Applications of state space models in finance

160 Appendix A
A.2 The consumption-based model
The most basic specification of the pricing equation presented in (A.1) can be derived
from the first-order condition of an investor’s marginal utility of consumption. The
marginal loss of utility of consuming less today to buy more assets, should be equal to a
higher degree of marginal utility gained from consuming more of the assets’ future payoff.
Hence, the value of an asset today should correspond to the expected discounted value
of its payoff. A stream of uncertain future cash flows can be valued by determining the
worth of this payoff to a typical investor using his or her utility function of consumption:
U(ct, ct+1) = u(ct) + δEt (u(ct+1)) , (A.3)
where ct is consumption at time t, ct+1 denotes random consumption at time t + 1,
and u(·) represents a concave and increasing utility function that reflects a declining
marginal value of more consumption. Investors’ typical preference of a steady stream of
consumption and their impatience for future cash flows are captured by the curvature of
the utility function and by δ, the subjective discount factor. Under the assumption that
the investor is able to freely buy and sell any amount of the payoff xt+1 at a price pt, the
basic pricing equation can be derived as the first-order condition for optimal portfolio
formation and consumption:
pt = Et
�
δ u′ (ct+1)
u ′ (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 := δ u′ (ct+1)
u ′ . (A.5)
(ct)
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 information 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 as p = 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 motivates alternative asset
pricing models to avoid the empirical shortcomings of the consumption-based approach.
Different functions for mt+1 have 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 direct modeling of marginal utility
based on alternative variables leads to factor pricing models, on which the analysis in
the empirical part of this thesis is grounded.