 2019).Note that the anterior half period contains the global financial crisis 2007-2008. As is seen in this table, the Port t outperforms the Port G in the both half periods. In particular, the difference of the annual return in the anterior halfperiod is larger than that in the posterior half period. It is said that the market volatility during the global financialcrisis intensively fluctuated whereby non-Gaussian nature clearly emerged in the global stock market. In such situation,the TPLVM is a consistent model to describe the intermittent volatility fluctuations. Thus, we can construct a robustportfolio by the TPLVM based minimum-variance portfolio. In the literature of Bayesian machine learning, the Gaussian process has been developed and utilized to the diverse areaincluding finance. It is. however, well known that the historical financial data follows non-Gaussian distributions. The6
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18, 2020Student’s t -process is proposed, as the generalization of the Gaussian process, to model the observed data followingthe non-Gaussian distributions with fat-tails.In this paper, we proposed the TPLVM by incorporating the latent variables into the Student’s t -process. The TPLVMcan be used to reduce the number of explanation variable following the non-Gaussian distributions with fat-tails. Thenonlinear correlation of the TPLVM is modelled by prescribed kernel functions. The hyperparameters of the TPLVMcan be determined by the method of maximum-likelihood. As a robust parameter optimization, we presented themethod of variational inference of the TPLVM, which utilize the information of prior distribution of latent variables.The problem of the portfolio optimization has been studied in both academia and industry. We applied the TPLVM intothe portfolio optimization with the use of the minimum-variance portfolio. To test the performance of the proposedportfolio, we implemented the empirical analysis for the global stock market data and compared the Port G with Port t . It was shown that the Port t outperforms the Port G in the whole test periods because Port t can capture thenon-Gaussian nature of the global stock market especially in the period of the global financial crisis.The TPLVM can be applied other risk-based portfolios such as risk parity [28], maximum risk diversification [29], andcomplex valued risk diversification [30]. These applications are expected to show high-performance compared withconventional ones. In addition, the TPLVM can be modified to a latent variable dynamical model to catch the natureof historical volatility fluctuations. These ways of research are our future works. References [1] Harry Markowitz. Portfolio selection.
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