Intertemporal Capital Asset Pricing Model

The CAPM has been challenged empirically in a series of papers. One of the alternatives offered was the ICAPM of Merton (1973). Based on Merton‟s (1973) approach and the assumption of rational expectations, Cox et al. (1985a) developed the partial differential equation for asset prices which is similar to Lucas‟ (1978) theory of asset pricing.

Lucas‟ (1978) paper is regarded as a theoretical examination of the stochastic behaviour of equilibrium asset prices in a one-good, pure exchange economy with identical consumers. The random output of these processes is exogenously determined and consumable. Assets are defined as claims to all or a part of the output of a process and the equilibrium determines the asset prices. A general method of constructing equilibrium prices is developed and applied to a series of examples.

Although the CAPM has had significant impact on the non-academic financial community, it is still subject to theoretical and empirical criticism. The model assumes that investors choose their portfolios according to Markowitz's mean-variance criterion. There are, however, theoretical objections to this criterion (Merton 1973).

An intertemporal model for the capital market is deduced from the portfolio selection behaviour by an arbitrary number of investors who act to maximize the expected utility of lifetime consumption and who can trade continuously in time. Explicit demand functions for assets are derived, and it is shown that, unlike the one period model, current demands are affected by the possibility of uncertain changes in the future investment opportunities. After aggregating demands and requiring market clearing, the equilibrium relationships among expected returns are derived, and contrary to the classical capital asset pricing model, expected return on risky assets may differ from the riskless rate even when they have no systematic or market risk (Merton 1973).

research. This theory employs some characteristics of Merton (1973) and Lucas (1978). The most significant feature of the model is the integration of real and financial variables. It is also fully consistent with the rational expectations and maximizing behaviour of all agents. It determines the stochastic process followed by the equilibrium price of any financial asset and the underlying real variables. The model can be extended in a number of ways and it is well suited to a wide variety of applications.

Cox, et al. (1985b) used the model to develop a theory of the term structure of interest rates. This article reviewed and applied the general equilibrium model of Cox et al. (1985a). It uses an intertemporal general equilibrium asset pricing model to study the term structure of interest rates. In this model, expectations, risk attitude, investment alternatives, and preferences play a role in determining bond prices. Many of the factors are influencing the term structure and are therefore included in a way which is consistent with maximizing behaviour and rational expectations hypothesis. The model develops specific formulas for bond pricing which are convenient for empirical testing.

Merton (1973) developed the ICAPM using utility maximization to get exact multifactor predictions of expected security returns. Fama (1996) built Merton's ICAPM on similar intuition. The ICAPM risk return relation is a natural generalization of the CAPM. It adds risk premiums for the sensitivities of Ri to the returns, Rs, s=1,…,S, on the (economic)

state- variable related portfolios. The ICAPM has the following form:

where, i and is, are the slopes from the multiple regression of Ri and Rmand Rs. As in the

CAPM, the relation between expected return and multifactor risks in the ICAPM is the condition on the weights for securities that holds in any multifactor-efficient portfolio.

Fama (1998) presented a study which aimed to determine the number of priced state ) 7 . 3 ( ) ( ) ( 1 s s it t st is t mt i i t it r R r R r u R

variables in the ICAPM. He tried to answer the questions that go to the heart of the economics of the ICAPM. Specifically, given ICAPM asset pricing, and given that there is a total of S state variables potentially of hedging concern to investors, i) how can we determine which of these state variables are in fact of hedging concern, and ii) in what sense do these state variables produce special risk premiums in expected returns. According to Fama (1998) it is possible to find the set of priced variables when the state variables are identified (named). When the number of state variables is known, but their names are not, confident conclusions about even the number of them that produce special risk premiums are probably impossible. However, the existing literature fails in identifying the specific state variables that produce risk in the context of ICAPM.

A study by Brennan et al. (2004) developed and estimated a simple valuation model. The investment opportunity set is completely described by the real interest rate and the maximum Sharpe ratio. Bond yields are linearly related to the state variables, the real interest rate and the maximum Sharpe ratio. Bond yields and expected inflation were used to estimate state variables and the parameters. The estimated real interest rate and the Sharpe ratio both show strong business cycle-related variations. The model parameters and time series of the state variables are estimated using U.S economic variables. Treasury bond yields and expected inflation from January 1952 to December 2000, and as predicted, the estimated maximum Sharpe ratio are related to the equity premium. In cross-sectional asset-pricing tests, both state variables have significant risk premium, which is consistent with Merton's ICAPM. Brennan et al (2004) claim that their results with ICAPM are encouraging further empirical investigation.

A recent study by Gerard and Guojun (2006) developed a simple ICAPM, estimated it and tested this model. They analysed the statistical and economic relevance of intertemporal risk in explaining the dynamics of the premium for holding stocks and bonds. They tested a

conditional asset pricing model that includes long-term interest rate risk as a priced factor for four asset classes: large stocks, small stocks, and long-term Treasury and corporate bonds. They found that the interest risk premium is the main component of the risk premiums for bond portfolios, while representing a small fraction of total risk premiums for equities. This suggests that stocks, especially small stocks, are hedges against variations in the investment opportunity set. They also estimate that, at average market volatility levels, investors earn annual premiums between 3.6% during expansions and 5.8% during recessions for bearing intertemporal risk alone.

The existing literature fails in identification of the specific state variables which are the likely sources of risk. There is no practical solution to manage the systematic risk and identify significant macroeconomic variables in the context of ICAPM, thus this gave rise to the usage of alternative multifactor assets pricing models such as the APT.