Factor model

In the economy ,there is more than one factor to identify the goal of security analysis.A formal statement of such a relationship is termed a factor model of security return.
SINGLE FACTOR MODEL
Single factor include the growth ratio of the gross national product and the growth rate in industrial production.
In equation single factor model as
$\tilde{R_{i}}=a_{i}+b_{i}+\tilde{F}+\tilde{e_{i}}$
Where
$\tilde{R_{i}}$=the (uncertain) return on security
$a_{i}b_{i}$=constants
$\tilde{F}$ = the (uncertain) value of the factor
$\tilde{e_{i}}$=the (uncertain) security specific
If the value of factor is zero,the return on the return on the security would equal $e_{i}$ by convention,the expected value of the security specific component of return is assumed to by zero.This nearly requires that $a_{i}$ include the expected portion of non-factor related return.This means that the expected return on security i,according to the single factor model can be written as
$E_{i}=a_{i}+b_{i}ef$
where
$E_{i}$=the expected return on security
Ef=the expected value of the factor where the standard deviation can be determined from
The characteristic  line can now be shown to be an example of single factor model where the factor is the return of the market portfolio.As the characteristic line expressed as
$\tilde{R}_{i}-T=\alpha&space;_{i}+\beta_{im}(\tilde{R})_{im}-T)+r_{i}$
where $\alpha&space;_{i}$ and $\beta&space;_{im}$ are the alpha  and beta of the security i, respectively and $\tilde{r}_{i}$is a random error term with an expected value of zero.

MULTIPLE -FACTOR MODELS
A multiple-factor model is needed in a complex world as the security returns are affected by a number of factors ,e.g. expectations about future levels of real GNP ,expectations about future real interest rates,expectation about future level of inflation etc.These factors ,while they impact on the return on the market portfolio, may impact on the returns on different securities differently.The general form of a factor model can be written as:
$\tilde{R}_{i}=a_{i}+b_{i1}\tilde{F}_{1}+b_{i2}\tilde{F}_{2}+...b_{im}\tilde{F}_{m}\tilde{e}_{i}$
Where:
m=the number of factor with the assumption that:
i) the expected value of each security -specific is zero.
ii)Security -specific returns are uncorrelated with factors.
iii) Security -specific returns are uncorrelated with each other.