Örebro University
School of Business and Economics Economics, advanced level, thesis, 30 hp Supervisor: Lars Hultkrantz
Examiner: Dan Johansson Fall 2015
Testing CAPM for the Swedish Stock Market
In Order to Capture the Price Expectations
–
A Comparison Between Conditional CAPM,
and
Unconditional CAPM
Åsa Grek 890727 Abdi Fatah Jimaale 881205
Abstract
The Capital Asset Pricing Model (CAPM) can be estimated by different econometric models. Since CAPM has formed, many econometric models have been introduced and many studies have tried to investigate which econometric model can estimate investor expectations. The purpose of the study is to investigate which econometric model that can capture investor expectations the most accurate using Swedish daily data for 28 stocks during 1995-2012. The study compares a relative new model – MGARCH DCC against Ordinary Least Square (OLS) and Rolling Window OLS (OLS-RW). The models were compared by analyzing the models Mean Squared Error (MSE). The most accurate model is estimated by OLS-RW with semi-short window to predict investor expectations.
Keywords: Capital Asset Pricing Model (CAPM), investor expectations, MGARCH DCC,
rolling window OLS (OLS-RW), OLS, unconditional CAPM, conditional CAPM, Swedish stock market.
!
Table of content
1. Introduction 1
2. Theoretical framework 4
2.1 Asset prices 4
2.2 Efficient market and Hypothesis 4
2.3 Theory on short and long term predictability 5
2.4 Behavioral asset pricing models 5
2.5 Different Asset Pricing Models 6
2.6 The Capital Asset Pricing Model 6
3. Literature Review 8
4. Data 10
5. Econometric models 12
5.1 Estimating beta with Ordinary Least Square (OLS) 12
5.2 Estimating beta with Multivariate Generalized Autoregressive 14
Conditional Heteroskedastic Dynamic Conditional Correlation Model
6. Empirical Results 16 7. Discussion 18 8. Conclusions 21 References 22 Appendix ! ! !
1. Introduction
The uncertainty regarding the risk involved in asset returns has given rise to the development of many asset-pricing models within the last fifty years. The aim of these asset-pricing models is to help investors to predict future returns and to take proper investment decisions. Among the most used asset pricing models in the financial market is the Capital Asset Pricing Model (CAPM) of Sharpe (1964), Lintner (1965) and Mossin (1966).!CAPM aims to predict future returns by assuming that the expectations investors have regarding their asset returns is the same among the investors. This means all investors mean, variance and covariance of expected returns are equal (Hansson, Hordahl, 1998).
To evaluate the predictive ability of CAPM with different approaches is crucial to practitioners as well as researchers in order to find the future value of stocks. The predictive ability depends on the econometric methods used, and a better method gives a more accurate prediction. Moreover, accurate prediction of future returns would help practitioners and researchers to take better investment decisions.
Fama and MacBeth (1973) suggest that the betas, which are time varying, should be used as risk measurement due to investor expectations shift between times. The conditional model with time-varying betas still assume that the expectations investors have regarding their asset returns is the same among the investors, but their expectations vary over time (Hansson, Hordahl, 1998), which means that the model allows an assets risk premium to vary over time. Ho and Hung (2009) reported in their study, that the CAPM model with dynamic betas, i.e. conditional CAPM, is more accurate than a CAPM with static betas, i.e. unconditional CAPM, in capturing the systematic risk of an asset.
Even though there is empirical evidence of time-varying betas, there is no clear method on how to model the time-varying component to capture the assets volatility in the conditional CAPM. Fama and MacBeth (1973) suggested that Rolling Window Ordinary Least Square (OLS-RW) should be used to estimate time-varying betas. However, there are assumptions behind the OLS model, which are violated when the model is used on financial data. The financial data have both heteroscedasticity and autocorrelation that affects the OLS estimator, which results in unbiased estimates but with too small standard deviations, i.e. the OLS is not the Best Linear Unbiased Estimator (BLUE). Therefore, the OLS estimates cannot be used to derive inference (Greene, 2012).
Since CAPM was built, new econometric models have been developed which are supposed to solve the problem with the OLS estimator, and as a result, different studies have attempted to
investigate which econometric models can be used to capture an asset's true beta. One of the most common approaches to capture the true beta is the! Generalized Autoregressive Conditional Heteroscedasticity (GARCH) (Hansson, Hordahl, 1998).
The GARCH approach uses an asset’s past innovation and conditional variances to generate an asset’s beta. There are different formulations of GARCH depending on how the conditional variance is specified. An approach, which has more advantages than a simple GARCH method, is the Dynamic Conditional Correlation Multivariate GARCH (MGARCH DCC). The MGARCH DCC predicts dynamic betas through previous conditional covariance and conditional variances (Engle, 2002). The relatively new model MGARCH DCC has not been tested in CAPM on the Swedish stock market. So a theoretical contribution of the study can therefore be additional knowledge regarding the performance of MGARCH DCC in CAPM to model the investor expectations on the Swedish stock market. So, the aim of the study is to investigate which models that best capture investor’s expectations, due to how well they capture beta.
According to theory, due to the fact that investor expectations vary over time, the unconditional CAPM should be worse than the conditional CAPM. This will be the first hypothesis:
!!:!!"#$"%&'&$"()!!"#$!!"!!"!!""#$!%&!!"!!!!"#$%&%"#'(!!"#$!!"!!ℎ!!!"#$%&ℎ!!"#$%!!"#$%&!
!ℎ!"!!"#!!"!!"#$%&'(!
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Since OLS is not BLUE due to heteroscedasticity and autocorrelation it should not be the best model to estimate CAPM. MGARCH DCC was developed to solve the problem with autocorrelation and heteroscedasticity in the data. So, MGARCH DCC should give the best predictions of investor expectations. This is the second hypothesis:
!!:!!"#!!"#ℎ!!"##$%&!!"#$%!!!"!!"##"$!!"!!"!!"#$%!!"!!""#$!%&!!"!!"#$%&!!""!!"!!ℎ!!
!"#$%&ℎ!!"#$%!!"#$%&!!ℎ!"!!"#!!"!!"#$%&'(
!!: !"##$%&!!"#$%!!!"#!!"!!"#$%!!ℎ!"!!"#$%&!!""!!"!!ℎ!!!"#$%&ℎ!!"#$%!!"#$%&! !ℎ!"!!"#!!"!!"#$%&'(!
To test the hypothesis, the unconditional CAPM will be used as a reference point and will be estimated through an OLS regression. The conditional CAPM will be estimated through a MGARCH DCC approach and OLS-RW of length 30 and 250 days. The results are based on the model that has the lowest prediction error (Mean Squared Error (MSE)).
The data selected in this study consist of daily stock prices listed in Nasdaq OMX Nordic and daily government borrowing rate listed in Riksgälden as a proxy for the risk-free rate in CAPM. Our sample period was 2nd of January 1995 to the 20th of March 2012.
The result showed that OLS-RW with 30 days was the best followed by MGARCH DCC, which was better than OLS-RW with 250 days. The worst model was the unconditional CAPM.
This study is organized as follows: Section 2 goes through relevant financial theories!and different pricing models and ends with the pricing model chosen in the study. Section 3 presents previous relevant research. Section 4 presents the data that are used in the study. Section 5 describes those selected econometric models and their assumptions. Section 6 shows the results which were obtained. Section 7 is a discussion about the results linked to the theory, previous studies and econometric models assumptions. Section 8 concludes the founding’s in the study.
2. Theoretical framework
2.1 Asset prices
An asset price can be interpreted as the equilibrium price of the market, where the price paid is the market price of the risk an individual takes (Sharpe, 1964). Understanding how prices in the capital market behave is essential for many investors and financial institutions, because it can provide vital information regarding their investment decisions. Some of the theories that have contributed to the understanding of asset price behavior are: The efficient market hypothesis, random walk and Shiller’s theory regarding long term prediction (Nobel Prize Committee, 2013).
2.2 Efficient market and Hypothesis
Fama (1965) defined an EM (Efficient Market), as a capital market where rational investors who are yield maximizers try to outperform one another. The investors compete with each other in an attempt to find future movements on asset prices, by using currently available information. In doing so, the information gets incorporated in the asset prices and prices go towards their equilibrium level.
According to Fama (1970), asset prices should take into account all existing information available in the financial market and the risk that an investor has to take.!Otherwise, investors would have arbitrage opportunities (Ross, 1976). In order to test the efficient market theory empirically, Fama (1970) formulated the efficient market hypothesis (EMH).
The efficient market hypothesis (EMH), was categorized as weakly, semi-strongly and strongly efficient market, based on the level of information available which is reflected by market prices. In a weakly efficient market, the present value of an asset reflects the information which can be derived by studying an asset’s previous prices. In a semi-strongly efficient market, company specific information and historical prices are incorporated in the asset’s present value. In a strongly efficient market the asset prices reflect the previous categories and inside information (Fama, 1970).
The Efficient Market Hypothesis (EMH) states that asset prices move randomly towards the equilibrium price. It is associated with the idea of random walk (Dupernex, 2007).A random walk is a process where asset prices are assumed to follow a random and unpredictable pattern (Malkiel, 1973). This means, the results of an investor’s attempt to outperform the market when the market is efficient, depends on pure chance.
A random walk is written as:
!! = !!!!+!!! !!~!!"(0, !!) (2.1) Where !! and !!!!!are the logarithmic prices at time ! and ! − 1, !! is random error term with a mean zero and constant variance. The price !! is thus conditional on its previous price and the direction of the movement is decided by the error term (Tsay, 2010 and Campbell et al. 1997). In other words, random walk theory also implies that in an efficient market, information contained in past prices has already been reflected in the current prices. Earlier studies by Fama (1965) support the random walk theory when predicting asset prices over a short-time horizon.
2.3 Theory on short and long term predictability
Shiller (1981) explains that changes of asset prices within short periods of time are small, for example, price changes within the same day. These small changes are lost in the disruptions that affect the market, hence it is difficult to predict over short-time horizons. This is consistent with Fama’s idea regarding short term prediction (Englund et al. 2013).
Shiller (1981) suggests that even if there are difficulties with short-terms predictions, there is still a possibility to do a long-term prediction. When an asset price is observed over a long period of time, it is possible to detect a movement from which the price fluctuates around. It simply means that a high price tends to follow after a low price and vice versa, which makes it possible to predict asset prices (Englund et al. 2013).
However, predictions are associated with errors and asset prices are volatile. The volatility varies between different assets. Also different asset prices have different covariation with other asset prices. To understand the risk embedded in financial assets, researchers and investors use various models, for instance the CAPM.
2.4 Behavioral asset pricing models
There are many different types of asset pricing models discussed in research literature, but they may differ from one another depending on what sort of study that has been performed. The foundation of asset pricing models is the Capital Asset Pricing Model (CAPM) developed by Sharpe (1964), Lintner (1965) and Mossin (1966). Sharpe used this model to determine assets expected return by calculating the assets price sensitivity in relation to the market portfolio (Fama and French, 2004). The sense behind CAPM is that investment has a risk-value component and a time-risk-value component. Thus the purchase of an asset should be reimbursed for the investment period and the risk an investor has to take, which mean there should be a risk premium for the investment (Sharpe, 1964).
2
.5 Different Asset Pricing Models
Besides CAPM which shall be further explained in the coming chapters, other variations of the CAPM model exist. One of these models is the Consumption-Based Asset Pricing Model (CBAPM), which was developed by Lucas (1978) and Breeden (1979). The CBAPM is a macro-economic model that explains the changes in asset prices and asset returns through aggregated consumption (Mehra, 2012). The investor’s view of CBAPM may differ from the traditional CAPM in the sense that the risk involved in assets creates uncertainties in future consumption and not in wealth as CAPM does.
Another model is the Arbitrage Pricing Theory (APT). This multifactor model was suggested as an alternative model to CAPM when Ross (1976) developed it. The arbitrage theory states that there are many factors, which influence an asset price other than the market portfolio, which are used in CAPM (Ross, 1979). The idea of this model is that, an asset return can be determined as a function of numerous independent macro-economic factors such as interest rate, industrial production index, inflation etc. (Huberman and Wang, 2005). The arbitrage theory assumes that assets are on occasion mispriced and that rational investors get arbitrage opportunities by using this model to find mispriced assets (Ross, 1979).
Another extension of CAPM is the Fama-French Three Factor Model. The three-factor model includes factors such as size premium and value premium other than the market premium, which are used in the CAPM. The size premium is the return an investor receive from a small market capitalization. The premium value is the excess return between high-book-to-market and low-book-to-market return. Fama and French (1992) developed this model in attempt to better the measure of an assets expected return. Fama and French (1992) argued that an asset risk is also related to premium value and firm size. When they are taken into consideration it often gives a more accurate prediction of asset prices.
2
.6 The Capital Asset Pricing Model
The CAPM of Sharpe (1964), Lintner (1965) and Mossin (1966) was built on the mean-variance efficient portfolio theory by Markowitz in 1952. The mean-mean-variance efficient portfolio assumes that the investors are risk averse and they want to minimize the risk (i.e. variance) for a given expected return or maximize expected return given a level of risk. According to the efficient market theory, the expected return of an asset should be related to the risk an investor has to take in the market, which is the systematic risk (Fama, 1970). The CAPM tries to examine the relationship between the expected return and the systematic risk
(β). The beta risk is the correlated volatility between an asset’s return and the market portfolio’s return. The beta value helps investors to see how volatile their investment is in comparison to the market (Sharpe, 1964).
The assumptions behind the CAPM are derived from Markowitz (1952) and Tobin (1958) studies.
i. Investors are risk-averse, i.e. investors want to obtain as high return as possible at the lowest risk.
ii. Investors expectations regarding future asset returns are the same.
iii. Investors have the possibility to lend out or borrow an unlimited amount of capital at a certain level of the risk-free rate.
iv. Investors can trade assets without market imperfection such as taxes and commission on a competitive market.
v. Investors have access to all information in the capital market at the same time and the information is free.
vi. Investors have the same investment horizon on their investments. Source: Hull (2010), Berk & Demarzo (2011).
The CAPM equation in time can be written as follows:
!(!!) = !!+ !! !(!!) − !! (2.2)
Where !(!!) is the expected rate of return of asset ! for ! = 1 … . , ! assets. The risk-free interest rate is denoted by !!, !! is the systematic risk for asset ! and !(!!) − !! is the market premium required for taking on a risk.
Fama and MacBeth (1973) developed a new model since investors are risk-averse and that the investors do not have only one portfolio decision for the whole time period. They found that the investors change their portfolio decisions from period ! to period!! + 1 in CAPM. As the portfolio decisions changes from one period to the next Fama and MacBeth (1973) derived a model, which includes a time-varying coefficient !! (see section 5.1).
The CAPM is the most frequently used pricing models in the financial market. A survey performed in Europe show that 80 percent of 356 evaluation experts used the CAPM when predicting their cost of equity (Bancel and Mittoo, 2014). The CAPM is also mentioned in financial literature because it models the risk-return relationship in a simple way as a linear equation, than the other models mentioned earlier.
3.
Literature Review
This section presents some selected studies which have focused on the forecast accuracy of beta using unconditional CAPM against conditional CAPM, i.e. the ability to capture the investor expectations.
The first study, which suggested a use of time-varying betas, was Fama and MacBeth (1973). They realized that the expected return at time ! + 1 depended on the information, which was available at that time ! (see 2.6 and 5.1). Fama together with French then continued the research on unconditional CAPM and found that since unconditional CAPM does not consider investor expectations it violates one assumption, that the investor expectation changes with time, which makes the unconditional CAPM achieve a weak forecast (Fama and French, 1992, 1993, 1996).
Fabozzi and Francis (1978) and Sunder (1980) were two of the earliest studies that compared unconditional CAPM with conditional CAPM. They find that a time varying coefficients in the conditional CAPM explains more of the total risk then an unconditional CAPM with a static coefficient.
Jagannathan and Wang (1996) found in their study using US data that conditional CAPM had a smaller pricing error, when predicting excepted asset return, than the unconditional CAPM. Jagannathan and Wang (1996) also fund that the dynamic models of the conditional CAPM often gives attractive results, but it is more difficult to use then other conditional models. Abdymomunov and Morley (2011) used value-weighted portfolios separated by B/M ratios on New York Stock Exchange (NYSE), AMEX and NASDAQ. They found that when conditional CAPM were used, it was able to explain the returns better than unconditional CAPM did.
The first study which used MGARCH DCC in a conditional CAPM model was done by Bali and Engle (2010) on the Dow Jones Industrial Average. According to them!the reason behind it was that the MGARCH DCC “has clear computational advantage over multivariate
GARCH models in that the number of parameters to be estimated in the correlation process is independent of the number of series to be correlated. Therefore, potentially very large correlation matrices can be estimated with the DCC approach” (Bali & Engle 2010, p.382).
They found that MGARCH DCC was able to estimate beta significantly.
A more recent study conducted by Godeiro, da Silva and Rodrigues (2013) used both MGARCH DCC and Kalman filter in CAPM on 28 stocks in the Brazilian stock market
between 1995-2012. They find that beta estimated by MGARCH DCC was smoother in time and more accurate then Kalman filter.
From these previous studies it is shown that unconditional CAPM is less accurate than conditional CAPM. The studies used different econometric models to estimate conditional CAPM. Most of them used OLS-RW, but some used GARCH. There are two studies mentioned that did use MGARCH DCC to estimate CAPM – Bali and Engle (2010) and also Godeiro, da Silva and Rodrigues (2013). The aim is not to examine all econometric models which can be used to estimate conditional CAPM. Instead the study will focus on the “new” way of estimating CAPM – MGARCH DCC and compare it with a highly used econometric model – OLS-RW. To know if the conditional CAPM is more efficient in the Swedish stock market, an unconditional CAPM will also be used and will be compared by the models MSE. According to pervious studies the unconditional CAPM is not better, but still it will be compared against the conditional CAPM in order to “know” that the econometric model, which is selected is the “best”.
4. Data
The data for this study consists of adjusted closing prices for 28 stocks with trading date from the 2nd January 1995 to the 20th March 2012. Adjusted closing prices means that the data is adjusted for stock splits, new issues and dividends. The data contains 4313 observations for each stock and a total of 120764 observations for all 28 stocks.
A study made by Godeiro, da Silva and Rodrigues (2013) (see literature review) used the same number of stocks under the same time period, to predict CAPM with MGARCH DCC against OLS. For that reason, this study now uses 28 stocks from the 2nd of January 1995 to the 20th of March 2012. Together with observations of the weekly government borrowing rate transformed into daily observations that gives 4313 observations as a proxy for the risk-free borrowing rate and 4313 observations for OMX30 as a proxy for the market portfolio. This period were selected because under this time period, Sweden experienced a financial crises until 1994 and then another at 2008 so this time period covers a whole financial cycle.
The stocks were selected by a simple random sampling from 319 stocks using a Random Number Generator (RNG). A stock was selected if the stock had a time series from the selected period. If not then a new sample was drawn. This means that there are only companies, which exists during this period and are not new or have been bankrupted. The companies that ended up in this study are relative stable and quite large. Companies that are smaller and more volatile are not included in this study.
OMX30 is a weighted index, which means that each company's weight in the index is equal to the proportion of a company's market value in relation to the total market value of all companies (Berk and Demarzo, 2011). The OMX30 was selected as a proxy for the market portfolio for the reason that it reflects the performance of the Swedish capital market, since OMX30 consists of the 30 most frequently traded stocks on NASDAQ OMX Stockholm. It is important to take into account that the government borrowings rate was calculated differently before 2004. It was then calculated as an un-weighted average, which means that the size of the bond loans had large effect on the government borrowings rate. After 2004 the calculation method changed, but it still reflects the current borrowing rate in the market (Riksgälden, 2015).
Table 1 is the descriptive statistics for the excess return of each stock and the market proxy (i.e. OMX 30). The means for each stock were negative, which indicates that the average return under the whole time series is negative. Skewness measures where the distribution is centered round (where the mean are) and kurtosis measures how fat the tales the distribution
has. High kurtosis mean standard normal distribution has a skewness value of 0 and kurtosis value of 3 (Tsay, 2010). In the data the skewness lies between -2.954824 to 2.078469, and kurtosis between 5.7371 to 82.05368, so none of the stocks returns are normally distributed. Table 1. Descriptive statistics for the excess return for each stock and the market proxy
Stock Mean Standard deviation Skewness Kurtosis
Assa Abloy -.0061689 .0249569 .1015498 7.383888 Atlas Copco B -.0062535 .0234374 .219999 6.696458 B&B Tool B -.0067813 .030608 .2699753 8.738582 Beijer Alma B -.0065425 .0209792 -.1200388 11.51144 Beijer Ref B -.0065898 .0255309 .2377901 9.599283 Bong -.0064009 .0194041 .1821819 7.204929 Concordia Maritime B -.0069774 .0277994 .1158091 7.558519 Doro -.0067076 .0202536 .7753479 12.83596 Electrolux B -.0064116 .0242988 .312735 7.886946 Elektra B -.0063725 .0273622 .2967594 10.82818 Ericsson B -.0067601 .0320312 -.3817307 10.57772 Fabege -.0066697 .0217694 .2080678 10.32336 Fast Partner -.0062802 .0243449 1.38961 28.04541 Getinge B -.0065955 .0202214 .1661199 8.113735 Gunnebo -.0066233 .0249132 .0465033 29.40896 Haldex -.0068231 .0271129 -2.954824 82.05368 Hexagon B -.0060913 .0240371 .2414143 8.055735 H&M B -.0061151 .0215276 -.9323242 25.65577 Intellecta B -.0065915 .0185248 .226045 8.5366 Midway B -.0069739 .0234291 -.6149742 17.52615 PEAB B -.0065091 .0228539 -.6747171 20.30533 SinterCast -.0071454 .0399248 2.078469 42.22238 Skanska B -.006614 .0204222 -.6128035 16.03803 SSAB B -.0066582 .025095 .0902307 7.138032 Svenska Handelsbanken B -.0063937 .019759 .2725669 5.7371 Svolder B -.0067271 .0201481 -.5108638 9.96508 Volvo B -.0065474 .0225717 .0368472 6.698253 ÅF B -.0066211 .0258765 .1486168 6.754085 Market -.0064967 .0161187 .1769014 6.183189
In the theory CAPM is built on the true market portfolio, but here a proxy for the market return are used (OMX 30). This had an impact on the results, but it is not clear how much effect it had.
5. Econometric models
In this section the econometric models that will be used to estimate CAPM will be presented. !
5.1 Estimating beta with Ordinary Least Square (OLS)
The CAPM that exist over time can be explained as a linear regression model!!,! = !!+ !!!!,!+ !!, !!!~!! 0, !!! (5.1) !!!!= !!+ !!, !!!~!! 0, !!!
!!!!= !!+ !!, !!!~!! 0, !!!
Where !!,! is the excess return of asset i ! !!,! −!!!,! 1 at time t and !
!,! is the excess return of the market ! !!,! −!!!,! at time t. The three error terms !!, !!, and !! are independent of each other. According to the expression CAPM is time varying and ! and ! ensures that CAPM should follow a random walk process (Tsay, 2010). The model can be written in a matrix form: !!!! !!!! = 1 00 1 !! !! + !! !! , !!,! = 1 !!,! !!! ! + !!, Which is equal to !! = !!,!′! + !! (5.2) Equation (5.2) will be estimated by using Ordinary Least Square (OLS). So,
!! = !!,!′! + !! (5.3)
The error term in the equation above can be estimated as the difference at time t !!= !!− !!,!′!
The least square minimize the squared error terms !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
1!The return !
! of an asset is the continuously compounded rate of return !! = !" !!/!!!! where !! is the closing price of an asset at time t and !" is the natural logarithm.
!!! !
!!! = !!!! !!− !!,!′! !
To be able to minimize the sum of squares ! has to be chosen according to min!! ! = !!′!! = !!− !!,!′! ′ !!− !!,!′!
Expanding the expression gives
!!′!!= ! !!′!!− !′!!,!′!!− !!′!!,!! + !′!!,!′!!,!!
= !!′!!− !!!′!!,!! + !!′!!,!!
The first order condition with respect to beta has to be fulfilled in order to find the minimum !" !
!" = −!!!,!′!!+ !!!,!′!!,!! = ! Then the equation is solved for ! ! = !!,!′!!,! !!!!,!′!!=!"# !!,!,!!!"# !
!,! (5.4)
From the equation above (5.4) ! is the covariance between the market returns and the asset returns dived by the variance of the market return. As can be seen, beta is usually estimated by Ordinary Least Squares (OLS) (Greene, 2012). To get unbiased consistent estimate of an OLS regression there are six assumptions that has to be fulfilled, those are:
1. Linearity: The model describes a linear connection between the dependent and independent variables 2. Full rank: There is not any perfect relationship (high correlation) between two or more independent
variables
3. Exogeneity of the independent variable: States that the expected value of error term cannot be explained by the independent variables
4. Homoscedasticity and nonautocorrelation: See below for an explanation
5. Data generation: Tells that the data can be consist of both constants and random variables
6. !!!~!! !, !!! : Claims that the error terms are normally distributed with the expected value of zero and that they have the same variance.
Source: William H. Greene, 2012.
When OLS are performed on financial time series data, assumption 4 can be violated.
Homoscedasticity is when the error term has the same variance. The opposite case when the disturbance has different variances is called heteroscedasticity. According to Greene (2012) heteroscedasticity typically “arises in volatile high-frequency time-series data such as daily
has high error terms, which makes it difficult to discover abnormalities from a random walk process. Shiller (1981) also argued about the difficulties this causes to do short term predictions but it would still be possible to do long term predictions (Englund et. al 2013). Autocorrelation means that a variable has inner correlation to itself, i.e. that a dataset remembers its previous values Greene (2012) claims that “economic time series often display
a memory in that variation around the regression function is not independent from one period to the next” (p.297). This is consistent to the market hypothesis according to Dupernex
(2007).
As has been shown above, OLS should not be used to estimate CAPM due to heteroscedasticity and autocorrelation. Therefore a model that adjusts for heteroscedasticity and autocorrelation should be used.
The unconditional CAPM assumes a single time horizon, which will be estimated using: !! = ! + !!!+ !, !!~!! 0, !!! (5.5) Where !! is the excess return of asset i ! !! −!!! , ! is the intercept, ! is the coefficient, !! is the excess return of the market ! !! −!!!,! and ! is the error term which has a normal distribution with mean zero and variance !!!. The estimate of !, denoted by ! is:
b = !!′!! !!!!′!!=!"#!"#!!!,!!
! (5.6)
Fama and MacBeth (1973) suggested that a rolling window ordinary least squares (OLS-RW) should be used to estimate CAPM. So, in this study both unconditional CAPM and conditional CAPM will be used in order to predict the investor expectation. The conditional CAPM with OLS-RW will use a semi-short time horizon of 30 days and a long horizon of 250 days.
5.2 Estimating beta with Multivariate Generalized Autoregressive Conditional
Heteroskedastic Dynamic Conditional Correlation Model
According to Hueng and Huang (2008) allowing for time varying betas investigates the asymmetrical relationship between risks and return.
Another way of investigate equation (5.1) according to Tse and Tsui (2002) MGARCH DCC with time-varying betas can be used. This approach allows estimation of dynamic betas form conditional covariance and conditional correlation varying in time.
Let !! still be the excess return ! !!" −!!!,! of a stock ! at time ! = 1, … , ! and let the market have k elements (stocks), where !! = (!!!, … , !!").
Then let Φ! be the set of information that exist at time t, the variance of !! given the information set !"# !! Φ! = Ω!.
The variance for each element that is located in matrix Ω! are denoted by σ!" for ! = 1, … , ! and the covariance are σ!"#, 1 ≤ ! ≤ ! ≤ !. Let !! be the diagonal of Ω! of size !×! where the diagonal elements are σ!"#. The standardized residual !! !! = !!!!!
! is assumed to be IID with !!!~!! 0, !! , where !! = !!"# are a posivive define matrix. The correlation of the excess return !! is denoted by !!=!!!!!!. The conditional variance in !! follows a Vech-diagonal developed by Bollerslev 1988 (see Bollerslev for more details) and are given by a GARCH (p,q) model:
!!"! = !! + !!!!!!!!!,!!!! + !!!!!!!!!,!!!! ,! = 1, … , !
Where the constants !!,!!!! and !!! are nonnegative and !!!! !!! + !!!!!!! < 1 for all ! = 1, … , !.
The time-varying conditional correlation matrix is estimated by
!! = 1 − !!− !! !+!!!!!!+ !!!!!!, where !! is a matrix whos elements are a function from the lagged values of !!. The parameters !! and !! ≥ 0 and !!+ !! ≤ 1.
In the case of estimating CAPM with MGARCH then some relaxations are done. !! is assumed to be a standalized measure and !!!! depends on the standalized lagged residuals !! and !! = !!"# . !!",!!! = !!!!!!!,!!!!!!,!!! !!,!!! ! !!! !!!!!!,!!! (5.7)
for 1 ≤ ! ≤ ! ≤ !. !!!! is the correlation matrix of !!!!, … , !!!! .
Let !!!!= !!!!, … , !!!! !. If !!!! is the diagonal of a matrix of the same size the ith element in the diagonal is ! !!,!!!!
!!! !/! for ! = 1, … , !. Then !!!! = !!!!!! !!!!!!!!! !!!!!!. The conditional CAPM will be generated by using both rolling-window OLS and MGARCH DCC. The MGARCH DCC outputs are used to calculate the conditional covariances and variances and it estimates the unconditional correlations. The conditional betas are calculated by dividing the conditional covariances of an asset and the market with the conditional variances of the market.
6. Empirical Results
Equation 5.4 are used to estimate equation 5.1 for OLS-RW, where t is the number of days in the rolling window, !! is the rate of return and !!,! is the return of the market proxy. The
unconditional CAPM are estimated by equation 5.6 for the whole time period.
The outputs from the MGARCH DCC are used to calculate the conditional variances and co-variances and unconditional correlations that are used to estimate CAPM. The different econometric methods are then evaluated by the mean squared error (MSE). MSE is the average squared difference between the real rate of return for each asset and estimated rate of return of the asset.
!"# =!! !!− !! ! = ! !! !
!!! (6.1)
A GARCH model cannot be built if there isn’t any ARCH-effect therefore an LM-test for ARCH-effect had to be done (see Appendix 3). The !!: !ℎ!"!!!"!!"!!"#$!!"!"#$ against
!!: !ℎ!"!!!"!!"!!"#$!!""!#$. From Appendix 3 it is shown that all stocks rejected the null hypothesis for all lags.
Since all the stocks has ARCH effect then the best GARCH DCC specification should be chosen and this was done by evaluating the AIC for different specifications (see Appendix 7). The model for each stocks rate of return that is recommended has a fat cursive style and the lower the AIC value is, the better, the model fits the data. There are two stocks, which did not get any MGARCH DCC model (Fast Partner and SinterCast). It depends on that the log-likelihood estimation of the model didn’t converge, which made it impossible to estimate a model. The parameter estimations of the MGARCH DCC models is found in Appendix 8. For Svolder B it can be seen that the parameters for !! and !! violates one assumption behind the MGARCH DCC model !!+ !! ≤ 1 .
The error terms of the MGARCH DCC models are estimated to see if they are white noise. If they are white noise then the model sufficiently capture the data. From Appendix 9 one can observe that for 16 models the error terms where white noise at five percent significance level.
In Table 2 it is shown the MSE for all models. It is observed that OLS-RW 30 is consistently the best model, followed by MGARCH DCC, OLS-RW 250 and finally OLS.
Table 2. The MSE from the four different methods
STOCK, MARKET Unconditional OLS OLS with rolling-window (30) OLS with rolling-window (250) MGARCH DCC
Assa Abloy B 0.000568 0.0002745 0.00042165 0.000354432 Atlas Copco B 0.000705 0.00049994 0.0005879 0.000547677 B&B Tools B 0,00051067 0.00047486 0.00050745 0.000490061 Beijer Alma B 0.00054721 0.000244731 0.00044696 0.00035168 Beijer Ref B 0.00078551 0.00058512 0.000667245 0.000623051 Bong 0.000818481 0.000412456 0.00048719 0.000590522 Concordia Maritime B 0000954 0.00071111 0.000795 0.000772831 Doro 0.000652114 0.00032167 0.000491518 0.00043895 Electrolux B 0.00074 0.000539731 0.000601 0.000580641 Elektra B 0,000907 0,000695 0,000855 0,000748933 Ericsson B 0.001201 0.000938 0.001091 0.001024678 Fabege 0.000624535 0.000431142 0.000488728 0.000472229
Fast Partner B 0.000649 0.00049 0.00055 No convergence
Getinge B 0.0004267 0.0003661 0.00041265 0.00037726 Gunnebo 0.0004357 0.000194791 0.000419313 0.000282125 Haldex 0.000895389 0.000666199 0.000788702 0.000735518 Hexagon B 0.000795 0.000521 0.000579 0.000572828 H&M B 0.000604852 0.000424194 0.000475973 0.000461601 Intellecta B 0.0005512 0.000311436 0.00054645 0.00042549 Midway B 0.00072 0.000502 0.000562 0.000547766 PEAB B 0.000761662 0.0004758 0.00069196 0.000519746 SinterCast 0.001787 0.001495 0.001584 No ARCH effect Skanska B 0.000586 0.000382 0.000477 0.000041 SSAB B 0.000795649 0.000572064 0.00067847 0.000625834 Svenska Handelsbanken B 0.000542 0.000358 0.000401 0.000389831 Svolder B 0.000640177 0.000370348 0.000492115 0.000402177 Volvo B 0.00066612 0.00045805 0.00051721 0.000506925 Åf B 0.000532 0.0003115 0.000487 0.00036470
7. Discussion
The purpose has been to investigate which models that best capture investor expectations; so two hypotheses have been tested:
The first hypothesis states “The unconditional CAPM is as accurate as conditional CAPM in
the Swedish stock market”.
!"#$%ℎ!"#"!1:
!!:!!"#$"%&'&$"()!!"#$!!"!!"!!""#$!%&!!"!!!!"#$%&%"#'(!!"#$!!"!!ℎ!!!"#$%&ℎ!!"#$%!!"#$%&!
!ℎ!"!!"#!!"!!"#$%&'(!
!!: !ℎ!"!!!"!!!!"##$!"#$"!!"#$""%!!"#$"%&'&$"()!!"#$!!"#!!ℎ!!!"#$%&%"#'(!!"#$! !"!!ℎ!!!"#$%&ℎ!!"#$%!!"#$%&!!ℎ!"!!"#!!"!!"#$%&'(!
The first hypothesis was tested by comparing the MSE for the unconditional OLS with three different conditional methods (OLS-RW with 30 and 250, together with MGARCH DCC). From table 2 it appears as all the three conditional methods are better than the unconditional CAPM. So, therefore !! is rejected and there is a difference between unconditional CAPM and conditional CAPM, the conditional CAPM is more precise than the unconditional CAPM in predicting the investor’s expectations.
This depends on that stocks are volatile and a model that uses one prediction for the whole series does not consider daily movements. To find one beta value that should fit the data better than daily betas seems quite impossible.
There are a number of studies that have found that unconditional CAPM performs poorly (Fama and French (1992, 1993, 1996)). According to Abdymomunov and Morley (2011) conditional CAPM should explain returns better than unconditional CAPM. If the time-period of estimated returns is longer, than the unconditional CAPM can give a better result, but not in this case even if the data consist of 4313 trading days. This is due to that Shiller (1981) states that if the market has full information then the stock price in each time period should be equal to the expected discounted value of future dividends (Englund et al. 2013).
The results are consistent with Fabozzi and Francis (1978) and Sunder (1980). Our conditional CAPM models have a lower MSE and therefore they explain investors expectations better, due to that the outcome becomes closer to the expected outcome.
MSE is the mean squared pricing error and according to Jagannathan and Wang (1996) conditional CAPM should have a smaller pricing error than the unconditional CAPM, which are seen in the results.
The second hypothesis states, a more sophisticated model (MGARCH DCC) is better than a more simple model (rolling window OLS), i.e. “MGARCH DCC has a more accurate
performance than rolling window OLS”.
!"#$%ℎ!"#"!2:
!!:!!"#!!"#ℎ!!"##$%&!!"#$%!!!"!!"##"$!!"!!"!!"#$%!!"!!""#$!%&!!"!!"#$%&!!""!!"!!ℎ!! !"#$%&ℎ!!"#$%!!"#$%&!!ℎ!"!!"#!!"!!"#$%&'(
!!: !"##$%&!!"#$%!!!"#!!"!!"#$%!!ℎ!"!!"#$%&!!""!!"!!ℎ!!!"#$%&ℎ!!"#$%!!"#$%&! !ℎ!"!!"#!!"!!"#$%&'(!
The second hypothesis is examined by comparing the MSE for the OLS-RW 30 and 250 against MGARCH DCC. Consistently OLS-RW 30 is better than MGARCH DCC, which is consistently better than OLS-RW 250. So, an OLS-RW with a semi-short rolling window is better than MGARCH DCC. When OLS-RW had a longer window that is approximately the number of trading days per year, then MGARCH DCC is not better.
According to the theory behind the econometric models, OLS should not be able to capture the movements due to heteroscedasticity and autocorrelation, which raises the question: why OLS is better?
As stated by Engle (2001), when there is heteroscedasticity in the data OLS estimates are still unbiased. Hence, OLS could therefore give a sufficient prediction in the CAPM. The heteroscedasticity that occurs in the data will affect the standard errors and make them smaller and consequently affect the confidence intervals of the estimates. A small confidence interval of the estimated coefficients makes the prediction seem more accurate than they really are. MGARCH DCC treats the heteroscedasticity as an additional variance to be estimated.
Engle (2001) continued his discussion about the accuracy of the predictions that the variance of the error terms is in some period large and in some period small. This typically arises when the dependent variable is asset returns. In Appendix 2 from the graphs over excess return it can be seen that some periods have more movements, i.e. higher volatility and some periods have less movements, i.e. smaller volatility. These periods does not rise independently of each other, which mean that autocorrelation exists. The OLS estimates are inconsistent but still
unbiased, which means that OLS can be used even if it is not the best linear unbiased estimator (BLUE). As with heteroscedasticity the variance is still under estimated and inference cannot be made.
There is a computational burden, i.e. a computational cost of MGARCH DCC estimation, which for example is shown in the estimation of the correlations (see equation 3.6). There are 4313 days and 28th stocks, which leads to 120736 correlations (Bali and Engle, 2010). A high computational burden requires an estimation technique that use both a computer with capacity and time. A high computational burden can even cause complications when prediction asset returns. Jagannathan and Wang (1996) state that dynamic models (like MGARCH DCC) are challenging to work with but it often gives nice results.
One problem with MGARCH DCC is that it can be over-parameterized. This means that the model fits the data too well so when the returns of an asset fluctuates highly the model does not follow as good as a more “generalized model” does, which means that the model can under or over estimates the daily returns (Chiang and Chen, 2015).
8. Conclusions
This paper has examined different models capacity to capture the investor expectations in the stock market. The research question was: “The unconditional CAPM is as accurate as
conditional CAPM in the Swedish stock market”. To answer this question, two hypotheses
was formulated: “The unconditional CAPM is as accurate as conditional CAPM in the
Swedish stock market and “MGARCH DCC has a more accurate performance than rolling window OLS”. Comparing the Mean Squared Error (MSE) tested the hypothesis; the lower
MSE the better the model could capture the investor expectations.
The results showed that the conditional CAPM is better than unconditional CAPM, which is the same result that earlier studies have found (see Jagannathan and Wang, 1996 and Abdymomunov and Morley, 2011). So, a model with time varying betas is needed to predict the CAPM sufficiently in the Swedish stock market.
The results also revealed that OLS-RW is a better method to predict CAPM with, than MGARCH DCC is. But this was the case when a small window is used. When OLS-RW had a longer window for the prediction of CAPM then MGARCH DCC gives a more precisely predictions of CAPM, which was the same result found by Godeiro, da Silva and Rodrigues (2013). The procedure to predict the MGARCH DCC was quite difficult. It did not work for all stocks and it were time consuming. For those reasons MGARCH DCC is not a recommended as a method for estimating CAPM.
The final conclusion is that OLS-RW with a semi-short window is the best way to predict the CAPM in the Swedish stock market. The result indicates that there is not any benefit in estimating beta by using a more complex model as MGARCH DCC compared to OLS-RW. The result showed that CAPM with MGARCH DCC is rather a more complex model with less forecast accuracy, i.e. less ability to capture the investor expectations, than OLS-RW. An improvement of the study could be to add Kalman filter as a method and extend the data so it also contains data for small and medium sized companies.
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Appendix
Appendix 1
Table 3. List of all selected stocksStock Ticker
Assa Abloy B ASSA B
Atlas Copco B ATCO B
B&B Tools B BBTO B
Beijer Alma B BEIA B
Beijer Ref B BEIJ B
Bong BONG
Concordia Maritime B CCOR B
Doro DORO
Electrolux B ELUX A
Elektra B EKTA B
Ericsson B ERIC B
Fabege FABG
Fast Partner FPAR
Getinge B GETI B Gunnebo GUNN Haldex HLDX Hexagon B HEXA B H&M B HM B Intellecta B ICTA B Midway B MDW B PEAB B PEAB B
! SinterCast SINT Skanska B SKA B SSAB B SSAB B Svenska Handelsbanken B SHB B Svolder B SVOL B Volvo B VOLV B Åf B AF B
!
!
Appendix 2
Graphs 1-28. Time Series over prices (a) and excess return (b)
1a Time series over Assa Abloy B prices 1b Time series over Assa Abloy B excess return
2a Time series over Atlas Copco prices 2b Time series over Atlas Copco prices excess return
3a Time series over B&B Tools B prices 3b Time series over B&B Tools B prices excess return
0 20 40 60 80 10 0 ASSA 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date ASSA ASSA -. 2 -. 1 0 .1 .2 Assa Ab lo yB 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date AssaAbloyB AssaAbloyB 0 50 10 0 15 0 AT C O B 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date ATCO B ATCO B -. 2 -. 1 0 .1 .2 At la sC op co B 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date AtlasCopcoB AtlasCopcoB 0 10 0 20 0 30 0 TO OL S 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date TOOLS TOOLS -. 2 -. 1 0 .1 .2 BBT o ol sB 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date BBToolsB BBToolsB
!
4a Time series over Beijer Alma B prices 4b Time series over Beijer Alma B prices excess return
5a Time series over Beijer Ref B prices 5b Time series over Beijer Ref B prices excess return
6a Time series over Bong prices 6b Time series over Bong prices excess return
0 50 10 0 15 0 20 0 BEI JER AL MA 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date
BEIJER ALMA BEIJER ALMA
-. 3 -. 2 -. 1 0 .1 .2 Be ije rAl ma B 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date BeijerAlmaB BeijerAlmaB 0 50 10 0 15 0 BEI JER R EF 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date
BEIJER REF BEIJER REF
-. 1 0 .1 .2 Be jie rR e fB 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date BejierRefB BejierRefB 0 20 40 60 80 BO N G 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date BONG BONG -. 4 -. 2 0 .2 .4 Bo n g 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date Bong Bong
!
!
7a Time series over Concordia Maritime B prices 7b Time series over Concordia Maritime B prices excess return
8a Time series over Doro prices 8b Time series over Doro prices excess return
9a Time series over Electrolux B prices 9b Time series over Electrolux B prices excess return
10 20 30 40 50 60 CC O R B 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date CCOR B CCOR B -. 3 -. 2 -. 1 0 .1 .2 C o nco rid aMa rt ime B 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date ConcoridaMartimeB ConcoridaMartimeB 0 10 0 20 0 30 0 DO RO 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date DORO DORO -. 4 -. 2 0 .2 .4 D o ro 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date Doro Doro 0 50 10 0 15 0 20 0 EL U X B 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date ELUX B ELUX B -. 2 -. 1 0 .1 .2 El e ct ro lu xB 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date ElectroluxB ElectroluxB
!
10a Time series over Elektra B prices 10b Time series over Elektra B prices excess return
11a Time series over Ericsson B prices 11b Time series over Ericsson B prices excess return
12a Time series over Fabege prices 12b Time series over Fabege prices excess return
0 20 40 60 80 EKT A B 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date EKTA B EKTA B -. 2 -. 1 0 .1 .2 .3 El e kt ra B 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date ElektraB ElektraB 0 20 0 40 0 60 0 80 0 ER IC B 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date ERIC B ERIC B -. 3 -. 2 -. 1 0 .1 .2 Eri csso n B 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date EricssonB EricssonB 20 40 60 80 10 0 F ABG 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date FABG FABG -. 2 -. 1 0 .1 .2 Fa be ge 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date Fabege Fabege
!
!
13a Time series over Fast Partner prices 13b Time series over Fast Partner prices excess return
14a Time series over Getinge B prices! 14a Time series over Getinge B prices excess return
15a Time series over Gunnebo prices 15b Time series over Gunnebo prices excess return
0 20 40 60 80 F AST PAR 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date
FAST PAR FAST PAR
-. 2 0 .2 .4 .6 F a st Pa rt n er 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date FastPartner FastPartner 0 50 10 0 15 0 20 0 G ET IN G E 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date GETINGE GETINGE -. 2 -. 1 0 .1 Ge tin g eB 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date GetingeB GetingeB 0 20 40 60 80 G U N N EBO 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date GUNNEBO GUNNEBO -. 2 -. 1 0 .1 .2 Gu nn eb o 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date Gunnebo Gunnebo
!
16a Time series over Haldex prices 16b Time series over Haldex prices excess return
17a Time series over Hexagon B prices 17b Time series over Hexagon B prices excess return
18a Time series over H&M B prices 18b Time series over H&M B prices excess return
0 50 10 0 15 0 HL DX 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date HLDX HLDX -. 6 -. 4 -. 2 0 .2 .4 Ha ld ex 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date Haldex Haldex 0 50 10 0 15 0 20 0 H EXA B 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date HEXA B HEXA B -. 2 -. 1 0 .1 .2 H e xa g on B 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date HexagonB HexagonB 0 50 10 0 15 0 20 0 25 0 H M B 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date HM B HM B -. 4 -. 2 0 .2 H MB 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date HMB HMB
!
!
19a Time series over Intellecta B prices! 19b Time series over Intellecta B prices excess return
20a Time series over Midway B prices 20b Time series over Midway B prices excess return
21a Time series over PEAB B prices 21b Time series over PEAB B prices excess return
0 50 10 0 15 0 IN T EL LEC T A 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date INTELLECTA INTELLECTA -. 2 -. 1 0 .1 .2 In te lle ct aB 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date IntellectaB IntellectaB 0 20 40 60 80 MI D W B 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date MIDW B MIDW B -. 3 -. 2 -. 1 0 .1 .2 Mi d w a yB 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date MidwayB MidwayB 0 50 10 0 15 0 PEAB B 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date PEAB B PEAB B -. 4 -. 2 0 .2 Pe a bB 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date PeabB PeabB
!
22a Time series over SinterCast prices 22b Time series over SinterCast prices excess return
23a Time series over Skanska B prices 23b Time series over Skanska B prices excess return
24a Time series over SSAB B prices 24b Time series over SSAB B prices excess return!
0 10 0 20 0 30 0 40 0 50 0 SI N T 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date SINT SINT -. 2 0 .2 .4 .6 .8 Si n te rC a st 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date SinterCast SinterCast 0 50 10 0 15 0 20 0 SKA B 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date SKA B SKA B -. 3 -. 2 -. 1 0 .1 .2 Ska n ska B 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date SkanskaB SkanskaB 0 10 0 20 0 30 0 SSAB B 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date SSAB B SSAB B -. 2 -. 1 0 .1 .2 SSABB 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date SSABB SSABB
!
!
25a Time series over Svenska Handelsbanken B prices 25b Time series over Svenska Handelsbanken B prices excess return!
26a Time series over Svolder B prices 26b Time series over Svolder B prices excess return!
27a Time series over Volvo B prices 27b Time series over Volvo B prices excess return!
0 20 40 60 80 SH B B 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date SHB B SHB B -. 1 -. 0 5 0 .0 5 .1 Sve n ska H an de lsb an ke nB 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date SvenskaHandelsbankenB SvenskaHandelsbankenB 20 40 60 80 10 0 SVO L B 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date SVOL B SVOL B -. 2 -. 1 0 .1 Svo ld e rB 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date SvolderB SvolderB 0 50 10 0 15 0 VO L V B 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date VOLV B VOLV B -. 2 -. 1 0 .1 .2 Vo lvo B 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date VolvoB VolvoB
!
28a Time series over Åf B prices 28b Time series over Åf B prices excess return!
0 20 40 60 80 ÅF 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date ÅF ÅF -. 2 -. 1 0 .1 .2 Af B 1/1/19941/1/19961/1/19981/1/20001/1/20021/1/20041/1/20061/1/20081/1/20101/1/2012 Date AfB AfB
!
!
Appendix 3
Table 4. LM test of ARCH effect
Stock Q(1) Q(5) Q(10) Q(15) Q(20) Market 128.475 (0.0000) 454.939 (0.0000) 540.230 (0.0000) 567.198 (0.0000) 593.981 (0.0000) Assa Abloy B 56.911 (0.0000) 117.944 (0,0000) 168.548 (0,0000) 186.180 (0,0000) 190.921 (0,0000) Atlas Copco B 61.951 (0.0000) 210.536 (0,0000) 352.352 (0,0000) 391.605 (0,0000) 426.368 (0,0000) B&B Tool B 108.204 (0.0000) 130.660 (0,0000) 174.174 (0,0000) 190.982 (0,0000) 194.946 (0,0000) Beijer Alma B 486.130 (0.0000) 530.656 (0.0000) 540.710 (0.0000) 628.328 (0.0000) 639.614 (0.0000) Beijer Ref B 138.421 (0.0000) 230.604 (0.0000) 242.726 (0.0000) 248.132 (0.0000) 253.437 (0.0000) Bong 29.109 (0.0000) 181.445 (0,0000) 508.351 (0,0000) 643.229 (0,0000) 669.372 (0,0000) Concordia Maritime B 86.187 (0.0000) 232.536 (0,0000) 326.498 (0,0000) 343.703 (0,0000) 356.824 (0,0000) Doro 150.647 (0.0000) 187.306 (0,0000) 216.159 (0,0000) 256.071 (0,0000) 261.133 (0,0000) Electrolux B 32.921 (0.0000) 120.567 (0.0000) 147.226 (0.0000) 172.014 (0.0000) 183.690 (0.0000) Elektra B 79.153 (0.0000) 132.570 (0,0000) 181.208 (0,0000) 206.097 (0,0000) 207.354 (0,0000) Ericsson B 174.425 (0.0000) 281.604 (0,0000) 339.852 (0,0000) 416.546 (0,0000) 440.616 (0,0000) Fabege 76.567 (0.0000) 255.618 (0.0000) 366.085 (0.0000) 382.756 (0.0000) 435.492 (0.0000) Fast Partner 31.028 (0.0000) 51.506 (0,0000) 55.233 (0,0000) 59.262 (0,0000) 81.667 (0,0000) Getinge B 66.682 (0.0000) 120.297 (0,0000) 144.484 (0,0000) 170.633 (0,0000) 171.786 (0,0000) Gunnebo 103.847 (0,0000) 290.408 (0,0000) 326.484 (0,0000) 356.223 (0,0000) 359.175 (0,0000) Haldex 0.523 (0.4698) 2.477 (0.7800) 3.417 (0.9698) 579.622 (0.0000) 579.227 (0.0000) Hexagon B 154.847 (0.000) 353.382 (0,0000) 429.775 (0,0000) 464.008 (0,0000) 529.257 (0,0000) H&M B 10.866 (0.0000) 31.931 (0,0000) 47.290 (0,0000) 55.545 (0,0000) 61.814 (0,0000) Intellecta B 341.789 (0.0000) 315.498 (0,0000) 419.795 (0,0000) 465.992 (0,0000) 488.793 (0,0000) Midway B 71.298 (0.0000) 80.340 (0,0000) 91.245 (0,0000) 92.321 (0,0000) 94.802 (0,0000) PEAB B 18.405 (0.0000) 85.034 (0,0000) 109.039 (0,0000) 117.740 (0,0000) 147.262 (0,0000) SinterCast 38.011 (0.0000) 49.721 (0,0000) 50.473 (0,0000) 92.461 (0,0000) 93.184 (0,0000) Skanska B 32.346 (0,0000) 91.956 (0,0000) 146.974 (0,0000) 165.494 (0,0000) 186.556 (0,0000)
! SSAB B 147.632 (0.0000) 429.585 (0,0000) 605.095 (0,0000) 628.294 (0,0000) 684.083 (0,0000) Svenska Handelsbanken B 186.864 (0.0000) 494.004 (0,0000) 574.635 (0,0000) 601.534 (0,0000) 641.792 (0,0000) Svolder B 269.465 (0.0000) 330.447 (0,0000) 415.936 (0,0000) 431.623 (0,0000) 433.886 (0,0000) Volvo B 138.339 (0.0000) 299.324 (0.0000) 445.700 (0.0000) 470.559 (0.0000) 489.994 (0.0000) Åf B 195.274 (0.0000) 225.618 (0,0000) 258.210 (0,0000) 279.503 (0,0000) 294.762 (0,0000)
