The Theory \ Details \ asset-pricing models \ general approach

Factor-based Asset Pricing Models

General approach

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

where

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:

Model Assumptions

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.

Practical Implications

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

Portfolio Statistics

Portfolio Beta
Consider portfolio

. Vector

of portfolio betas can be found in the following way:

Portfolio Variance
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 Statistics

Other useful formulae related to asset pricing models include:

Statistics

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