Question

We can calculate beta as the slope of the SCL(Security Characteristic Line) or as cov(ri, rM)/sM2. Which way is better and why?

Answer 1

Beta from security characteristic line is calculated as below

Beta for security I= E(Ri-Rf)/E(Rm-Rf)

Ri is the return from individual security i.

Rf is the risk free rate in the market.

Rm is the market rate of return.

Beta from Capital asset pricing model is calculated by

Beta_i=Cov(Ri,Rm)/Var(Rm)

If we observe closely beta from SCL is approximation of first moment such as mean and expected values whereas for CAPM we consider 2nd moment of distribution such as covariances and variance that helps to minimise squared errors done in prediction.

When we work with First moment beta that is from SCL we provide equal weight to each outcome of Ri-Rm and in case of CAPM magnitude of difference between Ri-Rf and Rm-Rf gets higher value that is the outlier will always gets higher value in calculation of CAPM beta . SCL works better when Return from security I is considered to be linear function of risk premium which is Rm-Rf but fails to provide justifiable outcome when non- linearity between Ri-Rf and Rm-Rf is seen whereas beta from CAPM works out well in the case of non linear relationship between Ri-Rf and Rm-Rf.

Hence looking at the irregular and less linear behavior of the market we most of the times should prefer beta from CAPM to beta from SCL