# Empirical Evidence in Capital Asset Pricing Model

According to the CAPM the expected return on a security is:

E(Ri) = Rf + βi [E (RM) – Rf ] ————Eq 1

The ideal way to test the CAPM would be to observe investors’ expectations of betas and expected returns on individual securities and the market portfolio and then compare the expected return on each security with its return as predicted by CAPM. Unfortunately, this procedure is not very practical since information on investor expectations is very sketchy.

In practice researchers have tested the CAPM using ex-post facto data, rather than ex-ante data. The commonly followed procedure involves three basic steps:

1. Set up he sample data
2. Estimate the security characteristics lines (SCLs)
3. Estimate the security market line (SML)

Set up the Sample Data: Suppose you are looking at a sample of 75 securities over a period of 60 monthly holding periods (five years). For each of the 60 holding periods, you have to collect the rates of returns on 75 securities. A market portfolio proxy and one-month (risk-free) Treasury bill. Your data set will thus consist of:

Rit: returns on 75 securities over the 60 month period (i=1,2,… 75 and t= 1,2,…60)

RMt: returns on market portfolio proxy over the 60 month period

Rft: risk free rates over the 60 month period.
This constitutes a total of 77 x 60 =4260 rates of return.

Estimate the Security Characteristics Lines: You have to estimate the beta for each security in the sample. The beta for each security is simply the slope of its security characteristic line. There are two ways in which security beta is estimated:

Rit = ai + bi RMt + eit ————– Eq 2

Rit – Rft = ai + bi ( RMt – Rft ) + eit ———- Eq 3

Note that in Eq 2 the return on security is regressed on the return on the market portfolio, whereas in Eq 3 the excess return on security is regressed on the excess return on market portfolio. It appears that Eq 3 is used more commonly.

Estimate the Security Market Line: Once you have the beta estimates of various securities you can estimate the security market line:
Ri = γ0 + γ1bi + ei = 1…. 75 —————- Eq 4

Comparing Eqs you can infer that if the CAPM holds:

1. The relationship should be linear. This means that terms like bi2, if substituted for bi, should not yield better explanatory power.
2. γ0, the intercept should not be significantly different from the risk–free rate, Rf
3. γ1, the slope coefficient should not be significantly different from RM – Rf
4. No other factors such as company size or total variance should affect Ri
5. The model should explain a significant portion of variation in returns among securities.

Numerous empirical studies have been conducted to test the CAPM. Without going into the details of the individual studies, let us note the following general conclusions that emerge firm these studies.

1. The relation appears to be linear
2. In general γ0 is a greater than the risk free rate and γ1 is less than RM – Rf. This means that the actual relationship between risk (as measured by beta) and return is flatter than what the CAPM says
3. In addition to beta, some other factors, such as standard deviation of returns and company size, too have a bearing on return.
4. Beta does not explain very high percentage of the variance in return among securities.