CAPM and the state of the market's environment : Analysis of alpha under changing market conditions

CAPM and the State of the

Market’s Environment

MASTER THESIS WITHIN: Business Administration NUMBER OF CREDITS: 15 ECTS

PROGRAMME OF STUDY: International Financial Analysis AUTHOR: Andrejs Zurbins and Angus Hawkins

JÖNKÖPING May 2020

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Acknowledgements

We would like to express our gratitude to our tutor Michael Olsson, PhD, for his useful and relevant inputs during the thesis writing process. We would also like to thank Andreas Stephan, PhD, for the interesting conversations that seeded this paper. The scope of this research and the huge amount of data processing involved was only possible thanks to the dedicated community of R developers who create packages to facilitate advanced statistical analysis. Finally, we would also like to thank Lea Rosskopf and Rebecka Rutersten for their contributions during the seminar sessions.

_________________________ __________________________

Angus Hawkins Andrejs Zurbins

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Master Thesis in Business Administration

Tutor: Michael Olsson, PhD Date: 2020-05-18

Key terms: CAPM, alpha, beta, risk-free rate, market return

Abstract

The omnipresence of the CAPM in calculating the cost of equity or the required rate of return means that any potential improvement in the accuracy of the model has highly prac-tical consequences in financial management questions. In this study we considered the pric-ing error in the CAPM model (alpha) and tested whether the mean value changes for differ-ent beta value ranges. These tests were performed for the US, UK and Japanese markets for the period January 1980- December 2019. We further compared the pricing errors with mar-ket state variables - marmar-ket return, marmar-ket volatility and risk-free rate - because conditional CAPM theory dictates that this delivers more accurate modelling. We investigated the corre-lation between these variables and the size and polarity of the pricing error. We concluded the study by comparing pricing errors between countries and controlling for beta brackets and market state. Our results show statistically significant variation in pricing errors both between beta brackets and changing market states. On average CAPM seems to overstate expected return for betas in the positive extreme of the spectrum and slightly understate in betas that are less than one. Means also vary significantly at 5% significance level within each beta bracket and depending on market conditions. The above findings were consistent within each country studied. These findings could be used by organizations and other practitioners looking to allocate capital. Incorporating these results will allow for a more accurate calcula-tion of the cost of equity or required return on equity.

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Table of Contents

1

Introduction ... 1

1.1 Setting a price to risk ... 1

1.2 Problem Discussion and Purpose ... 3

1.3 Scope of Study ... 4

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Theoretical Framework ... 5

2.1 Modern Portfolio theory ... 5

2.2 Capital Asset Pricing Model (CAPM) ... 6

2.2.1 Assumptions ... 6

2.3 Sources of failure of the CAPM ... 7

2.4 Multi-factor models ... 7

2.5 Intertemporal CAPM ... 8

2.6 Conditional CAPM ... 8

2.6.1 Risk captured by CCAPM – Time varying beta ... 9

2.7 Over specification or under specification ... 10

2.8 Alpha ... 11

2.9 Proxy for conditional MRP ... 11

2.10 Perspectives on Alpha ... 12

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Data and Method ... 13

3.1 Data ... 13

3.2 Approach ... 17

4

Results ... 20

4.1 Alpha means and factor loadings ... 20

4.2 Alpha means by market conditions and factor loadings ... 22

4.2.1 Japan ... 22

4.2.2 UK ... 25

4.2.3 USA ... 28

4.3 Pricing error comparison between countries ... 30

4.4 Market state effect on alpha mean from regression perspective ... 33

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Conclusions ... 35

5.1 Conclusions in relation to other related research ... 36

5.2 Limitations and future research ... 37

Reference list ... 38

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1 Introduction

In this part we cover why cost of equity is such an important factor in all financial systems, be they global, national, regional or individual organizations. We also introduce a few neces-sary high level concepts that are important for the big picture understanding prior to delving into a more focused review of econometric models pertinent to our field of research. Shall we invest time and money in this project? When will it turn a profit? These are two central questions that organizations have been asking and will continue to ask long into the future as they are central to their future profitability and by extension survival. Integral to determining whether to invest in a project or between projects running over different lengths of time is the cost of financing that project. Consequently, it has always been important to know the cost of the money that is going to be used to finance a project. On balance sheets this cost is itemized under WACC (Weighted Average Cost of Capital) and this itself is de-termined by two components, namely the cost of debt and the cost of equity. Actors on the financial markets, like individual organizations, are also interested in investing in companies and face the same fundamental question of where they should allocate capital to yield the best returns. Also, like companies they factor in future returns and risk level to determine the cost of money to the organization. The markets then supply the financing needed and at appropriate price either by; (1) buying bonds, thereby securing a fixed interest rate on the capital invested or (2) providing equity funding, thereby hoping to secure superior returns on capital invested. Given the ubiquity of this type of market behaviour and the huge sums of money involved this means this an area of acute market and academic interest.

An organization that needs to decide between various projects will determine the net present value (NPV) of project and use this as a basis for making a final decision. Clearly the WACC will be a vital factor determining its NPV. However, it is the markets that will determine the debt/equity (WACC) cost that the company has to pay. This means financial actors are faced with the challenge of knowing how much to charge for debt and the riskiness of the equity they are buying.

1.1 Setting a price to risk

Sharpe (1965) presented a working paper for a model that endeavoured to solve this highly practical issue of standardizing the pricing of risk. By standardizing the price of risk, a vital

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step was taken in enabling the markets to become more efficient in the allocation of capital. Given the essence of the problem it was trying to resolve it is perhaps unsurprising that it received the name Capital Asset Pricing Model, which usually goes by the abbreviation the CAPM. The model itself is simple and appeals to a mathematical desire for an efficient model, it states simply that excess return on a security is a function of market risk, the return of the market as a whole and the cost of risk-free funding.

Due to its importance in the functioning of the market CAPM has been the subject of much scrutiny conducted by academics interested in problematizing the proposed model. Black, Jensen, and Scholes (1972) found that intercept and beta coefficients were not static over the time, however, similar to the studies conducted by Fama and MacBeth (1973), Amihud, Christensen, and Mendelson (1992), among others, pointed out that there is a relationship between returns and beta coefficient. There is large body of research that argues that the model is inefficient. Fama and French (1992, 1996, 2004) dedicated extensive research on CAPM models and argue against the model and its usefulness. Lai and Stohs (2015) attempt to test CAPM and disprove the model algebraically by showing that it has either endogeneity or circularity problem.

The creation of the CAPM as a valuation and capital allocation tool has led to multiple at-tempts to create more accurate models to overcome CAPM’s perceived poor empirical per-formance. Portfolios constructed with varying company characteristics earn returns that dif-fer considerably from those predicted by CAPM (Da, Guo and Jagannathan 2012). Some of the anomalies that are problematic for the validity of the CAPM include size (Banz, 1981), earnings to price ratio (Basu, 1983), book to market value of equity (Rosenberg, Reid, and Lanstein, 1985), cash flow to price ratio and sales growth (Lakonishok, Shleifer, and Vishny 1994), past returns (DeBondt, and Thaler, 1985), and others.

This has resulted in the development of multifactor models to better capture the real cost of equity (CE) for companies. Lutzenberger (2017) surveyed eight competing single and multi-factor models in his CE estimation review in the European Union, which include Fama and French (1993, 1995) three factor and five factor models, Carhart (1997) four factor model and others, of which the Carhart model seemed to outperform the other ones by means of explanatory power of expected stock returns.

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Despite the many critiques of the CAPM and array of alternative models it continues to be the financial industry’s tool of preference for valuation and determining the cost of equity capital. As the figures show this may have declined somewhat since its inception. Bruner, Eads, Harris and Higgins (1998) found that it was the tool of choice for CE calculations with 85 percent of leading US companies using it. Graham and Harvey (2001) found that 73.5 percent of CFOs use it for ascertaining CE. In academia Welch (2008) found that 75 percent of professors advocated its usage and most recently Bancel and Mittoo (2014) found that 80 percent of 365 European Financial Experts use CAPM to calculate CE.

1.2 Problem Discussion and Purpose

Some of the principal reasons for the popular endurance of the model include: 1) the data for calculation of the alpha and beta are readily available and easily comparable with other companies and sectors, 2) the rational theory support makes the model more palatable for decision-makers and regulation authorities (Lutzenberger, 2017), 3) the CAPM builds on the rational theory of market equilibrium (Subrahmanyam, 2010), and 4) empirical evidence sug-gesting that there is little direct indication to warrant the rejection of CAPM for estimating CE (Da, Guo and Jagannathan, 2012).

The popular endurance is, however, linked to a modification of the original formula to one that includes an alpha factor, which is also referred to as Jensen’s Alpha – a performance metric introduced by Jensen (1968). It is the difference between realized return of an asset and the expected return given by the CAPM over the specific time frame. It can also be termed as a pricing error which could be used as a measure of whether realized expected returns are properly described by the pricing model (Lutzenberger, 2017).

One of the aspects, nevertheless, that possibly should be considered when using the model is that economic states are likely to change over time. For example, the economy might ex-perience high interest rates and low market return in one period and low interest rates and high market return after some years in the future. This was and still is one of the obstacles in performance analysis of funds and asset classes in general, which received much attention from academics, including Ferson and Schadt (1996). They showed that by conditioning al-pha and beta on some of the economic instruments, including short-term Treasury bill, im-proves performance of the funds, in terms of alpha. Later, Ferson and Qian (2004) extended their research by adding more variables, eleven in total, as proxies of the state of the economy

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which supported their findings in previous work. Nevertheless, some literature argues that the conditional CAPM does not describe abnormality of returns (Lewellen and Nagel, 2006). We will take a more detailed look at conditioned CAPM in the theoretical framework section, it is however relevant to point out that the academic world asserts some relationship between economic environment and performance of the asset pricing models either through increas-ing number of factors in the models or by conditionincreas-ing coefficients. This, on the other hand, raises the question of whether CAPM users are increasing the likelihood of overstatement or understatement of the cost of equity given the state of the market?

Our motivation for this paper was driven by the extensive usage of the CAPM in the current financial world and contributions made by Mukherji (2011) and French (2018). Mukherji gave guidelines on the choice of risk-free rate, while French focused on selection of appro-priate historical time frame for the CAPM estimates. We, on the other hand, would like to examine the model’s performance given the state of the market’s environment, namely, from the perspective of market return, market volatility, and interest rate. We will: 1) analyse the relationship between market return, market volatility, interest rate and alpha depending on a beta bracket, and 2) given the state of the market, examine potential increase or decrease of the risk of overstating or understating the cost of equity. We will study this to answer the question of whether a beta bracket has any influence on the alpha mean and to what extent market environment impacts, if at all, deviations of pricing error within each beta bracket. 1.3 Scope of Study

In this paper, we will analyse three markets – USA, UK and Japan. The principal reason for choosing these markets being that they offer differing interest rate environments and these are correlated with different market conditions (Barro, 1998). We chose the long period of 1980 to 2019 because we want to cover many market conditions on which we can base any conclusions and we are not aware of any research done on this scale in this field.

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2 Theoretical Framework

In this section we present a narrative review of the CAPM model, its underlying assumptions and investigations that have been made with a view to improving the reliability of the model for practitioners. We will present the historical context of the latest findings in the CAPM field and they key perspectives of the relevant researchers and thereby provide a solid theo-retical foundation for the later data analysis.

2.1 Modern Portfolio theory

Markowitz (1952) in “Portfolio Selection” outlined the relationship between the risk level of different securities and the expected returns. This relationship forms part of the backbone of modern financial theory and by extension is a fundamental concept for guiding investment decisions. In its simplest terms Markowitz demonstrated that the riskiness of an investment is a function of it expected return, historical mean and variance/standard deviation. The expected portfolio return is

𝐸(𝑅𝑝) = ∑ 𝑤𝑖 𝐸(𝑅𝑖) 𝑛

𝑖=1

(1) and the portfolio variance is

𝜎𝑝2= ∑ ∑ 𝑤𝑖𝑤𝑗𝐶𝑜𝑣(𝑅𝑖, 𝑅𝑗 ) 𝑛 𝑗=1 𝑛 𝑖=1 , (2)

where 𝑅𝑝 = return on portfolio, 𝑅𝑖 = return on individual asset, 𝑤𝑖 and 𝑤𝑗 = weighting of

asset i and j, respectively and 𝜎𝑝2 = variance of portfolio return.

An understanding of this fundamental assumption then allows investors, portfolio managers, project managers to gauge to either invest either according to a given risk profile or an ex-pected return profile. This paved the way to the strategy of selecting portfolio assets based on their change in price in relation to one another rather than as individual assets. By con-sidering the asset portfolio as a group it became feasible to target lowest possible risk for a given return or higher expected returns at certain risk levels. Ultimately, it is a strategy of diversification through a more complete understanding of the interrelationships between the component assets.

6 2.2 Capital Asset Pricing Model (CAPM)

Sharpe (1965), Lintner (1965) and Mossin (1996) further developed Markowitz’s work in the area of portfolio theory and created the CAPM as a means of evaluating the risk and return trade-off for individual stocks. In this model risk is devolved into two categories, diversifiable risk and non-diversifiable market risk. Certain firm specific risk can be diversified away using ideas central to Markowitz’s portfolio theory, however, a certain amount of risk is systematic or non-diversifiable and this is captured in the beta component of the CAPM equation. This means that risks equates to the covariance of returns in the volatility of the whole economy. The expected return on an individual asset, i, is

𝐸(𝑅𝑖) = 𝑅𝑓+ 𝛽𝑖(𝑅𝑚− 𝑅𝑓) (3)

and the systematic risk of individual asset is

𝛽_𝑖 = Cov(𝑅𝑖,𝑅𝑚)

Var(𝑅𝑚) , (4)

where 𝑅𝑓 = risk-free rate for the time period, 𝑅𝑚 = the realized return of the market

port-folio and 𝑅𝑚− 𝑅𝑓 = Market risk Premium (MRP).

In this model 𝛽𝑖 becomes a measure of the relative risk associated with a movement in the

market. A 𝛽𝑖 = 0.5 would mean that a 1% move in the market would lead to an expected

0.5% move of that asset in the same direction as the market. Depending on the nature of the asset 𝛽𝑖 can take on a of value of negative or positive. It is this firm specific 𝛽𝑖 value that

becomes important for investment decisions, cost of equity decisions as it is so widely used when making financial decisions. It is worth noting here, because it will be dealt with more exhaustively below, that textbooks tend to state this as a linear relationship, however, it is more correct to talk about the beta as an average value (Ardalan, 2000).

2.2.1 Assumptions

The CAPM assumes that market participants: maximize their economic utilities, are rational and risk averse, diversify across investments, are price takers, have unrestricted access to the risk free rate, do not incur transaction costs, all securities are highly divisible, have access to the same information at the same time and that the variance of past returns is a perfect proxy for the future risk of a given security.

7 2.3 Sources of failure of the CAPM

The CAPM model tries to capture all the systematic risk in one single parameter, beta. This according to its critics can be resolved in one of two ways. Either more factors need to be added to the model, because beta cannot realistically be the sole measure of systematic risk. Attempts to alleviate this problem add factors and we will consider this in the multi-factor model section as presented in the extensively cited study by Fame and French (1992). The other path is that CAPM should really be a conditional model (CCAPM) grounded in the time-varying nature of risk as it is unreasonable to assume the constant risk assumption in the hypothetical model economy of the CAPM model. (Jagannathan and Wang, 1996). Below we will review the literature concerning both ways to improve the reliability of the CAPM model. It is this CCAPM route that we will pursue in our research.

2.4 Multi-factor models

The empirical results from the above research have led to the development of multifactor models that try to better capture the real CE for companies. Some of these include: The Fama French 3 factor model (FF3) (extended with: size risk, value risk), Fama and French (1993), Carhart four factor model (extended with: momentum), Carhart (1997), Pástor Sham-baugh extension model (extended with: liquidity) , Pástor and StamSham-baugh (2003), Fama French 5 factor model (extended with: profitability, investment), Fama and French (1993) (FF5), Hahn Lee Model (extended with: changes in default spread and term spread as proxies for size and value ), Hahn and Lee (2006), Conditional CAPM Model (extended with: time -varying risk of betas for value and growth), Petkova (2006) and Bad Beta, Good beta model (extended with: future cash flows and market discount rates), Campbell and Vuolteenaho (2004). In the context of our paper these models are also attempts to refine the CAPM model for better reliability in the prediction of portfolio returns. The important factor, from this paper’s perspective, that distinguishes these multifactor models, with the exception of Petkova (2006), from a CCAPM is that beta is treated as static or that beta itself is factorized. Consequently, they are relevant from the perspective of teasing out the complexities of the CAPM model, however, in the context of CE and usage by practitioners the value is less clear especially as the CAPM and FF3 model return as reliable and sometimes more reliable results than the considerably more complicated later models with sometimes as many as eight factors (Lutzenberger, 2017).

8 2.5 Intertemporal CAPM

Merton (1973) formulated the term ICAPM, where the ‘I’ stands for intertemporal, as a means for factoring in state variables to help predict market returns when pricing of financial assets. ICAPM models have been empirically tested Campbell (1996), Campbell and Vuolteenaho (2004), Brennan (2004) and Guo (2006). Whilst these models do consider fac-tors that impact on beta risk, they do not factor in the time varying nature of beta. Despite the use of the term “intertemporal”, which could suggest changes over time this is not the same as “conditional” which has the strict meaning of factor betas/risk prices are time-var-ying. “Conditional” therefore refers to market conditions, which is also commonly referred to as the business cycle state. Research has also been done, Maio (2013) to combine intem-poral and conditional variables, called a “scaled ICAPM” model with favourable results. Clearly, such a scaled ICAPM would be a complicated proposition for a practitioner not familiar with such complexities. For this reason, it suffices to acknowledge that such models also exist, but do not coincide with our goal of simple application. Consequently, we now turn our attention to conditional CAPM.

2.6 Conditional CAPM

CCAPM theory contends that there is a fundamental flaw in the CAPM model, namely that it is a one-period model due to its grounding in the model of Markowitz (1952). CAPM’s basic assumption is that beta represents the non-diversifiable risk of a stock in relation to the market, where low beta assets are affected less than high beta assets when the market swings one way or the other. At an intuitive level it is easy to understand that the riskiness of assets change during the course of the business cycle and this fact is not considered in the CAPM. However, CCAPM is inclusive of the fact that these betas cannot be static as the appetite for risk changes depending on market conditions, i.e. risk tolerance is conditional. During booms investors will feel comfortable being exposed to more risk and the MRP will be lower, whilst the opposite is the case in recession. This in turn means that to calculate risk more accurately the MRP cannot simply be averaged as in the CAPM model but must instead be weighted according to the amount of time spent in the two states, expansion and contraction. Conse-quently, an understanding of the two concepts above could lead to CCAPM being expressed in the following manner. The return on an individual asset is

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where 𝑤𝑏 and 𝑤𝑟 are the weightings of business cycle spent in boom/recession (𝑤𝑏+ 𝑤𝑟 =

1), 𝛽_𝑖𝑏 and 𝛽_𝑖𝑟 are beta of asset in expansion/recession.

Advocates of the CCAPM model contend that this failure to account for changing risk ap-petites may be one of the reasons why CAPM cannot account for irregular returns. Indeed, Vendrame, Guermat & Tucker (2018) found strong support for conditional CAPM with beta explaining growing and contracting markets. Should the above be the case then we would expect to find this this pattern hidden in the discrepancies between predicted and observed data. We will consider this more carefully when looking at alpha. Breloer, Hühn & Scholz (2016) investigated the impact of time dependent market phases on the Jensen Alpha (JA) in the context of equity funds and found that JA is clearly affected and more especially so when considering shorter time spans. Campbell, Giglio, Polk & Turley (2018) included time varia-tion in stock returns volatility to find that stochastic volatility is an important feature in equity return time series. As such this confirms the usefulness investigating the nature of alpha and ours research to further develop the conditional nature of the JA.

2.6.1 Risk captured by CCAPM – Time varying beta

What are the theoretical considerations when trying to gauge how much of an impact a time-varying approach to betas will have? Phrasing this differently, can we expect the CCAPM to be substantially better than the CAPM.

If we compare the two models (3) and (5) we have already considered from the perspective that MRP will be higher in periods of recession and lower in periods of expansion/boom and that betas 𝛽𝑖𝑏 and 𝛽𝑖𝑟 will also vary, it soon becomes apparent that averaging these

dif-ferences, á la CAPM, could lead to considerably different predicted excess returns.

Below follows a simple worked example to clarify the rationale behind the model. Consider an example where for two stocks A and B we have the given conditions, the MRP in different market states, expansion or recession, and the betas for the two stocks: Using the simple CAPM averaging approach we see that 𝑅𝑖 remains constant at 7.8%, calculated using

(𝑤_𝑟𝑀𝑅𝑃 + 𝑤_𝑏𝑀𝑅𝑃) irrespective of changing betas, which are correlated to the amount of time spent in recession or expansion. However, if we include the effect of varying betas on the same MRPs calculated using (𝛽𝑖𝑟𝑤𝑟𝑀𝑅𝑃 + 𝛽𝑖𝑏𝑤𝑏𝑀𝑅𝑃 ) we get conditional risk

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premiums, Cond.RPs, which are different for the two stocks, 9.2% and 8.2%. Consequently we see that a conditional CAPM that takes into account the changing values of beta with identical MRP leads to different 𝑅𝑖 values than those calculated using the static

(non-condi-tional) CAPM model. This in turn means that the larger this difference is, the larger the impact on the reliability of CE decisions made on the basis of the model.

Stock A Stock B MRP % Betas Cond.RP % MRP % Beta Cond.RP % Recession 𝑤𝑟 = 20% 15 2 𝛽𝑖𝑟 30 15 1/3 𝛽𝑖𝑟 5 Expansion 𝑤_𝑏 = 80% 6 2/3 𝛽_𝑖𝑏 4 6 3/2 𝛽_𝑖𝑏 9 CCAPM (average) 7.8 0.933 9.2 𝑅𝑖 7.8 1.27 8.2 𝑅𝑖 CAPM (average) 7.8 0.933 7.8 𝑅_𝑖 7.8 1.27 7.8 𝑅_𝑖

2.7 Over specification or under specification

Lutzenberger (2017) provides evidence of over-specification of many CAPM multifactor models due to the equal or inferior results of more complicated models. However, we must still take seriously the question of whether the original CAPM model is still underspecified. If the model is underspecified, then evidence for under-specification will be found in the discrepancy between predicted and observed data, i.e. the residual errors. To reject the idea of under-specification we would expect to find a stationary stochastic process in the residual errors. This means that the residual errors should show constant mean, constant variance and constant covariance structure (ie covariance is not a function of time), that is to say the definition of a stationary process. These assumptions form the basis of the rational of the hypotheses we will test for. Expressed explicitly, if we were to find static mean, constant variance and constant covariance in the residual errors then we could conclude that CAPM is a perfect model that coincides with the efficient market hypothesis. However, if we find this is not the case we will need to investigate further the nature of the residual errors. To find a point of departure for our analysis we consequently find ourselves returning to the

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earliest addition to the CAPM model, which we deliberately excluded in the models above, and its first additional component, namely alpha because it is a clear expression of the issue with residual errors, indeed it is even called the pricing error.

2.8 Alpha

Jensen (1968) proposed a component called alpha as an addition to the CAPM model. This model proposes that expected returns are a function of beta, market returns, risk free and an unknown component alpha which represents the over or underperformance of an asset in relation to its CAPM expected return value, this is called the pricing error. Generally alpha is understood in the context of portfolio performance and often used as the term for quanti-fying the over or underperformance of an investment manager. In this study we are interested in studying the alphas of individual assets. Thus we modify equation (3) so that it now in-cludes alpha. Now expected return is

𝐸(𝑅_𝑖) = 𝛼 + 𝑅_𝑓 + 𝛽_𝑖(𝑅_𝑚− 𝑅_𝑓), (6)

where 𝛼 = deviation from CAPM model (pricing error).

Traditional assumptions of alpha hold that due to efficient market hypothesis assumptions the pricing error should be arbitraged out of the system and therefore approach zero over time.

2.9 Proxy for conditional MRP

Assuming that MRP varies over the business cycle means that we will need to find a variable that helps predict future economic conditions. Stock and Watson (1989) examined several such variables and concluded that the spread between the six-month commercial paper and six-month T-bill rates and the one and ten year Treasury bonds are amongst the best predic-tors of the business cycle. Bernanke (1990), a former FED chairman, concluded the single best variable was the difference between the Treasury bill and commercial paper rate first identified by Stock and Watson.

The non-linear behaviour of business cycles presents a challenge to researchers trying to model its behaviour. Should they use a parametric model or a non-parametric model. There

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are two common varieties of the parametric model that are used. One is the so called Markow switching model which uses the probability of states of the variable. The alternatives are threshold models which use threshold variables to switch the regime. Of this second kind of regime-switching mechanism the Threshold Autoregressive (TAR) model or Threshold Vec-tor AuVec-toregressive (TVAR) model (when all variables are endogenous) is one of two com-mon candidates for the proxy of the business cycle. There are two principal benefits of the TAR/TVAR models. Firstly, they allow for comparison of cross-country business cycle con-ditions. Second, they enable researchers to consider the economic significances of the thresh-old variables and associated values. The other candidate is the Current Depth of Recession (CDR) model, first devised by Beaudry and Koop (1993) in their attempts to analyse asym-metry in business cycles. One main attraction to this model is that its determining variable is real output (GDP), for which it is easier to get long data series than other macroeconomic variables. Lee and Wang (2012) modified the CDR to create the MCDR so that it could incorporated into the TAR model, thereby maintaining the advantages of both models.

2.10 Perspectives on Alpha

The alpha measure can due to its nature be considered as equivalent to an error term in a linear regression model, which may or may not include hidden variables. To wit. When we perform a linear regression on data, we obtain an equation that provides a best fit for that data. Implicit in this equation is the fact that the model is not a perfect fit. The differences between the observed data and the models predicted data are called the residual errors. By making this assumption about the nature of alpha we can also submit alpha or in our cases a large range of alphas to statistical analysis. In doing so we will investigate whether there might be any unobserved patterns in alpha means, which could then be investigated for each eco-nomic state. Any statistically significant relationship could ultimately prove to be useful in-formation for practitioners, i.e. improve the reliability of the model.

First, we will compare alpha means between market risk beta brackets in each country and see whether they are statistically different. Second, we investigate market state impact on the alpha mean within each beta bracket. Finally, we will examine whether alpha means between countries behave similarly within each beta bracket and market state.

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3 Data and Method

Following section details the data used in the analysis, describes methods and gives basic statistical overview of the key variables which will be used for more thorough analysis in the later parts.

3.1 Data

For our analysis we selected three markets – USA, UK and Japan – of which monthly prices of both active and inactive common stocks were obtained from Thomson Reuters Eikon Datastream. Stocks were selected from each country’s major stock exchange – New York Stock Exchange (NYSE), Tokyo Stock Exchange (TSE) and London Stock Exchange (LSE), for the period from January 1980 till December 2019. The reason for selection of such an extended period lies chiefly in the attempt to capture various market conditions. We decided to include inactive stocks as well to achieve as small deviation in number of stocks between rolling windows as possible. Altogether, total number of companies listed on each stock exchange were: 8,049 for NYSE, 4,110 for TSE, and 6,679 for LSE.

One of the questions we faced was the choice of the rolling window. As mentioned by French (2018), the common practice is to use five years, however, other time frames could also be considered, e.g., based on the average tenure of a CEO, which on average is about three years for all CEOs and nine years for the ones managing Fortune 500 companies. Since companies used in our study include both large and small size listings, we decided to run regressions over the 60 months rolling window. In addition, due to the fact that the list includes inactive companies, regressions were run over the continuous excess monthly re-turns of the companies that were listed during each 60 months window. For example, if the stock was delisted in any of the months of the rolling window, it was excluded from the regressions run in this particular time frame.

Another choice we had to make was related to the risk-free rate. Mukherji (2011) recom-mends using short term rather than long term Treasury rate of return for the CAPM since it has the lowest market risk. Thus, this was the reason for us picking short term rates for all three countries with their respective currencies. Since the short-term rates were annualized, we recalculated them into monthly continuous returns to match those of the stocks. For all

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three markets, three months rate was used - T-bills for UK and USA, and deposit rates for Japan.

The choice of the market proxy was another problem we faced. Stambaugh (1982) found that CAPM sensitivity is not related to the choice of the market proxy and for this reason the choice of it should not raise any empirical issues. On the other hand, Kamara and Young (2018) found that composition of market proxy has a substantial impact on the cost of equity estimate. Nevertheless, they could not identify the size of the effect. We will use Wil-shire5000, TOPIX500, and FTSE All-Share as market proxies for the USA, UK and Japan, respectively, since they are capitalization weighted and include largest companies of the re-spective nations, thus making comparison between countries more consistent. Market port-folio returns, as well as individual stock returns and risk-free rates, will be recalculated into continuously compounded monthly returns with risk free rate subtracted afterwards to ob-tain excess returns for the further calculations.

Table 1 shows summary statistics of key variables used in the study. Data was obtained from 141 rolling windows for the period under analysis, where each rolling window had its own mean for alpha, beta, market return, market volatility and risk-free rate, in total - 141 obser-vations for each variable. The approach is described in later section in a more detail. From the table we can see that mean monthly alpha varied about the zero mark in Japan with standard deviation of 0.5%. This suggests that at some periods, on average, CAPM tend to overestimate and at some underestimate cost of equity. This observation is true for all mar-kets studied. Interestingly, mean alpha was negative for the USA and UK, -0.2% and -0.5% respectively on a monthly basis, which shows overestimation of the cost of equity in the long run. Average factor loading in Japan and UK was very close to each other, at about 0.86, and just over one in the USA. In three countries, market returns were positive with the highest in the USA (0.7%) and the lowest in Japan (0.2%), while UK returned around 0.6%. Market volatility, expressed as standard deviation of monthly returns in each rolling window, was approximately equal in the USA and UK at about 4.4% and 4.5%. Japan, on the other hand, saw volatility one percent higher than that of UK. Finally, Japan has enjoyed lower interest rate environment for the longer period than the USA and UK. As can be seen from the table, mean risk-free rate in Japan was close to 0.2% per month, however the median figure indi-cates that at least half of the period was with the short-term interest rates close to zero.

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Table 1: Summary Statistics of Key Variables by Country.

Japan

Variable Minimum Maximum Mean Median St. dev

Alpha -0.014 0.009 0.000 0.001 0.005 Beta 0.434 1.151 0.864 0.875 0.177 Market Return -0.014 0.023 0.002 0.001 0.009 Market St. dev 0.036 0.077 0.055 0.054 0.010 Risk-Free Rate 0.000 0.006 0.002 0.000 0.002 UK

Variable Minimum Maximum Mean Median St. dev

Alpha -0.015 0.007 -0.005 -0.005 0.004 Beta 0.498 1.179 0.868 0.918 0.150 Market Return -0.008 0.021 0.006 0.005 0.006 Market St. dev 0.027 0.068 0.045 0.043 0.011 Risk-Free Rate 0.000 0.009 0.005 0.005 0.003 USA

Variable Minimum Maximum Mean Median St. dev

Alpha -0.009 0.010 -0.002 -0.002 0.004

Beta 0.631 1.375 1.041 1.053 0.202

Market Return -0.005 0.018 0.007 0.008 0.005

Market St. dev 0.023 0.069 0.044 0.042 0.012

Risk-Free Rate 0.000 0.009 0.003 0.004 0.002

Note: The data represents statistics related to rolling windows where each rolling window contains one mean for respective variable. 60 months window was rolled quarterly (in total 141 rolling windows) since January 1980 till December 2019.

Table 2 shows correlation matrix of the key variables obtained in the process as described above. It is possible to see that overall there is significant correlation between variables on a country basis. First of all, we can observe that there is significant negative correlation be-tween mean alpha and beta in Japan (-0.45) and the USA (-0.19), while that of UK being close to zero. Already from these results, we can expect to reject one of our hypotheses. We also can see interesting significant correlation between average alpha and average market return. Both Japan and UK have significant positive correlation, with 0.43 and 0.37 respec-tively, while the USA have significant negative correlation (-0.49). For all three markets, mar-ket volatility has no correlation with average alpha, however significant positive correlation with the average beta. Risk-free rate is significantly correlated (0.27) with mean alpha in Jap-anese market, however, it is not different from zero in the UK and USA. Nevertheless, risk-free rate has significant negative correlation with the average beta in the USA (-0.36) and Japan (-0.27), and significantly positive in the UK (0.36). To sum up, it is possible to see that correlations between variables by country are significant, however, they vary in signs

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depending on the market. For this reason, we will continue with analysis of mean alpha by beta bracket and market conditions in each country by following approach described below. These results are consistent with Cederburg and O’Doherty (2016) who in their analysis of beta-sorted portfolios and market states demonstrate that beta is positively correlated to market volatility.

Table 2: Correlation Matrix for Key Variables by Country.

Japan

Alpha Beta Market Return Market St. dev

Beta -0.45***

Market Return 0.43*** -0.68***

Market St. dev -0.07 0.57*** -0.59***

Risk-Free Rate 0.27*** -0.27*** 0.34*** 0.18**

UK

Alpha Beta Market Return Market St. dev

Beta 0.02

Market Return 0.37*** -0.25***

Market St. dev -0.06 0.53*** -0.09

Risk-Free Rate 0.11 0.36*** 0.54*** 0.53***

USA

Alpha Beta Market Return Market St. dev

Beta -0.19**

Market Return -0.49*** -0.14*

Market St. dev -0.09 0.3*** -0.54***

Risk-Free Rate -0.10 -0.36*** 0.27*** -0.08

Note: Data used in this table was obtained from the calculations and approach described in the note of Table 1. *** - significant at 1% level;

** - significant at 5% level; * - significant at 10% level.

17 3.2 Approach

The approach in our study can be divided into two parts: 1) aggregation of variables obtained from CAPM regressions and 2) statistical tests and comparisons of variables between market states and countries, all of which is performed in R. The variables used in latter analysis are alpha and beta, which represent an intercept (pricing error) and factor loading, respectively, of the linear regression (CAPM). Excess return is

𝑅𝑖 − 𝑅𝑓 = 𝛼𝑖 + 𝛽𝑖(𝑅𝑚− 𝑅𝑓). (7)

After the necessary variables were aggregated, we started with an analysis of means of alpha for each beta bracket. Obtained betas were divided into six groups - below 0, 0-0.5, 0.5-1, 1-1.5, 1.5-2, and above 2.

Thus, our hypothesis is:

𝐻1: Market risk premium factor loading has no impact on the pricing error of the CAPM,

namely:

𝛼₁ = 𝛼₂ = ⋯ = 𝛼_𝑛, where 𝛼𝑖 is mean pricing error in the beta bracket i.

We decided that the most appropriate methods for testing, since we are interested in market states, were one-way ANOVA and paired t-tests. Since ANOVA assumes homogeneity, nor-mality and equal amount of observations between groups, we first tested whether any of the assumptions are violated. By conducting Levene’s test for variance homogeneity, we found that variances of alpha between beta brackets are different at the extremely low significance level. Shapiro-Wilk test for normality also gave significant result at close to zero p-values. In addition, since the number of observations are different between each beta bracket, balance of observations is violated as well. Due to the fact that many assumptions were violated, the decision was made to perform non-parametric tests – Kruskal-Wallis, which is an alternative to the one-way ANOVA test, and Pairwise Wilcoxon tests, which is an alternative to paired Student’s t-tests, both allowing for aforementioned violations and base their tests on assigned ranks. Finally, since our data includes vast amount of observations, 40 years of data on over 18,000 companies, we decided to take sample means of observations in each beta bracket for some of our illustrations, chiefly to highlight differences, if any, in alpha means.

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We follow a similar approach in the following parts as described above to compare alpha means within beta bracket and country based on the market state.

Thus, our next hypotheses are:

𝐻₂: The market’s state has no impact on the mean pricing error: 𝛼_𝑖𝑗 = 𝛼_𝑖 ∀ 𝑗,

where 𝛼𝑖𝑗 is mean pricing error in the beta bracket i and market’s state j and

𝐻₃: The market’s state impact on the mean pricing error is not different between countries: 𝛼_𝑖𝑗1 = 𝛼_𝑖𝑗2= 𝛼_𝑖𝑗3,

where 𝛼𝑖𝑗𝑘 is mean pricing error in the beta bracket i, market’s state j and country k.

As for the market state we first examined the quantiles of variables of each market, which includes market’s return, volatility (expressed as standard deviation of returns) and risk-free rate (Table 3). We have chosen these parameters since they are readily available for practi-tioners who are using CAPM and the fact that market returns and risk-free rate are the only right hand side components of the model. From the table we can see that market conditions differ depending on the country. During the last 40 years Japan’s 33% of rolling 60 months’ returns were negative 4.5% and lower, while in the other markets it was positive, 3.4% and 6.8% for the UK and USA respectively. Volatility also showed different character, with the highest being in Japan and somewhat similar in the UK and USA. From the interest rate perspective, Japan has seen the longest period of low interest rate environment among three countries. In fact, two thirds of the observations for Japanese risk-free rate lay below 2.7% per annum, lower than the UK’s 33.3% quantile’s mark.

Since the key market variables were different among the selected countries, we decided not to split them by quantiles but rather by giving thresholds which are equal for all markets (Table 4). The thresholds were chosen arbitrarily by using quantile’s data in Table 3 as a reference. As Table 4 has three key variables and three market conditions theoretically there are 27 unique market states, however, we observed only 25 of them and we will use these in our analysis in later sections.

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Table 3: Quantiles for Key Variables Explaining Market's Conditions in the USA, UK and Japan: January 1980-December 2019 (Annualized from Monthly Figures).

Quantile UK Japan USA

Market's Return

33.3% 0.034 -0.045 0.068

66.7% 0.098 0.084 0.112

Market's Volatility (st. dev.)

33.3% 0.134 0.176 0.132

66.7% 0.167 0.200 0.180

Risk-Free Rate

33.3% 0.043 0.003 0.025

66.7% 0.066 0.027 0.050

Note: Data was obtained from the rolling 60 months window.

Table 4: Market’s State Based on the Key Variables’ Thresholds (for Continuous Annual Data).

Market’s State Market's Return Market's st. dev. Risk-Free Rate

Low Below 5% Below 13% Below 2%

Medium 5%-10% 13%-18% 2%-5%

High Above 10% Above 18% Above 5%

Finally, to supplement our findings from non-parametric tests for 𝐻2, we will conduct the

ordinary least squares (OLS) regression (Equation 8) to test whether there is any impact of the selected market variables on the alpha means for each beta bracket and country. To ac-count for possible heteroscedasticity in the residuals of the regression we used White’s stand-ard errors.

𝛼_𝑖 = 𝜃_𝑖0+ 𝜃_𝑖1𝑅_𝑚+ 𝜃_𝑖2𝑉_𝑚+ 𝜃_𝑖3𝑅_𝑓, (8)

where i is beta bracket, 𝛼𝑖 is mean pricing error, 𝜃𝑖0 is intercept, 𝜃𝑖1, 𝜃𝑖2 and 𝜃𝑖3 are factor

loadings for market return (𝑅𝑚), market volatility (𝑉𝑚) expressed as standard deviation of

monthly returns and risk-free rate (𝑅𝑓), respectively.

Thus, we further test 𝐻2 with the following equality:

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4 Results

In this section we present and analyse results obtained from statistical tests in relation to mean pricing error between market conditions, beta brackets and countries. First, we exam-ine differences of alpha means in each beta bracket and country, then we compare pricing error averages between market state within each beta bracket, and finally we compare whether alpha means are similar between countries while controlling for beta bracket and market condition.

4.1 Alpha means and factor loadings

Table 5 comprises statistical results for each beta bracket in the USA, UK and Japan. To better capture the differences between each beta bracket, illustrations can be seen in Figure 1. Both the statistical summary and illustration convey one important message – alpha means are influenced by beta bracket. Kruskal-Wallis tests gave significant results at extremely low confidence level in each country, which suggests that alpha means are different between beta brackets. By performing Pairwise Wilcoxon test (Table A1), we see that all but few pricing errors were significantly different between each beta bracket at a very low significance level. Some exceptions were for UK, where difference in alpha means were significant at 5% and insignificant at 1% level for “0-0.5” and “0.5-1” beta brackets, and for the USA, where dif-ference was insignificant at 10% level for below zero and “0-0.5” beta brackets. We can observe from Figure 1 that alphas’ distance from zero increases as beta gets larger than one. In addition, it seems that pricing error has a tendency to shift in the non-linear way from positive to negative, viz., the model understates and overstates the cost of equity, for betas ordered from the smallest to largest, respectively. Interestingly, on average CAPM in the UK market overstates CE regardless of the beta bracket and has the different directional shift in the alpha’s mean for negative betas as opposed to the USA and Japan. The model showed the smallest deviation from zero at the “0.5-1” beta bracket in the USA, Japan and UK, where this deviation was -0.42%, 0.22%, and -3.02% at annualized rate, respectively. In other words, in the UK practitioners who used CAPM would have overstated cost of equity on average by at least 3% which possibly could have led to rejection of some of the projects. As from the variance perspective, it follows similar changes as in alphas’ means, and tends to contract and expand as beta shifts in general from the “0-0.5” to both extremes, respectively. These results mean that we can reject the first hypotheses and conclude that alpha means are dif-ferent between beta brackets in each country.

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Table 5: Estimated CAPM Alpha Summary Statistics for Individual Stocks Listed on TSE, LSE and NYSE: January 1980-December 2019 Monthly Excess Returns (Grouped by Beta Bracket).

Japan

Beta Obs. Minimum Maximum Mean Median St. dev

(<0) 3,811 -0.09255 0.06114 0.00480 0.00401 0.01441 (0-0.5) 54,675 -0.11194 0.06976 0.00123 0.00085 0.01027 (0.5-1) 106,188 -0.14105 0.07184 0.00018 0.00004 0.01053 (1-1.5) 78,760 -0.12240 0.08072 -0.00134 -0.00110 0.01162 (1.5-2) 21,661 -0.14414 0.06172 -0.00274 -0.00242 0.01386 (>2) 4,125 -0.11426 0.08115 -0.00644 -0.00601 0.01992 Total 269,220 -0.14414 0.08115 -0.00032 -0.00025 0.01146 UK

Beta Obs. Minimum Maximum Mean Median St. dev

(<0) 5,666 -0.13550 0.06731 -0.00814 -0.00541 0.02427 (0-0.5) 26,209 -0.13683 0.06528 -0.00259 -0.00029 0.01804 (0.5-1) 48,085 -0.13958 0.06682 -0.00251 -0.00061 0.01683 (1-1.5) 32,616 -0.13218 0.06901 -0.00503 -0.00256 0.01816 (1.5-2) 9,551 -0.14630 0.05975 -0.00987 -0.00699 0.02171 (>2) 4,853 -0.14673 0.07694 -0.01869 -0.01505 0.02764 Total 126,980 -0.14673 0.07694 -0.00460 -0.00187 0.01905 USA

Beta Obs. Minimum Maximum Mean Median St. dev

(<0) 3,801 -0.09972 0.11957 0.00053 0.00061 0.01636 (0-0.5) 33,483 -0.10082 0.07711 0.00044 0.00078 0.01148 (0.5-1) 63,963 -0.16539 0.07098 -0.00035 0.00042 0.01254 (1-1.5) 59,569 -0.11873 0.08280 -0.00280 -0.00172 0.01405 (1.5-2) 25,814 -0.11500 0.07226 -0.00578 -0.00459 0.01670 (>2) 12,622 -0.11263 0.08971 -0.01032 -0.00917 0.02018 Total 199,252 -0.16539 0.11957 -0.00227 -0.00094 0.01439

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Figure 1: Distribution of Alpha by Country and Beta Bracket: January 1980-December 2019 Monthly Excess Returns.

Note: Data was aggregated in two steps: 1) quarterly rolled 60 months window CAPM regressions were run to obtain alpha and beta for each stock listed on TSE, LSE and NYSE and 2) 2000 sample means of 50 observations for each beta bracket and country were used to illustrate alpha means and their distributions.

4.2 Alpha means by market conditions and factor loadings

From the analysis made on alpha means grouped by beta bracket we have learned one im-portant thing – alpha means change, and they follow similar pattern in the USA, UK and Japan. In the following subsections we will look at market conditions in each country and examine their impact on the means of pricing error within the respective factor loading group. In order to not to repeat findings in each section, we would like to report that Kruskal-Wallis tests, for each beta bracket and country, showed significant differences in alpha means at exceptionally low significance level. Thus, we can already conclude, that market conditions effect average pricing error in every factor loading bracket and country. Nevertheless, Krus-kal-Wallis test indicates only on whether there are substantial differences in general and do not give any information on the effects from a specific market state. To assess effects on a more granular level we will continue our analysis by using Pairwise Wilcoxon tests. Below we detail how alpha means are different between various market conditions within each beta bracket and are thus able to reject the null hypotheses for the Japan, UK and USA market. 4.2.1 Japan

Table 6 comprises alpha means by each market state and beta bracket in Japan. We can see that Japan have experienced 13 different market conditions over the last 40 years based on the grouping as described in Table 4. To have a better grasp on the differences between pricing errors we advise to use Figure 2 as a supplement to Table 6. We can readily see that

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despite the fact the average pricing error on a monthly basis for negative betas was 0.48%, there were three market states when mean alphas were below zero. All three market condi-tions had low market return and high volatility, and suggest that during these market states, CAPM in Japan overstated the cost of equity while in the others understated. Nevertheless, we would like to remind that in the extreme betas – below zero and above two – the number of observations for some market conditions are at low double digit numbers and this might lead to increased possibility of errors when making any inferences from the given results (see Table A2 for more detailed summary statistics for Japan as well as Table A3 and Table A4 for UK and the USA markets, respectively). However, beta brackets that are closer to one have substantial amount of data and thus comparisons can be made without elevated risk of error. The highest mispricing of the cost of equity in “0-0.5” bracket occurred when market had high return, medium volatility and medium interest rate environment. In this market state, CAPM on average understated the pricing error by about 12.6% (1.5% multiplied by 12 months) in annualized terms. If we look at the other spectrum, CAPM overstated mean alpha by approximately 9.84% when market return was low, volatility – high, and risk-free rate at a medium level. This is a huge difference in annual terms if we compare both extremes, more precisely about 22.44%.

We will continue our discussion by focusing chiefly on “1-1.5” and “1.5-2” beta brackets as these are probably the most commonly used in the practice for cost of equity estimates, especially for valuation of new projects and investments. By looking at Figure 2, we can see that deviations from zero seem to be lower for “1-1.5” factor loadings. The model made the smallest error when market return was high, volatility – medium, and interest rates – low. Investors on average would have overstated the cost of equity by a mere 0.8% per annum. The highest absolute deviation from the perfect pricing was during high market return, me-dium volatility and risk-free rate. In this market condition, underestimation was about 5% per annum. An interesting observation can be made by looking at “1.5-2” beta bracket. While mean alpha for these factor loadings, without controlling for market states, was -0.24% per month, a significant opposite effect, also the highest in absolute terms, was during the low market return, high volatility and interest rate environment. During this period, monthly pricing error was 0.69%, or 8.3% in annualized rate.

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Table 6: Mean Alpha by Beta Bracket and Market Condition in Japan: January 1980-Decem-ber 2019 Monthly Excess Continuous Returns.

Market Condition (<0) (0-0.5) (0.5-1) (1-1.5) (1.5-2) (>2) MR(M)-MV(M)-RF(L) 0.0094 0.0053 0.0036 0.0009 -0.0026 -0.0061 MR(M)-MV(H)-RF(L) 0.0038 0.0037 0.0015 -0.0020 -0.0055 -0.0071 MR(M)-MV(H)-RF(H) 0.0097 0.0058 0.0020 0.0010 0.0010 No-Obs MR(L)-MV(M)-RF(L) 0.0010 0.0031 0.0035 0.0011 -0.0005 0.0002 MR(L)-MV(H)-RF(M) -0.0137 -0.0082 -0.0033 -0.0019 -0.0015 -0.0034 MR(L)-MV(H)-RF(L) -0.0073 -0.0036 -0.0034 -0.0041 -0.0042 -0.0079 MR(L)-MV(H)-RF(H) -0.0231 -0.0066 0.0009 0.0037 0.0069 0.0217 MR(H)-MV(M)-RF(M) 0.0119 0.0105 0.0081 0.0042 -0.0055 -0.0128 MR(H)-MV(M)-RF(L) 0.0150 0.0068 0.0033 -0.0007 -0.0044 -0.0091 MR(H)-MV(M)-RF(H) 0.0145 0.0064 0.0024 -0.0012 -0.0042 -0.0092 MR(H)-MV(L)-RF(H) 0.0023 -0.0013 -0.0015 -0.0011 -0.0022 -0.0020 MR(H)-MV(H)-RF(L) 0.0147 0.0070 0.0035 -0.0017 -0.0043 -0.0074 MR(H)-MV(H)-RF(H) 0.0129 0.0068 0.0040 0.0017 -0.0012 -0.0076 Note: Data derived and aggregated from quarterly rolled 60 months window CAPM regressions on a monthly excess continuous returns. MR = market return; MV = market volatility (expressed as standard deviation of excess monthly continuous returns); RF = risk-free rate. For Low(L)/Medium(M)/High(H) levels refer to Table 4.

Figure 2: Distribution of Alpha by Beta Bracket and Market Condition in Japan: January 1980-December 2019 Monthly Excess Continuous Returns.

Note: Data was aggregated in two steps: 1) quarterly rolled 60 months window CAPM regressions were run to obtain alpha and beta for each stock listed on TSE and 2) to exclude outliers only quantile data above 5% and below 95% was included in the illustration. Red line indicates pricing error of zero. MR = market return; MV = market volatility (expressed as standard deviation of excess monthly continuous returns); RF = risk-free rate. For Low(L)/Me-dium(M)/High(H) levels refer to Table 4.

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Table 7 gives the summary of the Pairwise Wilcoxon test. It shows significant pairwise dif-ferences as a proportion of total observations for each market condition in the specific factor loading group and total significant observations as proportion of total in each beta bracket. This table can be used as a supplement to Kruskal-Wallis test to better capture the extent of the significance of market condition effect on the mean pricing error. By looking at the total significance figures one can see that market’s state effect grows as beta gets closer to “0.5-1”, namely, proportion of significant paired differences is 0.94 of all pairs in that bracket. Although this also may be related to lower variance in general in aforementioned bracket as can be seen in Figure 2. This means that practitioners should expect more inconsistency in mispricing errors as market conditions change and thus be more aware of the period in which the model is used. In addition, low market return, high volatility and interest rates seem to have the highest impact on average alpha. From obtained results, we can infer that alpha means are in aggregate different depending on the market state within each beta bracket and thus reject our second hypothesis for Japanese market.

Table 7: Paired Significance Proportion of Mean Pricing Error by Beta Bracket and Market Condition in Japan: January 1980-December 2019 Monthly Excess Continuous Returns.

Market Condition (<0) (0-0.5) (0.5-1) (1-1.5) (1.5-2) (>2) MR(M)-MV(M)-RF(L) 0.75 0.92 0.92 0.83 0.67 0.45 MR(M)-MV(H)-RF(L) 0.83 1.00 0.92 0.83 0.67 0.18 MR(M)-MV(H)-RF(H) 0.42 0.67 0.92 0.83 0.75 No-Obs MR(L)-MV(M)-RF(L) 0.92 1.00 0.83 0.83 0.67 0.82 MR(L)-MV(H)-RF(M) 1.00 1.00 0.92 0.92 0.75 0.55 MR(L)-MV(H)-RF(L) 1.00 1.00 0.92 1.00 0.67 0.27 MR(L)-MV(H)-RF(H) 1.00 1.00 1.00 0.92 1.00 1.00 MR(H)-MV(M)-RF(M) 0.67 1.00 1.00 0.92 0.50 0.64 MR(H)-MV(M)-RF(L) 0.67 0.67 0.92 0.83 0.58 0.36 MR(H)-MV(M)-RF(H) 0.67 0.67 1.00 0.83 0.50 0.45 MR(H)-MV(L)-RF(H) 0.83 1.00 1.00 0.58 0.75 0.45 MR(H)-MV(H)-RF(L) 0.58 0.75 0.92 0.83 0.58 0.45 MR(H)-MV(H)-RF(H) 0.50 0.67 0.92 1.00 0.58 0.36 Total significance 0.76 0.87 0.94 0.86 0.67 0.50

Note: Data was aggregated in three steps: 1) quarterly rolled 60 months window CAPM regressions were run to obtain alpha and beta for each stock listed on TSE, 2) Pairwise Wilcoxon test was performed for each market state within each beta bracket and 3) proportion of significantly different (at 5% significance level) mean pricing errors for each market condition was estimated as a sum of significant differences over total paired comparisons. Total significance was estimated by taking all significant differences as proportion of all observations in a beta bracket.

4.2.2 UK

Table 8 shows mean alpha by beta bracket and market condition for the last 40 years in the UK. As one can notice, the number of different market conditions is higher than that in

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Japan. In total there were 20 unique periods which could have an impact on mean pricing error of the CAPM. As well, paired Wilcoxon tests have similarity in the nature of the signif-icance of the differences in pricing error averages within each beta bracket, although the highest total significance is for factor loadings in “0-0.5” bracket (Table 9). As before, we would like to look mainly at “1-1.5” beta bracket. From Table 5 we can remember that CAPM on average overstated cost of equity by approximately 0.5% on a monthly basis and this we can see for mostly all market conditions both in Table 8 and Figure 3. Nevertheless, all-medium and all-high market conditions understated the alpha by 0.02% and 0.06% per month, respectively. Remarkably, all-high market environment increases mean alpha in the positive direction, although, as was illustrated in Figure 1, the distance of the mean alpha for each positive beta bracket increases in the negative direction. This can also be observed in the Table 9 as proportion of significant differences is the highest in this market state. Finally, market state with high market return, on average, tend to have more impact on mean alpha as in comparison to medium and low market returns across all beta brackets. Therefore, to summarise results, we can reject the hypotheses of no market impact on alpha means within each beta bracket for UK market as well.

Table 8: Mean Alpha by Beta Bracket and Market Condition in the UK: January 1980-De-cember 2019 Monthly Excess Continuous Returns.

Market Condition (<0) (0-0.5) (0.5-1) (1-1.5) (1.5-2) (>2) MR(M)-MV(M)-RF(M) -0.0053 0.0023 0.0028 0.0002 -0.0066 -0.0157 MR(M)-MV(M)-RF(L) 0.0005 0.0036 0.0025 -0.0053 -0.0120 -0.0206 MR(M)-MV(M)-RF(H) -0.0111 -0.0059 -0.0051 -0.0072 -0.0149 -0.0329 MR(M)-MV(L)-RF(M) -0.0149 -0.0014 0.0006 -0.0020 -0.0061 -0.0171 MR(M)-MV(L)-RF(L) -0.0050 0.0023 -0.0015 -0.0075 -0.0151 -0.0383 MR(M)-MV(L)-RF(H) -0.0082 -0.0094 -0.0073 -0.0092 -0.0098 -0.0165 MR(M)-MV(H)-RF(H) -0.0170 -0.0043 -0.0055 -0.0067 -0.0110 -0.0260 MR(L)-MV(M)-RF(M) -0.0170 -0.0026 0.0003 -0.0031 -0.0077 -0.0126 MR(L)-MV(M)-RF(L) -0.0040 0.0007 0.0026 -0.0063 -0.0147 -0.0238 MR(L)-MV(M)-RF(H) -0.0175 -0.0064 -0.0055 -0.0064 -0.0068 -0.0107 MR(L)-MV(L)-RF(M) -0.0177 -0.0130 -0.0041 -0.0041 -0.0084 -0.0246 MR(L)-MV(L)-RF(L) -0.0096 -0.0022 -0.0028 -0.0077 -0.0176 -0.0327 MR(L)-MV(H)-RF(M) -0.0166 -0.0083 -0.0066 -0.0087 -0.0085 -0.0166 MR(L)-MV(H)-RF(L) -0.0167 -0.0058 -0.0058 -0.0102 -0.0125 -0.0214 MR(L)-MV(H)-RF(H) -0.0119 -0.0101 -0.0098 -0.0126 -0.0192 -0.0442 MR(H)-MV(M)-RF(L) 0.0053 0.0074 0.0036 -0.0021 -0.0116 -0.0160 MR(H)-MV(M)-RF(H) -0.0033 -0.0013 -0.0017 -0.0029 -0.0107 -0.0224 MR(H)-MV(L)-RF(M) -0.0077 0.0006 -0.0003 -0.0004 -0.0061 -0.0126 MR(H)-MV(L)-RF(H) -0.0033 -0.0032 -0.0041 -0.0062 -0.0094 -0.0198 MR(H)-MV(H)-RF(H) 0.0120 0.0035 0.0014 0.0006 0.0013 0.0111 Note: Data derived and aggregated from quarterly rolled 60 months window CAPM regressions on a monthly excess continuous returns. MR = market return; MV = market volatility (expressed as standard deviation of excess monthly continuous returns); RF = risk-free rate. For Low(L)/Medium(M)/High(H) levels refer to Table 4.

27

Figure 3: Distribution of Alpha by Beta Bracket and Market Condition in the UK: January 1980-December 2019 Monthly Excess Continuous Returns.

Note: Data was aggregated in two steps: 1) quarterly rolled 60 months window CAPM regressions were run to obtain alpha and beta for each stock listed on LSE and 2) to exclude outliers only quantile data above 5% and below 95% was included in the illustration. Red line indicates pricing error of zero. MR = market return; MV = market volatility (expressed as standard deviation of excess monthly continuous returns); RF = risk-free rate. For Low(L)/Me-dium(M)/High(H) levels refer to Table 4.

Table 9: Paired Significance Proportion of Mean Pricing Error by Beta Bracket and Market Condition in the UK: January 1980-December 2019 Monthly Excess Continuous Returns.

Market Condition (<0) (0-0.5) (0.5-1) (1-1.5) (1.5-2) (>2) MR(M)-MV(M)-RF(M) 0.42 0.84 0.84 0.84 0.47 0.53 MR(M)-MV(M)-RF(L) 0.74 0.89 0.84 0.74 0.47 0.47 MR(M)-MV(M)-RF(H) 0.58 0.95 0.79 0.74 0.68 0.68 MR(M)-MV(L)-RF(M) 0.37 0.74 0.79 0.79 0.58 0.42 MR(M)-MV(L)-RF(L) 0.58 0.89 0.84 0.74 0.63 0.74 MR(M)-MV(L)-RF(H) 0.16 0.89 1.00 0.89 0.11 0.00 MR(M)-MV(H)-RF(H) 0.21 0.89 0.89 0.63 0.42 0.32 MR(L)-MV(M)-RF(M) 0.58 0.89 0.89 0.74 0.58 0.74 MR(L)-MV(M)-RF(L) 0.47 0.79 0.84 0.32 0.32 0.32 MR(L)-MV(M)-RF(H) 0.63 0.95 0.74 0.68 0.58 0.74 MR(L)-MV(L)-RF(M) 0.42 0.84 0.84 0.53 0.26 0.53 MR(L)-MV(L)-RF(L) 0.58 0.84 0.95 0.58 0.63 0.68 MR(L)-MV(H)-RF(M) 0.53 0.89 0.79 0.68 0.37 0.53 MR(L)-MV(H)-RF(L) 0.53 0.89 0.79 0.63 0.47 0.53 MR(L)-MV(H)-RF(H) 0.05 0.84 1.00 0.95 0.84 0.79 MR(H)-MV(M)-RF(L) 0.84 1.00 0.84 0.68 0.11 0.26 MR(H)-MV(M)-RF(H) 0.58 0.95 0.95 0.84 0.58 0.47 MR(H)-MV(L)-RF(M) 0.37 0.84 0.84 0.84 0.58 0.74 MR(H)-MV(L)-RF(H) 0.63 0.89 0.74 0.63 0.53 0.42

28 Table 9: Continued

MR(H)-MV(H)-RF(H) 0.95 0.84 0.84 0.84 1.00 0.95

Total significance 0.51 0.88 0.85 0.72 0.51 0.54

Note: Data was aggregated in three steps: 1) quarterly rolled 60 months window CAPM regressions were run to obtain alpha and beta for each stock listed on LSE, 2) Pairwise Wilcoxon test was performed for each market state within each beta bracket and 3) proportion of significantly different (at 5% significance level) mean pricing errors for each market condition was estimated as a sum of significant differences over total paired comparisons. Total significance was estimated by taking all significant differences as proportion of all observations in a beta bracket.

4.2.3 USA

Table 10 shows mean pricing error by factor loading group and market condition in the USA and Figure 4 illustrates this. In total there were 21 unique periods which could have an impact on mean pricing error of the CAPM. We can see from the figure that, as observed in the UK and Japan market, there is variation in the alpha means within each beta bracket, which can be supported with proportion of significant deviations aggregated in Table 11. We can also observe that this proportion is higher at “0.5-1” beta bracket which fades as beta shifts to-wards its extremes. Alpha means in the below zero beta bracket seem to be less affected by changing market conditions, but we would like to repeat that number of observations in this spectrum is very low in some market conditions, which can cause wrong inferences. These differences can be explained by the fact that mean beta is correlated with market state, which we have observed in Table 2 in method section. From Figure 4 it is possible to see that low market return, medium volatility and risk-free rate state, in general, sees positive alpha means except for betas above two, where it turns negative. Nevertheless, we would like to point out that, although it drops below zero, this pricing error is the least overstated among other market states. We made a similar observation when we looked at the Japanese market, where market low market return, although with high volatility and interest rate environment in gen-eral have seen a positive alpha means. In aggregate, we can conclude that alpha means are different between various market conditions within each beta bracket and thus reject the null hypotheses for the USA market, too.

Table 10: Mean Alpha by Beta Bracket and Market Condition in the USA: January 1980-December 2019 Monthly Excess Continuous Returns.

Market Condition (<0) (0-0.5) (0.5-1) (1-1.5) (1.5-2) (>2)

MR(H)-MV(H)-RF(H) -0.0177 -0.0043 -0.0045 -0.0067 -0.0117 -0.0175 MR(H)-MV(H)-RF(L) 0.0048 0.0041 0.0030 -0.0023 -0.0054 -0.0118 MR(H)-MV(L)-RF(H) -0.0040 -0.0022 -0.0018 -0.0029 -0.0049 -0.0067 MR(H)-MV(L)-RF(L) 0.0008 0.0010 -0.0017 -0.0047 -0.0113 -0.0259