δu0(ct+1) u0(ct) xt+1
. (A.4)
The stochastic discount factor or marginal rate of substitution, which describes an investor’s willingness to intertemporal substitution of consumption, is defined as
mt+1 :=δu0(ct+1)
u0(ct) . (A.5)
According to Cochrane (2005, p. 6) Equation (A.4) represents “the central pricing equation”. Most of the asset pricing theory simply consists of rearrangements and specializations of this formula. The expectation is generally conditioned on infor-mation up to time t; the price and the payoff are always considered at times t and t+ 1, respectively. In the following the subscripts will be suppressed and (A.1) is simply stated asp=E(mx).
A.3 Alternative asset pricing models
Despite its theoretical appeal as a universal way to value any uncertain cash flow and security, the consumption-based model has not worked well in applied work.
Possible explanations for poor model performance relate, among others, to the use of wrong utility functions and measurement errors in consumption data. This mo-tivates alternative asset pricing models to avoid the empirical shortcomings of the
consumption-based approach. Different functions formt+1have been proposed. The alternative approaches either employ different utility functions or link asset prices not to consumption data but to other factors or macroeconomic aggregates. The di-rect modeling of marginal utility based on alternative variables leads tofactor pricing models, on which the analysis in the empirical part of this thesis is grounded.
Factor pricing models specify the stochastic discount factor as a linear function of the form
m =a+b0f, (A.6)
with free parameters a and b. The K ×1 vector of factors f is chosen as a proxy for an investor’s marginal utility growth. As demonstrated by Cochrane (2005, 6), (A.6) is equivalent to a multiple-beta model of expected returns:
E(Ri) =γ +λ0βi, (A.7)
with theK×1 vector βi containing the multiple regression coefficients of returnsRi onf for assets i= 1, . . . , N. This specification, usually referred to as a beta pricing model, states that each expected return is proportional to the asset specificβi, which is also known as the quantity of risk. The k×1 vector of free parameters λ, which is the same for all assets i, can be interpreted as the price of risk. In a world with existing risk-free assets, i.e. a zero-beta portfolio, the constant γ is usually assumed to be equal to the risk-free interest rate, denoted byrf; the economic model in (A.7) can be written in terms of returns in excess of the risk-free rate with γ being set to zero.
In order to identify the factors f that can serve as appropriate proxies for marginal utility growth, one looks for variables for which (A.6) approximately holds.
As consumption is economically linked to the state of the economy, macroeconomic variables, such as interest rates, GDP growth and broad-based portfolios, constitute the first set of factors. Consumption can be assumed to also depend on current newsflow that signals future income and consumption changes. Variables that ei-ther indicate changes in consumption and/ or oei-ther macroeconomic indicators, or predict asset returns directly, also qualify as potential factors; important variables include dividend yields, stock returns or the term premium (Cochrane, 2005, 9).
The most important factor pricing models include the single-factor Capital Asset Pricing Model (CAPM), the Intertemporal CAPM (ICAPM) and the Arbitrage Pricing Theory (APT). The latter two allow for multiple sources of systematic risk.
They all represent specializations of the consumption-based model, in which extra assumptions allow for the use of other variables to proxy for marginal utility growth.
They can be summarized as follows:
The CAPM, developed by Sharpe (1964) and Lintner (1965), is the first and still most widely used factor pricing model. It linearly relates the expected return of an asset to the return’s covariance with the return on the wealth portfolio. The return on total wealth is usually approximated by the return on a broad-market stock portfolio.
The ICAPM by Merton (1973) is grounded on equilibrium arguments where an investor tries to hedge uncertainty about future returns by demanding assets that do well on bad news. In equilibrium, expected returns depend on the covariation with current market returns and on the covariation with news that predict changes in the investment opportunity set of an investor. The ICAPM can be represented by (A.6) where each state variable that forecasts future market returns can be a factor.
The APT, introduced by Ross (1976) as an alternative to the CAPM, is based on arbitrage arguments. The starting point of the APT is a statistical char-acterization of the return covariance matrix. The idea is that the common variation in returns can be related to common components, or risk factors, that describe the covariance matrix of returns. In this setting, idiosyncratic movements in returns are not priced as they can be completely diversified away.
In contrast to the ICAPM, the factors f in (A.6) are assumed to provide a description of the return covariance matrix and to be IID and orthogonal (cf.
Cochrane, 2005, 9).
Overall, the ICAPM and the APT are more general than the CAPM as they allow for multiple risk factors. They both offer the advantage of not necessarily requiring the identification of the wealth portfolio. On the other hand, the nature and the number of factors are not specified by the underlying models. For more details on multifactor asset pricing models, their underlying assumptions and the various factor selection procedures, see, for example, Fama and French (1996), Campbell et al. (1997, 6) or Cochrane (2005, 9).
Figures
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Figure B.1: t-GARCH and stochastic volatility conditional betas.
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Figure B.2: Random walk and mean reverting conditional betas.
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Figure B.3: Moving mean reverting and generalized random walk conditional betas.
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Figure B.4: Markov switching and Markov switching market conditional betas.
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