asset allocation under threshold autoregressive models

ResearchArticle

Received 7 May 2010, Revised 27 February 2011, Accepted 27 February 2011 Published online in Wiley Online Library

(wileyonlinelibrary.com) DOI: 10.1002/asmb.897

Asset allocation under threshold autoregressivemodelsNaSonga∗†, TakKuenSiub‡,Wa-KiChinga,Howell Tongc §andHailiangYangd ¶

We discuss the asset allocation problem in the important class of parametric non-linear time series models called the thresholdautoregressive model in (J. Roy. Statist. Soc. Ser. A 1977; 140:34–35; Patten Recognition and Signal Processing. Sijthoffand Noordhoff: Netherlands, 1978; and J. Roy. Statist. Soc. Ser. B 1980; 42:245–292). We consider two specific forms, oneself-exciting (i.e. the SETAR model) and the other smooth (i.e. the STAR) model developed by Chan and Tong (J. TimeSer. Anal. 1986; 7:179–190). The problem of maximizing the expected utility of wealth over a planning horizon is consid-ered using a discrete-time dynamic programming approach. This optimization approach is flexible enough to deal with theoptimal asset allocation problem under a general stochastic dynamical system, which includes the SETAR model and theSTAR model as particular cases. Numerical studies are conducted to demonstrate the practical implementation of the proposedmodel. We also investigate the impacts of non-linearity in the SETAR and STAR models on the optimal portfolio strategies.Copyright © 2011 John Wiley & Sons, Ltd.

Keywords: asset allocation; SETAR model; STAR model; non-linearity; conditional heteroscedasticity; dynamical programming;stochastic dynamical system

1. Introduction

The asset allocation problem is of great importance from both theoretical and practical perspectives. The pioneering papersof Merton [1, 2] first establish the optimal portfolio allocation problem in a continuous-time economy and explore then thestate-of-the-art stochastic optimal control techniques to provide an elegant and theoretically sound solution to the optimalasset allocation problem. His work provides practical and plausible solutions to the optimal asset allocation problem. Undercertain assumptions on the asset price dynamics and the forms of the utility function, Merton was able to obtain closed-form solutions to the optimal asset allocation problem in a continuous-time setting. In the case when there are two assets,namely, a risk-free asset and a risky asset, Merton obtains a very simple and intuitive solution to the optimal asset allocationproblem under the assumptions of the lognormality of the returns from the risky asset and the power utility. In this case, theoptimal proportion invested in the risky asset is a constant. This is known as theMerton ratio. Interestingly, with appropriateconfigurations of some specimen values of the model parameters, the Merton ratio turns out to be remarkably close to 60%in stocks, which is a rule of thumb in the strategic asset allocations of final salary pension schemes [3].

Since the works of Merton [1, 2], there have been numerous works that extend the basic apparatus of Merton to variouscontinuous-time stochastic models. Some of these extensions were motivated by empirical analysis of the asset pricedynamics whereas others were motivated by theoretical convenience and interests. Jump-diffusion models, stochastic

aAdvanced Modeling and Applied Computing, Department of Mathematics, The University of Hong Kong, Hong Kong, Hong KongbDepartment of Actuarial Studies and Center of Financial Risk, Faculty of Business and Economics, Macquarie University, Sydney, AustraliacLondon School of Economics, U.K.dDepartment of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong∗Correspondence to: Na Song, Advanced Modeling and Applied Computing, Department of Mathematics, The University of Hong Kong, HongKong, Hong Kong.

†E-mail: [email protected]‡Associate Professor.§Chair Professor of Statistics.¶Professor of Actuarial Science.

Copyright © 2011 John Wiley & Sons, Ltd.

29 April 2011

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Appl. Stochastic Models Bus. Ind. 2012, 28 – 60 72

volatility models and regime-switching models are three important classes of asset price models used in the extensions ofthe Merton model. Interested readers may refer to [4--6] for jump-diffusion models, [7, 8] for stochastic volatility models,and [9, 10] for regime-switching models. It seems that the literature on the optimal asset allocation problemmainly focuseson continuous-time asset price models. There is a relatively small amount of work on the problem in a discrete-time frame-work. Samuelson [11] pioneers the optimal asset allocation problem in a discrete-time setting. His framework is similar toa discrete-time version of the model by Merton [1].

Many empirical studies on financial time series indicate that parametric non-linear time series models provide good fitsto financial returns data. Hsieh [12] performs empirical studies for some important non-linear time series models, includingthe (Generalized) Autoregressive Conditional Heteroscedastic ((G)ARCH) model of Engle [13] (Bollerslev [14] and Taylor[15]) and the Self-Exciting Threshold Autoregressive (SETAR) model first introduced in [16, 17] and later developed in[18--20]. The works of Hsieh [12] provide empirical evidence of the utility of parametric non-linear time series modelsin describing the non-linearity in the observed financial returns data, which is attributed to time-varying expected returnsand expected volatilities. In particular, he adopts the additive non-linear processes, such as the SETAR model, and themultiplicativenon-linear processes, such as theARCH-typemodels, to describe time-varying expected returns and expectedvolatilities, respectively. The monograph by Franses and van Dijk [21] provides a comprehensive discussion on the use ofvarious important parametric non-linear time series models for describing empirical behaviors of financial returns. Tong[22, Chapter 5] mentions that it might be possible that the share price follows an SETAR model where, in particular, theshare price dynamics are governed by one of the two linear dynamics depending on whether the change in the past shareprice is positive or not. He applies SETAR models to fit real financial data, such as IBM daily closing prices and HangSeng Index, and finds that the SETAR models provide promising results for the fitting. Tsay [23] proposes and providesevidence for the use of SETARmodels to describe asymmetric responses in volatility between positive and negative returns.In particular, he applies the SETAR model to describe asymmetric responses in volatility to the sign of the shocks in dailyIBM data. From Rakesh et al. [24], we find that when analyzing stock price over-reactions and under-reactions to events,there is a surprising lack of agreement between daily and weekly return horizons. By examining stock return behavior afterlarge price changes, [24] reconciles the conflict with the help of the SETAR model. Indeed, the SETAR model is not onlycapable of incorporating various characteristic of volatility, but also appears to be the only one which is able to distinguisha persistent shock of the market from an extraordinary shock. For some works on optimal asset allocation under non-lineartime series, see, for example, [25] for GARCH models, and [26, 27] for regime-switching models. However, the optimalasset allocation problems under the SETAR models and the STAR models remain unexplored.

In this paper, we explore the optimal asset allocation problem in a non-linear world, namely that which can bemodeled bythe SETARmodel and the STARmodel. The SETARmodel is different from and has different economic interpretations andimplications than the Markovian regime-switching model considered in [27]. The transitions of the regimes in the formerdepend on the past values of the financial returns, which are self-exciting, whereas those in the latter are driven by externaleconomic states or modes described by a hidden or unobservable Markov chain. The self-exciting feature of the SETARmodel for financial returns has interesting economic interpretations and implications. It might be one way to describe theconcept of reflexibility in financial markets proposed by George Soros, which concerns the feedback effect of the agents’trading behaviors on the asset prices. It provides a piecewise linear approximation to non-linear time series models byintroducing regimes via the threshold principle and has become a standard approach in the non-linear time series analysis.The SETAR model can incorporate both the non-linearity in the conditional mean and the conditional heteroscedasticity,where the latter is often overlooked (see [20, Equation (4.2)], [28, 372–374], and [29]). Through the state-dependency,the SETAR model can also incorporate the effect of volatility clustering, which is an important empirical ‘stylized’ factof financial time series; SETAR model does so discontinuously while the ARCH-type models do so continuously. Wealso consider the asset allocation problem under the Smooth Threshold Autoregressive (STAR) model introduced by Chanand Tong [30]. The STAR model provides a pertinent way to smooth through the regime-switching in the SETAR model.It also strengthens the link between the ARCH-type models and the SETAR model and provides a parsimonious way todescribe the conditional heteroscedasticity. We consider the problem of maximizing the expected utility of wealth overa planning horizon by exploiting a discrete-time dynamic programming approach. This optimization approach is flexibleenough to deal with the optimal asset allocation problem under a general stochastic dynamical system, which includes boththe SETAR model and the STAR model as special cases. Numerical results are provided to illustrate the practical imple-mentation of the model and the impacts of non-linearity in the SETARmodel and the STARmodel on the optimal portfoliostrategies.

This paper is structured as follows. The next section presents the two parametric non-linear time series models consideredin this paper and their economic interpretations and implications. Section 3 presents the asset allocation problem undera general stochastic dynamical system and their solutions. We shall also present the solutions for two parametric casesof the stochastic dynamical system, namely, the SETAR model and the STAR model. Section 4 presents the numericalexperiments. The final section summarizes the main findings of the paper and suggests some possible topics for furtherresearch.


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2. Model dynamics: the SETAR and STARmodels

In this section, we shall describe the two parametric non-linear time series models, namely, the SETARmodel and the STARmodel, for modeling the financial returns, and their economic interpretations and implications. We shall first consider theSETAR model and then the STAR model.

First, we fix a complete probability space (�,F,P), where P is a real-world probability measure. Here, we considera discrete-time financial model with time index set T :={0,1,2, . . .} and with two investment assets, namely, a risk-freebond B and a risky asset S. For each t ∈T, �t represents the information set containing all market information up to andincluding time t . The filtration � :={�t }t∈T on (�,F,P) specifies the information structure of the discrete-time model.Let {�t }t∈T denote a real-valued stochastic process on (�,F), with �0=0. For each t ∈T, �t represents the innovation orthe noise term in the return process from the risky asset S at time t . It is assumed that �t is known given�t and that {�t }t∈Tis a sequence of independent and identically distributed (i.i.d.) and �t ∈N (0,1), for each t ∈T, where N (0,1) is a standardnormal distribution. In general, we assume that for each t ∈T, �t ∈D(0,1), where D(0,1) is any distribution with mean 0and variance 1.

The rationale of using the SETAR model is to provide a piecewise linear approximation and parametric specification ofsome underlying non-linear autoregressive dynamics governing the log returns of the risky asset under consideration. First,let r be the constant continuously compounded risk-free interest rate of the risk-free asset B. For each t ∈T, let Bt and Stdenote the prices of B and S at time t , respectively. Then, we suppose that the price dynamics of B are governed by:

Bt = Bt−1(1+r ), t=1,2, . . . . (1)

Let Yt := ln(St/(St−1)), which represents the log return from S in the period [t−1, t]. Then, we assume that, underP, thedynamics of the log returns {Yt}t∈T from S satisfy the following k-regime SETAR(k; p1, p2, . . . , pk):

Yt =k∑

i=1

(�(i)+

pi∑j=1

�(i)j Yt− j +�i�t

)I{ri−1<Yt−d�ri }, t=1,2, . . . , (2)

where

1. d is the delay parameter, which is a positive integer.2. For each i =1,2, . . . ,k, pi is the autoregressive order in the i th regime of the model.3. The threshold parameters satisfy the constraint −∞=r0<r1< · · ·<rk<∞.4. IA is the indicator function of the event A. It determines in which regime the process of log returns falls.5. �2i is the conditional variance of Yt given �t−1 in the i th regime of the model.6. The index i represents a state of the world or regime of the model.7. The regime of the model at each time t depends on the observable history of the log returns {Yt }t∈T. In particular, the

regime at each time t is determined by the value of Yt−d , hence the term an SETAR model [20].

Under the SETAR model, the dynamics of the log returns Yt are assumed to be governed by a linear autoregressive timeseries model within each regime. The SETAR model is locally linear in each regime and globally non-linear or piecewiselinear. The central tenet of the threshold idea underlying the SETAR model is ‘divide and rule’; in that any complicatednon-linear system can be divided into several simple and easily managed linear subsystems. It provides a pertinent solutiontomodeling time series whenwe leave the linearworld for amore complicated non-linear world, where there is an infinitudeof possible models. The threshold principle behind the SETAR model also provides a systematic way to group and classifya number of important parametric non-linear time series models. This strengthens the link between different parametricnon-linear time models in the literature.

It has been mentioned in [20, Equation (4.2)], [28, pp. 372–374] and [29] that the SETARmodel can incorporate both thenon-linearity in the conditional mean and the conditional heteroscedasticity. Unfortunately, the latter is often overlooked.In fact, it provides a parametric way other than the ARCH-type models to describe the conditional heteroskedasticity.The conditional variance of Yt given �t−1 depends on the past value of the process Y , say Yt−d . For each i =1,2, . . . ,N ,when ri−1<Yt−d�ri , the conditional variance of Yt given �t−1 in the i th regime of the model is �2i . Through expressingthe state-dependency discontinuously in the SETAR model, one can also incorporate the effect of volatility clustering,which is an important empirical ‘stylized’ fact of financial time series and is expressed continuously in the ARCH-typemodels. Besides incorporating conditional heteroscedasticity and volatility clustering, the SETAR model can also capturethe leptokurticity or heavy-tailed nature of the distribution of financial returns. Tong [29] illustrates this point by consideringa simple threshold autoregressive white noise model. Like most of the non-linear time series models, the class of SETARmodels is time-irreversible in general. That is, the probabilistic properties of the model are different when it is investigatedbackward through time. Recall that univariate Gaussian time series models are time-reversible. Chan et al. [31] provided


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an exposition of time-reversibility of multivariate time series models. Taylor [15] adopts the concept of time-reversibilityto distinguish different financial time series.

The SETAR model for financial returns entails different economic interpretations of the Markovian regime-switchingmodel for financial returns, in which the dynamics of financial returns switches over time according to a finite-stateMarkovchain. The concept of regime-switching in the time series analysis has its originality in [16, 19]. Econometric applicationsof discrete-time Markovian regime-switching time series models were detailed by Hamilton [32]. See, for example, [23],for further discussion. In the Markovian regime-switching models, the dynamics of financial returns switches over timeaccording to the states of external economic factors, which might be hidden or unobservable. In the SETAR model, thedynamics of financial returns switches over time according to the past values of the financial returns, which are observableby market participants. The economic interpretations of the latter are that the trading decisions of the market practitionersare affected by the observed past and current prices of the assets and that the aggregate trading decisions and behaviors of themarket practitioners, in turn, affect the future behaviors of the asset prices. This feedback effect of the market practitionerson themarket has a point of contact with the concept of reflexibility in financialmarkets proposed by Soros [33]. The conceptof reflexibility posits that the market practitioners or observers of market prices affect the prices they observe. This is notunlike the basic assumption in quantum physics, which postulates that observers of an experiment affect the observationsor results from the experiment.

The self-exciting feature of the SETAR model is a type of state-dependency. With the property of state-dependency, thedynamics of future prices depend on the current and the past prices, which are regarded as observable state variables. Basedon the economic interpretations described above, market participants react on the observed state variables and their tradingbehaviors cause non-constancy of the statistical properties of the future prices, in particular, the non-constancy or state-dependency of the conditional volatility. This may provide one possible way to explain the conditional heteroscedasticity.

Besides the SETAR model and the ARCH-type model, another possible model to describe the property of state-dependency is the STAR model introduced by Chan and Tong [30]. It provides a device to smooth the regime-switchingin the SETAR model. The switching of regimes are abrupt or discontinuously in the SETAR model. These abrupt changesover the regimes are expressed through the indicator functions in (2). The key idea of the STAR model is to introduce a‘smooth’ function, say F(·) to replace the indicator functions in the SETAR model. The ‘smooth’ function F(·) serves as adevice to smooth the discontinuities in the SETAR model. The choice of F(·) is very flexible. We only require F(·) to becontinuous and non-decreasing. A typical choice of F(·) is a cumulative distribution function, such as the standard normaldistribution and the logistic distribution. When F(·) is chosen to be the logistic distribution, F(·) is related to neurons inneural network models. One can also choose F(·) to be different link functions in the Generalized Linear Models (GLIMs)rather than the standard normal distribution and the logistic distribution. To illustrate the key idea of the STAR model, wefirst set d=1 and k=2; in other words, there are two regimes in the model for financial returns. Then we consider thefollowing SETAR(2; p, p) model for financial returns:

Yt =(

�(1)+p∑

j=1�(1)j Yt− j +�1�t

)(1− Ir1 (Yt−1))+

(�(2)+

p∑j=1

�(2)j Yt− j +�2�t

)Ir1 (Yt−1), (3)

where Ir1 (y) is an indicator function with value 1 (0) when y>r1(y�r1).Suppose the first (second) regime represents a ‘Bad’ (‘Good’) economy. If the return from the risky asset in the (t−1)th

period is high enough (i.e. above the threshold level r1), the evolution of the return from the risky asset in the next periodis governed by the regime corresponding to the ‘Good’ economy. If the return from the risky asset in the (t−1)th periodis low (i.e. below the threshold level r1), the evolution of the return from the risky asset in the next period is governed bythe regime corresponding to the ‘Bad’ economy. The switching from the ‘Bad’ economy regime to the ‘Good’ one, or viceversa, occurs abruptly or suddenly and is determined by the past value of the return process itself. At each time period, theevolution of the return is governed by only one economic regime, but not both.

Then, the STAR version of the SETAR(2; p, p) model for financial returns is given by

Yt =(

�(1)+p∑

j=1�(1)j Yt− j +�1�t

)(1−F

(Yt−1−r1

�

))+(

�(2)+p∑

j=1�(2)j Yt− j +�2�t

)F

(Yt−1−r1

�

), (4)

where �>0 is a scale parameter and the magnitude of � tells us how abrupt the changes over regimes are. For example, as�→0, F((Yt−1−r1)/�)→ Ir1 (Yt−1). That is, the STAR model coincides with the SETAR model. For more discussion onthe convergence of the STARmodel to the SETARmodel, see for instance [28, Section 4.2.13, pp. 183–186]. This problemis associated with finding stationary distributions of time series models.

We also suppose that the first (second) regime represents a ‘Bad’ (‘Good’) economy. In the STAR model, the switchingfrom the ‘Bad’ economy to the ‘Good’ one, or vice versa, occurs smoothly or gradually and is governed by the continuousfunction F(·) and the past value of the return process itself. At each time period, the evolution of the return is governed by a


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convex combination of the two economic regimes. The proportion of the evolution of the return attributed to each of the twoeconomic regimes depends on the smooth function F(·) and the past value of the return process itself. In particular, if thereturn Yt−1 is higher (lower), F((Yt−1−r1)/�) increases (decreases) and the weight given to the ‘Good’ economic regimeis larger (smaller). It is interesting to note that the STAR model resembles a self-exciting version of the mixture of twolinear AR models introduced byWong and Li [34]. The mixing distribution in the STARmodel is self-exciting in the sensethat it depends on the past value of the return process itself. Whereas, the mixing distributions in the mixture AR models in[34] seem to be governed by some external auxiliary mixing variables. The relationship between the STAR model and themixture AR model of [34] is not unlike that between the SETAR model and the Markovian regime-switching model.

In the SETAR model and STAR model described above, when the return in the (t−1)th period is higher, the ‘Good’economic regime plays a dominant role in describing the evolution of the return in the next period. This corresponds tothe price momentum phenomenon. The SETAR model and the STAR model can also be used to describe price reversalphenomenon. This can be achieved by interchanging the positions of 1− Ir1 (Yt−1) and Ir1 (Yt−1) in the case of the SETARmodel and interchanging the positions of 1−F(·) and F(·) in the case of the STAR model.

Let

G(Yt−1,Yt−2, . . . ,Yt−p)=(

�(1)+p∑

j=1�(1)j Yt− j

)(1−F

(Yt−1−r1

�

))+(

�(2)+p∑

j=1�(2)j Yt− j

)F

(Yt−1−r1

�

), (5)

and

H (Yt−1)=�1

(1−F

(Yt−1−r1

�

))+�2F

(Yt−1−r1

�

). (6)

Then, the STAR model can be written in the following form:

Yt =G(Yt−1,Yt−2, . . . ,Yt−p)+H (Yt−1)�t . (7)

Here, the state-dependency of the conditional mean and the conditional volatility are described by G and H , respectively.The STAR model can be viewed as a stochastic dynamical system with skeleton G and stochastic noise H�t . It relates tothe stochastic chaos or noisy chaos described in [35].

By considering the stochastic noise part of the STAR model,

�t H (Yt−1)=�t

[�1

(1−F

(Yt−1−r1

�

))+�2F

(Yt−1−r1

�

)], (8)

we can see the connection between the ARCH model and the STAR model.Note that under a zero-mean ARCH(1) model for financial returns,

Yt =�t

√�+�Y 2

t−1. (9)

The connection between the STARmodel and the ARCHmodel can be seen by comparing H (Yt−1) to√

�+�Y 2t−1, both of

which are functions of the state variable Yt−1. This connection is also highlighted in [29]. Given the link between the STARmodel and the SETAR model, the STAR model can also strengthen the link between the ARCH model and the SETARmodel (see [29] for detail).

The STAR model can, by sharing the same threshold, provide a more parsimonious way to incorporate both the state-dependency of the conditional mean and the conditional volatility.

3. The asset allocation problems and their solutions

In this section, we shall first present a general framework for the optimal asset allocation problem of an investor whowishesto maximize the expected utility of his/her wealth over a finite planning horizon.We shall describe a discrete-time dynamicprogramming approach,which is flexible enough to dealwith the optimal asset allocation problemunder a general stochasticdynamical system. The stochastic dynamical system is general enough to include the linear AR model, the SETAR model,the STARmodel, linear and non-linear ARCH-type models, the second-generationmodels introduced in [28, 36], and otherimportant non-linear time series models. We shall develop the solution to the optimal asset allocation problem under thisgeneral setup. This solution is applicable to all linear and non-linear time series models nested in the stochastic dynamicalsystem. Then, we consider the asset allocation problems under the two particular cases of the stochastic dynamical system,namely, the SETAR model and STAR model, and outline their solutions.


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We consider an investorwhowishes to allocate his/her wealth rationally among two primary assets, namely, the risk-freeasset B and the risky asset S. The price process of the bond B is given by (1). Let F1 :�p →� and F2 :�q →� denote tworeal-valued functions, where p and q are positive integers. The dynamics of the log returns {Yt}t∈T are governed by thefollowing general stochastic dynamical system:

Yt = F1(Yt−1,Yt−2, . . . ,Yt−p)+F2(Yt−1,Yt−2, . . . ,Yt−q)�t . (10)

Numerous linear and non-linear time series models can be written in the form of the general stochastic dynamical systemby suitable choices of the functions F1 and F2. [28, Chapter 4], [35, Chapter 3] and [29] provide detailed discussion on theprobabilistic structure and properties of the above general stochastic dynamical system and its connection to the concept ofstochastic chaos.

The objective of the investor is to maximize the expected utility of his/her wealth over a finite-time horizon [0,T ].We suppose that the investor does not consume his/her wealth in the planning horizon [0,T ]. Here, we represent the riskpreference of the investor via constant relative risk aversion (CRRA) utility function with the following form:

U (W )= W �

�, ��1, � �=0, (11)

or

U (W )= ln(W ), �=0. (12)

Here, W is the wealth of the investor. (11) is the limiting case of (10) as �→0. This is not unlike the limiting case of theBox–Cox transformation in statistics. The power utility has a constant CRRA of 1−�, where � represents an index of riskpreference. An investor can choose a value of � subjectively according to his/her risk preference. The level of risk aversiondecreases as � increases. When �=1, the level of risk aversion is the lowest. In general, there is no restriction that ��0.When � takes negative values, the level of risk aversion becomes higher.

Let denote an impatient factor or a discount factor, where >0.We suppose that the investor makes his/her investmentdecision at the beginning of each time period. Let t0 :=max(p,q)−1, where p,q�1. Then, at each time point t= t0, t0+1, . . . ,T −1, the investor decides the proportiont of his/herwealth to be invested in the risky asset S. Then, 1−t representsthe proportion of his/her wealth to be invested in the risk-free asset B.Wt represents the total wealth of the investor at timet . In the asset allocation problem, the objective of the investor is to choose t to maximize

J (t,Wt ,t ) := E

[t+1∑i=1

(1+)−iU (Wi )|Ft

], t= t0, t0+1, . . . ,T −1, (13)

whereFt represents the information generated by the market prices up to and including time t .In other words, the investor decides t so as to maximize the expected discounted utility of his/her wealth over the

planning horizon, for each t= t0, t0+1, . . . ,T −1.Let Rt := St/(St−1), for each t= t0+1, t0+2, . . . ,T . Then, the evolution of the wealth process of the investor is governed

by the following stochastic difference equation:

Wt+1=Wt [(1−t )(1+r )+t Rt+1]. (14)

Then, we can state the asset allocation problem of the investor more precisely as follows:

max{t }

J (t,Wt ,t ) (15)

subject to the constraint:

Wt+1=Wt [(1−t )(1+r )+t Rt+1], (16)

with a given initial wealth Wt0 =w. This is a recursive asset allocation problem, in which the investor updates his assetallocation decision when new information comes. Initially, the investor decides the proportion t0 of his/her wealth Wt0invested in the risky asset and invests the rest of his/her wealth in the risk-free asset. At time t0+1, the value of the returnfrom the risky asset Rt0+1 is realized and Wt0+1 is known exactly; the investor then uses this piece of information to makehis/her asset allocation decision t0+1 at time t0+1, and so on.


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We shall derive a forward recursion formula for the solution of the optimal asset allocation problem. First, we considerthe following set of functions:

J (t,Wt ,t ) := E

[t+1∑i=1

(1+)−iU (Wi )|Ft

], t= t0, t0+1, . . . ,T −1. (17)

At time t= t0, we have

J (t0,Wt0 ,t0 )=t0∑i=1

(1+)−iU (Wi )+E[(1+)−t0−1U (Wt0+1)|Ft0 ]. (18)

Then,

J (t0,Wt0,t0 )=t0∑i=1

(1+)iW �i

�+ (1+)−t0−1W �

t0

�E{[(1−t0 )(1+r )+t0 Rt0+1]

�|Ft0}. (19)

Now, our goal is to find t0 so as to maximize J (t0,Wt0 ,t0 ). That is, we consider the maximization of the next period’sexpected utility given the current and past information. This is a single-period optimization problem. DifferentiatingJ (t0,Wt0 ,t0 ) with respect to t0 and setting the derivative equal to zero, we get the following first-order condition for theoptimal asset allocation problem at time t0:

E{[(1−t0 )(1+r )+t0Rt0+1]�−1[Rt0+1−(1+r )]|Ft0}=0, (20)

from which we can solve for the optimal asset allocation t0 at time t0.For other time periods, say t= t0+1, t0+2, . . . ,T −1, we determine the optimal asset allocation strategies (t0+1, . . . ,

T−1) by solving the same single-period optimization problem with updated value of the controlled state process fromthe non-linear time series models. These optimal asset allocation strategies are determined by the same first-order condi-tion for the optimal asset allocation problem at time t0, except that the updated value of the controlled state process isused. That is, the optimal asset allocation decisions (t0+1, . . . , T−1) can be obtained by solving the following recursiveformula:

E{[(1−t )(1+r )+t Rt+1]�−1[Rt+1−(1+r )]|Ft }=0, t= t0+1, . . . ,T −1. (21)

We have assumed that the dynamics of the log returns {Yt}t∈T are governed by the following general stochastic dynamicalsystem:

Yt = F1(Yt−1,Yt−2, . . . ,Yt−p)+F2(Yt−1,Yt−2, . . . ,Yt−q)�t , (22)

where F1 :�p →� and F2 :�q →� denote two real-valued functions, the constants p and q are positive integers. Then,(t0, t0+1, . . . , T−1) satisfies the following recursive integral equation:

∫�

[(1−t )(1+r )+t exp

(F1(Yt ,Yt−1, . . . ,Yt+1−p)+F2(Yt ,Yt−1, . . . ,Yt+1−q)y

)]�−1

[exp

(F1(Yt ,Yt−1, . . . ,Yt+1−p)+F2(Yt ,Yt−1, . . . ,Yt+1−q)y

)−(1+r )]�(y)dy=0,

t= t0+1, . . . ,T −1, (23)

where �(·) denotes the probability density function of a standard normal distribution.Note that the above recursive integral equation for the optimal asset allocation decisions under the general stochastic

dynamical system is highly non-linear due to the multiplicity of the non-linearity induced by the combined effect of thenon-linearity of the power utility function and the non-linear multivariate functions Fi (·), i =1,2. These recursive integralequation can be simplified, or perhaps, partially linearized if we consider an SETAR model for the financial returns ratherthan the general stochastic dynamical system.

Now, we present the solution to the optimal asset allocation problem under the SETAR model described in (2)with the delay parameter d=1 and pi = p, for all i =1,2, . . . ,k. In this case, the optimal asset allocation decisions


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N. SONG ET AL.

(p−1, p, . . . , T−1) can be obtained by solving the following recursive integral equation:

k∑i=1

⎧⎨⎩∫

�

[(1−t )(1+r )+t exp

(�(i)+

p∑j=1

�(i)j Yt+1− j +�i y

)]�−1

[exp

(�(i)+

p∑j=1

�(i)j Yt+1− j +�i y

)−(1+r )

]�(y)dy

}I{ri−1�Yt�ri } =0, (24)

t= p−1, . . . ,T −1.The recursive integral equationunder theSETARmodel seems tobe less complicated than that under thegeneral stochastic

dynamical system. The non-linear functions Fi (·)(i =1,2) in the recursive integral equation are piecewise-linearized if theSETAR model for financial returns is used. In other words, the optimal asset allocation problem can be simplified by thepiecewise linear approximation provided by the SETAR model to a general stochastic dynamical system or a non-linearautoregressive model for financial returns.

Then, we present the solution to the optimal asset allocation problem under the STAR model corresponding to theSETAR(2, p, p) model described in (4). In this case, the optimal asset allocation decisions (p−1, p, . . . , T−1) can beobtained by solving the following recursive integral equation:

∫�

{(1−t )(1+r )+t exp

{(�(1)+

p∑j=1

�(1)j Yt+1− j

)(1−F

(Yt −r1

�

))

+(

�(2)+p∑

j=1�(2)j Yt+1− j

)F

(Yt −r1

�

)+[�1

(1−F

(Yt −r1

�

))+�2F

(Yt −r1

�

)]y

}}�−1

×{exp

{(�(1)+

p∑j=1

�(1)j Yt+1− j

)(1−F

(Yt −r1

�

))+(

�(2)+p∑

j=1�(2)j Yt+1− j

)F

(Yt −r1

�

)

+[�1

(1−F

(Yt −r1

�

))+�2F

(Yt −r1

�

)]y

}−(1+r )

}�(y)dy=0, (25)

where t= p−1, p, . . . ,T −1.The recursive integral equation under the STARmodel seems to be more complex than that under the SETARmodel with

k=2.Hence, with the same autoregressive orders, the SETARmodel seems to give the simplest solution to the optimal assetallocation problem compared with the STAR model and the general stochastic dynamical system.

4. Numerical experiments

In this section, we conduct numerical experiments to illustrate the practical implementation of the proposed models andthe impacts of non-linearity in the SETAR model and the STAR model on the optimal asset allocation strategies. First,we note that the optimal asset allocation decisions over time are themselves (non-linear) time series. They are endogenoustime series, which are derived from the optimal asset allocation problem under the exogenous (non-linear) time series forfinancial returns. Through the numerical experiments, we shall see how the different forms of non-linearity in the SETARmodel and the STAR model are inherent from the exogenous time series for financial returns and the endogenous timeseries for temporal optimal asset allocation decisions. For illustration, we shall consider a two-regime first-order SETARmodel and two first-order STAR models, one with the standard normal distribution as a smooth function and another withthe logistic distribution as a smooth function. We shall compare the temporal behaviors of the optimal portfolio strategiesobtained from the SETAR model and the two STAR models to that arising from a first-order linear AR model, which actsas a benchmark for comparison.We shall highlight some important features of and economic implications for the temporalbehaviors of the optimal portfolio strategies arising from the non-linearity and conditional heteroscedasticity described bythe SETAR and STAR models. All computations in this section were done by MATLAB codes.

With fixed t and Wt (t =1,2, . . . ,T ), the function J (t,Wt ,t ) is a differentiable, concave, function of t defined on theinterval [0, 1]. We employ Newton’s method to solve numerically the optimal allocation problem in the interval [0,1]. Weset T =100 and consider the first-order AR model, the first-order SETAR model and the corresponding STAR models.This process is repeated until a sufficiently accurate value is attained. The proportion t takes a value between 0 and 1.


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N. SONG ET AL.

0 20 40 60 80 100

0

0.01

0.02

0.03

0.04

0.05

t

A simulated sample path for Model I

0 20 40 60 80 100

0

0.01

0.02

0.03

0.04

0.05

t

A simulated sample path for Model II

0 20 40 60 80 100

0

0.01

0.02

0.03

0.04

0.05

t

A simulated sample path for Model III

0 20 40 60 80 100

0

0.01

0.02

0.03

0.04

0.05

t

A simulated sample path for Model IV

Figure 1. Simulated sample path for Model I (top left), Model II (top right), Model III (bottom left) and Model IV (bottom right).

Consequently, if the approximation of the solution obtained by Newton’s method is greater than 1, we record ‘1’ as theoptimal allocation; also we record ‘0’ if the approximation solution is less than 0.

First, the first-order SETAR model for the log returns {Yt}t∈T from the risky asset S considered here is presented asfollows:

Yt =(�(1)+�(1)Yt−1+�1�t

)(1− Ir1 (Yt−1))+

(�(2)+�(2)Yt−1+�2�t

)Ir1 (Yt−1), t=1,2, . . . , (26)

where r1 is the threshold parameter and {�t } is a sequence of i.i.d. random variables having a standard normal distribution.That is, �t ∈N (0,1), for each t ∈T. The time index t represents the t th trading day.

The corresponding STAR model for the log returns {Yt }t∈T from the risky asset S is given by

Yt =(�(1)+�(1)Yt−1+�1�t

)(1−F

(Yt−1−r1

�

))+(�(2)+�(2)Yt−1+�2�t

)F

(Yt−1−r1

�

), t=1,2, . . . , (27)

where � is the scale parameter as in the one defined in Section 2 and F(·) can be a standard normal distribution or a logisticdistribution.


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N. SONG ET AL.

0 20 40 60 80 100 1200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 5.5:Optimal Portfolio Strategiesarising from Model I

t

The

pro

port

ion

of th

e w

ealth

inve

sted

in th

e ris

ky a

sset

S

The

pro

port

ion

of th

e w

ealth

inve

sted

in th

e ris

ky a

sset

S

The

pro

port

ion

of th

e w

ealth

inve

sted

in th

e ris

ky a

sset

S

The

pro

port

ion

of th

e w

ealth

inve

sted

in th

e ris

ky a

sset

S

0 20 40 60 80 100 1200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 5.6:Optimal Portfolio Strategiesarising from Model II

t

0 20 40 60 80 100 1200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 5.7:Optimal Portfolio Strategiesarising from Model III

t

0 20 40 60 80 100 1200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 5.8:Optimal Portfolio Strategiesarising from Model IV

t

Figure 2. Optimal portfolio strategies for Model I (top left), Model II (top right), Model III (bottom left) and Model IV (bottom right).

The SETAR and STARmodels presented above have two regimes. In the first regime, the parameters are �(1), �(1) and �1.Whereas, in the second regime, the parameters are �(2), �(2) and �2. As before, the first regime represents a ‘Bad’ economywhereas the second regime represents a ‘Good’ economy.

The benchmark first-order linear AR model is given by:

Yt =�(2)+�(2)Yt−1+�2�t , t=1,2, . . . . (28)

Hence, the first-order linear AR model corresponds to the ‘Good’ economic regime of the SETAR and STAR models. It isnested in both the SETAR and STARmodels.When r1→−∞, the SETARmodel becomes the first-order linear ARmodel.When F(y)=1, for all y∈�, the STAR model reduces to the first-order linear AR model.

We shall consider some specimen values of the model parameters and assume that the risk-free interest rate r =0.0002;�(1)=0.0004, �(2)=0.0012, �(1)=0.02, �(2)=0.006, �(1)=0.1, �(2)=0.3, r1=0 and �=0.1 and �=0.5.

Figure 1 depicts a simulated sample path for each of the first-order linear AR model (Model I), (see Equation (28)), thefirst-order SETARmodel (Model II), (26), the corresponding STARmodel with a standard normal smooth function (ModelIII), (27), and the STAR model with a logistic smooth function (Model IV), (27).

From Figure 1, we see that the simulated returns from linear AR model (Model I) are less volatile than those from thenon-linear time series models (Models II, III and IV). The non-linear time series models give more extreme returns than thelinear AR model. Among the non-linear time series models, the SETAR model, (Model II), seems to give the most volatile


69


N. SONG ET AL.

0 10 20 30 40 500.44

0.46

0.48

0.5

0.52

0.54

0.56

0.58

0.6

The number of the simulated data sets

The

pro

port

ion

of th

e w

ealth

inve

sted

in th

e ris

ky a

sset

S

The averaged results of the optimalallocation for model I

0 10 20 30 40 500.44

0.46

0.48

0.5

0.52

0.54

0.56

0.58

0.6

The number of the simulated data setsT

he p

ropo

rtio

n of

the

wea

lth in

vest

ed in

the

risky

ass

et S

The averaged results of the optimalallocation for model II

0 10 20 30 40 500.44

0.46

0.48

0.5

0.52

0.54

0.56

0.58

0.6


The

pro

port

ion

of th

e w

ealth

inve

sted

in th

e ris

ky a

sset

S

The averaged results of the optimalallocation for model III

0 10 20 30 40 500.44

0.46

0.48

0.5

0.52

0.54

0.56

0.58

0.6


The

pro

port

ion

of th

e w

ealth

inve

sted

in th

e ris

ky a

sset

S

The averaged results of the optimalallocation for model IV

Figure 3. The averaged results of the optimal allocation based on 50 sets of simulated data for Model I (top left), Model II (top right),Model III (bottom left) and Model IV (bottom right).

and extreme simulated returns. This may be explained by the fact that the structural changes in the model dynamics in theSETAR model are abrupt whereas those in the STAR models (Models III and IV) are gradual.

Figure 2 depicts plots of the optimal portfolio strategies arising fromModels I–IV.These optimal strategies are the optimalproportions invested in the risky asset over time.

From Figure 2, we see that the non-linear features of the time series models for returns are inherent to the time series ofoptimal portfolio strategies. The endogenous time series of optimal strategies arising from the linear AR model (Model I)is the least volatile and extreme one. Again among the non-linear time series models, the endogenous time series of optimalportfolio strategies from the SETARmodel (Model II) is themost volatile and extremeone. These happen since the economicagent reacts rationally to the variations of returns from different non-linear time series models.

From the numerical results, we see that the optimal asset allocation strategies are quite sensitive to the choice of aparametric form of a time series model for financial returns (Figure 3). An important problem about how to select the


70


N. SONG ET AL.

parametric form of the time seriesmodel arises. Here, we discuss this problem by focusing on the choice between the SETARmodel and the STAR model. The SETAR model can describe abrupt structural changes in model dynamics of financialreturns. These structural changes may be attributed to changes in economic conditions, market catastrophes (e.g. financialcrises), and changes in political regimes. In the SETAR model for financial returns, it is assumed that these changes arereflected in the past values of the returns,which, in turn, determine the dynamicsof future returns.Consequently, if the goal ofa fund manager is to develop a prudent or conservative asset allocation policy, which takes into account the adverse effect ofthemarket and economic catastrophes on financial returns, themanagermay consider the SETARmodel for financial returnsin developing the asset allocation policy. The SETARmodel also has potential applications in modeling hedge fund returnswith a view to developing asset allocation policies with investment opportunities including hedge funds [37]. The STARmodel seems a half-way house between the SETARmodel and the ARCH-type models. It incorporates state-dependency inboth the conditionalmean and conditional volatility. If one focuses onmodeling the impact of conditional heteroscedasticityon asset allocation, one can use either the STAR model or the ARCH-type model. If one wishes to incorporate the non-linearity in the conditional mean as well, the STAR model seems more appropriate than the ARCH-type models. Differentfrom the SETAR model, changes in the model regimes in the STAR model are gradual. The changes may be attributed tovariation in the level of market activities, such as the variation of the buy–sell forces and emergence of market news orevents which have marginal, or modest, effects on financial returns.

5. Concluding remarks

We considered the asset allocation problem under two important classes of non-linear time series models, the SETARmodel and the STARmodel. The former model provides a piecewise linear approximation to non-linear time series modelsby introducing regimes via the threshold principle whereas the latter one provides a pertinent way to smooth through theregime-switching in the SETARmodel. The STARmodel can also highlight the connection between theARCH-typemodelsand the SETAR model. The problem of maximizing the expected utility of wealth over a planning horizon was consideredby exploiting the discrete-time dynamic programming approach. The optimization approach was flexible and general. Wederived the recursive integral equations for the optimal asset allocation decisions under a general stochastic dynamicalsystem and its particular cases, the SETAR model and the STAR model, using the optimization approach. The recursiveintegral equation is the simplest in the case of the SETAR model due to its piecewise linearity.

Acknowledgements

This research is supported in part by RGC Grants 7017/07P, HKU Strategic Research Theme Fund on Computational Sciences. Theauthors thank the associate editor and the two anonymous reviewers for their helpful comments and suggestions.

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asset allocation under threshold autoregressive models

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