Applications of state space models in finance

A.3 Alternative asset pricing models 161
Factor pricing models specify the stochastic discount factor as a linear function of the
form
m = a + b ′ f, (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 + λ ′ β i, (A.7)
with the K × 1 vector β i containing the multiple regression coefficients of returns Ri on
f 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 specific β i, 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 by r f ; 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 either indicate changes in consumption
and/ or other 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.