**Section 4**

**Finance and Economics** 

388 Risk Management for the Future – Theory and Cases

Yu, C., & Bi, X. (2008). Survival Analysis on Information Technology Adoption of Chinese

*Mobile Computing (WiCOM'08),* pp. 1–5, 2008

Enterprises, *4th International Conference on Wireless Communications, Networking and* 

**0**

**17**

*Australia*

**Risk, Return and Market Condition: From a**

<sup>1</sup>*Section of Statistics, School of Mathematical Sciences, The University of Adelaide* <sup>2</sup>*Department of Mathematics and Statistics, Curtin University of Technology, Perth*

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

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

where *Ri* is the return on security *i*, *Rm* is the return on the market portfolio and *β<sup>i</sup>* is the measure of security *i*'s non-diversifiable risk relative to that of the market portfolio. Here the return on individual security *Ri* can be decomposed into the specific return, including expected specific return *α<sup>i</sup>* and random specific return *εi*, and the systematic return, *βiRm*, owing to the common market return *Rm*. In this model, the quantity *β<sup>i</sup>* is of particular importance, which is an alternative measure of the risk that an investor has to bear owing

In the traditional CAPM, *β<sup>i</sup>* is assumed to be constant. This assumption has been widely documented to be untrue in the literature. Blume (1971) was among the first to consider the time-varying beta market model, which showed that the estimated beta tended to regress toward the mean; see also (Blume 1975). Earlier studies that attempted to apply random coefficient model to beta include, among others, (Sunder 1980) and (Simonds, LaMotte and McWhorter 1986) who suggested a random-walk coefficient model, and (Ohlson and Rosenberg 1982) and (Collins, Ledolter and Rayburn 1987) who proposed an ARMA(1,1) model for the beta coefficient. More recent literature has widely recognized that the systematic risk of asset changing over time may be due to both the microeconomic factors in the level of the firm and the macroeconomic factors; see (Fabozzi and Francis 1978; Bos and Newbold

beta in the presence of large market fluctuations (c.f., Estrada and Vargas 2011).

**1. Introduction**

expressed in a general non-expected form as

to the systematic market movement.

**Three-Beta to a Functional-Beta Capital**

**Asset Pricing Model**

Zudi Lu<sup>1</sup> and Yuchen Zhuang<sup>2</sup>

*Ri* = *α<sup>i</sup>* + *βiRm* + *εi*, (1)

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

Zudi Lu<sup>1</sup> and Yuchen Zhuang<sup>2</sup>

<sup>1</sup>*Section of Statistics, School of Mathematical Sciences, The University of Adelaide* <sup>2</sup>*Department of Mathematics and Statistics, Curtin University of Technology, Perth Australia*

#### **1. 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

$$R\_{\dot{i}} = \alpha\_{\dot{i}} + \beta\_{\dot{i}} R\_m + \varepsilon\_{\dot{i}\prime} \tag{1}$$

where *Ri* is the return on security *i*, *Rm* is the return on the market portfolio and *β<sup>i</sup>* is the measure of security *i*'s non-diversifiable risk relative to that of the market portfolio. Here the return on individual security *Ri* can be decomposed into the specific return, including expected specific return *α<sup>i</sup>* and random specific return *εi*, and the systematic return, *βiRm*, owing to the common market return *Rm*. In this model, the quantity *β<sup>i</sup>* is of particular importance, which is an alternative measure of the risk that an investor has to bear owing to the systematic market movement.

In the traditional CAPM, *β<sup>i</sup>* is assumed to be constant. This assumption has been widely documented to be untrue in the literature. Blume (1971) was among the first to consider the time-varying beta market model, which showed that the estimated beta tended to regress toward the mean; see also (Blume 1975). Earlier studies that attempted to apply random coefficient model to beta include, among others, (Sunder 1980) and (Simonds, LaMotte and McWhorter 1986) who suggested a random-walk coefficient model, and (Ohlson and Rosenberg 1982) and (Collins, Ledolter and Rayburn 1987) who proposed an ARMA(1,1) model for the beta coefficient. More recent literature has widely recognized that the systematic risk of asset changing over time may be due to both the microeconomic factors in the level of the firm and the macroeconomic factors; see (Fabozzi and Francis 1978; Bos and Newbold

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

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

1 *i f σ*<sup>2</sup>

1 *i f σ*<sup>2</sup>

neutral market condition, *IHt* represents high market condition. By investigating empirically on the single factor CAPM *Rit* = *α<sup>i</sup>* + *βiRmt* + *εit* , to estimate the betas in the low, neutral and high volatility markets, Galagedera and Faff extended the market model given in (1), in the

where *βiL*, *βiN*, *βiH* 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*

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

1 *i f σ*<sup>2</sup>

*Mt* < *<sup>σ</sup>*<sup>2</sup> *L*

*<sup>H</sup>* <sup>&</sup>lt; *<sup>σ</sup>*<sup>2</sup> *Mt*

*<sup>H</sup>* are constants; *ILt* represents the low market condition, *INt* represents the

*Rit* = *α<sup>i</sup>* + *βiL*(*ILtRMt*) + *βiN*(*INtRMt*) + *βiH*(*IHtRMt*) + *εi*, (5)

*Mt* <sup>&</sup>lt; *<sup>σ</sup>*<sup>2</sup> *H*

*<sup>L</sup>* <sup>&</sup>lt; *<sup>σ</sup>*<sup>2</sup>

*Mt* into one of three market volatility regimes,

393

<sup>0</sup> *i f otherwise* (2)

<sup>0</sup> *i f otherwise* (3)

<sup>0</sup> *i f otherwise*. (4)

over time.

Here *σ*<sup>2</sup>

form:

*<sup>L</sup>* and *<sup>σ</sup>*<sup>2</sup>

Faff 2005) classify daily market volatility *σ*<sup>2</sup>

Risk, Return and Market Condition:

From a Three-Beta to a Functional-Beta Capital Asset Pricing Model

using appropriately defined indicator functions:

*ILt* =

*IHt* =

*securities' responses to the market vary depending on the changing market volatility.*

*INt* =

1984). Considerable empirical evidences have suggested that beta stability assumption is invalid. The literature is abundant, see, for example, (Kim 1993), (Bos and 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 2 Will-be-set-by-IN-TECH

1984). Considerable empirical evidences have suggested that beta stability assumption is invalid. The literature is abundant, see, for example, (Kim 1993), (Bos and Ferson 1992, 1995), (Wells 1994), (Bos, Ferson, Martikainen and Perttunen 1995), (Brooks, Faff and Lee 1992) and

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

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

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

(Cheng 1997).

in detail by using the Kalman filter approach.

the classification of up and down markets is more pronounced.

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*<sup>2</sup> *Mt* into one of three market volatility regimes, using appropriately defined indicator functions:

$$I\_{Lt} = \begin{cases} 1 & \text{if} \quad \sigma\_{Mt}^2 < \sigma\_L^2 \\ 0 & \text{if} \quad \text{otherwise} \end{cases} \tag{2}$$

$$I\_{Nt} = \begin{cases} 1 & \text{if } \quad \sigma\_L^2 < \sigma\_{Mt}^2 < \sigma\_H^2\\ 0 & \text{if } \quad otherwise \end{cases} \tag{3}$$

$$I\_{Ht} = \begin{cases} 1 & \text{if} \quad \sigma\_H^2 < \sigma\_{Mt}^2 \\ 0 & \text{if} \quad \text{otherwise.} \end{cases} \tag{4}$$

Here *σ*<sup>2</sup> *<sup>L</sup>* and *<sup>σ</sup>*<sup>2</sup> *<sup>H</sup>* are constants; *ILt* represents the low market condition, *INt* represents the neutral market condition, *IHt* represents high market condition. By investigating empirically on the single factor CAPM *Rit* = *α<sup>i</sup>* + *βiRmt* + *εit* , 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:

$$\mathcal{R}\_{\dot{t}t} = \mathfrak{a}\_{i} + \mathfrak{f}\_{iL}(I\_{Lt}\mathbb{R}\_{Mt}) + \mathfrak{f}\_{iN}(I\_{Nt}\mathbb{R}\_{Mt}) + \mathfrak{f}\_{iH}(I\_{Ht}\mathbb{R}\_{Mt}) + \mathfrak{e}\_{i\nu} \tag{5}$$

where *βiL*, *βiN*, *βiH* 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

**2.2 Estimation of functional-beta CAPM: nonparametric method**

From a Three-Beta to a Functional-Beta Capital Asset Pricing Model

unobservable market volatility *σ*<sup>2</sup>

Risk, Return and Market Condition:

try to estimate *σ*<sup>2</sup>

market volatility *σ*<sup>2</sup>

close to *v*, then *β*(*σ*<sup>2</sup>

unknown local parameters. Therefore, replacing *σ*<sup>2</sup>

be made by setting *α*

take

where

at *v* as follows:

*AT* =

*T* ∑ *t*=1 ⎛ ⎝

(*σ*ˆ <sup>2</sup>

Given the historical observations (*Rit*, *RMt*), *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

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

*al*. 2008) for a comprehensive review on realized volatility. In the following we assume the

local linear modelling technique (c.f. Fan and Gijbels, 1996). The basic idea of least squares

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

where *h* = *hT* → 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,

Therefore, we have three unknown local parameters *α*, *β*<sup>0</sup> , *β*<sup>1</sup> . Applying partial differentiation, we get the expression of the estimators of the three unknown local parameters

<sup>⎠</sup> <sup>=</sup> *<sup>A</sup>*−<sup>1</sup>

*Mt <sup>R</sup>*<sup>2</sup>

*Mt* (*σ*<sup>ˆ</sup> <sup>2</sup>

(*Rit* <sup>−</sup> [*<sup>α</sup>* + (*β*<sup>0</sup> <sup>+</sup> *<sup>β</sup>*1(*σ*<sup>ˆ</sup> <sup>2</sup>

*<sup>K</sup>*(*x*) = *<sup>φ</sup>*(*x*) = <sup>1</sup>

⎞

1 *RMt RMt*(*σ*ˆ <sup>2</sup>

*Mt* <sup>−</sup> *<sup>v</sup>*)*R*<sup>2</sup>

⎛ ⎝ �*α* �*β*0 �*β*1

*RMt R*<sup>2</sup>

*Mt* <sup>−</sup> *<sup>v</sup>*)*RMt* (*σ*<sup>ˆ</sup> <sup>2</sup>

*Mt* <sup>−</sup> *<sup>v</sup>*) <sup>≡</sup> *<sup>β</sup>*<sup>0</sup> <sup>+</sup> *<sup>β</sup>*1(*σ*<sup>2</sup>

169). Alternatively, we can use realized market volatility as an estimate of *σ*<sup>2</sup>

local linear modelling technique with *<sup>β</sup>*(·) can be described as follows. When *<sup>σ</sup>*<sup>2</sup>

*Mt*) can be expressed or approximated by

(*v*)(*σ*<sup>2</sup>

*Rit* <sup>≈</sup> *<sup>α</sup>* + (*β*<sup>0</sup> <sup>+</sup> *<sup>β</sup>*1(*σ*<sup>2</sup>

*Mt* has been estimated, denoted by *<sup>σ</sup>*<sup>ˆ</sup> <sup>2</sup>

We will estimate the unknown functional *β*(*v*) at the market volatility *σ*<sup>2</sup>

*β*(*v*) + *β*�

where though we can also assume *α* depending on *σ*<sup>2</sup>

*Mt* by *<sup>σ</sup>*<sup>ˆ</sup> <sup>2</sup>

*L*(*α*, *β*0, *β*1) =

�(*v*) = �*<sup>α</sup>* and *<sup>β</sup>*

*T* ∑ *t*=1

Locally at *v*, the model can then be approximately expressed as:

*Mt*. Using the market returns *RMt*, *t* = 1, 2, ··· , *T*, we can

*Mt*, *t* = 1, 2, ··· , *T*.

*Mt*, the least squares local linear estimate of *α* and *β*(·) in (9) can

*Mt* <sup>−</sup> *<sup>v</sup>*)*RMt*)])2*K*(

*Mt* − *v*)

*Mt* − *v*)

*Mt*

⎞ ⎠ *K* � *σ*ˆ 2 *Mt* − *v h*

*Mt*(*σ*<sup>ˆ</sup> <sup>2</sup>

*Mt* <sup>−</sup> *<sup>v</sup>*)2*R*<sup>2</sup>

�(*v*) = �*β*0, where (�*α*, �*β*0, �*β*1) minimizes:

<sup>√</sup>2*<sup>π</sup> e* − *x*2 *Mt*; see (Allen *et*

395

*Mt* is equal or

*<sup>h</sup>* ), (12)

� ,

*Mt* = *v* by least squares

*Mt* − *v*) (10)

*Mt* − *v*))*RMt* + *εt*, (11)

*Mt* in model (9) and apply local linear

*σ*ˆ 2 *Mt* − *v*

*<sup>T</sup> BT*, (14)

<sup>2</sup> . (13)

*Mt* in various ways. A simple way is to apply the econometric models of

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.

#### **2. 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.

#### **2.1 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

$$R\_{\rm it} = \mathfrak{a}\_{\rm i} + (\beta\_{\rm iL} I\_{\rm Lt} + \beta\_{\rm iN} I\_{\rm Nt} + \beta\_{\rm iH} I\_{\rm Ht}) R\_{\rm Mt} + \varepsilon\_{\rm it} \equiv \mathfrak{a}\_{\rm i} + \beta\_{\rm iI} R\_{\rm Mt} + \varepsilon\_{\rm it} \tag{6}$$

which is a time-varying beta model, with

$$
\beta\_{\rm it} = \beta\_{\rm iL} I\_{\rm Lt} + \beta\_{\rm iN} I\_{\rm Nt} + \beta\_{\rm iH} I\_{\rm Ht}.\tag{7}
$$

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*<sup>2</sup> *Mt*. Therefore *βit* is a simple functional of the market volatility *σ*<sup>2</sup> *Mt*, that is

$$\beta\_{\rm if} = \begin{cases} \beta\_{\rm iL} & \text{if } \quad \sigma\_{\rm Mt}^2 < \sigma\_{\rm L}^2 \\ \beta\_{\rm iN} & \text{if } \quad \sigma\_{\rm L}^2 \le \sigma\_{\rm Mt}^2 < \sigma\_{\rm H'}^2 \\ \beta\_{\rm iH} & \text{if } \quad \sigma\_{\rm Mt}^2 \ge \sigma\_{\rm H}^2 \end{cases} \tag{8}$$

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:

$$R\_{it} = \mathfrak{a}\_i + \beta\_i(\sigma\_{Mt}^2) R\_{Mt} + \varepsilon\_{it\prime} \tag{9}$$

where as before, *Rit* is the return of financial asset *i* at time *t*, *RMt* is the market return at time *t*, *σ*<sup>2</sup> *Mt* is the market volatility at time *t*, *α<sup>i</sup>* is the conditional expected specific return, *εit* is random specific return, and *β<sup>i</sup>* is the coefficient of the contribution due to the market factor, changing with the market volatility. Here *βi*(·) is particularly important, which is the systematic risk functional, in capital asset pricing modelling. We may also treat *α<sup>i</sup>* as varying with *σ*<sup>2</sup> *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<sup>i</sup>* and *βi*(·) in (9).

#### **2.2 Estimation of functional-beta CAPM: nonparametric method**

Given the historical observations (*Rit*, *RMt*), *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*<sup>2</sup> *Mt*. Using the market returns *RMt*, *t* = 1, 2, ··· , *T*, we can try to estimate *σ*<sup>2</sup> *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*<sup>2</sup> *Mt*; see (Allen *et al*. 2008) for a comprehensive review on realized volatility. In the following we assume the market volatility *σ*<sup>2</sup> *Mt* has been estimated, denoted by *<sup>σ</sup>*<sup>ˆ</sup> <sup>2</sup> *Mt*, *t* = 1, 2, ··· , *T*.

We will estimate the unknown functional *β*(*v*) at the market volatility *σ*<sup>2</sup> *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 *<sup>β</sup>*(·) can be described as follows. When *<sup>σ</sup>*<sup>2</sup> *Mt* is equal or close to *v*, then *β*(*σ*<sup>2</sup> *Mt*) can be expressed or approximated by

$$
\beta(v) + \beta'(v)(\sigma\_{Mt}^2 - v) \equiv \beta\_0 + \beta\_1(\sigma\_{Mt}^2 - v) \tag{10}
$$

Locally at *v*, the model can then be approximately expressed as:

$$\mathcal{R}\_{\rm if} \approx \alpha + (\beta\_0 + \beta\_1(\sigma\_{Mt}^2 - v))\mathcal{R}\_{Mt} + \varepsilon\_{t\prime} \tag{11}$$

where though we can also assume *α* depending on *σ*<sup>2</sup> *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*<sup>2</sup> *Mt* by *<sup>σ</sup>*<sup>ˆ</sup> <sup>2</sup> *Mt*, the least squares local linear estimate of *α* and *β*(·) in (9) can be made by setting *α* �(*v*) = �*<sup>α</sup>* and *<sup>β</sup>* �(*v*) = �*β*0, where (�*α*, �*β*0, �*β*1) minimizes:

$$L(\boldsymbol{a}, \boldsymbol{\beta}\_{0}, \boldsymbol{\beta}\_{1}) = \sum\_{t=1}^{T} (\mathbb{R}\_{it} - [\boldsymbol{a} + (\boldsymbol{\beta}\_{0} + \boldsymbol{\beta}\_{1}(\boldsymbol{\sigma}\_{Mt}^{2} - \boldsymbol{v})\mathbb{R}\_{Mt})])^{2} \text{K}(\frac{\hat{\sigma}\_{Mt}^{2} - \boldsymbol{v}}{\hbar}),\tag{12}$$

where *h* = *hT* → 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

$$K(\mathbf{x}) = \phi(\mathbf{x}) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}.\tag{13}$$

Therefore, we have three unknown local parameters *α*, *β*<sup>0</sup> , *β*<sup>1</sup> . Applying partial differentiation, we get the expression of the estimators of the three unknown local parameters at *v* as follows:

$$
\begin{pmatrix}
\widehat{\vec{\alpha}} \\
\widehat{\beta\_0} \\
\widehat{\beta\_1}
\end{pmatrix} = A\_T^{-1} B\_{T\prime} \tag{14}
$$

where

4 Will-be-set-by-IN-TECH

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

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

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)

We note that the volatility of market returns is partitioned into three regimes in (2)–(4), which

*Mt*, that is

*βiL i f σ*<sup>2</sup>

*βiN i f σ*<sup>2</sup>

*βiH i f σ*<sup>2</sup>

*Rit* = *α<sup>i</sup>* + *βi*(*σ*<sup>2</sup>

So the three-beta CAPM proposed by (Galagedera and Faff 2005) is a simple functional beta

In this chapter, we will extend the model (5) that was suggested by (Galagedera and Faff 2005)

where as before, *Rit* is the return of financial asset *i* at time *t*, *RMt* is the market return at

*εit* is random specific return, and *β<sup>i</sup>* is the coefficient of the contribution due to the market factor, changing with the market volatility. Here *βi*(·) is particularly important, which is the systematic risk functional, in capital asset pricing modelling. We may also treat *α<sup>i</sup>* as varying

*Mt* is the market volatility at time *t*, *α<sup>i</sup>* is the conditional expected specific return,

*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<sup>i</sup>* and

are the functions of the size of the conditional market volatility, say, *σ*<sup>2</sup>

⎧ ⎪⎨

⎪⎩

*βit* =

and propose a general functional-beta model as follows:

*Rit* = *α<sup>i</sup>* + (*βiL ILt* + *βiN INt* + *βiH IHt*)*RMt* + *εit* ≡ *α<sup>i</sup>* + *βitRMt* + *εit*, (6)

*Mt* <sup>&</sup>lt; *<sup>σ</sup>*<sup>2</sup> *L*,

*Mt* <sup>≥</sup> *<sup>σ</sup>*<sup>2</sup> *H*.

*Mt* < *<sup>σ</sup>*<sup>2</sup> *H*,

*Mt*)*RMt* + *εit*, (9)

*<sup>L</sup>* <sup>≤</sup> *<sup>σ</sup>*<sup>2</sup>

*βit* = *βiL ILt* + *βiN INt* + *βiH IHt*. (7)

*Mt*. Therefore *βit* is a

(8)

**2. Methodology: From a Three-Beta CAPM to a Functional-Beta CAPM**

Section 5.

**2.1 Model**

model.

time *t*, *σ*<sup>2</sup>

with *σ*<sup>2</sup>

*βi*(·) in (9).

can be expressed as

Functional-Beta CAPM in Subsection 2.2.

which is a time-varying beta model, with

simple functional of the market volatility *σ*<sup>2</sup>

$$A\_{T} = \sum\_{t=1}^{T} \begin{pmatrix} 1 & R\_{Mt} & R\_{Mt}(\hat{\sigma}\_{Mt}^{2} - v) \\ R\_{Mt} & R\_{Mt}^{2} & R\_{Mt}^{2}(\hat{\sigma}\_{Mt}^{2} - v) \\ (\hat{\sigma}\_{Mt}^{2} - v)R\_{Mt} & (\hat{\sigma}\_{Mt}^{2} - v)R\_{Mt}^{2} & (\hat{\sigma}\_{Mt}^{2} - v)^{2}R\_{Mt}^{2} \end{pmatrix} K \left(\frac{\hat{\sigma}\_{Mt}^{2} - v}{h}\right) \mathcal{I}$$

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.

According to ASX Indices (including All Ordinaries Index, ASX 200 GICS Sectors Index), we take sample size 986, from August 2nd 2004 to August 8th 2008, for an illustration. The sectors indexes include ASX 200 GICS Energy, ASX 200 GICS Materials, ASX 200 GICS Health Care, ASX 200 GICS Financials, ASX 200 GICS Finance-v-property trusts. Moreover, we also take a group of individual stock data which is ANZ bank group limited as survey sample of individual stock analysis. An introduction to individual and market return series is presented in Subsection 3.1; estimation of the market volatility that we need in estimating functional-beta

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 *Rt* (for individual sector index or for individual security), can be expressed

where *Pt* represents the closing price of individual sector index in day *t*. The daily market

and *PMt* represents the closing price of all ordinaries index in day *t*, both of which are plotted

The market volatility that can not be observed directly needs to be estimated by using the market return series *RMt*. 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

*RMt* = *<sup>a</sup>*<sup>0</sup> + *<sup>a</sup>*1*RM*,*t*−<sup>1</sup> + *�<sup>t</sup>*

*<sup>t</sup>*−<sup>1</sup> <sup>+</sup> *<sup>β</sup>*1*σ*<sup>2</sup>

*Mt*,

*Mt*

*Mt* = *<sup>α</sup>*<sup>0</sup> + *<sup>α</sup>*1*�*<sup>2</sup>

In addition to market return data, *RMt*, we also need the daily market volatility, *σ*<sup>2</sup>

to the return series of All Ordinaries Index. In the GARCH(1,1) model

*�<sup>t</sup>* = *etσ*<sup>2</sup>

⎧ ⎪⎨

⎪⎩

*σ*2

*Rt* = (log *Pt* − log *Pt*−1) × 100, (17)

*RMt* = (log *PMt* − log *PM*,*t*−1) × 100, (18)

*Mt* based on the market return data, *RMt*,

*Mt*, which is

397

(19)

CAPM is detailed in Subsection 3.2.

Risk, Return and Market Condition:

From a Three-Beta to a Functional-Beta Capital Asset Pricing Model

as:

in Figure 3.

in the next subsection.

**3.2 Market volatility**

**3.1 Individual and market return series**

return data, *RMt* , can be expressed as:

unobservable directly. We discuss how to estimate *σ*<sup>2</sup>

$$B\_T = \begin{pmatrix} \sum\_{t=1}^T R\_{it} K(\frac{\delta\_{Mt}^2 - v}{h}) \\ \sum\_{t=1}^T R\_{Mt} R\_{it} K(\frac{\delta\_{Mt}^2 - v}{h}) \\ \sum\_{t=1}^T (\hat{\sigma}\_{Mt}^2 - v) R\_{Mt} R\_{it} K(\frac{\delta\_{Mt}^2 - v}{h}) \end{pmatrix}.$$

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:

$$\mathcal{CV}(h) = \sum\_{s=1}^{T} \{\mathcal{R}\_{\rm is} - \hat{\mathfrak{a}}\_{-s}(\boldsymbol{\vartheta}\_{\rm Ms}^{2}) - \hat{\mathfrak{f}}\_{-s}(\boldsymbol{\vartheta}\_{\rm Ms}^{2})\mathcal{R}\_{\rm Ms}\}^{2} \tag{15}$$

where (*α*<sup>ˆ</sup> <sup>−</sup>*s*(·), *<sup>β</sup>*ˆ−*s*(·)) are the estimators of (*α*(·), *<sup>β</sup>*(·)) obtained by minimizing (12) with the term *t* = *s* removed from the sum of (12). We select the *hopt* that minimizes *CV*(·) over *h* ∈ [*hL*, *hU*], where 0 < *hL* < *hU* are appropriately given. To simplify the computation, a partition of [*hL*, *hU*] into *q* points *h*1, *h*2, ··· , *hq* is applied. We adapt an empirical rule for selecting a bandwidth by (Fan *et al*. 2003) to determining the bandwidth *h*; see also (Lu *et al*. 2009): Up to first order asymptotics, the optimal bandwidth is *hopt* <sup>=</sup> {*c*2/(4*Tc*1)}1/5, minimising *CV*(*h*) = *<sup>c</sup>*<sup>0</sup> + *<sup>c</sup>*1*h*<sup>4</sup> + *<sup>c</sup>*<sup>2</sup> *Th* <sup>+</sup> *oP*(*h*<sup>4</sup> <sup>+</sup> *<sup>T</sup>*−1*h*−1). In practice, the coefficients *<sup>c</sup>*0, *<sup>c</sup>*<sup>1</sup> and *c*<sup>2</sup> will be estimated from *CV*(*hk*), *k* = 1, 2, ··· , *q*, via least squares regression,

$$\min\_{\mathcal{L}\_0, \mathcal{L}\_1, \mathcal{L}\_2} \sum\_{k=1}^q \left\{ \mathbb{C}V\_k - c\_0 - c\_1 h\_k^4 - c\_2 / \left( Th\_k \right) \right\}^2 \tag{16}$$

where *CVk* <sup>=</sup> *CV*(*hk*) obtained from (15). We let *hopt* <sup>=</sup> {*c*ˆ2/(4*Tc*ˆ1)}1/5 when both *<sup>c</sup>*ˆ1 and *<sup>c</sup>*ˆ2, the estimators of *c*<sup>1</sup> and *c*2, are positive. In the likely event that one of them is nonpositive, we let *hopt* = arg min1≤*k*≤*<sup>q</sup> CV*(*hk*). 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.

#### **3. 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). Ball, Brown and Officer (1976) may be the first published test of the CAPM using Australian data. They employed the basic univariate testing methodology and found evidence supporting their model. Therefore using the Australian data set will help us to better understand the varying-coefficient nature of CAPM. Secondly, it can be seen that a relatively few, very large companies dominate the Australian market. For example, around 40 per cent of market capitalization and trading value is produced by just 10 stocks, whereas a similar number of the largest US stocks constitute only about 15 per cent of the total US 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.

According to ASX Indices (including All Ordinaries Index, ASX 200 GICS Sectors Index), we take sample size 986, from August 2nd 2004 to August 8th 2008, for an illustration. The sectors indexes include ASX 200 GICS Energy, ASX 200 GICS Materials, ASX 200 GICS Health Care, ASX 200 GICS Financials, ASX 200 GICS Finance-v-property trusts. Moreover, we also take a group of individual stock data which is ANZ bank group limited as survey sample of individual stock analysis. An introduction to individual and market return series is presented in Subsection 3.1; estimation of the market volatility that we need in estimating functional-beta CAPM is detailed in Subsection 3.2.

#### **3.1 Individual and market return series**

6 Will-be-set-by-IN-TECH

*<sup>t</sup>*=<sup>1</sup> *RitK*(

*<sup>t</sup>*=<sup>1</sup> *RMtRitK*(

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*<sup>ˆ</sup> <sup>−</sup>*s*(·), *<sup>β</sup>*ˆ−*s*(·)) are the estimators of (*α*(·), *<sup>β</sup>*(·)) obtained by minimizing (12) with the term *t* = *s* removed from the sum of (12). We select the *hopt* that minimizes *CV*(·) over *h* ∈ [*hL*, *hU*], where 0 < *hL* < *hU* are appropriately given. To simplify the computation, a partition of [*hL*, *hU*] into *q* points *h*1, *h*2, ··· , *hq* is applied. We adapt an empirical rule for selecting a bandwidth by (Fan *et al*. 2003) to determining the bandwidth *h*; see also (Lu *et al*. 2009): Up to first order asymptotics, the optimal bandwidth is *hopt* <sup>=</sup> {*c*2/(4*Tc*1)}1/5,

*CVk* <sup>−</sup> *<sup>c</sup>*<sup>0</sup> <sup>−</sup> *<sup>c</sup>*1*h*<sup>4</sup>

where *CVk* <sup>=</sup> *CV*(*hk*) obtained from (15). We let *hopt* <sup>=</sup> {*c*ˆ2/(4*Tc*ˆ1)}1/5 when both *<sup>c</sup>*ˆ1 and *<sup>c</sup>*ˆ2, the estimators of *c*<sup>1</sup> and *c*2, are positive. In the likely event that one of them is nonpositive, we let *hopt* = arg min1≤*k*≤*<sup>q</sup> CV*(*hk*). This bandwidth selection procedure is computationally

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

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). Ball, Brown and Officer (1976) may be the first published test of the CAPM using Australian data. They employed the basic univariate testing methodology and found evidence supporting their model. Therefore using the Australian data set will help us to better understand the varying-coefficient nature of CAPM. Secondly, it can be seen that a relatively few, very large companies dominate the Australian market. For example, around 40 per cent of market capitalization and trading value is produced by just 10 stocks, whereas a similar number of the largest US stocks constitute only about 15 per cent of the total US

*σ*ˆ 2 *Mt*−*v <sup>h</sup>* )

*Mt* <sup>−</sup> *<sup>v</sup>*)*RMtRitK*( *<sup>σ</sup>*<sup>ˆ</sup> <sup>2</sup>

*σ*ˆ 2 *Mt*−*v <sup>h</sup>* )

*Ms*) <sup>−</sup> *<sup>β</sup>*ˆ−*s*(*σ*<sup>ˆ</sup> <sup>2</sup>

*Mt*−*v <sup>h</sup>* )

*Th* <sup>+</sup> *oP*(*h*<sup>4</sup> <sup>+</sup> *<sup>T</sup>*−1*h*−1). In practice, the coefficients *<sup>c</sup>*0, *<sup>c</sup>*<sup>1</sup> and

�2

*<sup>k</sup>* − *c*2/(*Thk* )

⎞

⎟⎟⎠ .

*Ms*)*RMs*}2, (15)

, (16)

∑*T*

{*Ris* <sup>−</sup> *<sup>α</sup>*<sup>ˆ</sup> <sup>−</sup>*s*(*σ*<sup>ˆ</sup> <sup>2</sup>

*c*<sup>2</sup> will be estimated from *CV*(*hk*), *k* = 1, 2, ··· , *q*, via least squares regression,

efficient as *q* is moderately small, i.e. we only need to compute *q* CV-values.

∑*T*

*BT* =

*CV*(*h*) =

min *c*0,*c*1,*c*<sup>2</sup>

**3. Data sets from Australian stock market**

*q* ∑ *k*=1 �

minimising *CV*(*h*) = *<sup>c</sup>*<sup>0</sup> + *<sup>c</sup>*1*h*<sup>4</sup> + *<sup>c</sup>*<sup>2</sup>

next section.

⎛

⎜⎜⎝

∑*T <sup>t</sup>*=1(*σ*<sup>ˆ</sup> <sup>2</sup>

*T* ∑ *s*=1

> 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 *Rt* (for individual sector index or for individual security), can be expressed as:

$$R\_l = (\log P\_l - \log P\_{l-1}) \times 100\,\tag{17}$$

where *Pt* represents the closing price of individual sector index in day *t*. The daily market return data, *RMt* , can be expressed as:

$$R\_{Mt} = (\log P\_{Mt} - \log P\_{M,t-1}) \times 100,\tag{18}$$

and *PMt* 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, *RMt*, we also need the daily market volatility, *σ*<sup>2</sup> *Mt*, which is unobservable directly. We discuss how to estimate *σ*<sup>2</sup> *Mt* based on the market return data, *RMt*, in the next subsection.

#### **3.2 Market volatility**

The market volatility that can not be observed directly needs to be estimated by using the market return series *RMt*. 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

$$\begin{cases} \mathcal{R}\_{Mt} = a\_0 + a\_1 \mathcal{R}\_{M, t-1} + \varepsilon\_t \\ \varepsilon\_t = \varepsilon\_t \sigma\_{Mt}^2 \\ \sigma\_{Mt}^2 = a\_0 + a\_1 \varepsilon\_{t-1}^2 + \beta\_1 \sigma\_{Mt'}^2 \end{cases} \tag{19}$$

**Energy sector index**

From a Three-Beta to a Functional-Beta Capital Asset Pricing Model

**Finance sector Index**

399

Time

**Materials sector Index**

Time

**ANZ stock price**

Time

0 200 400 600 800 1000

0 200 400 600 800 1000

0 200 400 600 800 1000

return

return

return

−10

ordinaries index

 0 5

−4 0 4

−10 0 5

Risk, Return and Market Condition:

Time

**Healthcare sector Index**

Time

**Fin−X−Prop sector Index**

Time

market is quite volatile with large extreme values.

0 200 400 600 800 1000

0 200 400 600 800 1000

0 200 400 600 800 1000

return

return

return

Fig. 2. The return series of the 5 ASX 200 GICS Sectors Indexes and ANZ daily closing price

garch-normal -0.019310 -0.022673 0.033037 0.110890 0.871949 3.248890

(p-value) (0.3363) (0.6457) (0.0360) (1.14e-06) (2e-16) (7.50e-05)

(p-value) (0.3536) (0.5460) (0.0267) (3.94e-07) (2e-16) (2e-16)

Table 1. Estimated GARCH models under different innovations for the return of all

garch-t -0.032541 -0.015214 0.031242 0.114034 0.873597 10.00000 3.248322

garch-ged -0.031626 -0.020063 0.031619 0.111393 0.872692 1.682406 3.244669

on the market return and volatility are provided in Table 2, from which we can see that the

in Australia from August 2nd 2004 to August 8th 2008. Sample size =985.

(p-value) (0.5731) (0.4995) (0.0146) (4.53e-08) (2e-16)

−15 0 10

*a*<sup>0</sup> *a*<sup>1</sup> *α*<sup>0</sup> *α*<sup>1</sup> *β*<sup>1</sup> shape AIC

−10 0 5

−5 0 5

Fig. 1. The 5 ASX 200 GICS Sectors Indexes and ANZ daily closing price in Australia from August 2nd 2004 to August 8th 2008. Sample size =986.

we use R fGarch package to calculate the parameters under *et* satisfying *Eet* = 0 and var(*et*) = 1. In order to examine the impacts of distribution for *et* on the estimation of the volatility, we tried different distributions for *et* 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*<sup>0</sup> and *a*<sup>1</sup> 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 *et* 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 8 Will-be-set-by-IN-TECH

price

price

price

Fig. 1. The 5 ASX 200 GICS Sectors Indexes and ANZ daily closing price in Australia from

we use R fGarch package to calculate the parameters under *et* satisfying *Eet* = 0 and var(*et*) = 1. In order to examine the impacts of distribution for *et* on the estimation of the volatility, we tried different distributions for *et* 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*<sup>0</sup> and *a*<sup>1</sup> 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 *et* 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

15

 25

8000 14000

3000

 6000 **Finance sector Index**

Time

**Materials sector Index**

Time

**ANZ stock price**

Time

0 200 400 600 800 1000

0 200 400 600 800 1000

0 200 400 600 800 1000

**Energy sector index**

price

price

price

3000

 6000

7500 9000

12000

 18000

Time

**Healthcare sector Index**

Time

**Fin−X−Prop sector Index**

Time

August 2nd 2004 to August 8th 2008. Sample size =986.

0 200 400 600 800 1000

0 200 400 600 800 1000

0 200 400 600 800 1000

Fig. 2. The return series of the 5 ASX 200 GICS Sectors Indexes and ANZ daily closing price in Australia from August 2nd 2004 to August 8th 2008. Sample size =985.


Table 1. Estimated GARCH models under different innovations for the return of all ordinaries index

on the market return and volatility are provided in Table 2, from which we can see that the market is quite volatile with large extreme values.

**volatility of all ordinaries index under garch−ged**

<sup>401</sup> Risk, Return and Market Condition:

From a Three-Beta to a Functional-Beta Capital Asset Pricing Model

volatility

1234

0.0

**4.1 Nonparametric evidence**

bandwidths as follows in Table 3:

estimates under different GARCH models

 0.4

 0.8

Density

Time

**volatility of all ordinaries index**

garch−ged garch−normal garch−t

1234

N = 985 Bandwidth = 0.08461

Fig. 4. The estimated volatility of All Ordinaries Index and its kernel probability density

market volatility, rather than three or more simple constant beta's.

4.1 and further investigation into the parametric evidence in Subsection 4.2.

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

We will present our empirical evidence based nonparametric estimation method in Subsection

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

0 200 400 600 800 1000

Fig. 3. All Ordinaries Index (sample size =986) and its return series (sample size=985) in Australia from August 2nd 2004 to August 8th 2008.


Table 2. Some statistics data for the Australia index

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.

#### **4. 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, 10 Will-be-set-by-IN-TECH

price

return

in this section.

Australia from August 2nd 2004 to August 8th 2008.

Table 2. Some statistics data for the Australia index

**4. Functional-beta CAPM: Empirical evidences**

3000

 5000

 7000 **All Ordinaries Index**

Time

**All Ordinaries Index**

Time

Fig. 3. All Ordinaries Index (sample size =986) and its return series (sample size=985) in

Market return 0.04388 2.0214 0.4596 6.5235 -0.0099 -5.3601 8.5536 Market volatility 1.3089 0.3169 2.1493 8.7173 1.1729 0.6023 4.3905

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

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,

0 200 400 600 800 1000

mean standard deviation skewness kurtosis median min max

0 200 400 600 800 1000

**volatility of all ordinaries index under garch−ged**

Fig. 4. The estimated volatility of All Ordinaries Index and its kernel probability density estimates under different GARCH models

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.

#### **4.1 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:

Fig. 5. For Energy sector index:

Fig. 6. For Finance sector index:

Fig. 7. For Health care sector index:

1234

1234

v

**FinXProp**

functional−beta CAPM: nonparametric functional−beta CAPM: parametric constant−beta CAPM

1234

v

**ANZ**

functional−beta CAPM: nonparametric functional−beta CAPM: parametric constant−beta CAPM

1234

v

**Materials**

functional−beta CAPM: nonparametric functional−beta CAPM: parametric constant−beta CAPM

0.0

0.0

−1.0 −0.5 0.0 0.5 1.0 1.5 2.0

beta(.)

 0.5

 1.0

beta(.)

 1.5

 2.0

 0.5

 1.0

beta(.)

<sup>403</sup> Risk, Return and Market Condition:

 1.5

 2.0

v

**FinXProp**

1234

v

**ANZ**

functional−beta CAPM: nonparametric functional−beta CAPM: parametric constant−beta CAPM

1234

v

Fig. 10. For ANZ bank group limited:

Fig. 9. For Financial-v-Properties Trusts sector index:

Fig. 8. For Materials sector index:

functional−beta CAPM: nonparametric functional−beta CAPM: parametric constant−beta CAPM

**Materials**

From a Three-Beta to a Functional-Beta Capital Asset Pricing Model

functional−beta CAPM: nonparametric functional−beta CAPM: parametric constant−beta CAPM

0.0

0.0

−1.0 −0.5 0.0 0.5 1.0 1.5 2.0

beta(.)

 0.5

 1.0

beta(.)

 1.5

 2.0

 0.5

 1.0

beta(.)

 1.5

 2.0

12 Will-be-set-by-IN-TECH

0.0

0.0

0.0

0.5

1.0

beta(.)

1.5

0.5

beta(.)

1.0

1.5

 0.5

 1.0

beta(.)

 1.5

 2.0

1234

v

**Finance**

functional−beta CAPM: nonparametric functional−beta CAPM: parametric constant−beta CAPM

1234

v

**Healthcare**

functional−beta CAPM: nonparametric functional−beta CAPM: parametric constant−beta CAPM

1234

v

**Energy**

functional−beta CAPM: nonparametric functional−beta CAPM: parametric constant−beta CAPM

1234

v

**Finance**

functional−beta CAPM: nonparametric functional−beta CAPM: parametric constant−beta CAPM

1234

v

**Healthcare**

1234

v

Fig. 7. For Health care sector index:

Fig. 6. For Finance sector index:

functional−beta CAPM: nonparametric functional−beta CAPM: parametric constant−beta CAPM

Fig. 5. For Energy sector index:

**Energy**

functional−beta CAPM: nonparametric functional−beta CAPM: parametric constant−beta CAPM

0.0

0.0

0.0

0.5

1.0

beta(.)

1.5

0.5

beta(.)

1.0

1.5

 0.5

 1.0

beta(.)

 1.5

 2.0

Fig. 8. For Materials sector index:

v

Fig. 9. For Financial-v-Properties Trusts sector index:

Fig. 10. For ANZ bank group limited:

beta(.)

v

(i) Traditional CAPM: *Rit* = *α<sup>i</sup>* + *βiRmt* + *εit β<sup>i</sup>* = *βi*<sup>0</sup>

From a Three-Beta to a Functional-Beta Capital Asset Pricing Model

�

*Rit* = *α<sup>i</sup>* + *βit*,*L*(*ILtRMt*) + *βit*,*N*(*INtRMt*) + *βit*,*H*(*IHtRMt*) + *εit*

⎧ ⎪⎨

⎪⎩

⎧ ⎪⎨

⎪⎩

(iv) Three-regime (two-threshold) CAPM:

�*σ*<sup>2</sup> <sup>=</sup> <sup>1</sup> *n*

*n* ∑ *t*=1

(*Rit* − *R*

relationship between market return and individual return.

�*it*)�

method.

point *σ*<sup>2</sup>

volatility *σ*<sup>2</sup>

(ii)Linear-beta CAPM: *Rit* = *α<sup>i</sup>* + *βitRmt* + *εit βit* = *βi*<sup>0</sup> + *βi*1*σ*<sup>2</sup>

*βit*,*<sup>L</sup>* = *βi*<sup>0</sup> + *βi*1*σ*<sup>2</sup>

*βit*,*<sup>H</sup>* = *βi*<sup>2</sup> + *βi*3*σ*<sup>2</sup>

*βit*,*<sup>L</sup>* = *βi*<sup>0</sup> + *βi*1*σ*<sup>2</sup>

*βit*,*<sup>N</sup>* = *βi*<sup>2</sup> + *βi*3*σ*<sup>2</sup>

*βit*,*<sup>H</sup>* = *βi*<sup>4</sup> + *βi*5*σ*<sup>2</sup>

(iii) Two-regime (one-threshold) CAPM: *Rit* = *α<sup>i</sup>* + *βit*,*L*(*ILtRMt*) + *βit*,*H*(*IHtRMt*) + *εit*

<sup>405</sup> Risk, Return and Market Condition:

*Mt*, *<sup>σ</sup>*<sup>2</sup>

*Mt*, *<sup>σ</sup>*<sup>2</sup>

*Mt*, *<sup>σ</sup>*<sup>2</sup>

*Mt*, *<sup>σ</sup>*<sup>2</sup>

*Mt*, *<sup>σ</sup>*<sup>2</sup>

*Mt* <sup>≤</sup> *<sup>σ</sup>*<sup>2</sup> *L*

*<sup>L</sup>* <sup>&</sup>lt; *<sup>σ</sup>*<sup>2</sup>

Here all the *αi*, *βi*0, *βi*1, *βi*2, *βi*3, *βi*4, *βi*<sup>5</sup> are constants to be estimated by linear regression

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*), where *m* 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 *Ri* = (*Ri*1, ··· , *RiT*)� = *X***b** + (*εi*1, ··· ,*εiT*)� by suitably defining a *T* × *m* matrix *X*, where *m* is the number of parameters in each of the five CAPMs (See Table 4). Then we can define

> *AIC*(*m*) = *<sup>T</sup>* log �*σ*<sup>2</sup> <sup>+</sup> *<sup>T</sup>*(2*m*) *R*�*<sup>i</sup>* = *HRi H* = *X*(*X*�

> > (*Rit* − *R*

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

The two-stepwise beta functional in Model (iii) estimated by using the common changing

*<sup>L</sup>* 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

*Mt* (c.f., Figure 4), the results of nonparametric estimation are poor and unreliable at

�*it*) = <sup>1</sup> *n R*� *i* (*I* − *H*)�

*Mt* <sup>&</sup>gt; *<sup>σ</sup>*<sup>2</sup> *H*.

(v)Three-beta CAPM: *Rit* = *α<sup>i</sup>* + *βit*,*L*(*ILtRMt*) + *βit*,*N*(*INtRMt*) + *βit*,*H*(*IHtRMt*) + *εit*

*βit*,*<sup>L</sup>* = *βi*0, *σ*<sup>2</sup>

*βit*,*<sup>N</sup>* = *βi*2, *σ*<sup>2</sup>

*βit*,*<sup>H</sup>* = *βi*4, *σ*<sup>2</sup>

*Mt* <sup>≤</sup> *<sup>σ</sup>*<sup>2</sup> *L*

*Mt* > *<sup>σ</sup>*<sup>2</sup> *L*.

*Mt* <sup>≤</sup> *<sup>σ</sup>*<sup>2</sup> *L*

*<sup>L</sup>* < *<sup>σ</sup>*<sup>2</sup>

*Mt* > *<sup>σ</sup>*<sup>2</sup> *H*.

*Mt* <sup>≤</sup> *<sup>σ</sup>*<sup>2</sup> *H*

*Mt* <sup>≤</sup> *<sup>σ</sup>*<sup>2</sup> *H* *Mt*

*X*)−1*X*� (20)

(*I* − *H*)*Ri*. (21)


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.

#### **4.2 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*<sup>2</sup> *<sup>L</sup>* and *<sup>σ</sup>*<sup>2</sup> *<sup>H</sup>* 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*<sup>2</sup> *<sup>L</sup>* = 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*<sup>2</sup> *<sup>L</sup>* = 1.4 and *σ*<sup>2</sup> *<sup>H</sup>* = 2.8. (v)The fifth model is the three-beta CAPM of (Galagedera and Faff 2005), with *σ*2 *<sup>L</sup>* = 1.4 and *<sup>σ</sup>*<sup>2</sup> *<sup>H</sup>* = 2.8. Specifically,

14 Will-be-set-by-IN-TECH

Bandwidth 0.45 0.56 0.58 0.38 0.53 0.78

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

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

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

With reference to the results of non-parametric estimates in Figures 5–10, the market volatility

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.

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*<sup>2</sup>

*<sup>H</sup>* = 2.8. (v)The fifth model is the three-beta CAPM of (Galagedera and Faff 2005), with

*<sup>H</sup>* are quite amazingly very close to 1.4 and 2.8, respectively,

*<sup>L</sup>* = 1.4 as the

*<sup>L</sup>* = 1.4

general flexible functional-beta model which fits the financial data more adaptively.

From the nonparametric estimates in Figures 5–10, it looks reasonable to set *σ*<sup>2</sup>

*<sup>L</sup>* and *<sup>σ</sup>*<sup>2</sup>

*<sup>H</sup>* = 2.8. Specifically,

Table 3. Bandwidth Selection

subsection.

functional.

and *σ*<sup>2</sup>

*<sup>L</sup>* = 1.4 and *<sup>σ</sup>*<sup>2</sup>

*σ*2

**4.2 Parametric analysis**

changing regime points *σ*<sup>2</sup>

Energy Finance Healthcare Materials Fin-V-Prop ANZ

(i) Traditional CAPM: *Rit* = *α<sup>i</sup>* + *βiRmt* + *εit β<sup>i</sup>* = *βi*<sup>0</sup> (ii)Linear-beta CAPM: *Rit* = *α<sup>i</sup>* + *βitRmt* + *εit βit* = *βi*<sup>0</sup> + *βi*1*σ*<sup>2</sup> *Mt* (iii) Two-regime (one-threshold) CAPM: *Rit* = *α<sup>i</sup>* + *βit*,*L*(*ILtRMt*) + *βit*,*H*(*IHtRMt*) + *εit*

$$\begin{cases} \beta\_{\dot{t}t,L} = \beta\_{\dot{t}0} + \beta\_{\dot{t}1} \sigma\_{Mt\prime}^2 & \sigma\_{Mt}^2 \le \sigma\_L^2\\ \beta\_{\dot{t}t,H} = \beta\_{\dot{t}2} + \beta\_{\dot{t}3} \sigma\_{Mt\prime}^2 & \sigma\_{Mt}^2 > \sigma\_L^2. \end{cases}$$

(iv) Three-regime (two-threshold) CAPM: *Rit* = *α<sup>i</sup>* + *βit*,*L*(*ILtRMt*) + *βit*,*N*(*INtRMt*) + *βit*,*H*(*IHtRMt*) + *εit*

$$\begin{cases} \boldsymbol{\beta}\_{\boldsymbol{it},\boldsymbol{L}} = \boldsymbol{\beta}\_{\boldsymbol{i}0} + \boldsymbol{\beta}\_{\boldsymbol{i}1} \boldsymbol{\sigma}\_{\boldsymbol{Mt}\prime}^{2} & \boldsymbol{\sigma}\_{\boldsymbol{Mt}}^{2} \le \boldsymbol{\sigma}\_{\boldsymbol{L}}^{2} \\ \boldsymbol{\beta}\_{\boldsymbol{it},\boldsymbol{N}} = \boldsymbol{\beta}\_{\boldsymbol{i}2} + \boldsymbol{\beta}\_{\boldsymbol{i}3} \boldsymbol{\sigma}\_{\boldsymbol{Mt}\prime}^{2} & \boldsymbol{\sigma}\_{\boldsymbol{L}}^{2} < \boldsymbol{\sigma}\_{\boldsymbol{Mt}}^{2} \le \boldsymbol{\sigma}\_{\boldsymbol{H}}^{2} \\ \boldsymbol{\beta}\_{\boldsymbol{it},\boldsymbol{H}} = \boldsymbol{\beta}\_{\boldsymbol{i}4} + \boldsymbol{\beta}\_{\boldsymbol{i}5} \boldsymbol{\sigma}\_{\boldsymbol{Mt}\prime}^{2} & \boldsymbol{\sigma}\_{\boldsymbol{Mt}}^{2} > \boldsymbol{\sigma}\_{\boldsymbol{H}}^{2} . \end{cases}$$

(v)Three-beta CAPM: *Rit* = *α<sup>i</sup>* + *βit*,*L*(*ILtRMt*) + *βit*,*N*(*INtRMt*) + *βit*,*H*(*IHtRMt*) + *εit*

$$\begin{cases} \beta\_{\mathrm{i}t,L} = \beta\_{\mathrm{i}0\prime} & \sigma\_{\mathrm{M}t}^2 \le \sigma\_L^2\\ \beta\_{\mathrm{i}t,N} = \beta\_{\mathrm{i}2\prime} & \sigma\_L^2 < \sigma\_{\mathrm{M}t}^2 \le \sigma\_H^2\\ \beta\_{\mathrm{i}t,H} = \beta\_{\mathrm{i}4\prime} & \sigma\_{\mathrm{M}t}^2 > \sigma\_H^2. \end{cases}$$

Here all the *αi*, *βi*0, *βi*1, *βi*2, *βi*3, *βi*4, *βi*<sup>5</sup> are constants to be estimated by linear regression method.

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*), where *m* 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 *Ri* = (*Ri*1, ··· , *RiT*)� = *X***b** + (*εi*1, ··· ,*εiT*)� by suitably defining a *T* × *m* matrix *X*, where *m* is the number of parameters in each of the five CAPMs (See Table 4). Then we can define

$$\begin{aligned} AIC(m) &= T \log \widehat{\sigma}^2 + T(2m) \\ \widehat{R}\_i &= HR\_i \qquad H = X(X^\prime X)^{-1}X^\prime \end{aligned} \tag{20}$$

$$
\widehat{\sigma}^2 = \frac{1}{n} \sum\_{t=1}^n (\mathcal{R}\_{it} - \widehat{\mathcal{R}\_{it}})'(\mathcal{R}\_{it} - \widehat{\mathcal{R}\_{it}}) = \frac{1}{n} \mathcal{R}\_i'(I - H)'(I - H)\mathcal{R}\_i. \tag{21}
$$

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*<sup>2</sup> *<sup>L</sup>* 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*<sup>2</sup> *Mt* (c.f., Figure 4), the results of nonparametric estimation are poor and unreliable at

*α<sup>i</sup> βi*<sup>0</sup> *βi*<sup>1</sup> *βi*<sup>2</sup> *βi*<sup>3</sup> *βi*<sup>4</sup> *βi*<sup>5</sup> AIC

*α<sup>i</sup> βi*<sup>0</sup> *βi*<sup>1</sup> *βi*<sup>2</sup> *βi*<sup>3</sup> *βi*<sup>4</sup> *βi*<sup>5</sup> AIC

Threshold CAPM (2 step) 3643.563

<sup>407</sup> Risk, Return and Market Condition:

Threshold CAPM (3 step) 3645.087 Estimate 0.0355 1.1928 -0.6484 -0.2079 0.3883 0.8428 0.0377 T statistic 0.7187 3.6012 -2.2237 -0.8730 3.3893 1.3094 0.1965 p-value 0.4725 0.0003 0.0264 0.3829 0.0007 0.1907 0.8442 Three-beta CAPM (3 step) 3655.554

Estimate 0.0255 0.4660 0.5790 0.9690 T statistic 0.5174 8.6687 10.667 11.805 p-value 0.6050 1.8e-17 3.3e-25 3.6e-30

and (v) for Finance Sector Index

and (v) for Health care Sector Index

Table 6. Estimate, T-statistic and p-value for *α*ˆ and each component of *β*ˆ in Models (iii), (iv)

Threshold CAPM (2 step) 3318.096

Threshold CAPM (3 step) 3320.025 Estimate 0.0082 0.5475 -0.2469 0.1961 0.0198 -0.0505 0.1589 T statistic 0.1969 1.9494 -0.9990 0.9716 0.2043 -0.0926 0.9764 p-value 0.8440 0.0515 0.3181 0.3315 0.8381 0.9263 0.3291 Three-beta CAPM (3 step) 3316.039

Estimate 0.0054 0.2707 0.2365 0.4784 T statistic 0.1308 5.9824 5.1772 6.9238 p-value 0.8960 3.1e-09 2.7e-07 7.9e-12

Table 7. Estimate, T-statistic and p-value for *α*ˆ and each component of *β*ˆ in Models (iii), (iv)

Estimate 0.0080 0.5473 -0.2468 -0.0246 0.1382 T statistic 0.1909 1.9487 -0.9983 -0.1924 2.7342 p-value 0.8486 0.0516 0.3183 0.8475 0.0064

Estimate 0.0332 1.1911 -0.6470 -0.0248 0.2982 T statistic 0.6728 3.5952 -2.2182 -0.1642 5.0018 p-value 0.5013 0.0003 0.0268 0.8696 6.7e-07

From a Three-Beta to a Functional-Beta Capital Asset Pricing Model


Table 4. *AIC*(*m*) and The Type of CAPM Chosen

extreme market volatility, while the parametric results of two-stepwise or three-stepwise beta functionals provide reasonable outcomes in Figures 5–10. Under moderate market volatility, both nonparametric and parametric outcomes are pretty consistent.

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 *βi*<sup>1</sup> = *βi*<sup>3</sup> = *βi*<sup>5</sup> = 0. To examine their work, here we test the significance of *βi*1, *βi*3, *βi*<sup>5</sup> by applying T-statistics:

$$H\_0: \mathcal{J}\_{i1} = \mathcal{J}\_{i3} = \mathcal{J}\_{i5} = 0 \tag{22}$$

Applying linear regression method we get **<sup>b</sup>** , and residuals *<sup>r</sup>*:

$$\hat{\mathbf{b}} = (X^\prime X)^{-1} X^\prime \mathbb{R}\_{i\prime} \text{ where } \mathbf{b} = (\mathfrak{a}\_{i\prime} \mathfrak{z}\_{i0\prime} \mathfrak{z}\_{i1\prime} \mathfrak{z}\_{i2\prime} \mathfrak{z}\_{i3\prime} \mathfrak{z}\_{i4\prime} \mathfrak{z}\_{i5})^{\prime},\tag{23}$$

and let *<sup>δ</sup>* stand for the standard deviation of *<sup>r</sup>* <sup>=</sup> *Ri* <sup>−</sup> *<sup>X</sup>***<sup>b</sup>** . Then the T-statistics value of each estimated parameter is *Tj* <sup>=</sup> *bj*/{(*XXT*)−<sup>1</sup> *jj <sup>δ</sup>*2}1/2, where *bj* represents the *<sup>j</sup>*th element of **<sup>b</sup>** . 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.


Table 5. Estimate, T-statistic and p-value for *α*ˆ and each component of *β*ˆ in Models (iii), (iv) and (v) for Energy Sector Index

16 Will-be-set-by-IN-TECH

Model (i): m=2 (ii): m=3 (iii): m=5 (iv): m=7 (v): m=4 chosen CAPM Energy 3869.947 3799.963 3778.340 3779.698 3805.315 (iii) Finance 3677.854 3650.706 3643.563 3645.087 3655.554 (iii) HealthCare 3321.021 3318.320 3318.096 3320.025 3316.039 (v) Materials 4137.633 4066.287 4044.999 4026.751 4077.107 (iv) F-v-P Trusts 3778.767 3755.012 3746.462 3747.036 3758.679 (iii) ANZ bank 4577.802 4571.992 4558.787 4558.482 4578.409 (iv)

extreme market volatility, while the parametric results of two-stepwise or three-stepwise beta functionals provide reasonable outcomes in Figures 5–10. Under moderate market volatility,

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 *βi*<sup>1</sup> = *βi*<sup>3</sup> = *βi*<sup>5</sup> = 0. To examine their

and let *<sup>δ</sup>* stand for the standard deviation of *<sup>r</sup>* <sup>=</sup> *Ri* <sup>−</sup> *<sup>X</sup>***<sup>b</sup>** . Then the T-statistics value of each

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

Threshold CAPM (2 step) 3778.340

Threshold CAPM (3 step) 3779.698 Estimate -0.0192 1.5024 -1.0196 -0.5792 0.5527 0.5988 0.1933 T statistic -0.3625 4.2365 -3.2654 -2.2719 4.5061 0.8689 0.9408 p-value 0.7171 2.5e-05 0.0011 0.0233 7.4e-06 0.3851 0.3470 Three-beta CAPM (3 step) 3805.315

Estimate -0.0336 0.3597 0.5410 1.2432 T statistic -0.6315 6.2018 9.2367 14.035 p-value 0.5278 8.2e-10 1.5e-19 6.4e-41

Table 5. Estimate, T-statistic and p-value for *α*ˆ and each component of *β*ˆ in Models (iii), (iv)

Estimate -0.0222 1.5001 -1.0176 -0.4794 0.5093 T statistic -0.4199 4.2287 -3.2583 -2.9629 7.9788 p-value 0.6746 2.6e-05 0.0012 0.0031 4.1e-15

*Ri*, *where* **b** = (*αi*, *βi*0, *βi*1, *βi*2, *βi*3, *βi*4, *βi*5)�

*H*<sup>0</sup> : *βi*<sup>1</sup> = *βi*<sup>3</sup> = *βi*<sup>5</sup> = 0 (22)

*jj <sup>δ</sup>*2}1/2, where *bj* represents the *<sup>j</sup>*th element of **<sup>b</sup>** .

*α<sup>i</sup> βi*<sup>0</sup> *βi*<sup>1</sup> *βi*<sup>2</sup> *βi*<sup>3</sup> *βi*<sup>4</sup> *βi*<sup>5</sup> AIC

, (23)

Table 4. *AIC*(*m*) and The Type of CAPM Chosen

**b**ˆ = (*X*�

estimated parameter is *Tj* <sup>=</sup> *bj*/{(*XXT*)−<sup>1</sup>

sector index.

and (v) for Energy Sector Index

both nonparametric and parametric outcomes are pretty consistent.

Applying linear regression method we get **<sup>b</sup>** , and residuals *<sup>r</sup>*:

*X*)−1*X*�

work, here we test the significance of *βi*1, *βi*3, *βi*<sup>5</sup> by applying T-statistics:




Table 7. Estimate, T-statistic and p-value for *α*ˆ and each component of *β*ˆ in Models (iii), (iv) and (v) for Health care Sector Index

*α<sup>i</sup> βi*<sup>0</sup> *βi*<sup>1</sup> *βi*<sup>2</sup> *βi*<sup>3</sup> *βi*<sup>4</sup> *βi*<sup>5</sup> AIC

*<sup>M</sup>*, which is more flexible and adaptive to the changing

Threshold CAPM (2 step) 4558.787

<sup>409</sup> Risk, Return and Market Condition:

Threshold CAPM (3 step) 4558.482 Estimate 0.0903 2.5183 -2.0740 0.1687 -0.0282 2.4172 -0.7560 T statistic 1.1504 4.7822 -4.4736 0.4455 -0.1547 2.3621 -2.4779 p-value 0.2503 2.0e-06 8.6e-06 0.6560 0.8771 0.0184 0.0134 Three-beta CAPM (3 step) 4578.409

Table 10. Estimate, T-statistic and p-value for *α*ˆ and each component of *β*ˆ in Models (iii), (iv)

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<sup>i</sup>* changing

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

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

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.

Estimate 0.0453 0.1929 0.1149 -0.0954 T statistic 0.5749 2.2468 1.3247 -0.7278 p-value 0.5655 0.0249 0.1856 0.4669

and (v) for ANZ bank group limited

functionally with the market volatility *σ*<sup>2</sup>

**5. Closure and future work**

structure of financial systems.

fluctuations.

selection.

Estimate 0.0851 2.5144 -2.0707 0.4902 -0.1826 T statistic 1.0836 4.7693 -4.4614 2.03844 -1.9251 p-value 0.2788 2.1e-06 9.1e-06 0.0418 0.0545

From a Three-Beta to a Functional-Beta Capital Asset Pricing Model


Table 8. Estimate, T-statistic and p-value for *α*ˆ and each component of *β*ˆ in Models (iii), (iv) and (v) for Materials Sector Index


Table 9. Estimate, T-statistic and p-value for *α*ˆ and each component of *β*ˆ in Models (iii), (iv) and (v) for Financial-x-trusts Sector Index

18 Will-be-set-by-IN-TECH

Threshold CAPM (2 step) 4044.999

Threshold CAPM (3 step) 4026.751 Estimate -0.0140 1.8769 -1.3594 -1.0294 0.8800 2.6250 -0.3886 T statistic -0.2341 4.6686 -3.8409 -3.5619 6.3288 3.3600 -1.6685 p-value 0.8150 3.5e-06 0.0001 0.0004 3.8e-10 0.0008 0.0955 Three-beta CAPM (3 step) 4077.107

Estimate -0.0392 0.3529 0.7542 1.3335 T statistic -0.6415 5.3006 11.218 13.115 p-value 0.5214 1.4e-07 1.5e-27 2.5e-36

and (v) for Materials Sector Index

and (v) for Financial-x-trusts Sector Index

Table 8. Estimate, T-statistic and p-value for *α*ˆ and each component of *β*ˆ in Models (iii), (iv)

Threshold CAPM (2 step) 3746.462

Threshold CAPM (3 step) 3747.036 Estimate 0.0267 1.3202 -0.7441 -0.2572 0.4141 1.0547 -0.0197 T statistic 0.5133 3.7850 -2.4231 -1.0259 3.4323 1.5560 -0.0975 p-value 0.6078 0.0002 0.0156 0.3052 0.0006 0.1200 0.9224 Three-beta CAPM (3 step) 3758.679

Estimate 0.0146 0.4861 0.5820 0.9900 T statistic 0.2780 8.5826 10.176 11.445 p-value 0.7797 3.6e-17 3.4e-23 1.5e-28

Table 9. Estimate, T-statistic and p-value for *α*ˆ and each component of *β*ˆ in Models (iii), (iv)

Estimate 0.0238 1.3180 -0.7423 -0.0411 0.3083 T statistic 0.4574 3.7760 -2.4154 -0.2579 4.9091 p-value 0.6475 0.0002 0.0159 0.7966 1.1e-06

Estimate -0.0213 1.8714 -1.3549 -0.2381 0.4821 T statistic -0.3518 4.6074 -3.7889 -1.2850 6.5962 p-value 0.7251 4.6e-06 0.0002 0.1991 6.9e-11

*α<sup>i</sup> βi*<sup>0</sup> *βi*<sup>1</sup> *βi*<sup>2</sup> *βi*<sup>3</sup> *βi*<sup>4</sup> *βi*<sup>5</sup> AIC

*α<sup>i</sup> βi*<sup>0</sup> *βi*<sup>1</sup> *βi*<sup>2</sup> *βi*<sup>3</sup> *βi*<sup>4</sup> *βi*<sup>5</sup> AIC


Table 10. Estimate, T-statistic and p-value for *α*ˆ and each component of *β*ˆ in Models (iii), (iv) and (v) for ANZ bank group limited

#### **5. 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<sup>i</sup>* changing functionally with the market volatility *σ*<sup>2</sup> *<sup>M</sup>*, 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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#### **6. 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.

#### **7. References**


20 Will-be-set-by-IN-TECH

The work of this research was partially supported by the Discovery Project and Future

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**6. Acknowledgements**

**7. References**

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˚

*Econometrics* **31**, 307–327.


**18** 

*France* 

Hayette Gatfaoui *Rouen Business School,* 

**Linking U.S. CDS Indexes with the U.S. Stock Market: A Multidimensional Analysis with the** 

**Market Price and Market Volatility Channels** 

The recent mortgage subprime crisis and the partly resulting global financial crisis shed light on the weaknesses and required enhancements of prevailing risk management practices (e.g. Basel 2 limitations and frailties). Among the most important enforcements, liquidity concerns, counterparty credit risk, the correlation between various risks (e.g. contagion risk) and model stress testing as well as related scenario analysis have been highlighted by the Basel Committee on Banking Supervision under the Basel 3 framework (Basel Committee on Banking Supervision, 2010). With regard to liquidity, various liquidity measures coupled sometimes with a stressed scenario analysis are proposed at both the level of financial assets and the bank level in in- and off-balance sheet prospects (Blundell-Wignall and Atkinson, 2010; Lumsdaine, 2011; Van Den End, 2011). On the correlation viewpoint, the risk of correlation between risks (e.g. impacts of liquidity risk on market risk and *vice versa*) refers to the linkages between asset classes, and linkages between banks/financial institutions among others (Embrechts *et al.*, 2002, 2003; Hull and White, 2006; Lumsdaine, 2011). On the stress testing and scenario viewpoints, the mitigation of potential model risk and measurement errors is targeted in risk assessment and risk forecast

prospects (Basel Committee on Banking Supervision, 2010; Ferrari *et al.*, 2011).

Current research suggest that the credit market, as represented by credit default swaps or corporate bond spreads, is highly sensitive to the stock market trend and/or the corresponding market volatility as represented by equity volatility (Dupuis *et al.*, 2009; Gatfaoui, 2010; Scheicher, 2009; Vaz de Melo Mendes and Kolev, 2008). Under the Basel 3 setting, we focus concurrently on the correlation risk between credit default swap (CDS) spreads and referenced market risk components (Norden and Weber, 2009). Specifically, CDS spreads represent a credit risk proxy whereas market risk is envisioned with respect to two dimensions, namely a market price risk and a market volatility risk (Gatfaoui, 2010). The market price risk illustrates the impact of the global market trend (i.e. common general link within the stock market) whereas the market volatility risk represents the magnitude of global market moves (i.e. volatility feedback, liquidity concerns). In this lens, we study the asymmetric linkages between CDS spreads and both market price and market volatility risks, and we analyze the interaction between such linkages through a three-dimensional

**1. Introduction** 

