Econometric specifications University of Pavia March 2, 2007
Outline 1 Introduction 2 3 of Excess Returns
DAPM is refutable empirically if it restricts the joint distribution of the observable asset prices or returns under study.
s The goal is to price a given cash-flow stream based on agent s optimal consumption and investment decisions then a fully articulated specification of agents preferences, technologies and other constraints. If the concern is to derive prices as discounted cash flows, subject only to the constraint that there are no arbitrage opportunities in the economy, then it may be sufficient to specify how the relevant discount factors depend on the underlying risk factors affecting security prices, along with the joint distribution of these factors. Testing the restrictions implied by a particular equilibrium condition arising out of an agent s consumption/investment decision.
s The goal is to price a given cash-flow stream based on agent s optimal consumption and investment decisions then a fully articulated specification of agents preferences, technologies and other constraints. If the concern is to derive prices as discounted cash flows, subject only to the constraint that there are no arbitrage opportunities in the economy, then it may be sufficient to specify how the relevant discount factors depend on the underlying risk factors affecting security prices, along with the joint distribution of these factors. Testing the restrictions implied by a particular equilibrium condition arising out of an agent s consumption/investment decision.
s The goal is to price a given cash-flow stream based on agent s optimal consumption and investment decisions then a fully articulated specification of agents preferences, technologies and other constraints. If the concern is to derive prices as discounted cash flows, subject only to the constraint that there are no arbitrage opportunities in the economy, then it may be sufficient to specify how the relevant discount factors depend on the underlying risk factors affecting security prices, along with the joint distribution of these factors. Testing the restrictions implied by a particular equilibrium condition arising out of an agent s consumption/investment decision.
P s set of payoffs Payoffs at date s > t are to be priced at date t, for s > t by an economic model. Pricing function π t : P s R DAPM mantains the assumption of no arbitrage opportunities q t+1 P t+1 Pr {q t+1 0} = 0 Pr ({π t (q t+1 0)} {q t+1 > 0}) = 0
Non-negative payoffs at t + 1 that are positive with positive probability have positive prices at date t. The absence of arbitrage opportunities on a set of payoffs P s is essentially equivalent to the existence of a special payoff, a pricing kernel q s Pr(q s > 0) = 1 and represents the pricing function π t as π t (q s ) = E[q s q t I t ] q s P s when the uncertainty is generated by discrete random variables.
Overidentifying restrictions are obtained by restricting the functional form of q s or the joint distribution of the pricing environment (P s, q s, I t ).
Classification of DAPM Classification of DAPM according to the types of restrictions they impose on the distributions of the elements of (P s, q s, I t ): Arbitrage-free pricing models Beta representations of excess portfolio returns Linear asset pricing relations.
Classification of DAPM Classification of DAPM according to the types of restrictions they impose on the distributions of the elements of (P s, q s, I t ): Arbitrage-free pricing models Beta representations of excess portfolio returns Linear asset pricing relations.
Classification of DAPM Classification of DAPM according to the types of restrictions they impose on the distributions of the elements of (P s, q s, I t ): Arbitrage-free pricing models Beta representations of excess portfolio returns Linear asset pricing relations.
Classification of DAPM Classification of DAPM according to the types of restrictions they impose on the distributions of the elements of (P s, q s, I t ): Arbitrage-free pricing models Beta representations of excess portfolio returns Linear asset pricing relations.
The preference-based models parameterize agent s intertemporal consumption and investment decision problem. The economy is supposed to be comprised of a finite number of infinitely lived agents with identical endowments, information, and preferences in an uncertain environment. A t agent s information set at time t The agent ranks consumption sequences using a von Neumann-Morgestern utility functional [ ] E β t U(c t ) A 0 t=0 the preferences are assumed to be time separable with period utility function U and the subjective discount factor β (0, 1).
If the representative agent can trade the assets with payoffs P s and their asset holdings are interior to the set of admissible portfolios, the prices of these payoffs in equilibrium are given by where π t (q s ) = E[m s t s q s A t ] m s t s = β U (c s ) U (c t ) is the intertemporal marginal rate of substitution of consumption between t and s. For a given utility parameterization a preference-based DAPM allows the association of the pricing kernel q s with m s t s.
To determine the prices π t (q s ) requires a parametric assumption about the agent s utility function U(c t ) and sufficient economic structure to determine the joint, conditional distribution of ms s t and q s. Prices are part of the determination of an equilibrium in goods and securities markets. The modeler must specify a variety of features of an economy outside of securities markets in order to undertake preference-based pricing. Data limitations: some of the theoretical constructs appearing in utility functions or budget constraints do not have readily available, observable counterparts.
For these reasons the preference-based DAPM have attempted to evaluate whether, for a particular choice of utility function U(c t ) the pricing function π t (q s ) does in fact price the payoffs in P s. Given observations on a candidate ms s t and data on asset returns The condition R s {q s P s : π t (q s ) = 1} π t (q s ) = E[m s t s q s A t ] implies testable restrictions on the joint distribution of R s, π t (q s ) and elements of A t.
For each s-period return r s E[m s t s r s 1 A t ] = 0 r s R s An immediate implication of this moment restriction is that E[(m s t s r s 1)x t ] = 0 x t A t These unconditional moment restrictions can be used to construct method-of-moments estimators of the parameters governing ms s t and to test whether or not ms s t prices the securities with payoffs in P s. There will be no arbitrage opportunities in equilibrium.
Alternative pricing based on the presumption that asset prices are such that no arbitrage opportunities can arise. No equilibrium behavior is needed to determine asset prices. No arbitrage principle used by Black and Scholes (1973), Merton (1973), Ross (1978), Harrison and Kreps (1979). Under weak regularity conditions pricing can proceed as if agents are risk neutral. Continuous-time, trade of default-free bond that matures an instant in the future and pays the (continuously compounded) rate of return r t, discounting for risk-neutral pricing is done by the default-free roll-over return e ( R s t rudu)
If uncertainty about future prices and yields is generated by a continuous-time Markov process Y t (so the conditioning information set I t is generated by Y t ), then the price of the payoff q s is given equivalently by π t (q s ) = E[q s q s Y t ] = E Q [e R s t rudu q s Y t ] E Q is the expectation with regard to the risk-neutral conditional distribution of Y. The risk attitudes of investors are implicit in the exogenous specification of q as a function of the state Y t and hence in the change of probability measure under the risk-neutral representation.
The direct parameterization of the distribution of q facilitates the computation of security prices. The parameterization of (P s, q, Y t ) is chosen so that the expectation can be solved either analytically or numerically for This is facilitated by π t (q s ) = P(Y t ) the adoption of continuous time (continuous trading) special structure on the conditional distribution of Y constraints on the dependence of q on Y so that the expectation E Q is easily computed.
The knowledge of the risk-neutral distribution of Y t is sufficient for pricing, but this knowledge is typically not sufficient for econometric estimation. For the purpose of estimation using historical price or return information associated with the payoffs P s, we also need information about the distribution of Y under its DGP. The difference between actual and risk-neutral distributions of Y are adjustments for the market price of risk. If the conditional distribution of Y t Y t 1 is known (derivable form the continuous-time specification of Y ) then so is the conditional distribution of the observed market prices π t (q s ).
The completeness of the specification of the pricing relations (the distribution of Y and the functional form of P(Y t )) in this case implies that one can use ML to estimate the unknown parameters. This is possible using market price alone even though the risk factors Y may be latent. Key to this strategy is the presumption that the burden of computing π t (q s ) = P(Y t ) is low. For many specifications of the distribution of the state Y t, the pricing relation P(Y t ) must be determined by numerical methods.
The computational burden of solving for P(Y t ) while simultaneously estimating the parameters can be formidable, especially as the dimension of Y gets large. Alternative methods: GMM estimation Simulated ML estimation.
of Excess Returns CAPM static model: expected returns in terms of a security s beta with a benchmark portfolio. The key insights of the CAPM carry over to richer stochastic environments in which agents optimize over multiple periods. Intertemporal CAPM: s = t + 1, benchmark return from the pricing condition r t+1 = q t+1 π t (q t+1 ) π t (q t+1 ) = E[q t+1 q t+1 I t ] = E[q t+1 r t+1π t (q t+1) I t ]
of Excess Returns 1 π t (q t+1 ) = E[r t+1r t+1 I t ] π t (q t+1) = E[q 2 t+1 I t ] For all r t+1 R t+1 E[q 2 t+1 I t ] 1 = E[r t+1 r t+1 I t ] since r is one such term the benchmark return satisfies E[q 2 t+1 I t ] 1 = E[r 2 t+1 I t ] E[r t+1(r t+1 r t+1) I t ] = 0 r t+1 R t+1
of Excess Returns Several implications for the role of rt+1. One is that r t+1 is a benchmark return for a single-beta representation of excess returns ) E[r j,t+1 I t ] rt f = β jt (E[r t+1 I t ] rt f where β jt = E[r j,t+1, r t+1 I t] Var[r t+1 I t] and rt f is the interest rate on one-period riskless loans issued at date t.
of Excess Returns It turns out that the beta representation, together with r f t = 1 E[q t+1 I t] constitute exactly the same information as the basic pricing relation.
Much econometric analysis of DAPMs has focused on linear pricing relations. An important example of a linear DAPM is the version of the ICAPM obtained by assuming that β jt = β j Under this additional assumption, β j is the familiar β, where the CAPM is extended to allow both expected returns on stocks and the riskless interest rate to change over time. The mean of u j,t+1 (r j,t+1 r f t ) β j (r β t+1 r f t ) for all admissible r j. E[u j,t+1 I t ] = 0
The moment conditions: E[u j,t+1 x t ] = 0, x t I t can be used to construct estimators of the β j and tests of ICAPM.