The question is not fully clear:

I assume you mean the excess return of an individual share price?

And excess return: the return in excess of the riskfree rate, or the return in excess of the market return, or the return in excess of the required rate of return (i.e. the stock's alpha)?

Dear Beca,

What I understood is:

Rs = Rf + (Rm-Rf)*b

where Rs = return of a security

            Rf = risk free return

            Rm = market return, and 

             b = beta

Rs - Rf = (Rm - Rf )*b

ERs = ERm *b

where ERs = excess returns of a security 

            ERm = excess returns of market

                 b = market sensitivity

we can say that beta is ratio of stock excess returns to market excess returns, ie

b = ERs / ERm

hope it works! 

If stock excess returns are about 11% per year what can I say about beta?

Dear Mr. Frommel,

The question was about stock excess returns. If stock excess returns are about 11% per year what can I say about beta?

Thank you!

Mare stock excess return can not capture beta. As I mention excess stock returns is the function of excess market return, hence beta is the coefficient.

ERs = f(ERm)

Y = b*X  regression without intercept.

Dear Marușa Beca,

The postulates of the CAPM argue the linearity between risk and return. The excess return, the residue is not explained by beta and therefore could be captured by other factors that could explain the nonlinearity. For example, the three-factor model, the four-factor model or the five-factor model.

Best,

Fabrizio

I am attaching my article. I hope it can be useful.

Beta measures the covariance between the returns on a particular share with the returns on the market as a whole.

Neutral :  Beta = 1  

A 1% change in the market index return -> 1% change in the return on the share

Defensive : 0

at first , I think you should say excess return of what over risk-free rate ??

As you know, beta is excess return of security over risk-free rate divided by excess return of market over risk-free rate. 

 Expected rate of return in the derivation of the CAPM is assumed to be given and it is a constant parameter calculated from the density function of the asset rate of return (a random variable), How can a constant of expected rate of return depend on the other variable such as beta?

The beta coefficient is a measure of risk for the stock. If the stock return, risk free rate and market return are known you can find beta under the assumption of CAPM environment. 

Under the market model, beta is a coefficient of market rate of return. Do you believe a student's GPA (stand for asset's rate of return) depends on the whole class's GPA (stands for market rate of return)? 

Dear Tsong-Yue Lai, 

If you think the beta can not be effectively and properly calculated by the market model, it seems to me that you implicitly affirms the non-validity of the CAPM. If not, I apologize because I probably misunderstood your answer?

Best Regards,

Fabrizio

The expected rate of return is calculated from its density function and is a constant parameter in Statistics. How can a constant parameter depends on other factor (such as the beta)? Please see the proof in the attached file of of "The CAPM is Dead".

Dear Tsong-Yue Lai,

Thanks for your article, I read it as soon as possible with much pleasure. However, I am not a supporter of the CAPM. My modest contribution empirical, on the Italian stock market, questions the beta as the only factor that explains the risk-return relationship.

Best,

Fabrizio

I attach my modest contribution.

If the risk is measured by the covariance or variance, then the expected value (or so-called "mean" in Probability) must be given before the risk can be calculated. That is, without the mean (or expected value), one cannot calculate the covariance or variance.  In other words, we cannot use the risk ( here means covariance or variance) to explain the expected value (or expected rate of return) because the expected value already exists before the risk been calculated.

The vector of the beta is calculated (or defined as) by the product of the mean-variance efficient portfolio(the market portfolio) and the covariance matrix divided by the market variance. However, the mean-variance efficient portfolio (the optimal decision variable in Mathematical Programming of Portfolio section in mean-variance framework) depends on the means (the expected rate of return on assets). So, if the CAPM is true (valid), then the expected rate of return depends on the beta, which depends on the optimal portfolio, which depends on the expected rate of return. That results in the expected rate of return depends on the expected rate of return.   

The fact is very cruel ! The true is CAPM doesn't make any sense from perspectives in Calculus, Logics, Mathematic, Statistics!  

1. In Mathematics, a constant (e.g., expected rate of return) is a constant and should not be affected by any factor such as the beta.

2. In Statistics, the expected value (e.g., expected rate of return on an asset) should be given (or calculated) before the covariance or variance can be calculated. So, the expected value exists before the covariance being calculated. 

3. In Calculus, the first order condition to solve the optimal portfolio solution(e.g., the optimal portfolio=market portfolio under the assumption of the demand =supply ), which must depend on the parameters (e.g., mean, covariance), not vice versa. For example, minimize the f(x)=3x^2-5x, the first order condition is 6x-5=0 and the optimal solution for the minimum of f(x) is at x*=5/6, you cannot say 5 depends on x* even though 5=6x*= 6x*=6(5/6)=5 because 5 is a constant.

4. In Logics, to use the posted calculated covariance (related to the Beta) to explain the prior already existing expected value (expected rate of return on an asset) makes no sense. 

2. In Statistics, the expectation of an unconditional random variable (e.g., a rate of return on an asset) is a constant parameter as set by the density function and should not be affected by other factors (e.g., Beta).

Beta is caclulated by using linear regression on X: market return, Y: security return, e.g.using daily return values as samples. It is recommended to calculate the returns over the risk free rate. Beta is the slope of the regression line and tells you the relative risk of security to the market. If beta of the security is greater than 1, then it is more risky than the market, otherwise is less risky.

It is also recommended to calculate the intersect of the regression line, that called Jensen-alpha. Jensen-alpha captures the abnormal performance of security, that cannot explained by beta.

Return of the security can be estimated by the following:

rs ~ rF + as + bs*rM, where

- rs: the expected return of the security s

- rF: risk free rate

- as: Jensen-alpha of security s

- bs: beta of the security s

- rM: return of the market

I hope this helps, David

Beta is useful to measure the risk under the assumption that CAPM is valid.If CAPM is not valid, the beta is useless. 

If the Beta is calculated by the linear regression (i.e., the market index model), then you must believe the individual student's GPA must depend the whole class's GPA. In reality, that is not true because the whole class's GPA must be calculated after all individual student's GAP are given, the same as the market index model.

Please see the proof of CAPM is Dead in the INTERNATIONAL JOURNAL OF BUSINESS, 20(2), 2015 ISSN: 1083-4346 " Yes, CAPM Is Dead"

Hi,

You could have a look at this recent paper,

Time-varying continuous and jump betas: The role of firm characteristics and periods of stress

Journal of Empirical Finance, January 2017, Pages 1-19

Vitali Alexeev, Mardi Dungey, Wenying Yao

Best regards

So, you still believe individual student's GPA (e.g., an asset rate of return) depends on his whole class GPA (e.g., market portfolio rate of return)? (Please see equations (1) and (2) in your cited paper). I thought market portfolio is constructed by all individual securities in the market. Therefore, using the market portfolio rate of return as an independent variable to explain the individual security's rate of return  (i.e., dependent variable) is not convincing (at least to me).

Hi,

You may find this paper interesting:

Asset Pricing When 'This Time Is Different'.

By: Collin-Dufresne, Pierre; Johannes, Michael; Lochstoer, Lars A. Review of Financial Studies. Feb2017, Vol. 30 Issue 2, p505-538. 34p. 

Best regards.