UNIVERSITEIT VAN AMSTERDAM
Does Liquidity Risk
Influence Asset Pricing?
A discussion under a Best-beta framework
Shen, Cheng (10650423)7/7/2014
Master thesis in Finance Supervisor: Dr. Liang Zou
Abstract:
This thesis explores the liquidity adjusted Best-beta framework based on the approach in Archarya & Pedersen (2005). In liquidity adjusted Best-beta framework, expected return of an asset results from not only market return but also the liquidity risk. Generally, an illiquid asset needs a higher return so as to compensate the potential loss generated by illiquidity issue. This thesis also contains an empirical test. In the empirical results, a discussion is addressed to capture both statistical significance and economic significance. The findings show the liquidity adjusted Best-beta framework is a good alternative to the standard version, especially when the liquidity risk is large.
1 | P a g e
1. Introduction
The concept of liquidity risk is elusive. Nikolaou (2009) distinguishes the market liquidity risk from funding liquidity risk. This thesis mainly explores the market liquidity risk in asset pricing. Market liquidity risk1 is the term that describes the difficulties that are encountered in trading a security. An investor may be much more willing to pay a higher price for a liquid asset (CFA institution, 2013). Liquidity risk has attracted much attention from researchers (Davis & Steil, 2001; Amihud, Mendelson & Wood, 1990; Pastor & Stambaugh, 2003; amongst others) since the 1987 stock market crash. Several models have been proposed to measure liquidity risk and to link such risk with asset prices in financial markets (e.g., Amihud, 2002; Amihu, Mendelson & Pedersen. 2005; Gerhold, Guasoni, Muhle-Karbe & Schachemayer. 2014; Archarya & Pedersen, 2005; amongst others).
Amongst these scholars, Archarya & Pedersen (2005) have achieved an impressive research outcome. The authors impose a liquidity adjustment on the CAPM model, leading to a 20% more suitability for the asset pricing.
Recently, Zou (2005) and Zou (2006) have undertaken the same proposal of improving accuracy of asset pricing but has ultimately established methods that point in a different direction. Zou’s articles aim at improving the asset-pricing model by adjusting the beta-return relation. Best-beta CAPM (Zou, 2006) turns out to predict return 20% (on average) more accurate than CAPM. The Dichotomous Asset Pricing Model (also known as DAPM) (Zou, 2005) is more accurate than Best-beta CAPM in theory. However, there is no published empirical evidence regarding the validity of DAPM.
Few researches have thus far affirmed attempts to discover the potential improvement in asset pricing regarding the new models, as the Best-beta CAPM and DAPM are newly developed models. This deficiency yields a question: is it possible to integrate the two approaches above together? In other words, does the liquidity adjustment influence asset pricing under the Best-beta framework (the DAPM and the Best-Best-beta CAPM)?
This thesis will contribute to existing literature on best-beta framework (Zou, 2006 and Zou, 2005) by providing a supplementary Best-beta framework with liquidity adjustment and a
1 This thesis only discusses the market liquidity risk. In order to distinguish the market liquidity risk from the
market liquidity risk of the market portfolio (a variable mentioned in following sections), the market liquidity risk is called the liquidity risk directly throughout this thesis.
2 | P a g e corresponding empirical test. A hypothetical world where all securities are perfectly liquid is proposed in this thesis. For every risky asset in the real world, there is one and only one corresponding homogenous risky asset in the hypothetical world. Liquidity Premium, a concept in Gerhold et al. (2014), is employed to construct the relationship between a real-world risky asset and its corresponding homogenous risky asset in the hypothetical real-world. The Best-beta framework (Best-beta CAPM and DAPM) is used to price the hypothetical homogeneous asset. By replacing the return of homogenous asset in hypothetical world with the return of the corresponding real-world asset and its liquidity premium, liquidity adjusted Best-beta framework is generated. In liquidity adjusted Best-beta framework, there are three liquidity betas and one market beta. These liquidity betas in best-beta CAPM, up-market DAPM and down-market DAPM are:
𝐸𝐸(𝐿𝐿𝐿𝐿𝑃𝑃𝑃𝑃𝑖𝑖,[τ,τ +a]𝐿𝐿𝐿𝐿𝑃𝑃𝑃𝑃𝑚𝑚,[τ,τ +a]) 𝐸𝐸�(𝑥𝑥𝑚𝑚,[τ,τ +a]−𝐿𝐿𝐿𝐿𝑃𝑃𝑃𝑃𝑚𝑚,[τ,τ +a])2�; 𝐸𝐸(𝐿𝐿𝐿𝐿𝑃𝑃𝑃𝑃𝑖𝑖,[τ,τ +a]𝐿𝐿𝐿𝐿𝑃𝑃𝑃𝑃𝑚𝑚,[τ,τ +a]) 𝐸𝐸�(𝑥𝑥𝑚𝑚,[τ,τ +a]−𝐿𝐿𝐿𝐿𝑃𝑃𝑃𝑃𝑚𝑚,[τ,τ +a])2�; 𝐸𝐸(𝐿𝐿𝐿𝐿𝑃𝑃𝑃𝑃𝑖𝑖,[τ,τ +a]𝐿𝐿𝐿𝐿𝑃𝑃𝑃𝑃𝑚𝑚,[τ,τ +a]) 𝐸𝐸�(𝑥𝑥𝑚𝑚,[τ,τ +a]−𝐿𝐿𝐿𝐿𝑃𝑃𝑃𝑃𝑚𝑚,[τ,τ +a])2�, synergy of liquidity
premium of individual asset and market liquidity premium; 𝐸𝐸(𝑥𝑥𝑖𝑖,[τ,τ +a]𝐿𝐿𝐿𝐿𝑃𝑃𝑃𝑃𝑚𝑚,[τ,τ +a]) 𝐸𝐸�(𝑥𝑥𝑚𝑚,[τ,τ +a]−𝐿𝐿𝐿𝐿𝑃𝑃𝑃𝑃𝑚𝑚,[τ,τ +a])2�; 𝐸𝐸(𝑥𝑥𝑖𝑖,[τ,τ +a]𝐿𝐿𝐿𝐿𝑃𝑃𝑃𝑃𝑚𝑚,[τ,τ +a]) 𝐸𝐸�(𝑥𝑥𝑚𝑚,[τ,τ +a]−𝐿𝐿𝐿𝐿𝑃𝑃𝑃𝑃𝑚𝑚,[τ,τ +a])2�; 𝐸𝐸(𝑥𝑥𝑖𝑖,[τ,τ +a]𝐿𝐿𝐿𝐿𝑃𝑃𝑃𝑃𝑚𝑚,[τ,τ +a]) 𝐸𝐸�(𝑥𝑥𝑚𝑚,[τ,τ +a]−𝐿𝐿𝐿𝐿𝑃𝑃𝑃𝑃𝑚𝑚,[τ,τ +a])2�, synergy of return
of individual asset and market liquidity premium; and, 𝐸𝐸(𝐿𝐿𝐿𝐿𝑃𝑃𝑃𝑃𝑖𝑖,[τ,τ +a]𝑥𝑥𝑚𝑚,[τ,τ +a]) 𝐸𝐸�(𝑥𝑥𝑚𝑚,[τ,τ +a]−𝐿𝐿𝐿𝐿𝑃𝑃𝑃𝑃𝑚𝑚,[τ,τ +a])2�; 𝐸𝐸(𝐿𝐿𝐿𝐿𝑃𝑃𝑃𝑃𝑖𝑖,[τ,τ +a]𝑥𝑥𝑚𝑚,[τ,τ +a]) 𝐸𝐸�(𝑥𝑥𝑚𝑚,[τ,τ +a]−𝐿𝐿𝐿𝐿𝑃𝑃𝑃𝑃𝑚𝑚,[τ,τ +a])2�; 𝐸𝐸(𝐿𝐿𝐿𝐿𝑃𝑃𝑃𝑃𝑖𝑖,[τ,τ +a]𝑥𝑥𝑚𝑚,[τ,τ +a]) 𝐸𝐸�(𝑥𝑥𝑚𝑚,[τ,τ +a]−𝐿𝐿𝐿𝐿𝑃𝑃𝑃𝑃𝑚𝑚,[τ,τ +a])2�, synergy of liquidity
premium of individual asset and market return.
This thesis explores the time series regression of the Best-beta framework by using the credit-ranked NYSE stocks from 2003 to 2012. Then, a cross sectional regression is performed to investigate whether the liquidity betas have a significant impact on asset pricing. In contrast with the existing literature (Archarya & Pedersen, 2005; Lee, 2011), this thesis explores the specification test further. Besides tests on the different specifications of risk premia
(premium), Hansen’ J (known as J-test) is employed to test whether zero intercept model in cross sectional regression is applicable. Liquidity premium is measured by the method developed by Gerhold et al. (2014). Liquidity premium in this measurement is automatically normalized and doesn’t require other approaches of normalization. This research shows that the liquidity adjusted Best-beta framework performs better than the standard Best-beta framework in terms of squared-R and constant term for the cross sectional regression. However, the improvement is small in most cases except the Best-beta CAPM by using the data of low credit stocks. Hansen’ J cannot reject the zero-intercept cross sectional regression
3 | P a g e modeling for liquidity adjusted Best-beta CAPM with the data of low credit stocks. however, also using the data of low credit stocks, Hansen’ J rejects the zero-intercept cross sectional regression modeling for standard Best-beta CAPM. By adding the liquidity adjustment, the constant term in cross sectional regression for Best-beta CAPM of low credit stocks is eliminated and the suitability (squared-R) experiences a sharp increase. In the rest cases, Hansen’ J either accepts or rejects zero-intercept cross sectional regression modeling for both the standard Best-beta framework and the liquidity adjusted version at the same time.
Moreover, after liquidity adjustment, the suitability (squared-R) increases by a small amount as well in the rest cases.
The thesis confirms the results from previous literature (Archaya & Pedersen, 2005; Lee, 2011; Pastor & Stambaugh, 2003) as well. First, the multiple premia (namely duo-risk-premia and four-risk-premia) are not applicable in the cross sectional. As liquidity adjusted CAPM in (Archaya & Pedersen, 2005; Lee, 2011), the liquidity adjusted Best-beta framework also implies a single risk premium. When removing the model restriction of the single risk premium, risk premia are statistically different from each other. Indeed, the problem of multicollinearity also exists across these risk premia. Therefore, multi-risk-premia cross sectional regression modeling is rejected, which submits to the analysis in Acharya & Pedersen (2005) and Lee (2011). Second, the thesis explores the effect of liquidity risk. The combined effect of liquidity betas and liquidity premium is on average 0.746% monthly for low credit stocks and 0.201% monthly for high credit stocks. In contrast with the result of 4.6% (annually) in Archarya & Pedersen (2005) and the result of 7.5% (annually, excluding
liquidity premium) in Pastor & Stambaugh (2003), the estimations in this thesis share a common orders of magnitude with these results. The differences across the results may come from the components portfolios, the measurement of liquidity risk and sample period.
The thesis is formulated as follows. In Section 2, the thesis will detail background
information that enlightens this study. This section of the paper will principally introduce the highlights of previous papers that offer the core guidance of this paper. In specific, this section will cover the definition of liquidity premium, best-beta framework and the liquidity adjusted CAPM. In Section 3, the paper will derive the Best-beta framework with liquidity adjustment based on Archarya & Pedersen’s model (2005). The economic intuition in this model will thereafter be demonstrated. In Section 4, the paper will focus on the methodology, including the data that has been used in this research, and further data processing and
4 | P a g e regression by Generalized Method of Moments, known as GMM. The paper attempt to set different constraints on parameters and find out the most fit regression models. These models will then be applied to judge the suitability of liquidity adjustment and its corresponding economic implication. In Section 6, the composition of market portfolio will be changed. The thesis will trial different weight methods to check the robustness of the model. In Section 7, a conclusion will be made to have to an established an overview of this research.
2. Background
2.1. Assumptions
This paper assumes an overlapping economy. A new generation of investors appears in time t and leave in time t+1. Before an investor engages in the economy, he can choose between a real world economy and the hypothetical economy where no liquidity problem incurs. Once the decision is made, this investor can no longer change his position. At the initial status t, the investor has an endowment of one security and must sell it out at t+1. During the period in which he stays in the market, he can trade the stock for as many times as he wishes. However, he is required to possess and only possess one same security at any moment until the end. The absolute return for every transaction is recorded as the difference between end-up market price and initial market price within the sub-holding-period.
The thesis will assume the stock return in any case caters to the normal distribution in order to simplify this analysis. This assumption is fundamentally acknowledged in the literature (Hull, 2005: Bodie, 2011). While some other experts support a distribution style like Paretian distribution (Fama, 1965; Mandelbrot’s, 1963), the distribution models that are suggested by these almost have the same function in this paper. The thesis will select the simple and straightforward normal distribution in order to make the thesis comprehensible.
2.2. The Definition of Liquidity Premium
There are different ways to define the cost of liquidity. Ernst et al. (2009) argue that the cost on liquidity should contain three factors: the direct transaction cost, the cost caused by order
5 | P a g e size and the cost of delayed transaction. However, the more components that are added
generate the risk of more potential errors when the premium is determined. On the other hand, Amihud (2002) establish a liquidity measurement by linking it to the security volume.
However, trading frequency is expected to influence the outcome when these methods are used. Lee K.H. & Kim S.H. (2014) also argue that the empirical evidence may not support the Liquidity adjusted CAPM by using many popular liquidity measurements, including the measurement that was introduced by Amihud (2002).
Therefore, this thesis would like to apply the definition adopted in Gerhold et al. (2013), which defines liquidity premium as the extra return one is willing to give up to trade a homogenous risky asset but without the transaction cost. According to this definition, this thesis will set the return and price of risky asset i at time t as 𝑅𝑅𝐿𝐿,𝑡𝑡 and 𝑃𝑃𝐿𝐿,𝑡𝑡 respectively. Correspondingly, the returns and price of hypothetical homogenous risky asset are ℜ𝐿𝐿,𝑡𝑡 and ℘𝐿𝐿,𝑡𝑡. Two criteria must be declared before further explanation. Firstly, the return of
hypothetical homogenous risky asset is unobservable so that Eq. 1-5 is not the measurement of liquidity premium. This is presented here in order to introduce the concept of liquidity premium in this thesis, and to demonstrate how to normalize the liquidity premium according to this definition. The detailed measurement of liquidity premium will be displayed in the methodology section of this thesis. Secondly, and according to investigation of Gerhold et al. (2013), the absolute return of real world risky asset is simply the difference between the normal prices or middle prices at two time points, regardless of the liquidity factors that may decrease the return further.
The Liquidity premium, 𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝐿𝐿,𝑡𝑡 between any two-time points (namely n and k, n>k) follows:
𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝐿𝐿,𝑘𝑘 =𝑃𝑃𝐿𝐿,n 𝑃𝑃− 𝑃𝑃𝐿𝐿,k 𝐿𝐿,k −
℘𝐿𝐿,n− ℘𝐿𝐿,k ℘𝐿𝐿,k
(1)
So in unit observation period [τ, τ+a] (e.g. 1 day or 1 month), the one-off trade liquidity premium is: 𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝐿𝐿,τ =𝑃𝑃𝐿𝐿,τ+a 𝑃𝑃− 𝑃𝑃𝐿𝐿,τ 𝐿𝐿,τ − ℘𝐿𝐿,τ+a− ℘𝐿𝐿,τ ℘𝐿𝐿,τ (2)
Next, the investor is assumed to transact N times in the same unit observation period. The stock price at Nth transaction is noted as 𝑃𝑃𝐿𝐿,τ+n with n no larger than a. The last transaction
6 | P a g e occurs in the end of observation period when the stock price is 𝑃𝑃𝐿𝐿,τ+n or 𝑃𝑃𝐿𝐿,τ+a . Moreover, reinvestment in other securities (e.g. risk-free bonds) is prohibited and this investor is required to buy the same security immediately after selling one. Therefore, the return rate in unit observation period is:
𝑅𝑅𝐿𝐿,τ =𝑃𝑃𝐿𝐿,τ+a − 𝑃𝑃𝐿𝐿,τ+n−1 + 𝑃𝑃𝐿𝐿,τ+n−1 𝑃𝑃− 𝑃𝑃𝐿𝐿,τ+n−2 + ⋯ + 𝑃𝑃𝐿𝐿,τ+1− 𝑃𝑃𝐿𝐿,τ 𝐿𝐿,τ (3) ℜ𝐿𝐿,τ= ℘𝐿𝐿,τ+a −℘𝐿𝐿,τ+n−1 +℘𝐿𝐿,τ+n−1 −℘𝐿𝐿,τ+n−2 + ⋯ +℘𝐿𝐿,τ+1−℘𝐿𝐿,τ ℘𝐿𝐿,τ (4)
Then, the liquidity premium in this unit observation period becomes: 𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝐿𝐿,τ= 𝑅𝑅𝐿𝐿,τ− ℜ𝐿𝐿,τ =𝑃𝑃𝐿𝐿,τ+a − 𝑃𝑃𝐿𝐿,τ+n−1 + 𝑃𝑃𝐿𝐿,τ+n−1 𝑃𝑃− 𝑃𝑃𝐿𝐿,τ+n−2 + ⋯ + 𝑃𝑃𝐿𝐿,τ+1− 𝑃𝑃𝐿𝐿,τ 𝐿𝐿,τ −℘𝐿𝐿,τ+a −℘𝐿𝐿,τ+n−1 +℘𝐿𝐿,τ+n−1 ℘−℘𝐿𝐿,τ+n−2 + ⋯ +℘𝐿𝐿,τ+1−℘𝐿𝐿,τ 𝐿𝐿,τ =𝑃𝑃𝐿𝐿,τ+a 𝑃𝑃− 𝑃𝑃𝐿𝐿,τ 𝐿𝐿,τ − ℘𝐿𝐿,τ+a− ℘𝐿𝐿,τ ℘𝐿𝐿,τ (5)
From the equation (2) and equation (5), the liquidity premium in a certain time period is independent with the transaction frequency. The only factor that affects the value of liquidity premium is the selected period, where the start and end datum determine the difference in initial price and terminal price. Therefore, it is clear that the transaction frequency now is merely a noise when estimating the liquidity premium.
2.3. The Best-beta Framework
Best-beta framework presents not only a single model but also a series of models associated with the best-beta. In the research undertaken in this paper, the framework contains best-beta CAPM (Zou, 2006) and DAPM (Zou, 2005). In section 2.3.1 and 2.3.2, this thesis will briefly discuss the essential content of Best-beta CAPM and DAPM.
7 | P a g e 2.3.1. Best-beta CAPM
Fama & French (2003) empirically test the regular CAPM model and discover that the CAPM prediction fails to match the actual asset pricing. In Zou’s later work (2006), the CAPM is further utilized and modified. The modified CAPM, which predicts better than the regular CAPM model by 20% on average, is called the Best-beta CAPM. Much like the CAPM, the Best-beta CAPM also focuses on expected return- beta relation. The only difference is the beta. The Best-beta CAPM is indeed a non-intercept CAPM.
In regular CAPM model,
𝛽𝛽𝑀𝑀𝑀𝑀 = 𝐶𝐶𝐶𝐶𝐶𝐶(𝑥𝑥, 𝑥𝑥𝑚𝑚) 𝐶𝐶𝑉𝑉𝐿𝐿(𝑥𝑥𝑚𝑚)
(6)
𝑥𝑥 and 𝑥𝑥𝑚𝑚 stand for individual excess return and market excess return respectively.
In Best-beta CAPM, 𝛽𝛽𝐵𝐵 =𝐸𝐸(𝑥𝑥𝑥𝑥𝑚𝑚) 𝐸𝐸(𝑥𝑥𝑚𝑚2) = 𝐶𝐶𝐶𝐶𝐶𝐶(𝑥𝑥, 𝑥𝑥𝑚𝑚) + 𝐸𝐸(𝑥𝑥)𝐸𝐸(𝑥𝑥𝑚𝑚) 𝐸𝐸(𝑥𝑥𝑚𝑚2) = 𝐶𝐶𝑉𝑉𝐿𝐿(𝑥𝑥𝑚𝑚) 𝐸𝐸(𝑥𝑥𝑚𝑚2) 𝛽𝛽 𝑀𝑀𝑀𝑀 +𝐸𝐸(𝑥𝑥𝑚𝑚)2 𝐸𝐸(𝑥𝑥𝑚𝑚2) 𝛽𝛽 =𝐸𝐸(𝑥𝑥𝑚𝑚𝐸𝐸(𝑥𝑥2) − 𝐸𝐸(𝑥𝑥𝑚𝑚)2 𝑚𝑚2) 𝛽𝛽 𝑀𝑀𝑀𝑀+𝐸𝐸(𝑥𝑥𝑚𝑚)2 𝐸𝐸(𝑥𝑥𝑚𝑚2) 𝛽𝛽 = (1 −𝐸𝐸(𝑥𝑥𝐸𝐸(𝑥𝑥𝑚𝑚)2 𝑚𝑚2))𝛽𝛽 𝑀𝑀𝑀𝑀 +𝐸𝐸(𝑥𝑥𝑚𝑚)2 𝐸𝐸(𝑥𝑥𝑚𝑚2) 𝛽𝛽 (7)
𝛽𝛽 is the real beta and it cannot be predicted. From the equation above, the error of real beta and betas in two models is demonstrated below:
𝛽𝛽𝐵𝐵− 𝛽𝛽 = �1 −𝐸𝐸(𝑥𝑥𝑚𝑚)2 𝐸𝐸(𝑥𝑥𝑚𝑚2)� 𝛽𝛽 𝑀𝑀𝑀𝑀 + �𝐸𝐸(𝑥𝑥𝑚𝑚)2 𝐸𝐸(𝑥𝑥𝑚𝑚2) − 1� 𝛽𝛽 = �1 − 𝐸𝐸(𝑥𝑥𝑚𝑚)2 𝐸𝐸(𝑥𝑥𝑚𝑚2)� (𝛽𝛽 𝑀𝑀𝑀𝑀− 𝛽𝛽) (8)
Since the value of the term �1 −𝐸𝐸(𝑥𝑥𝑚𝑚)2
𝐸𝐸(𝑥𝑥𝑚𝑚2)� always stays between 0 and 1, therefore the error of
best-beta is consistently smaller than the regular beta. With the improvement of the accuracy of beta, this model is anticipated to outperform what regular CAPM performs.
8 | P a g e 2.3.2 The DAPM
In comparison with the Best-beta CAPM, the DAPM model progresses further. Zou (2005) attempts to split the aggregate market into two sub-markets, the up-market and the down-market respectively. The up-down-market implies the situation where the down-market return exceeds the risk-free interest rate (e.g. short-run T-bill). By contrast, the down-market demonstrates the situation where market return remains smaller than the risk-free interest rate. Consistently with Best-beta CAPM, the up-market beta and down-market are measured by best-beta. In addition, when mean-variance efficiency and gain-loss efficiency are both satisfied, then the DAPM model holds (Zou, 2005.) To illustrate mathematically, during the period [τ, τ +a], we have: Up-market: 𝐸𝐸�𝑥𝑥𝐿𝐿,[τ,τ +a]� = 𝛽𝛽+𝐸𝐸(𝑥𝑥𝑚𝑚[τ,τ +a]) (9) Down-market: 𝐸𝐸�𝑥𝑥𝐿𝐿,[τ,τ +a]� = 𝛽𝛽−𝐸𝐸(𝑥𝑥𝑚𝑚,[τ,τ +a]) (10) Where, 𝛽𝛽+ = 𝐸𝐸�𝑥𝑥𝐿𝐿,[τ,τ +a]𝑥𝑥𝑚𝑚,[τ,τ +a]�𝐺𝐺�𝑃𝑃(𝐺𝐺) 𝐸𝐸(𝑥𝑥𝑚𝑚,[τ,τ +a]2|𝐺𝐺)𝑃𝑃(𝐺𝐺) = 𝐸𝐸�𝑥𝑥𝐿𝐿,[τ,τ +a]𝑥𝑥𝑚𝑚,[τ,τ +a]�𝐺𝐺� 𝐸𝐸(𝑥𝑥𝑚𝑚,[τ,τ +a]2|𝐺𝐺) , 𝐺𝐺: 𝑥𝑥𝑚𝑚 > 0 (11) 𝛽𝛽− =𝐸𝐸�𝑥𝑥𝐿𝐿,[τ,τ +a]𝑥𝑥𝑚𝑚,[τ,τ +a]�𝐿𝐿�𝑃𝑃(𝐿𝐿) 𝐸𝐸(𝑥𝑥𝑚𝑚,[τ,τ +a]2|𝐿𝐿)𝑃𝑃(𝐿𝐿) = 𝐸𝐸�𝑥𝑥𝐿𝐿,[τ,τ +a]𝑥𝑥𝑚𝑚,[τ,τ +a]�𝐿𝐿� 𝐸𝐸(𝑥𝑥𝑚𝑚,[τ,τ +a]2|𝐿𝐿) 𝐿𝐿: 𝑥𝑥𝑚𝑚 ≤ 0 (12)
The two betas are equal to its beta of the Best-beta CAPM when mean variance efficiency and gain-loss efficiency are met (Zou, 2005). The DAPM model will indeed actually outperform the best-beta CAPM in the empirical analysis that follows in this paper.
9 | P a g e
2.4. Liquidity-adjusted CAPM
Essentially, Zou (2005) and Zou (2005, 2006) improve the asset pricing model by modifying the return-beta relation. However, scholars have also attempted to find other sources that have contributed to individual asset pricing. Among them, the Multifactor asset-pricing model that is proposed by Fama & French (1996) is the most well-known. Similarly, Archarya & Pedersen (2005) discover that liquidity risk plays an important role in asset pricing. However, in their paper, liquidity term is not treated as an external term as what Fama & French’s (1996) work. Instead, liquidity is treated as the position similar to risk-free rate. Archarya & Pedersen (2005) attempt to price perfect liquidated assets based on regular CAPM in order to obtain the corresponding homogenous real world asset. As a result, during the period [τ, τ + a], for any security i and corresponding benchmark portfolio, the Liquidity-adjusted CAPM is described mathematically below:
𝐸𝐸�𝑥𝑥𝐿𝐿,[τ,τ +a]− 𝐿𝐿𝐿𝐿,[τ,τ +a]� = 𝜆𝜆[τ,τ +a]𝐶𝐶𝐶𝐶𝐶𝐶(𝑥𝑥𝐿𝐿,[τ,τ +a]𝐶𝐶𝑉𝑉𝐿𝐿(𝑥𝑥−𝐿𝐿𝐿𝐿,[τ,τ +a], 𝑥𝑥𝑚𝑚[τ,τ +a]− 𝐿𝐿𝑚𝑚,[τ,τ +a]) 𝑚𝑚[τ,τ +a]− 𝐿𝐿𝑚𝑚,[τ,τ +a])
(13)
L here denotes a liquidity cost with a slightly different definition to the definition that is adopted in this thesis. The other variables are consistent with what has been defined before.
Then, if the net beta𝐶𝐶𝐶𝐶𝐶𝐶(𝑥𝑥𝐿𝐿,[τ,τ +a]−𝐿𝐿𝐿𝐿,[τ,τ +a] ,𝑥𝑥𝑚𝑚[τ,τ +a]−𝐿𝐿𝑚𝑚,[τ,τ +a])
𝐶𝐶𝑉𝑉𝐿𝐿(𝑥𝑥𝑚𝑚[τ,τ +a]−𝐿𝐿𝑚𝑚,[τ,τ +a]) is extended, there are four betas where
one market beta and three liquidity betas are included. Beta 𝐶𝐶𝐶𝐶𝐶𝐶(𝑥𝑥𝑖𝑖,[τ,τ +a],𝑥𝑥𝑚𝑚[τ,τ +a])
𝑀𝑀𝑉𝑉𝑃𝑃(𝑥𝑥𝑚𝑚[τ,τ +a]−𝐿𝐿𝑚𝑚,[τ,τ +a]) is the
market beta. It is comparable to the beta of regular CAPM model. Beta 𝐶𝐶𝐶𝐶𝐶𝐶(𝐿𝐿𝐿𝐿,[τ,τ +a],𝐿𝐿𝑚𝑚,[τ,τ +a]) 𝐶𝐶𝑉𝑉𝐿𝐿(𝑥𝑥𝑚𝑚[τ,τ +a]−𝐿𝐿𝑚𝑚,[τ,τ +a]) is
the one of the liquidity betas.Tthis means one needs an ‘allowance’ to hold a risky asset with a liquidation problem when the whole market is not liquid (Sadka, 2006).
Beta 𝐶𝐶𝐶𝐶𝐶𝐶(𝑥𝑥𝐿𝐿,[τ,τ +a],𝐿𝐿𝑚𝑚,[τ,τ +a])
𝐶𝐶𝑉𝑉𝐿𝐿(𝑥𝑥𝑚𝑚[τ,τ +a]−𝐿𝐿𝑚𝑚,[τ,τ +a]) is the liquidity beta as well. This beta has a negative influence on the
expected excess return. It purports that investors are expected to allow a lower return when making a trade-off between high return but high market illiquidity and low return with high market liquidity (Holmstrom and Tirole, 2000; Lustig & Chien, 2005). Similarly, Beta
𝐶𝐶𝐶𝐶𝐶𝐶(𝐿𝐿𝐿𝐿,[τ,τ +a],𝑥𝑥𝑚𝑚,[τ,τ +a])
10 | P a g e would like to possess a security with a good liquidity in a bear market even when the return is not high (Lee, 2011).
Liquidity adjusted CAPM improves the original CAPM significantly. According to Archarya & Pendersen (2005), the Liquidity-adjusted CAPM predicts more accurate than original CAPM by 20% empirically.
3: Derive the Liquidity-adjusted Best-beta Framework
Zou (2005) and Zou (2006) improve the asset pricing model in term of adjusting the expect return-beta relation. Archarya & Pendensen (2005) discover that the liquidity risk is a source that may determine the asset pricing as well. Indeed, both of these directions perform well and lead to a more precise prediction. This thesis will attempt to merge their ideas together and derive the liquidity adjusted best-beta framework. In section 3.1, the best-CAPM with liquidity risk is performed while the DAPM with liquidity risk is illustrated in Section 3.2.
3.1. The Best-beta CAPM with Liquidity Risk
The traditional asset-pricing model (e.g. the DAPM, the CAPM, the Best-beta CAPM) does not consider the liquidity premium. Hence, it is not apposite to utilize the traditional models directly to price asset. Similar to Archarya & Pedersen (2005), in the first stage, this thesis will construct the zero-transaction-cost securities based on their real world origins. These zero-transaction-cost securities are thus suitable for the traditional asset-pricing model. As is shown in section 2.2, the return on hypothetical non-transaction-cost security 𝔛𝔛𝐿𝐿 can be expressed as the difference between its homogenous real world asset return 𝑥𝑥𝐿𝐿and liquidity premium𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝐿𝐿.
Based on equation (1), it is obtained:2
2 Nyawata (2012) argues that the risk-free assets, especially T-bills, are superior liquid. Therefore, in this paper,
the liquidity premium for risk-free asset is treated as 0. Then, the risk-free returns in real world and non-transaction-cost world are equal.
11 | P a g e 𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝐿𝐿,[τ,τ +a] = 𝑅𝑅𝐿𝐿,[τ,τ +a]− ℜ𝐿𝐿,[τ,τ +a]
= �𝑥𝑥𝐿𝐿,[τ,τ +a]+ 𝐿𝐿𝑓𝑓,[τ,τ +a]� − �𝔛𝔛𝐿𝐿,[τ,τ +a]+ 𝐿𝐿𝑓𝑓,[τ,τ +a]� = 𝑥𝑥𝐿𝐿,[τ,τ +a]− 𝔛𝔛𝐿𝐿,[τ,τ +a]
(14)
𝔛𝔛𝐿𝐿,[τ,τ +a]= 𝑥𝑥𝐿𝐿,[τ,τ +a]− 𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝐿𝐿,[τ,τ +a] (15)
Then, the return of non-transaction-cost asset is priced by best-beta CAPM as follows:
𝐸𝐸(𝔛𝔛𝐿𝐿,[τ,τ +a]) = 𝛽𝛽𝐸𝐸(𝔛𝔛𝑚𝑚,[τ,τ +a]) (16)
𝛽𝛽 =𝐸𝐸(𝔛𝔛𝐿𝐿,[τ,τ +a]𝔛𝔛𝑚𝑚,[τ,τ +a]) 𝐸𝐸(𝔛𝔛𝑚𝑚,[τ,τ +a]2)
(17)
The next step is to substitute the hypothetical return rate with the real world one net its corresponding liquidity premium. This yields:
𝐸𝐸(𝑥𝑥𝐿𝐿,[τ,τ +a]− 𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝐿𝐿,[τ,τ +a])
=𝐸𝐸((𝑥𝑥𝐿𝐿,[τ,τ +a]− 𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝐿𝐿,[τ,τ +a])(𝑥𝑥𝑚𝑚,[τ,τ +a]− 𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝑚𝑚,[τ,τ +a])
𝐸𝐸�(𝑥𝑥𝑚𝑚,[τ,τ +a]−𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝑚𝑚,[τ,τ +a])2� 𝐸𝐸(𝑥𝑥𝑚𝑚,[τ,τ +a] −𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝑚𝑚,[τ,τ +a])
(18)
If the net beta is extended, comparable with the LCAPM, there are four betas regarding market risk and liquidity risk.
𝐸𝐸(𝑥𝑥𝐿𝐿,[τ,τ +a]) = 𝐸𝐸�𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝐿𝐿,[τ,τ +a]� + (𝛽𝛽1+ 𝛽𝛽2− 𝛽𝛽3− 𝛽𝛽4)𝐸𝐸(𝑥𝑥𝑚𝑚,[τ,τ +a] −𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝑚𝑚,[τ,τ +a]) (19) 𝛽𝛽1 = 𝐸𝐸(𝑥𝑥𝐿𝐿,[τ,τ +a]𝑥𝑥𝑚𝑚,[τ,τ +a]) 𝐸𝐸�(𝑥𝑥𝑚𝑚,[τ,τ +a]−𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝑚𝑚,[τ,τ +a])2� (20) 𝛽𝛽2= 𝐸𝐸(𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝐿𝐿,[τ,τ +a]𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝑚𝑚,[τ,τ +a]) 𝐸𝐸�(𝑥𝑥𝑚𝑚,[τ,τ +a]−𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝑚𝑚,[τ,τ +a])2� (21)
12 | P a g e 𝛽𝛽3= 𝐸𝐸(𝑥𝑥𝐿𝐿,[τ,τ +a]𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝑚𝑚,[τ,τ +a]) 𝐸𝐸�(𝑥𝑥𝑚𝑚,[τ,τ +a]−𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝑚𝑚,[τ,τ +a])2� (22) 𝛽𝛽4 = 𝐸𝐸(𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝐿𝐿,[τ,τ +a]𝑥𝑥𝑚𝑚,[τ,τ +a]) 𝐸𝐸�(𝑥𝑥𝑚𝑚,[τ,τ +a]−𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝑚𝑚,[τ,τ +a])2� (23)
In the same manner as the LCAPM, there is one market beta and three liquidity beta with respect to three different kinds of liquidity risks. Since the structure of the betas is similar to the ones specified in LCAPM, the economic interpretation of the betas in the LCAPM is also valid in this model. 𝛽𝛽1 is the market beta and functions as the beta in regular Best-beta CAPM. 𝛽𝛽2 evaluates the liquidity risk that lead investors to ask for compensation for hold risky asset that become illiquid once the market is illiquid. 𝛽𝛽3 states that the investors prefer low return with high market liquidity rather than high return but low market liquidity. 𝛽𝛽4 implies that the investors are willing to achieve a lower return in order to earn more liquidity in a bear market.
3.2. DAPM with Liquidity Risk
The way to determine liquidity-adjusted Best-beta CAPM is accessible for liquidity-adjusted DAPM as well. The first step is to plant the hypothetical non-transaction-cost risky asset 𝔛𝔛𝐿𝐿,. Depending on the value of 𝔛𝔛𝑚𝑚, the whole market is segmented into up-market (𝔛𝔛𝑚𝑚 > 0) and down-market (𝔛𝔛𝑚𝑚 ≤ 0).
For up-market, there is:
𝐸𝐸�𝔛𝔛𝐿𝐿,[τ,τ +a]� = 𝛽𝛽+𝐸𝐸(𝔛𝔛𝑚𝑚,[τ,τ +a]) (24)
𝛽𝛽+= 𝐸𝐸�𝔛𝔛𝐿𝐿,[τ,τ +a]𝔛𝔛𝑚𝑚,[τ,τ +a]�𝐺𝐺�
𝐸𝐸(𝔛𝔛𝑚𝑚,[τ,τ +a]2|𝐺𝐺) , 𝐺𝐺: 𝔛𝔛𝑚𝑚 > 0
(25)
And for down-market, there is:
13 | P a g e 𝛽𝛽− = 𝐸𝐸�𝔛𝔛𝐿𝐿,[τ,τ +a]𝔛𝔛𝑚𝑚,[τ,τ +a]�𝐿𝐿�
𝐸𝐸(𝔛𝔛𝑚𝑚,[τ,τ +a]2|𝐿𝐿) 𝐿𝐿: 𝔛𝔛𝑚𝑚 ≤ 0
(27)
To replicate what has been done for liquidity-adjusted Best-beta CAPM, 𝔛𝔛𝐿𝐿 is replaced with 𝑥𝑥𝐿𝐿 − 𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝐿𝐿. Then the outcome is:
For up-market:
𝐸𝐸�𝑥𝑥𝐿𝐿,[τ,τ +a]� = 𝐸𝐸�𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝐿𝐿,[τ,τ +a]� + (𝛽𝛽1++ 𝛽𝛽2+− 𝛽𝛽3+− 𝛽𝛽4+)𝐸𝐸(𝑥𝑥𝑚𝑚,[τ,τ +a] − 𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝑚𝑚,[τ,τ +a]) (28) 𝛽𝛽1+ = 𝐸𝐸(𝑥𝑥𝐿𝐿,[τ,τ +a]𝑥𝑥𝑚𝑚,[τ,τ +a]|𝐺𝐺) 𝐸𝐸�(𝑥𝑥𝑚𝑚,[τ,τ +a]−𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝑚𝑚,[τ,τ +a])2|𝐺𝐺�, 𝐺𝐺: 𝑥𝑥𝑚𝑚− 𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝑚𝑚 > 0 (29) 𝛽𝛽2+ = 𝐸𝐸(𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝐿𝐿,[τ,τ +a]𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝑚𝑚,[τ,τ +a]|𝐺𝐺) 𝐸𝐸�(𝑥𝑥𝑚𝑚,[τ,τ +a]−𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝑚𝑚,[τ,τ +a]|𝐺𝐺)2�, 𝐺𝐺: 𝑥𝑥𝑚𝑚− 𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝑚𝑚 > 0 (30) 𝛽𝛽3+ = 𝐸𝐸(𝑥𝑥𝐿𝐿,[τ,τ +a]𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝑚𝑚,[τ,τ +a]|𝐺𝐺) 𝐸𝐸�(𝑥𝑥𝑚𝑚,[τ,τ +a]−𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝑚𝑚,[τ,τ +a]|𝐺𝐺)2�, 𝐺𝐺: 𝑥𝑥𝑚𝑚− 𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝑚𝑚 > 0 (31) 𝛽𝛽4+ = 𝐸𝐸(𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝐿𝐿,[τ,τ +a]𝑥𝑥𝑚𝑚,[τ,τ +a]|𝐺𝐺) 𝐸𝐸�(𝑥𝑥𝑚𝑚,[τ,τ +a]−𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝑚𝑚,[τ,τ +a])2|𝐺𝐺�, 𝐺𝐺: 𝑥𝑥𝑚𝑚− 𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝑚𝑚 > 0 (32) For down-market:
𝐸𝐸�𝑥𝑥𝐿𝐿,[τ,τ +a]� = 𝐸𝐸�𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝐿𝐿,[τ,τ +a]� + (𝛽𝛽1−+ 𝛽𝛽2−− 𝛽𝛽3−− 𝛽𝛽4−)𝐸𝐸(𝑥𝑥𝑚𝑚,[τ,τ +a] − 𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝑚𝑚,[τ,τ +a]) (33) 𝛽𝛽1− = 𝐸𝐸(𝑥𝑥𝐿𝐿,[τ,τ +a]𝑥𝑥𝑚𝑚,[τ,τ +a]|𝐿𝐿) 𝐸𝐸�(𝑥𝑥𝑚𝑚,[τ,τ +a]−𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝑚𝑚,[τ,τ +a])2|𝐿𝐿�, 𝐿𝐿: 𝑥𝑥𝑚𝑚− 𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝑚𝑚 ≤ 0 (34) 𝛽𝛽2− = 𝐸𝐸(𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝐿𝐿,[τ,τ +a]𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝑚𝑚,[τ,τ +a]|𝐿𝐿) 𝐸𝐸�(𝑥𝑥𝑚𝑚,[τ,τ +a]−𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝑚𝑚,[τ,τ +a]|𝐿𝐿)2�, 𝐿𝐿: 𝑥𝑥𝑚𝑚− 𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝑚𝑚 ≤ 0 (35)
14 | P a g e 𝛽𝛽3− = 𝐸𝐸(𝑥𝑥𝐿𝐿,[τ,τ +a]𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝑚𝑚,[τ,τ +a]|𝐿𝐿) 𝐸𝐸�(𝑥𝑥𝑚𝑚,[τ,τ +a]−𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝑚𝑚,[τ,τ +a]|𝐿𝐿)2�, 𝐿𝐿: 𝑥𝑥𝑚𝑚− 𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝑚𝑚 ≤ 0 (36) 𝛽𝛽4− = 𝐸𝐸(𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝐿𝐿,[τ,τ +a]𝑥𝑥𝑚𝑚,[τ,τ +a]|𝐿𝐿) 𝐸𝐸�(𝑥𝑥𝑚𝑚,[τ,τ +a]−𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝑚𝑚,[τ,τ +a])2|𝐿𝐿�, 𝐿𝐿: 𝑥𝑥𝑚𝑚− 𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝑚𝑚 ≤ 0 (37)
The separation of the market is essential in the DAPM framework. In comparison with the traditional DAPM, the breakeven point of market segmentation is visibly distinct. Logically, given a series of market returns, being up-market in L-DAPM is a sufficient but not necessary condition for being up-market in regular DAPM model. In contrast, being down-market in L-DAPM is a necessary but not sufficient condition for being down-market in regular L-DAPM.
The first three betas in DAPM model have a similar meaning as those in Best-beta CAPM. However, the fourth beta is slightly different because of the definition of up/down-market. In L-CAPM (as well as Best-beta CAPM) the market condition is not specified (Archarya & Pedersen, 2005; Lee, 2011.) Nonetheless, in L-DAPM, the market condition has been determined so that it should be distinguished in up-market and down-market respectively. Under the circumstance that the market performs poorly, investors request for less
compensation in return or even would like to pay a premium when synergy of individual liquidity premium and market return, which is denoted by 𝛽𝛽4−. Moreover, this effect also submits to the explanation proffered by Archarya & Pedersen (2005). However, Archarya & Pederson (2005) did not investigate the situation that occurs in up-market. In L-DAPM, it is expected that if the market stays at a good condition, investors are willing to accept a lower return when the synergy of market return and individual liquidity premium is high.
4: Methodology 4.1. Data
This section of the thesis will firstly observe the stocks credit ranked by S&P via
COMPUSTAT. The thesis will retain the stocks with a credit rank that did not fluctuate on a large scale from the period of December 2002 to December 2012. By recording the Cusip code of these stocks (the only code that both the CRSP and COMPUSTAT have for each
15 | P a g e stock), the thesis will access all of the available monthly data with respect to volume, share outstanding, price, bid price, ask price, total return (including dividends), as well as the T-bill month return during period of December 2002 to December 2012.3 This data is derived from CRSP. In order to make the analysis reliable, the thesis will remove the stock that has missing data in any of the described. After ruling out the stocks with missing data, there are 78 high credit stocks and 75 low credit stocks in the sample. According to CRSP, if the price of stock is not accessible, CRSP will set the price as the mid-price of the bid price and the ask price, and will add a negative sign (-) before the value. Therefore, this thesis will use the absolute value of variable to avoid the negative price. In some rare cases, the difference in the ask price and the bid price (the bid-ask price) is lower than zero. This thesis will set the difference as 0 as long as it is negative number in order to prevent such a violation in the bid-ask price.
4.2. Measure Liquidity Premium
Archaya & Pedersen (2005) applies the measurement of liquidity premium developed by Amihud (2002). However, this measurement is established basically on empirical evidence. In this approach, illiquidity cost can be largely suitable for most stocks (Amihud, 2002). However, the theory established by empirical evidence may be confronted by the problem of timing validity. Even if this approach has been proven effective in 20th century, it may fail to match the current situation in the period (that has been designated for this study) of between 2002 and 2012.
As a result, this paper intends to apply the approach suggested by Gerhold et al. (2013). These authors go further and design a solution to liquidity premium based on the definition of liquidity premium that was introduced in section 2.2 of this paper. In this method, liquidity premium is determined purely theoretically by using mathematical tools. It gives the solution:
𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝐿𝐿,𝑡𝑡 ≈ 0.75 ∗ 𝑆𝑆ℎ𝑇𝑇𝑇𝑇𝐿𝐿,𝑡𝑡∗ 𝜀𝜀𝐿𝐿,𝑡𝑡 (38)
Here 𝑆𝑆ℎ𝑇𝑇𝑇𝑇 means the share turnover, which is the quotient of trade volume divided by share outstanding. 𝜀𝜀 is the relative bid-ask spread. In the measurement adopted in this thesis, the
3 In order to derive the market return and liquidity premium of market portfolio in January 2003, the data of
16 | P a g e amount is not normalized in the same manner as Archarya & Pedersen (2005) This approach has been adopted for the following reasons:
Firstly, based on the definition of liquidity premium, the initial setting has already excluded the problem that resulted from the transaction frequency. More precisely, the liquidity
premium is existing throughout the holding period, rather than occurring at the moment when individuals trade the stocks. In addition, the mathematical demonstration that was displayed in section 2.2 has already proven this it. Therefore, it has nothing to do with trading
frequency now.
Secondly, Archarya & Pedersen predict the amount of liquidity cost by assuming an
autocorrelation in liquidity cost. However, in this thesis, the liquidity premium is equivalent to the return of a portfolio constructed by longing a real-world asset and its hypothetical non-transaction-cost homogenous assets. By assuming that 𝑅𝑅𝐿𝐿,𝑡𝑡 and ℜ𝐿𝐿,𝑡𝑡 are normally distributed, the corresponding liquidity premium also submits to a normal distribution as a normal distribution has the property of a stable distribution. Moreover, Fama (1970) has suggested that the stock return turn out to be very low autocorrelated. Therefore, this yields
𝐶𝐶𝐶𝐶𝐶𝐶�𝑅𝑅𝐿𝐿,𝑡𝑡, 𝑅𝑅𝐿𝐿,𝑡𝑡−𝑗𝑗� = 0 and 𝐶𝐶𝐶𝐶𝐶𝐶�ℜ𝐿𝐿,𝑡𝑡, ℜ𝐿𝐿,𝑡𝑡−𝑗𝑗� = 0. Also, this paper further assumes that 𝑅𝑅𝐿𝐿,𝑡𝑡 can be written as a linear equation as 𝑅𝑅𝐿𝐿,𝑡𝑡 = 𝛼𝛼 + 𝛽𝛽ℜ𝐿𝐿,𝑡𝑡+ 𝜖𝜖 with error term independent of returns and lagged returns in hypothetical world. So the co-movements between the return in real world and one period lagged return in hypothetical world becomes 𝐶𝐶𝐶𝐶𝐶𝐶�𝑅𝑅𝐿𝐿,𝑡𝑡, ℜ𝐿𝐿,𝑡𝑡−𝑗𝑗� = 𝐶𝐶𝐶𝐶𝐶𝐶�𝛼𝛼 + 𝛽𝛽ℜ𝐿𝐿,𝑡𝑡+ 𝜖𝜖, ℜ𝐿𝐿,𝑡𝑡−𝑗𝑗� = 0. And the co-movement between return in hypothetical world and one period lagged return in real world is 𝐶𝐶𝐶𝐶𝐶𝐶�ℜ𝐿𝐿,𝑡𝑡, 𝑅𝑅𝐿𝐿,𝑡𝑡−𝑗𝑗� = 𝐶𝐶𝐶𝐶𝐶𝐶�ℜ𝐿𝐿,𝑡𝑡, 𝛼𝛼 + 𝛽𝛽ℜ𝐿𝐿,𝑡𝑡−𝑗𝑗+ 𝜖𝜖 � = 0. This proves that return in real world/return in hypothetical world is independent from the lagged return in hypothetical world/return in real world. Under these circumstances, the autocorrelation in liquidity premium can be derived as follows:
𝐶𝐶𝐶𝐶𝐶𝐶�𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝐿𝐿,𝑡𝑡, 𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝐿𝐿,𝑡𝑡−𝑗𝑗� = 𝐶𝐶𝐶𝐶𝐶𝐶�𝑅𝑅𝐿𝐿,𝑡𝑡− ℜ𝐿𝐿,𝑡𝑡, 𝑅𝑅𝐿𝐿,𝑡𝑡−𝑗𝑗− ℜ𝐿𝐿,𝑡𝑡−𝑗𝑗� = 𝐶𝐶𝐶𝐶𝐶𝐶�𝑅𝑅𝐿𝐿,𝑡𝑡, 𝑅𝑅𝐿𝐿,𝑡𝑡−𝑗𝑗� − 𝐶𝐶𝐶𝐶𝐶𝐶�𝑅𝑅𝐿𝐿,𝑡𝑡, ℜ𝐿𝐿,𝑡𝑡−𝑗𝑗� − 𝐶𝐶𝐶𝐶𝐶𝐶�ℜ𝐿𝐿,𝑡𝑡, 𝑅𝑅𝐿𝐿,𝑡𝑡−𝑗𝑗� + 𝐶𝐶𝐶𝐶𝐶𝐶�ℜ𝐿𝐿,𝑡𝑡, ℜ𝐿𝐿,𝑡𝑡−𝑗𝑗� = 0
(39)
The empirical evidence also suggests that the AR prediction can only explain maximal 27% and on average less than 10%. Therefore, it is neither reasonable nor necessary to normalize
17 | P a g e the liquidity premium in this instance. Moreover, this thesis treats the result of Eq. (38) exactly as the price of liquidity risk.
4.3. Investigation on Betas
In the empirical analysis adopted in this paper, individual securities are utilized as the basic unit to investigate the asset pricing. This method differs from that adopted in many similar studies that construct portfolios as the basic unit (see, for example, Archarya & Pedersen, 2005; Fama & French, 1992). These studies have constructed portfolios as the basic unit for two reasons. Firstly, a portfolio is probably anticipated to omit the information that could be disclosed in individual security. Secondly, the amount of the data is not so sufficient to assign portfolio since many stocks, ranked by COMPUSTAT cannot be obtained from CRSP.
Barber et al. (2008) conclude that retained investors purchase stocks with good status in volume, return, and news. Therefore, the credit ranking is a signal to investors when they desire to participate in the financial market. Those stocks with a high credit ranking may possess the superior liquidation ability than those with a poor ranking. Relying on potential liquidation ability, this thesis divides the stocks into two groups. One group is ranked at level A+, A and A-, while the other is ranked at B-, C and D. Each group has its own benchmark portfolio. The benchmark portfolio is the average weight of the stocks contained in each group. Meanwhile, the value-weighted portfolios are constructed as the robustness check. Then the way to compute market return and market liquidity premium is shown as:
For average weighted portfolio:
𝑅𝑅𝑚𝑚,𝑡𝑡 =∑ 𝑃𝑃𝐿𝐿∑ 𝑃𝑃𝐿𝐿,𝑡𝑡−1∗ 𝑅𝑅𝐿𝐿,𝑡𝑡 𝐿𝐿,𝑡𝑡−1 𝐿𝐿 (40) 𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝑚𝑚,𝑡𝑡 =∑ 𝑃𝑃𝐿𝐿 𝐿𝐿,𝑡𝑡−1∑ 𝑃𝑃∗ 𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝐿𝐿,𝑡𝑡 𝐿𝐿,𝑡𝑡−1 𝐿𝐿 (41)
For value weight portfolio:
𝑅𝑅𝑚𝑚,𝑡𝑡= ∑ 𝑅𝑅𝐿𝐿𝑁𝑁𝐿𝐿,𝑡𝑡
18 | P a g e 𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝑚𝑚,𝑡𝑡 = ∑ 𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝐿𝐿 𝑁𝑁 𝐿𝐿,𝑡𝑡
(43)
Where N is the number of the stock in this portfolio.
This thesis will only display the summary results for betas of each stock for the sake of brevity. However, the analysis that follows ultimately considers all the available stocks. The betas are figured out by relying on Eq. (20-23; 29-32 and 34-37). Table 1 displays the betas in the DAPM and the Best-beta CAPM model for low credit ranked and high credit ranked stocks. There are two markets under the DAPM framework. The left 4 columns under up-market represent the up-up-market betas and the middle 4 columns under down-up-market stand for the down-market betas. The betas for the best-beta CAPM framework are located in the four right columns.
The first commonality of high ranked and low ranked stocks is the determinate power of market beta. In Table 1, the market beta is globally larger than any one of the liquidity betas regardless of the differentials in credit ranking. This phenomenon connotes that the market factor remains the major influence that liquidity risk cannot have on asset pricing. Indeed, the value of these betas caters to previous works that show that the dominant position of market return (Archarya & Perdersen, 2005; Lee, 2011). However, there remain differences in high ranked stocks and low ranked stocks. That is, the quotient of each liquidity beta against market beta for low ranked stocks is generally somewhat higher than that for high ranked stocks.
The second commonality is that the third beta (𝛽𝛽3+/𝛽𝛽3−) and fourth beta (𝛽𝛽4+/𝛽𝛽4−) generally have a different sign in up-market and down-market, a positive in up-market and a negative in down-market. In addition, in the best-beta CAPM, 𝛽𝛽3 and 𝛽𝛽4 have an ambiguous sign that implies the difference in sign results from the splits of market. It is easy to interpret this phenomenon mathematically. The denominator of the expression for 𝛽𝛽3 and 𝛽𝛽4 are non-negative value in both down-market and up-market. Theoretically, the numerator of the expression for 𝛽𝛽3 and 𝛽𝛽4 in up-market and down-market has no particular restriction with respect to the sign. Nonetheless, some factors have lead to the situation that has occurred in this analysis. As the liquidity premium in any case cannot drop to a negative value, the sign of 𝛽𝛽3+/𝛽𝛽3− is actually determined by 𝑥𝑥𝐿𝐿 in two markets while 𝛽𝛽4+/𝛽𝛽4− depends on 𝑥𝑥𝑚𝑚,. For
19 | P a g e 𝛽𝛽3+/𝛽𝛽3−, if the individual return (𝑥𝑥𝐿𝐿), in the observation periods, generally follows the trend of market return (in up-market, 𝑥𝑥𝐿𝐿 > 0 ; in down-market, 𝑥𝑥𝐿𝐿 < 0), then numerator of 𝛽𝛽3+ is expected to stay above 0 while numerator of 𝛽𝛽3− should be negative. For 𝛽𝛽4+/𝛽𝛽4−, 𝑥𝑥𝑚𝑚 in any case must be a positive number in the up-market. And the liquidity premium is non-negative as explained. Under the circumstance of an up-market, the numerator is surely non-negative. In the down-market, the condition of a positive 𝛽𝛽4− is that average value of the product of individual liquidity premium and market return must exceeds 0. That is, in the majority of the observation periods, the value of 𝑥𝑥𝑚𝑚 has to keep between 0 and the corresponding market liquidity premium. But in reality, this condition is difficult to hold since the liquidity
premium is usually very small. Consequently, the interval that meets the condition is narrow either. This is the reason why the betas (𝛽𝛽4+/𝛽𝛽4− and 𝛽𝛽3+/𝛽𝛽3−) have a positive sign in the up-market while they are negative in down-up-market.
In economic intuition, 𝛽𝛽3+ is interpreted as that lower return that is accepted if the synergy of market liquidity premium and individual return is large when the market preforms well. In contrast, 𝛽𝛽3− implies less compensation is required if the synergy of market liquidity premium and individual return when the market performs poorly.
𝛽𝛽4+ denotes that in an up-market, large synergy of individual liquidity premium and market return results in low demanded return. At the same time, 𝛽𝛽4− implies that large synergy of individual liquidity premium and market return leads to less compensation in return.
In fact, the meanings of betas (𝛽𝛽4+/𝛽𝛽4− and 𝛽𝛽3+/𝛽𝛽3−) that have been evaluated in this thesis are principally consistent with the interpretation adopted in previous works. The only difference is that the thesis has extended this interpretation into two markets. But the aggregate trend is similar: that the smaller co-movement of market liquidity premium and individual
return/market return and liquidity premium, the more return one is expected to obtain.
5: Cross-sectional Regression
5.1. Group 1: Low Ranked Stocks
In this section, the thesis will present the influence of liquidity risk on excess return. This thesis adopts the suggestion of Cochrane (2001) and Archarya & Pedersen (2005) to use
20 | P a g e generalized method of moments (‘GMM’) to carry out a cross-sectional regression on the two groups of securities. Following Fama & French (1992) and Lee (2010), with non-zero
intercept permitted, the thesis establishes the cross-sectional regression model for the Best-beta CAPM, the up-market of DAPM and the down-market of DAPM:
𝐸𝐸(𝑥𝑥𝐿𝐿) = 𝛼𝛼 + 𝜇𝜇𝐸𝐸(𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝐿𝐿) + 𝜆𝜆1𝛽𝛽1,𝐿𝐿+ 𝜆𝜆2𝛽𝛽2,𝐿𝐿+ 𝜆𝜆3𝛽𝛽3,𝐿𝐿+ 𝜆𝜆4𝛽𝛽4,𝐿𝐿 (44) 𝐸𝐸(𝑥𝑥+
𝐿𝐿) = 𝛼𝛼 + 𝜇𝜇𝐸𝐸(𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿+𝐿𝐿) + 𝜆𝜆1𝛽𝛽1,𝐿𝐿++ 𝜆𝜆2𝛽𝛽2,𝐿𝐿++ 𝜆𝜆3𝛽𝛽3,𝐿𝐿++ 𝜆𝜆4𝛽𝛽4,𝐿𝐿+ (45) 𝐸𝐸(𝑥𝑥−
𝐿𝐿) = 𝛼𝛼 + 𝜇𝜇𝐸𝐸(𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿−𝐿𝐿) + 𝜆𝜆1𝛽𝛽1,𝐿𝐿−+ 𝜆𝜆2𝛽𝛽2,𝐿𝐿−+ 𝜆𝜆3𝛽𝛽3,𝐿𝐿−+ 𝜆𝜆4𝛽𝛽4,𝐿𝐿− (46) 𝑥𝑥𝐿𝐿 , 𝑥𝑥+𝐿𝐿 and 𝑥𝑥−𝐿𝐿 in this regression model are the average excess return of stock i in best-beta CAPM, the up-market DAPM and the down-market DAPM during observation period (Jan. 2003 - Dec. 2012). With the same purpose, 𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝐿𝐿, 𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿+𝐿𝐿 and 𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿−𝐿𝐿 are specified as the average liquidity premium of the whole market, the up-market and the down-market in the observation period (Jan. 2003- Dec. 2012).
Simultaneously, the second-pass regression (cross-sectional regression) for traditional Best-beta CAPM and DAPM. To compare with the best-Best-beta framework, the conventional version is more straightforward than the regression model and only contains dependent variable 𝑥𝑥𝐿𝐿, independent variable 𝛽𝛽𝐿𝐿 and a non-zero intercept 𝛼𝛼. The regressions of the conventional version are: 𝐸𝐸(𝑥𝑥𝐿𝐿) = 𝛼𝛼 + 𝜆𝜆𝛽𝛽𝐿𝐿 (47) 𝐸𝐸(𝑥𝑥+ 𝐿𝐿) = 𝛼𝛼 + 𝜆𝜆𝛽𝛽𝐿𝐿+ (48) 𝐸𝐸(𝑥𝑥− 𝐿𝐿) = 𝛼𝛼 + 𝜆𝜆𝛽𝛽𝐿𝐿− (49)
First, the thesis sets a strict constraint on the model. One criterion of the restriction is that both liquidity betas and market betas share a single premium jointly. This discipline leads to 𝜆𝜆1 = 𝜆𝜆2= 𝜆𝜆3 = 𝜆𝜆4 = 𝜆𝜆 in every regression model. If the model can predict the individual return exactly, there is no intercept in these regression models. Hence, the thesis manually sets the value of 𝛼𝛼 to be zero. The last criterion is to fix the value of 𝜇𝜇 to be one. Within a certain period, one is supposed to undertake and only undertake the corresponding liquidity
21 | P a g e premium exactly, rather than the scaled amount. In this thesis, the liquidity premium has been already measured as the monthly specification. This approach differs from that adopted by Archarya & Pedersen (2005). It is not suitable to correct the error between the estimation period and the holding period by imposing a multiple on the value of liquidity cost. Consequently, relying on the best-beta framework, 𝜇𝜇 is forced to be one.
Nevertheless, as a non-intercept model, excess moment conditions exist over the number of parameters. As a result, it is compulsory to inspect the problem of over-identification. Hansen’s J (Hansen L. P., 1982) offers a solution to judge whether the model is
over-identified. Hansen’s J inspects whether the data and estimators match the moment conditions. Similar to the T-test and F-test, it also has two hypothesis: 𝐻𝐻0: 𝑚𝑚(𝜃𝜃0) = 0, [𝑉𝑉 𝐶𝐶𝑉𝑉𝑣𝑣𝐿𝐿𝑣𝑣 𝑚𝑚𝐶𝐶𝑣𝑣𝑚𝑚𝑣𝑣] and 𝐻𝐻1: 𝑚𝑚(𝜃𝜃0) ≠ 0, [𝑉𝑉 𝐿𝐿𝑖𝑖𝐶𝐶𝑉𝑉𝑣𝑣𝐿𝐿𝑣𝑣 𝑚𝑚𝐶𝐶𝑣𝑣𝑚𝑚𝑣𝑣]. When the J-test result is smaller than the critical value, corresponding quantile of 𝜒𝜒𝑘𝑘−𝑙𝑙2 distribution, then it subjects to H0 and confirms the modeling. Otherwise, if the J-test result is larger than the critical value, then it should be rejected, implying an invalid modeling (Hansen L. P., 1982). In addition, compared with squared –R or the T-test for each parameter, the J-test is prior since it directly decides the validity of the modeling.
The regression outcomes of Eq. (44-46) under the restriction above are reported in row 1, 10 and 19 for the up-market (DAPM), the down-market (DAPM) and the Best-beta CAPM respectively. In the up-market, the p-value is almost zero so that the hypothesis deserves to be rejected. Thus, the model with such restrictions turns out to be failure in the up-market model. Nonetheless, the situation in the down-market as well as the Best-CAPM model demonstrates a different picture. High p-value of Hansen’s J suggests that the model with these constraints is effective in the market and the best-beta CAPM. The risk premium for the down-market is negative, consistent with the condition for the down-down-market. Indeed, the amount is significant at 0.1% confidence level, thus confirming model further. With similar implication, the risk premium for the best-beta CAPM is positive, with a significant level at 0.1%, this further supports this model. What about the conventional Best-beta framework? Without the consideration on liquidity risk, in an up-market, the no-intercept cross-sectional regression also fails with p-value of Hansen’s J being zero. In the down-market, the Hansen’s J accepts the no-intercept cross sectional regression with a high p-value of 0.6206. Then, comparing square-R, the thesis finds that the difference is small, 0.9464 for the model with liquidity risk and 0.9398 for the model excluding liquidity risk. In the Best-beta CAPM without liquidity
22 | P a g e risk, Hansen J rejects non-intercept modeling at a 5% level but accepts it at 1% level with p-value of 0.0251. For the sake of further investigation, the thesis will temporarily accept this and compute the squared-R. Then, the thesis finds that the conventional best-beta CAPM with squared-R of 0.4902 outperforms the version with liquidity risk (squared-R of 0.1601). However, it remains difficult thus far to offer any evaluation with respect to these two versions of the best-beta CAPM. Because the result of the J-test is ambiguous and there is no other support to evaluate whether the cross sectional regression with these constraints is acceptable. If the imposed constraints prove to be misleading, then result of the high squared-R cannot have further economic intuition.
Therefore, I loosens the constraints that the constant term 𝛼𝛼 is permitted; whilst other
imposed restrictions are left unchanged. Under these circumstances, the number of moments conditions matches the number of parameters, and thus the over-identification problem no longer exists. Further, in table 3, the J-test result for 𝛼𝛼-allowed regressions is not reported. When giving up the zero intercept, the result is shown in row 2, 11 and 20 for the up-market, the down-market and the best-beta CAPM respectively. In the up-market, both the liquidity premium and constant term are positive and significant at 0.1% confidence level. This result confirms the Hansen’s J in row 1. The non-intercept regression is rejected due to over-identification. Meanwhile, the intercept-allowed regression points demonstrate that the intercept is significant from zero. This suggests that the intercept may be a source of over-identification. In the down-market, the risk premium is almost the same as that in the non-intercept model (-0.0246 and -0.0247 respectively). But the constant term is insignificant with t-statistics of -0.03. This result is also implied by Hansen’s J in row 10. In non-intercept version, there is no over-identification problem, insinuating that the data matches the moment condition well. And the result here has the same statement. In contrast, in the best-beta CAPM, neither the liquidity premium nor the constant term is significant. Then, the thesis utilizes squared-R to contrast which version functions well, the conventional version or the version with liquidity risk. In the up-market, the conventional DAPM with squared-R of 0.7332 can explain almost as much as what the DAPM with liquidity risk explains. Indeed, the constant term and the liquidity premium are also significant in the traditional DAPM. In the down-market, the intercept-allowed regression shows a similar confrontation result as no-intercept regression. The only difference is that the conventional DAPM and the liquidity adjusted DAPM under non-intercept cross-sectional regression explain the individual return better. Both the conventional and the liquidity-adjusted DAPM here have a squared-R around
23 | P a g e 0.74. The situation in best-beta CAPM is somewhat more complicated. Neither the version with liquidity risk nor the version without liquidity adjustment can fit well. In addition, the t-value (2.28) of constant term in the traditional version gives rise to the unsolved problem that is associated with Hansen’s J . In the non-intercept regression, as is illustrated, the Hansen’s J gives p-value of 0.0251, resulting in a trade-off of whether to reject the H0.
The thesis has thus far developed two reasons to reject it after the intercept-allowed cross-sectional regression. Firstly, the t-value here proves it is a value significantly different from zero at 5% level as common critical amount. This implies it is necessary to include a constant term in the cross-sectional regression. And without sufficient evidence regarding significance of intercept, it is not appropriate to allow a non-intercept version. Secondly, the jump of squared-R from the intercept-allowed regression to the non-intercept regression under the traditional best-beta CAPM is significantly unusual, with the latter (0.4902) ten times as much as the former (0.0471). Therefore, either of these two results must be misled. By
referring to the comparable outcomes (see, for example, Archarya & Pedersen, 2005; Fama & French, 1992), the squared-R of regular CAPM is least likely to be not as large as 50%. Thus, this thesis contends that the cross-sectional regression with intercept for the traditional beta CAPM is to be preferred than that without intercept. And for the liquidity-adjusted best-beta CAPM, the non-intercept cross-sectional regression is obviously more suitable as the other obtains insignificance in liquidity premium and constant term. As a result, the traditional CAPM in this case can only explain 0.0471 of the individual return whilst the liquidity-adjusted version can explain 10% more (0.1601).
Before getting into the economic significance, this model may ultimately be s able to improve asset pricing in the field of statistics. The thesis will reduce the imposed constraints gradually in light of this proposal. In row 3, 12 and 21, the thesis reports the results of cross-sectional regression with open 𝛼𝛼 and 𝜇𝜇 for best-beta framework. However, the outcomes state that there is little improvement in the DAPM (both in the up-market and in the down-market). Squared-R in the regression with an open 𝜇𝜇, compared with a fixed 𝜇𝜇 of 1, experiences a slight drop. In the best-beta CAPM, squared-R increases from 0.1601 (row 19, no-intercept cross-sectional regression) to 0.1643. One the one hand, the progress is not remarkable as only 0.32% improvement is achieved. On the other hand, such a small improvement is acquired at the cost of insignificance of the liquidity premium. Consequently, this result is insufficient to have a meaningful conclusion.
24 | P a g e More constraints can be unlocked in order to explore these issues more deeply. Now, the risk premium in liquidity betas can differ from the market betas. But liquidity betas still share a single risk premium λ𝑙𝑙. Then, the thesis will replicate the previous work (Archarya &
Pedersen, 2005; Lee, 2011). Firstly, the no-intercept regression with fixed 𝜇𝜇 is completed (see row 4, 13 and 22, Table 3). Secondly, an intercept-allowed version is provided (see row 5, 14 and 23, Table 3). Finally, the regression with free 𝜇𝜇 is obtained in row 6, 15 and 24 of table 3. Separating the net beta into two parts basically generates higher squared-R within the Best-beta framework. However, such modification ultimately results in the usual estimator of liquidity premium 𝐸𝐸(𝐿𝐿𝐿𝐿𝑃𝑃𝐿𝐿𝐿𝐿). Under single risk premium conditions, the estimators of liquidity premium remain positive even though some of these premiums are insignificant due to
different constraints. It nevertheless turns out to be negative and significant under duo-risk-premium in this instance.
Lastly, the thesis undertakes the empirical analysis more completely. In this instance, not only the risk premium between liquidity betas and market beta can differ from each other, but also the risk premia within liquidity betas are allowed to differ. Therefore, four different risk premia are generated, completely consistent with Eq. (44-46). Like the paragraph above, this specification is treated under three conditions: fixed 𝜇𝜇 with zero-𝛼𝛼, fixed 𝜇𝜇 with free 𝛼𝛼 and free 𝜇𝜇 with free 𝛼𝛼. With a completely split risk premium, there is a great improvement in squared-R. In the up-market, the thesis achieves squared-R up to 0.9217 under the condition of free 𝛼𝛼 and free 𝜇𝜇. In the down-market, squared-R is 7% lower but remains as high as 0.8579 under the same condition. The progress made in the Best-beta CAPM is dramatically high, increasing up to 0.6833 with free 𝛼𝛼 and 𝜇𝜇. Nevertheless, it is also notable that the side effect of this specification is even more severe than duo-risk-premium specification. In this case, the risk premium for liquidity betas is over large throughout the Best-beta framework. And in the best-beta CAPM, the constant term proves to be zero in single risk premium specification and duo-risk-premia specification. However, non-intercept modeling is rejected by Hansen’ J under zero-𝛼𝛼 setting when I allow four different risk premia. Consequently, whilst the squared-R is higher, the advantage of zero constant term is not applicable under this circumstance.
25 | P a g e The analysis on high credit stocks will also performed in comparison to low credit stocks. Regressions are conducted according to Eq. (44-46,) as has been completed in section 5.1. First of all, the most strict constraint is imposed that the constant term 𝛼𝛼 is forced to be 0 and the estimator of liquidity premium is set to be 1. The result for such a restriction under single-premium condition is reported in row 1, 10 and 19 in table 7 for the up-market, the down-market and the best-beta CAPM. In single-premium specification, the J-test rejects the modeling of the up-market (p = 0.0001) and the best-beta CAPM (p = 0.0009) but cannot reject the modeling of the down-market (p=0.0991). If constant term is allowed, row 2, 11 and 20 show the consistence with respect to the J-test. In the best-beta CAPM and the up-market DAPM, the constant term is significant, suggesting an unsuitability of non-intercept modeling. While constant term in the down –market is insignificant, which confirms the result of corresponding J-test. Since the non-intercept model definitely has a higher squared-R than the regular model, the non-intercept model in down-market (row 19) performs better than regular one (row 20). Also, to compare it with the regular DAPM, in the down-market, the squared-R of the liquidity adjusted version (non-intercept model) is 85 basis points higher than regular version. As the intercept-allowed model is less suitable in the down-market, the thesis will not compare the intercept-allowed model with the model in the regular version of DAPM. In the up-market and the best-beta CAPM, constant terms are reduced and squared-Rs are higher in contrast with the regular version. Finally, a free 𝜇𝜇 is unlocked. In row 3, 12 and 21, the results are illustrated. The Best-beta CAPM have a significant 𝜇𝜇 and generates a great improvement in squared-R, 12% higher than modeling with a fixed 𝜇𝜇. In the up-market, the improvement in squared-R is very small and 𝜇𝜇 becomes insignificant. In the down-market, the improvement is the least significant. Squared-R does not surpass the one in the modeling of fixed 𝜇𝜇 and non-intercept. Indeed, 𝜇𝜇 is insignificant either.
This section will hereafter attempt to replicate the steps above for duo-risk-premia
specification and quad-risk-premia specification in order to maximize the number of possible outcomes in the field of statistics.
Under duo-risk-premia specification, the squared-Rs are around 2% more than those under single-risk-premium specification for each constraint on 𝜇𝜇 and 𝛼𝛼 in up-market (see row 2 and 5; row 3 and 6). In addition, in this case, Hansen’ J also rejects the non-intercept modeling in the up-market (row 4). When it comes to the significance of estimators, results state that both
26 | P a g e the risk premium of liquidity beta and that of market beta are significant. However, the difference between the two risk premiums is large. This phenomenon will be discussed in the following section. The Down-market unravels a similar story. Hansen’ J cannot reject the non-intercept modeling as single-risk-premium specification. The squared-Rs under three different constraints are slightly (roughly dozens of basis points) larger than those with
single-risk-premium specification. And now the risk premium of liquidity beta is insignificant, whilst the risk premium of market beta remains significant.
For the purposes of improvement in squared-R, the best-beta CAPM display a different picture. The squared-Rs increase 19% on average for each constraint by allowing an
independent risk premium of liquidity beta. Indeed, in this case, two risk premiums are both significant.
Further, the thesis extends the risk premium of net liquidity beta into three-risk premia of different liquidity betas, which yields four-risk premia (quad-risk-premia specification.). Generally, the squared-Rs increase slightly in the up-market and the down-market for a corresponding constraint when it is compared it with duo-risk-premia. In the up-market, Hansen’ J can reject the non-intercept modeling which is not accepted by Hansen’ J in the down-market. This outcome is in keeping with the previous single-risk-premium
specification and duo-risk-premia specification. At the same time, four-risk premia are basically significant in both the up-market and the down-market. However, the situation of best-beta is ultimately different. Firstly, squared-Rs have a remarkable increase of more than 15%, in contrast with duo-risk-premia specification. Secondly, risk premia of beta 2 and beta 4 are not significant under any restriction on 𝜇𝜇 and 𝛼𝛼. Therefore, it seems that extending the number of liquidity premia has both benefits (namely increased squared-R) and cost (namely insignificance of risk premia). In the next section, a benefit-cost analysis with respect to specification of risk premium (premia) will be computed under the economic significance.
5.3. Economic Significance
In this section, the thesis will bring the statistical results into economic significance. The most important issue is: which specification of risk premium (premia) is optimal is in the field of financial economics? On the one hand, as was derived in the statistical analysis, three different types of specification will be attempted: single risk premium, duo-risk-premia and
27 | P a g e quad-risk-premia. On the other hand, the Best-beta framework with liquidity risk is not the multi-factor model suggested by Fama & French (1992), which allows multiple risk premia. In fact, this model only installs some adjustments in traditional Best-beta framework
regarding liquidity risk. Consequently, it ought to obey the modeling of Best-beta framework that there is only one common risk premium in each model (or each market in DAPM). The specification that violates the rule of single risk premium is expected to be invalid. There is no doubt the single risk premium never violates this rule. Thus, the thesis will only
investigate the specification of duo-risk-premia and four-risk-premia. It is almost implausible to require risk premia to be exactly same because each risk premium has a confidence
interval and values in this interval are considered to be reliable. The adopted approach is straightforward: if the 95% confidence intervals of these risk premia have a common
overlapping area, then the specification is approved. Otherwise, the specification fails to meet the economic significance. Under this discipline, specification of four-liquidity-premia (row 7-9, 16-18 and 25-27, table 3) under any constraint, and in any model, never matches the economic significance. Meanwhile, the multicollinearity problem is also serious across the Best-beta framework. It brings noise in the regression model and consequently leads to usual estimators. Therefore, the thesis removes the four-risk-premia from consideration. Then, a discussion of Duo-risk-premia specification of low credit stock is settled. Duo-risk-premia specification is not suitable under any condition in the Best-beta CAPM. However, it is accepted under any constraint in the down-market DAPM and under a fixed 𝜇𝜇 and a free 𝛼𝛼 in the up-market DAPM. What about high credit stock? In the up-market, with each constraint, the risk premium of liquidity beta is unusually larger than that of market beta. And certainly, they never share an overlapping area in 95% confidence interval. In the down-market, risk premium of liquidity beta is insignificant under any constraint. Thus, it is ultimately
incomprehensible to explore whether they share an overlapping area. In the best-beta CAPM, the result is similar to that of low credit stocks that risk premium of liquidity beta and risk premium of market beta has no overlapping area. The outcome of high credit stock suggests that the existence of overlapping area in two-risk premia in the low credit stocks group is just a coincidence, rather than a strong evidence. Even though a combined beta may cover the original information brought by separated betas, multi-risk-premia specification generate outcomes which violate the essential economic significant. To make a trade-off, the thesis will keep the single-risk-premium specification and drop duo-risk-premia specification in order to fit the initial setting of the asset-pricing model. Moreover, it remains necessary to explore whether an open 𝜇𝜇 is acceptable in this instance. According to the liquidity adjusted
