Risk, Return and Market Condition: From a Three-Beta to a Functional-Beta Capital Asset Pricing Model

In this chapter, we are concerned with how a time-varying beta is linked to market condition in capital asset pricing model (CAPM). This is a question that is related to the recent interest in these large and unexpected swings in asset values, revived after the publication of Taleb’s (2007) book, “The Black Swan: The Impact of the Highly Improbable", to explore the merits of beta in the presence of large market fluctuations (c.f., Estrada and Vargas 2011).


Introduction
In this chapter, we are concerned with how a time-varying beta is linked to market condition in capital asset pricing model (CAPM). This is a question that is related to the recent interest in these large and unexpected swings in asset values, revived after the publication of Taleb's (2007) book, "The Black Swan: The Impact of the Highly Improbable", to explore the merits of beta in the presence of large market fluctuations (c.f., Estrada and Vargas 2011).
It is well known that capital asset pricing model due to (Sharpe 1964) and (Lintner 1965) conveys important information that individual securities are priced so that their expected return will compensate investors for their expected risk. Symbolically, CAPM can be expressed in a general non-expected form as 1984). Considerable empirical evidences have suggested that beta stability assumption is invalid. The literature is abundant, see, for example, (Kim 1993), Ferson 1992, 1995), (Wells 1994), (Bos, Ferson, Martikainen and Perttunen 1995), (Brooks, Faff and Lee 1992) and (Cheng 1997).
The time-varying beta models have also been investigate by using Australian stock market data sets. Brooks, Faff and Lee (1992), and (Faff, Lee and Fry 1992) were among the first to investigate the time-varying beta models. Faff, Lee and Fry (1992) employed a locally best invariant test to study the hypothesis of stationary beta, with evident finding of nonstarionarity across all of their analysis. The random coefficient model was further suggested by (Brooks, Faff and Lee 1994) as the preferred model to best describe the systematic risk of both individual shares and portfolios. However, (Pope and Warrington 1996) reported that random coefficient model was appropriate only for a bit more than 10% companies in their studies. Faff, Lee and Fry (1992) investigated the links between beta's nonstationarity and the three firm characteristics: riskiness, size and industrial sector, without finding the strong pattern between firm size or industry sector and nonstationarity. Faff and Brooks (1998) modelled industrial betas by different regimes based on market returns and volatility of the risk-free interest rate, their univariate and multivariate tests providing mixed evidence concerning the applicability of a time-varying beta model which incorporates these variables. Groenewold and Fraser (1999) argued that the industrial sectors could be divided into two groups: one of them has volatile and non-stationary betas and the other group has relatively constant and generally stationary beta. Other recent studies include (Gangemi, Brooks and Faff 2001), (Josev, Brooks and Faff 2001), and others. An interesting study recently made by (Yao and Gao 2004) investigated the problem of choosing a best possible time-varying beta for each individual industrial index using the state-space framework, including the random walk models, random coefficient models and mean reverting models, which were examined in detail by using the Kalman filter approach.
When testing the validity of asset pricing models, many studies account for market movements, defined as up and down markets. For example, (Kim and Zumwalt 1979) used the average monthly market return, the average risk-free rate and zero as three threshold levels; when the realized market return is above (below) the threshold level the market is said to be in the up (down) market state. Crombez and Vennet (2000) conducted an extensive investigation into the risk-return relationship in the tails of the market return distribution; they defined up and down markets with two thresholds: zero and the risk-free rate. Further, to define three regimes for market movements, that is substantially upward moving, neutral and substantial bear, different threshold points were used, such as: the average positive (negative) market return, the average positive (negative) market return plus (less) half the standard deviation of positive (negative) market returns, and the average positive (negative) market return plus (less) three-quarters of the standard deviation of positive (negative) market returns. The conditional beta risk-return relation has been found to be stronger if the classification of up and down markets is more pronounced. Galagedera and Faff (2005) has recently argued as in the finance literature and media that high volatility leads to high returns. High volatility in equity prices in many situations has been related to negative shocks to the real economy. On one hand, the volatility of macro-economic variables may partially explain the equity market price variation. On the other hand, the volatility in equity market prices may also be entrenched more in financial market disturbances. In particular, when the market volatility becomes extreme, it could have 392 Risk Management for the Future -Theory and Cases www.intechopen.com Risk, Return and Market Condition: From a Three-Beta to a Functional-Beta Capital Asset Pricing Model 3 an impact on financial markets. Some securities are more susceptible to market volatility than others. Two interesting questions that arise in this setting were posed by (Galagedera and Faff 2005): (i) Does the beta risk-return relationship depend on the various market volatility regimes? (ii) Are the betas corresponding to these volatility regimes priced? There have been empirical evidences raising concern about the ability of a single beta to explain cross-sectional variation of security and portfolio returns. Security or portfolio systematic risk is known to vary considerably over time, as documented in the literature in the above. It is further well known that the volatility of financial time series, particularly in high frequency data, changes over time.
In their pioneering work of three-beta CAPM, (Galagedera and Faff 2005) made an assumption that the market conditions can play an important part in explaining a changing beta and could be divided into three states specified as "low", "neutral" or "high" market volatility. First, they fit a volatility model for daily market returns and obtain the estimates for conditional variance. Then, based on the magnitude of these estimates, (Galagedera and Faff 2005) classify daily market volatility σ 2 Mt into one of three market volatility regimes, using appropriately defined indicator functions: Here σ 2 L and σ 2 H are constants; I Lt represents the low market condition, I Nt represents the neutral market condition, I Ht represents high market condition. By investigating empirically on the single factor CAPM R it = α i + β i R mt + ε it , to estimate the betas in the low, neutral and high volatility markets, Galagedera and Faff extended the market model given in (1), in the form: ( 5 ) where β iL , β iN , β iH are three constants defined as the systematic risks corresponding to the low, neutral and high market volatility regimes, respectively. This model is a richer specification than the traditional single factor CAPM. It is a three-state regime-switching model with the percentiles of market volatility used as threshold parameters. This model raises an interesting question: How is the beta risk linked to the market condition measured by the market volatility? Our main objective in this chapter is to investigate whether and how the securities' responses to the market vary depending on the changing market volatility.
We are making a careful investigation into this question in the following sections. In Section 2.1, we shall extend the three-beta model (5) to a more general functional-beta CAPM framework. Nonparametric estimation of the beta functional will be introduced in Section 2.2. We will use the data sets from the Australian stock market to empirically examine the evidences of functional-beta structure in CAPM. An introduction into the data will be provided in Section 3, where an estimation of the unobserved market volatility will be established. Section 4 will carefully examine the linkage of the beta to the market volatility by using both nonparametric and parametric approaches. Based on the findings from the 393 Risk, Return and Market Condition: From a Three-Beta to a Functional-Beta Capital Asset Pricing Model www.intechopen.com nonparametric estimation, we can suggest reasonable regime switching thresholds, by which a regime-switching threshold CAPM will be proposed and investigated. We will conclude in Section 5.

Methodology: From a Three-Beta CAPM to a Functional-Beta CAPM
In this section, we will first propose a Functional-Beta CAPM that is a generalization of the Three-Beta CAPM suggested by (Galagedera and Faff 2005) in Subsection 2.1, and then introduce a nonparametric method to estimate the unknown functional beta in the Functional-Beta CAPM in Subsection 2.2.

Model
Following the idea of (Galagedera and Faff 2005), we consider a new more general structural framework to incorporate market movements into asset pricing models by including the changes in the conditional market volatility. We achieve this by noting that the model (5) can be expressed as which is a time-varying beta model, with We note that the volatility of market returns is partitioned into three regimes in (2)-(4), which are the functions of the size of the conditional market volatility, say, σ 2 Mt . Therefore β it is a simple functional of the market volatility σ 2 Mt ,thatis So the three-beta CAPM proposed by (Galagedera and Faff 2005) is a simple functional beta model.
In this chapter, we will extend the model (5) that was suggested by (Galagedera and Faff 2005) and propose a general functional-beta model as follows: where as before, R it is the return of financial asset i at time t, R Mt is the market return at time t, σ 2 Mt is the market volatility at time t, α i is the conditional expected specific return, ε it is random specific return, and β i is the coefficient of the contribution due to the market factor, changing with the market volatility. Here β i (·) is particularly important, which is the systematic risk functional, in capital asset pricing modelling. We may also treat α i as varying with σ 2 Mt although its value is usually rather small to be often assumed as a constant. We call (9) the functional-beta CAPM. For our objective, we need to estimate the unknown α i and β i (·) in (9).

Estimation of functional-beta CAPM: nonparametric method
Given the historical observations (R it , R Mt ), t = 1, 2, ··· , T, we are concerned with how to estimate the unknown functional beta. First of all, we need some way to estimate the unobservable market volatility σ 2 Mt . Using the market returns R Mt , t = 1, 2, ··· , T,w ec a n try to estimate σ 2 Mt in various ways. A simple way is to apply the econometric models of ARCH of (Engle 1982) or GARCH of (Bollersleve 1986), as done in (Galagedera and Faff 2005). More involved stochastic volatility models can also be applied (c.f., Gao, 2007, page 169). Alternatively, we can use realized market volatility as an estimate of σ 2 Mt ; see (Allen et al. 2008) for a comprehensive review on realized volatility. In the following we assume the market volatility σ 2 Mt has been estimated, denoted byσ 2 Mt , t = 1, 2, ··· , T. We will estimate the unknown functional β(v) at the market volatility σ 2 Mt = v by least squares local linear modelling technique (c.f. Fan and Gijbels, 1996). The basic idea of least squares local linear modelling technique with β(·) can be described as follows. When σ 2 Mt is equal or close to v,thenβ(σ 2 Mt ) can be expressed or approximated by Locally at v, the model can then be approximately expressed as: where though we can also assume α depending on σ 2 Mt in model (9) and apply local linear idea to α(·), the estimation of α(·) is of less interest in capital asset pricing modelling, which is very close to zero, therefore in (11) α is treated as a local constant to reduce the number of unknown local parameters. Therefore, replacing σ 2 Mt byσ 2 Mt , the least squares local linear estimate of α and β(·) in (9) can be made by setting α(v)= α and β(v)= β 0 ,where( α, β 0 , β 1 ) minimizes: where h = h T → 0 is a bandwidth that controls the length of the local neighborhood of v in which the observations locally used fall, K(x) is a kernel function, which may, for example, take Therefore, we have three unknown local parameters α, β 0 , β 1 . Applying partial differentiation, we get the expression of the estimators of the three unknown local parameters at v as follows: where It is well-known (c.f., Fan and Gijbels, 1996) that the bandwidth h plays an important role in the process of estimation. Therefore, how to choose the bandwidth becomes an important step in the estimation. A popular method is the cross-validation (CV) selection of bandwidth (cf., Stone 1974). We here apply a leave-one-out CV, defined below, which is relatively computationally less intensive in comparison with other more involved CV principles: where (α −s (·),β −s (·)) are the estimators of (α(·), β(·)) obtained by minimizing (12) with the term t = s removed from the sum of (12). We select the h opt that minimizes . In practice, the coefficients c 0 , c 1 and c 2 will be estimated from CV(h k ), k = 1, 2, ··· , q, via least squares regression, where CV k = CV(h k ) obtained from (15). We let h opt = {ĉ 2 /(4Tĉ 1 )} 1/5 when bothĉ 1 andĉ 2 , the estimators of c 1 and c 2 , are positive. In the likely event that one of them is nonpositive, we let h opt = arg min 1≤k≤q CV(h k ). This bandwidth selection procedure is computationally efficient as q is moderately small, i.e. we only need to compute q CV-values.
Before we can apply the estimation method introduced in this section to examine the empirical evidence of the functional-beta CAPM, we need to introduce the data sets that we will use, in next section.

Data sets from Australian stock market
We will use a set of the stocks data collected from Australian stock market to explore the evidence of functional-beta CAPM in this chapter. The reason why we use the Australian data is because we believe an Australian dataset is ideal for this task. Firstly, the Australian evidence regarding the CAPM is well studied by (Ball, Brown and Officer 1976); (Faff 1991); (Wood 1991); (Faff 1992); (Brailsford and Faff 1997); and (Faff and Lau 1997) as well as (Yao and Gao 2004 market. Moreover, there are typically prolonged periods in which many smaller Australian companies do not trade. Therefore the market risk may be a significant factor that impacts the risk of the individual stock or index measured by the beta in CAPM. Thirdly, despite the above argument, the Australian equity market belongs to the dominant group of developed markets. For instance, as at the end of 1996 it ranked tenth among all markets in terms of market capitalization at about $US312,000 million. Interestingly, this is not greatly dis-similar from the size of the Canadian market which ranked sixth (Faff, Brooks, Fan 2004). Therefore the Australian data may be of some typical properties that the other markets may share.

Individual and market return series
At first we review the market return of Australia Index from August 2nd 2004 to August 8th 2008. The time series plot of the 5 sector daily indices and the ANZ stock daily closing price are depicted in Figure 1 together with their return series in Figure 2, where the daily return data denoted as R t (for individual sector index or for individual security), can be expressed as: where P t represents the closing price of individual sector index in day t. The daily market return data, R Mt , can be expressed as: and P Mt represents the closing price of all ordinaries index in day t, both of which are plotted in Figure 3.
In addition to market return data, R Mt , we also need the daily market volatility, σ 2 Mt ,whichis unobservable directly. We discuss how to estimate σ 2 Mt based on the market return data, R Mt , in the next subsection.

Market volatility
The market volatility that can not be observed directly needs to be estimated by using the market return series R Mt . A popular method to estimate the market volatility in the literature is by using the family of GARCH models proposed by (Engle 1982) and (Bollersleve 1986); in particular, we produce the market volatility by the most popular GARCH(1,1) model applied to the return series of All Ordinaries Index. In the GARCH(1,1) model  we use R fGarch package to calculate the parameters under e t satisfying Ee t = 0andvar(e t )= 1. In order to examine the impacts of distribution for e t on the estimation of the volatility, we tried different distributions for e t in the fGarch package, including normal, t and generalized error distribution and their skewed versions, and found that the non-skewed distributions are more acceptable according to their AIC values (Akaike, 1974), with the results listed in Table 1: Quite obviously it follows from the p-value in Table 1 that a 0 and a 1 in the AR part are close to zero while the GARCH parameters are all away from zero at the significance level of 5%. Also by the AIC values, these three GARCH models are quite close to each other and well fitted to the market return data set, with the GARCH-GED (i.e., with e t of generalized error distribution) preferred. The estimated volatility series under GARCH-GED is plotted Figure 4, where the kernel density estimators of the estimated volatility series under different GARCH models are also displayed, confirming that the estimated volatility under different models are very close. In the following we will use the estimated volatility from the GARCH-GED model as the volatility series in the estimation of the functional-beta CAPM. The summary statistics  In the next section, we shall explore the functional form of the beta risk associated with the market volatility in CAPM with the stocks data sets from Australian stock market introduced in this section.

Functional-beta CAPM: Empirical evidences
To carefully examine the evidences of functional-beta model (9), we are using the methodology introduced in Section 2, under the semiparametric beta structure that is estimated by local linear fitting introduced in Section 2.2. The advantage of this semiparametric method lies in that the data will determine the relationship of the beta coefficient associated with the market volatility, without pre-specifying a parametric structure to avoid model mis-specification. Based on the findings from the semiparametric method,  we shall clearly see whether the beta coefficient associated with the market volatility is of three-or multi-beta structure or not. Differently from the three-or multi-beta CAPM in (Galagedera and Faff 2005), our new findings will indicate that the functional beta may probably be parameterized as threshold (regime-switching) stepwise linear functionals of the market volatility, rather than three or more simple constant beta's.
We will present our empirical evidence based nonparametric estimation method in Subsection 4.1 and further investigation into the parametric evidence in Subsection 4.2.

Nonparametric evidence
Referred to Section 2.2, when we apply local linear fitting method to estimate the unknown beta function in model (9), we need to use the real data to choose the ideal bandwidth for each Sector index. The values of CV are calculated against 40 points of the bandwidth h for eight groups of data (with bandwidth ranging from 0.2 to 0.6 with partition interval of length 0.01). Hence using the CV calculation procedure given in Section 2.2, we can have the chosen bandwidths as follows in Table 3:   Table 3. Bandwidth Selection Based on the chosen bandwidth in Table 3, the results of nonparametric estimation of beta functional can be plotted in graphs. For each of eight groups of data (mentioned in Section 3), we can have a curve of beta function plotted in the solid line in Figures 5-10, respectively. As most of the beta functions except for ANZ are positive, it means that the market return has positive effects on all individual sector return. Moreover, the time changing of the beta factor is obvious; it also shows that the individual returns are influenced by the market returns under conditions of market volatility at different levels. We will examine the findings of regime-switching phenomena from the nonparametric estimation more carefully by considering different parametric beta structures in CAPM, which are studied in the next subsection.

Parametric analysis
In this part we focus on further parametric investigation according to the previous work of nonparametric outcomes. How to specify parametric structures for functional beat? In a recent pioneering work of three-beta CAPM, (Galagedera and Faff 2005) made an assumption that the market conditions can play an important part in explaining a changing beta and could be divided into three states specified as -"low", "neutral" and "high". The nonparametric outcomes in Section 4.1 provide us with some possible ways of parametrization of the beta functional.
To capture the findings of regime-switching phenomena in functional-beta CAPM, we need to suitably specify the switching regimes of market condition in a parametric analysis of functional-beta. The difficult choice of the specific switching regimes of market condition can be suggested in accordance with the nonparametric outcomes of the functional beta, by which we can select reasonable changing points (thresholds) that are needed in parametric estimation. The problem of how many thresholds we should choose in the functional-beta model will be solved by Akaike's information criterion (AIC). This way, we shall have a general flexible functional-beta model which fits the financial data more adaptively.
With reference to the results of non-parametric estimates in Figures 5-10, the market volatility changing regime points σ 2 L and σ 2 H are quite amazingly very close to 1.4 and 2.8, respectively, for all the individual sector indexes and the ANZ stock, if we would apply a three-regime (two-threshold) CAPM as done in the three-beta CAPM of (Galagedera and Faff 2005). We therefore run the comparisons of the following five types of parametric models to examine which one appears more flexible and better fitted to the real data. (i) The first one is the traditional CAPM with a constant beta as coefficient. (ii) The second one is similar to the first one but it has a linear functional beta. (iii) In the third one, we divide the market volatility into two regimes and the beta functional can be parameterized as a two stepwise linear function. From the nonparametric estimates in Figures 5-10, it looks reasonable to set σ 2 L = 1.4 as the threshold. (iv) In the fourth one, we divide the market volatility into three regimes and the beta functional can be parameterized as a three stepwise function, where we take σ 2 L = 1.
One important problem in practice is the model selection, that is, which model is the best suitable for a real data set among Models (i)-(v). In order to verify which type of CAPM best suits each group of data respectively, Akaike's information criterion, AIC, is to be applied in this part by minimizing the value of AIC(m),wherem is the number of unknown parameters in the model. Note that all 5 models (i)-(v) can be expressed in a linear model in the form R i =(R i1 , ··· , R iT ) ′ = Xb +(ε i1 , ··· , ε iT ) ′ by suitably defining a T × m matrix X,wherem is the number of parameters in each of the five CAPMs (See Table 4). Then we can define The results in Table 4 show that except for the health care sector index selecting Model (v), all the other data sets select either Model (iii), Model (iv), which means the beta functional could be divided into two or three regimes. Models (i) and (ii) may be too simple to describe the relationship between market return and individual return.
The two-stepwise beta functional in Model (iii) estimated by using the common changing point σ 2 L for each data set is plotted in dashed line in the left panel of Figures 5-10, respectively, and the three-stepwise beta functional in Model (iv) in dashed line in the right panel of Figures 5-10, respectively. Obviously, due to the sparseness of highly extreme market volatility σ 2 Mt (c.f., Figure 4), the results of nonparametric estimation are poor and unreliable at In (Galagedera and Faff 2005), the functional beta is assumed as three constants over three regimes, which is a special case of Model (iv) with β i1 = β i3 = β i5 = 0. To examine their work, here we test the significance of β i1 , β i3 , β i5 by applying T-statistics: Applying linear regression method we get b ,andresiduals r: and let δ stand for the standard deviation of r = R i − X b . Then the T-statistics value of each estimated parameter is T j = b j /{(XX T ) −1 jj δ 2 } 1/2 ,w h e r e b j represents the jth element of b . In a standard normal distribution, only 5% of the values fall outside the range plus-or-minus 2. Hence, as a rough rule of thumb, a t-statistic larger than 2 in absolute value would be significant at the significance level of 5%. The outcomes of the T-statistics with p-values for Models (iii), (iv) and (v) are listed in Tables 5-10, respectively, which indicate that the (Galagedera and Faff 2005)' three-beta model is basically rejected except for the health care sector index.

Closure and future work
In this chapter, we have suggested a functional-beta single-index CAPM, extending the work of three-beta CAPM (Galagedera and Faff, 2004) that takes into account the condition of market volatility. Differently from the three-beta CAPM, we allow systematic risk β i changing functionally with the market volatility σ 2 M , which is more flexible and adaptive to the changing structure of financial systems.
By using a set of stocks data sets collected from Australian stock market, empirical evidences of the functional-beta CAPM in Australia have been carefully examined under both nonparametric and parametric model structures. Differently from the three-or multi-beta (constant) CAPM in (Galagedera and Faff 2005), our new findings have convincingly demonstrated that the functional beta can be reasonably parameterized as threshold (regime-switching) linear functionals of market volatility with two or three regimes of market condition, taking as a special case the three-beta model of (Galagedera and Faff 2005) which was mostly rejected except for the health care sector index. In the condition of extreme market volatility, a parametric threshold functional-beta CAPM is found useful, which is of potential interest in exploring the Black Swan effect of the merits of beta in the presence of large market fluctuations.
The CAPM provides a usable measure of risk that helps investors determine what return they deserve for putting their money at risk. Our new model is no doubt helpful to better understand the relationship between risk and return under different market conditions. It can be potentially applied widely, for example, it may be useful both for market investors and financial risk managers in their investment/management decision-making, such as portfolio selection.
In addition, as done in (Galagedera and Faff 2005), it is interesting to investigate how the functional beta systematic risk is priced in the real financial assets.
We shall leave the above questions for future work.

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Risk, Return and Market Condition: From a Three-Beta to a Functional-Beta Capital Asset Pricing Model

Acknowledgements
The work of this research was partially supported by the Discovery Project and Future Fellowship grants from the Australian Research Council, which are acknowledged.