1 Literature review and introduction
2 Theoretical foundation
2.1 ThetraditionalCAPM
2.2 The flaws of the traditional CAPM
2.3 The Conditional CAPM
2.4 The (G)ARCH model foundations
2.5 The GARCH-M extension
3 Empirical analysis
3.1 Econometricapproachandmethods
3.2 Data description
3.3 Descriptive analysis and tests
3.4 Modelestimates
4 Conclusions
The Capital Asset Pricing Model (CAPM) of Sharpe (1964) and Lintner (1965) is one of the most popular models to evaluate the pricing of risky assets, in particular stocks. Doolan/Smith (2017) conclude, the CAPM provides a single state, single factor, general equilibrium theory of the risk-return relation. However, in the 1960s, Mandelbrot (1963) already observed stock returns to have a very peaked distribution with heavy tails and also periods of persistent volatility, which contradicts the CAPM. In response to these observations, the Conditional CAPM (C-CAPM) has been discussed many times by several authors, beginning with Merton (1973) and his inter-temporal and multi-beta CAPM version, Bodie et al. (1983, 1984) and, among others, the articles by Jagannathan/Wang (1996), Bollerslev et al. (1988) and Harvey (1989, 1991) where C-CAPM models are described, estimated and tested. In a C-CAPM investors can price an asset or portfolio conditional on the available information at a point in time. This is done by replacing the unconditional by conditional moments of returns. Statistically, processes of ”Generalized Autoregressive Conditional Heteroscedasticity” (GARCH) can capture the so called ”stylized facts”, some observed by Mandelbrot (1963). GARCH models were developed by Engle (1982) and Bollerslev (1986) and try to model time-varying second moments of asset returns. If a GARCH process is assumed for the disturbance term in a C-CAPM, a GARCH-in-mean model (GARCH-M) can be estimated, where the conditional variance or covariance impacts the conditional expectation of (excess) returns. The GARCH-M extension has been proposed by Engle, Lilien, Robins (1987) and Bollerslev, Engle, Wooldridge (1988). The GARCH-M can model time-varying conditional moments, but also time-varying risk premia and the implied beta factor. Linnenbrink (1998) and Hansson/Höordahl (1998) present this approach appealingly, also various tests and extensions have been made by e.g. Hodgson/Vorkink (2003) and Doolan/Smith (2017).
As for this seminar paper, I mostly follow the comprehensive dissertation ”Das CAPM mit zeitabhöangigen Beta-Faktoren” of Linnenbrink (1998) and the paper of Bollerslev et al. (1988). First, the theoretical foundations of the CAPM, the C-CAPM, GARCH processes and the GARCH-M extension are presented. Then, in the empirical part, I estimate a (univariate) GARCH-M representation of the C-CAPM.The goal is to compare its performance to a traditional CAPM with a single stock portfolio of an investor (Tesla, Inc.). Throughout this seminar paper, I will use a consistent notation of symbols, no matter what the original notation of the cited source is.
The traditional Sharpe-Lintner-CAPM provides a functional relationship between the expected return of an asset and the respective systematic risk, the so-called beta factor. In describing the CAPM, I will follow Linnenbrink (1998) and present the most important extracts of the CAPM for the purpose of this seminar paper. She extensively describes the crucial assumptions of the CAPM in 3 categories, namely regarding assumptions of the portfolio selection approach, the investor's investment behaviour and regarding the capital market. However, I want to focus only on some key points that are relevant here.
The CAPM is based on the portfolio selection approach by Markowitz (1959), who requires investment decisions to be made based on probability distributions of returns with expected returns = E(Ri) and variance a[2] = Var(Ri), which represents a measure of risk in the CAPM. Investors consider a one-period horizon in their portfolio optimization and behave risk-averse. Asset returns Rt are normally distributed and all investors expect the same first and second moments.
An investor builds his portfolio from a set of possible efficient portfolios, i.e. from the set of minimum variance portfolios. In equilibrium, every investor holds a combination of the risk-free asset and the market portfolio and this portfolio is on the capital market line (CML). This optimal portfolio is called tangential portfolio. Now, the CAPM model equation is able to reveal a relationship between expected returns and risk of a single asset or portfolio. Linnenbrink (1998) distinguishes the ex-ante and ex-post CAPM. The ex-ante-CAPM model looks as follows:
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At a glance, we see that the expected return of an asset in market equilibrium consists of a risk-free return and the expected market risk premium E(Rm) — Rf. Next, we can also formulate the model in terms of ”excess returns”, i.e. risk premia:
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The risk premium of a stock over the risk-free rate Yi = Ri — Rf is often referred to as ”equity risk premium” (ERP) and Ym refers to the market risk premium (MRP), which is Rm — Rf. We will need this formulation (2) later when we follow Bollerslev et al. (1988) who estimate a C-CAPM based on a risk premia formulation. We can see that the MRP is proportional to the ratio of the covariance of the asset return with the market portfolio return and the variance of the market portfolio return.
This relationship is referred to as the beta factor:
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The beta expresses the main point of the CAPM: a linear relationship between the ERP and the systematic risk. Beta became an established instrument for investors to evaluate market-related risk of an asset (Linnenbrink, 1998). So, the beta of a potential investment is a measure of how much market-related risk the investment will add to a portfolio. The total risk of an asset consists of the systematic and the so-called idiosyncratic risk on the other side. Idiosyncratic risk can be reduced by diversification, systematic risk is nondiversifiable as it ”moves with the market”. In the CAPM, the investors therefore pay a risk premium to bear the systematic risk. Now, we turn to the ex-post model equation of the CAPM, which is used for empirical application of the CAPM. In contrast to the ex-ante version (1), the ex-post version is based on a time series of returns, that are realizations of a stochastic process {Rt}. So, the model transitions into its empirically verifiable ex-post equation:
Rit = Rft + ßi(Rmt — Rft) + ^it ([4])
For the moments of the probability distribution of returns not to change over time, the additional assumption of weak stationarity of the stochastic process of returns has to be made. Weak stationarity requires that first and second moments do not depend on t, i.e. they are constant over time. One of the consequences of this additional assumption is that the CAPM has constant beta factors.
In general the assumptions of the CAPM are very restrictive. Linnenbrink (1998) summarizes six main violations of the assumptions: Non-existence of a risk-free return, imperfect capital markets, heterogeneous expectations of investors, different investment horizons, time-continuous trading and contradictory characteristic properties of returns (”stylized facts”). She demonstrates, that for the most part a violation of the assumptions can be encountered by modifications of the CAPM model equation, so that the statements of the model can be preserved. The last violation regarding the characteristics refers to two things: the return distribution and to their autocorrelation structure, i.e. the time-dependence of returns. Regarding the return distribution, the assumption of a normal distribution of returns is doubtful. Already Mandelbrot (1963) pointed out some observed characteristics of the empirical distribution of returns. Observed statistical characteristics of returns are considered as ”stylized facts”. Mandelbrot states that return distributions are usually heavy tailed and stronger ”peaked” compared to normal distributions. This characteristic is referred to as a leptokurtic probability distribution. Regarding the autocorrelation structure, Mandelbrot (1963) also emphasizes that ”...large changes tend to be followed by large changes - of either sign - and small changes tend to be followed by small changes”. This is the observation of volatility clustering and suggests some kind of autocorrelation. It is a well observed stylized fact that the absolute and squared daily returns show relatively high autocorrelation (Schmid/Trede, 2005) and a slow decay. (G)ARCH processes are suitable to explain volatility clusters, because they allow for conditional heteroscedasticity, i.e. the volatility is not constant, but can change over time.
Just like Bollerslev et al. (1988), I focus on the possibility that investors may have ”common expectations on the moments of returns, but that these are conditional expectations and therefore random variables rather than constants”. So, the beliefs of the investors are based on conditional information they receive, in particular the conditional moments. These aspects are taken into account by the C-CAPM in the next section.
In contrast to the CAPM, the C-CAPM allows for time-varying moments of a conditional probability distribution of returns (expected values, variances and covariances). For the C-CAPM, I will follow Linnenbrink (1998), Hansson/Hördahl (1998) and Hodgson/Vorkink (2003). As the crucial representations of the CAPM have already been displayed in the section 2.1, we simply need to enrich the formulas by replacing unconditional by conditional moments. For the purpose of this seminar paper and with respect to the empirical section I highlight the C-CAPM formulation in terms of risk premia, corresponding to equation (2):
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Notice, that we now account for the variability in t of the beta coefficient, which is now ßi = ß it . Hansson and Hordahl (1998) and also Linnenbrink (1998) comment on the C-CAPM model equation as the (conditional) expected (excess) return (i.e. the ERP) to be dependent on time-varying risk, which is the main point of the C- CAPM. The C-CAPM considers the time-varying systematic risk (Beta) and with that risk also the conditional ERP is time-varying. From the setting, we see that we need an additional assumption for the C-CAPM, namely that the returns now must be conditionally normally distributed.
The model equation (5) can be re-written to consider an aggregate measure of risk aversion (Bollerslev et al., 1988), often assumed to be the market price of risk. Merton (1980) analyses the conditional expected returns dependent on the conditional variance within an inter-temporal equilibrium model and finds a positive, but constant relationship between the expected returns and the conditional variance. This measure of aggregate relative risk aversion will be denoted as S and can also be seen as a constant relative risk aversion (CRRA) of a representative investor:
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Thus, conditioned on the information It-1 available in the market at time t - 1, an increase in the conditional variance of the market return (denumerator) induces an equal increase in the expected conditional MRP (numerator). So, S is a parameter to identify the trade-off between risk and expected (excess) return as we see, when we consider (7) together with equation (5):
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Linnenbrink (1998) and Hodgson/Vorkink (2003) explicitly formulate the two versions (6) and (8). Instead, Bollerslev et al. (1988) only focus on depicting the version of (8), stating however, that the implied betas are time-varying as well. As Bodurtha and Mark (1991) put it, ”the conditional CAPM provides a convenient way to incorporate the time-varying conditional variances and covariances that other researchers have found to be important in financial time series”. By all means, the Conditional CAPM considers the these stylized facts. Bollerslev et al. (1988) and Hansson/Hördahl (1998) state that this formulation is consistent with the inter-temporal investment decision making and could be considered as a statistical implementation of the inter-temporal CAPM.
In this section, the ARCH and GARCH models are introduced in general, before we transition to the GARCH-M model extension in the next section. Engle (1982) first came up with ARCH, Bollerslev (1986) with a GARCH model. In this section, I will also make use Taylor (1994) and Linnenbrink (1998) and the survey of Bollerslev, Chou, Kroner (1992). GARCH models can explain the leptokurtitc and heavy-tailed distributional characteristics of returns as well as volatility clusters that we have mentioned earlier. So, the unconditional (long-term) distributions of returns are leptokurtic and the variance is constant when used in the static CAPM. But conditional distributions have time-varying moments and a statistical model is needed to estimate them in a C-CAPM. GARCH models are that type of model, designed for processes with conditional heterogeneous variance. Essentially, the volatility of a time series firstly depends on the absolute value of that series at preceding times (lagged squared residuals) and secondly on the volatility by itself at preceding times (lagged conditional variance/standard deviation). These two factors gave rise to the ARCH and GARCH models. ARCH models take into account the squared residuals and GARCH uses both, the squared residuals and the lagged volatility. Hence, the models are able to explain volatility persistence within certain time periods of an asset. Engle (1982) proposed an ARCH(p) model, where the conditional variance is a linear function of past squared residuals. For, say, an ARCH(1) process, the residuals are serially uncorrelated with E(et) = 0 and have a conditional variance Var(s t | I t-1) that will be denoted by σ2 t:
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The process {εt} is said to be of order p, ARCH(p), in this case ARCH(1). Here, the last period's absolute fluctuations have a direct impact on current fluctuations and the conditional variance equation also includes a constant a0. Bollerslev (1986) proposed a generalization of the ARCH model (GARCH) that allows for an even more persistent volatility process: in addition, we have a component that represents the conditional volatility itself in t — 1, namely σ2t−1. A GARCH process of order p and q, GARCH(p,q), is defined as:
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Usually a GARCH(1, 1) is estimated, where we have:
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If α1 = β1 = 0 we have the conditional variance of = a0, i.e. a constant as in typical AR(MA) models. As soon as α1, β1 > 0 we can speak of volatility clustering and volatility persistence. If even α1 + β1 ≈ 1 we would have a very strong persistence of volatility. The observation of volatility clustering requires autocorrelation in the variance of returns. A basic property of GARCH processes is the autocorrelation of ε2t, which is an estimator for the conditional variance. Autocorrelated squared residuals however, contradict the EMH. On the other hand, unconditional first moments are not autocorrelated, so weakly stationary (Linnenbrink, 1998). Hence, they are unpredictable which matches to the efficient market hypothesis (EMH). According to Bollerslev (1986) it suffices for stationarity that α1 + β1 < 1. Regarding the volatility persistence one can say: the larger the αi's, the larger the fluctuations will be and the larger the βj's, the wider the volatility clusters will be. In summary, GARCH models are designed to model time-varying volatility, or specifically: heterogeneous variance.
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