Factor-based Asset Pricing Models
Factor-based Asset Pricing Models (asset pricing models for short) help imposing the additional structure on parameters and of the analytical model. Restrictions, imposed by an asset pricing model, reduce the number of model parameters that need to be estimated and force “rational” investors to allocate wealth among riskless asset and selected factors only, ignoring the rest of the risky assets. Particular cases of asset pricing models are CAPM and Fama-French 3-Factor Model.
Linear Regression Equation
Assume that the financial market consists of financial instruments, first of which we denote as assets, the others we denote as factors1. It is supposed that the dynamics of the market satisfies the conditions of the analytical model with excess Mu vector and covariance matrix . The given parameters can be decomposed according to assets and factors:
Let us denote through vectors of simple rates of return for assets and factors respectively on the interval , where is small enough.
Dependence of from can be expressed by the linear regression equation
is the vector of mispricing terms (Jensen’s instantaneous alphas);
is the matrix of factor sensitivities (Betas);
is random vector of residuals with zero average and such a covariance matrix that components of vectors and are uncorrelated.
The elements of matrix are known as residual variances. Their roots are referred to as standard errors of forecast, implied by the regression.
Let us denote sample estimates of and through and
Estimates of parameters and are found by means of ordinary least-squares using the following formulas:
The asset pricing model imposes the following restrictions, which must be satisfied by the above regression equation:
All mispricing terms in regression equation must be equal to zero.
- is diagonal.
Regression residuals must be uncorrelated.
If selected asset pricing model holds true, then the main implication for a CRRA investor (an investor with CRRA utility function) is that there is no sense to invest in individual assets. Corresponding optimal portfolio would contain only riskless asset and portfolio factors as its components. Of course, this is true only when selected factors are available for an investment (i.e. they are at least tradable financial instruments).
Model-implied Estimates of and
Assuming that the asset pricing model holds true, leads to the modified estimates for and . Model-implied estimates and are given by the following formulas:
where the matrix is diagonal and
Consider portfolio . Vector of portfolio betas can be found in the following way:
Under the selected set of factors the portfolio variance admits the following decomposition
into systematic risk (first item) and diversifiable (nonsystematic) risk (second one).
Other useful formulae related to asset pricing models include:
is vector, whose elements take values in . It is called determination coefficient. The more the value is closer to unity, the better the selected asset pricing model describes the dynamics of -th asset. Calculation formula:
Student’s -statistics for vector and matrix
-statistics are used in linear regression for testing of hypotheses about regression coefficients beingequal to zero. Formulas for the calculation of -statistics and are presented below.
Assume that is the sample size on which the regression coefficients were estimated, and is a number of years contained in sample. Then
The critical two-sided probabilities for Student’s -distribution with degrees of freedom are obtained based on the absolute values of the calculated statistics. If , then one may use normal distribution instead of -distribution. The values obtained in this way are probabilities of the corresponding regression coefficients being equal to zero.
Note. The following rule of thumb can be proposed: if the absolute value of statistics is more than 3, and the number of degrees of freedom is not less than 10, then with high probability (about 99%) the corresponding regression coefficient is significant.
1 Readers shall not confuse the subscript used in this section with the riskless rate symbol .