Question

The CAPM is a one- factor asset- pricing model- it assumes that stocks’ returns are determined by returns on the market plus random factors that affect individual stocks. However, some analysts and professionals argue that multi-factor models describe investor behavior better than the CAPM. What is a multi-factor model, and how could one test such models against the CAPM? (Note: Multi-factor models have been tested to see if they work better than the pure CAPM, but the results have been inconclusive.)

Answer 1

First you need to know when do we use Capital Asset Pricing Model (CAPM)

CAPM : It is a model which essentially predicts the relationship between risk and return for an individual security.

It is used when you want to know about the relationship between the risk of an asset and its expected return.

Equation of CAPM :

E(R_i) = R_f + [{ E(R_M) – R_f }/ Var_M]* (Covar_iM)          OR

         = R_{f +} [ E(R_M) – R_f ] * β_i

Where;

E(R_i) : Expected return on security i

R_{f :} Risk free return

E(R_M) : expected return on market portfolio

Var_M: variance of market portfolio

Covar_iM: Covariance of return between the security i and market portfolio

β_i: Covar_iM/ Var_M, it reflects the sensitivity of the security to market movements

In CAPM , we calculate the expected return of a security after adjusting for the risk associated with the security. In order to use CAPM we undertake a lot of assumptions. One of them is that an investor is compensated only for the market risk he/she bears and not for the diversifiable risk, which can be avoided by creating a sufficiently diversified portfolio, which is why according to CAPM investor gets the market risk premium = E(R_M) – R_f , in addition to R_f, for bearing market risk.

Multi-factor Model : This model stemmed for the argument that CAPM, being a single factor model in the sense that it assumes that investors should only be compensated for bearing market risk, doesn’t capture risk adequately.

It is a model which enables you to consider security or firm specific characteristics in addition to market risk as factors while calculating the expected return of a security.

This model considers that investors need to be compensated even for certain security or firm specific factors in addition to the market risk borne by investors.

In this approach, you choose, based on theoretical knowledge, the exact number and identity of risk factors which affect the expected return on a security.

R_it = a_i + [b_i1 * F_1t + b_i2 * F_2t+.......+ b_ik * F_kt] + e_it

Where :

R_it : return on security i in period t

a_i: zero beta return

b_i1: risk sensitivity relating to factor 1

F_jt: return associated with jth risk factor ; where j = 1,2,...k

e_it: random error term unique to security i in period t

Suppose you are calculating the return of a security ,then to use multifactor model, you need definite number of factors that you think would affect the return on that security. Suppose you decide that those factors are : industrial production ( measured by industrial production index) , change in inflation ( measured by Consumer price index)

Then you need measure the risk sensitivity of the security based on both of these factors (b_i1& bi2) and the return associated with both of these risk factors (F_1t& F_2t). After measuring all these, you can substitute these values in the above equation along with [ E(R_M) – R_f ] * β_i,

Which is on of the factors you need to account for ( compensation for market risk).

This is an example of how you can use a multi factor model.

Even though this model seems better than CAPM, there is a lot of subjectivity in estimation of the risk sensitivity of the factors and the return associated with those factors. So even though you can take into account a lot of factors which affect the expected return on a security, the subjectivity related to those factors cancels out the above advantage of multi factor models. This is why even today, CAPM is the most popular and simplistic measure used .

First you need to know when do we use Capital Asset Pricing Model (CAPM)

CAPM : It is a model which essentially predicts the relationship between risk and return for an individual security.

It is used when you want to know about the relationship between the risk of an asset and its expected return.

Equation of CAPM :

E(R_i) = R_f + [{ E(R_M) – R_f }/ Var_M]* (Covar_iM)          OR

         = R_{f +} [ E(R_M) – R_f ] * β_i

Where;

E(R_i) : Expected return on security i

R_{f :} Risk free return

E(R_M) : expected return on market portfolio

Var_M: variance of market portfolio

Covar_iM: Covariance of return between the security i and market portfolio

β_i: Covar_iM/ Var_M, it reflects the sensitivity of the security to market movements

In CAPM , we calculate the expected return of a security after adjusting for the risk associated with the security. In order to use CAPM we undertake a lot of assumptions. One of them is that an investor is compensated only for the market risk he/she bears and not for the diversifiable risk, which can be avoided by creating a sufficiently diversified portfolio, which is why according to CAPM investor gets the market risk premium = E(R_M) – R_f , in addition to R_f, for bearing market risk.

Multi-factor Model : This model stemmed for the argument that CAPM, being a single factor model in the sense that it assumes that investors should only be compensated for bearing market risk, doesn’t capture risk adequately.

It is a model which enables you to consider security or firm specific characteristics in addition to market risk as factors while calculating the expected return of a security.

This model considers that investors need to be compensated even for certain security or firm specific factors in addition to the market risk borne by investors.

In this approach, you choose, based on theoretical knowledge, the exact number and identity of risk factors which affect the expected return on a security.

R_it = a_i + [b_i1 * F_1t + b_i2 * F_2t+.......+ b_ik * F_kt] + e_it

Where :

R_it : return on security i in period t

a_i: zero beta return

b_i1: risk sensitivity relating to factor 1

F_jt: return associated with jth risk factor ; where j = 1,2,...k

e_it: random error term unique to security i in period t

Suppose you are calculating the return of a security ,then to use multifactor model, you need definite number of factors that you think would affect the return on that security. Suppose you decide that those factors are : industrial production ( measured by industrial production index) , change in inflation ( measured by Consumer price index)

Then you need measure the risk sensitivity of the security based on both of these factors (b_i1& bi2) and the return associated with both of these risk factors (F_1t& F_2t). After measuring all these, you can substitute these values in the above equation along with [ E(R_M) – R_f ] * β_i,

Which is on of the factors you need to account for ( compensation for market risk).

This is an example of how you can use a multi factor model.

Even though this model seems better than CAPM, there is a lot of subjectivity in estimation of the risk sensitivity of the factors and the return associated with those factors. So even though you can take into account a lot of factors which affect the expected return on a security, the subjectivity related to those factors cancels out the above advantage of multi factor models. This is why even today, CAPM is the most popular and simplistic measure used .