Capital asset pricing model series#

Capital asset pricing model series#

Least Squares has appealing theoretical and practical (numerical) properties, but obtained estimates are often unstable in the presence of extreme observations which are common in financial time series ( Chan and Lakonishok, 1992 Knez and Ready, 1997 Genton and Ronchetti, 2008). The standard method to obtain parameter estimates in a linear regression is Least Squares. When we re-interpret the CAPM as a one-factor model ( Luenberger, 1998, Chapter 8), however, the β estimates become useful measures of a stock's general correlation with the market, which may be used to construct variance–covariance matrices ( Chan et al., 1999). Enrico Schumann, in Numerical Methods and Optimization in Finance (Second Edition), 2019 Robust/resistant regressionĮmpirical evidence has shown that the Capital Asset Pricing Model (CAPM) explains asset returns in the cross-section rather poorly ( Fama and French, 1993, 2004).

The author is grateful to Steve Shreve for illuminating discussions and information. A natural discretization of the continuous time model introduces a situation of incompleteness. However, the analogue of the continuous time model when the randomness is modelled by a finite number of Wiener processes is not. The situation of a finite number of random events at each period is compatible with an assumption of complete markets. We present here a detailed theory of what can be done in discrete time. In discrete time, although natural, the corresponding treatment is not widely available in the literature (see Shreve for a partial treatment in the case of the binomial model). The theory has been further developed in continuous time, using properties of martingale generated by Wiener processes (see Karatzas and Shreve ). Curiously, this dynamic version is in continuous time and not in discrete time. A dynamic version of this model, called the intertemporal capital asset pricing model, has been introduced by Merton. This fact is well known for a single-period problem. When there are only a finite number of states of nature and when the assumption of complete markets is valid, these prices play a significant role in obtaining the optimal consumption and portfolio investment policies in a general setup. The concept of price of state of nature is a concept of stochastic economics introduced by Arrow and Debreu.

The capital asset pricing model is a major theory in modern finance. The beta coefficient of asset i, β i, is interpreted as a measure of the riskiness of the asset and is defined by, where σ iM is the covariance of the return on asset i and the return on the market portfolio.Alain Bensoussan, in Handbook of Numerical Analysis, 2009 1 Introduction The SML applies to individual assets and requires that, in equilibrium, the expected return on asset i satisfies Where r f is the risk-free interest rate, r̄ p is the expected return on the portfolio, r̄ M is the expected return on the market portfolio, and σ M the standard deviation of the return on the market portfolio. For every efficient portfolio, p, the CML states that Under these assumptions the equilibrium of the financial market is described by the Capital Market Line (CML) and the Security Market Line (SML). It also assumes that all investors have a one-period investment horizon, hold the same expectations about asset returns, have mean-variance preferences, and are able to borrow and lend at a risk-free rate of interest. The CAPM assumes the infinite divisibility of assets, no transaction costs, and no taxes.

A model of equilibrium in financial markets that generates very precise predictions about the structure of returns on risky assets.