R-Squared and Jensen's alpha

What is R-Squared

As Investopedia states, it is a statistical measure that represents the percentage of a fund or security's movements that can be explained by movements in a benchmark index.

Thus it denotes the portion of a risk associated with a security that is market related.

Beta and R-Squared

While Beta measure how sensitive a security is to the Index movements, R-Squared denotes the co-relation between Index movements and Security price movements. Thus, Value of R-Squared equal to 1 denotes perfect correlation between Index and Security Price movements.

Beta in isolation is a useless number. It is essential to take a look at R-squared along with beta. The R-squared value shows how reliable the beta number is. Higher the R-Squared more reliable Beta is (Exceptions always possible).

Both Beta and R-Squared are directly affected by choice of index made. Hence it is always advisable to have a Broad based Index like S&P500 while calculating.

Jensen's alpha

Jensen's alpha is used to determine the return of a security or portfolio of securities over the theoretical expected return. Theoretical Return is calculated by models like CAPM etc.

In capital Asset Pricing Model (CAPM) the expected return on asset/investment is calculated as,

Expected Return = Riskfree Rate + Beta_Asset (Equity Risk Premium)

To calculate Jensen’s alpha one requires the following inputs:

• realized return (on the security/portfolio),
• market return,
• risk-free rate of return, and
• beta of the security/portfolio.

Jensen's alpha = Security or Portfolio Return − [Risk Free Rate + Security or Portfolio Beta * (Market Return − Risk Free Rate)]

Thus it can clearly be seen that CAPM takes into account the relative risk of an investment/security and calculate the Expected theoretical return while, Jensen’s Alpha calculates returns over and above theoretical returns. Thus, Positive Jensen’s Alpha means performance of Security above expectations while, negative Jensen’s Alpha means below expectations performance by the security.

Jensen’s Alpha is derived from purely historical numbers and hence essentially it should not affect the future investment decision. Still it is widely used to judge performance of Mutual Funds.

Example

Statistics Summary	Tata Power	NTPC	Reliance Power	Adani Power	Power Grid Corp.
R Square	0.5866	0.4809	0.6987	0.0213	0.4197
Beta	1.1881	0.7145	1.4971	0.1251	0.6694
CAPM Estimated Returns	14.90%	10.82%	17.56%	5.76%	10.44%
Actual Returns over the Period in consideration	917%	121%	-21%	3.99%	-32%
Period in Consideration for the Stock (Years)*	10.33	6.58	3.00	1.83	3.5833
Annulised Returns  ((1 + Rate of Return)1/N) - 1	25%	13%	-7%	2%	-10%
Jensen's Alpha	10.27%	2.00%	-24.93%	-3.60%	-20.61%
*Refer Raw Data Sheet

Thus above example depicts the fact that investment in Tata Power has yielded more returns that theoretical estimations or say, expectations (of mine off course), historically.

Working Excel can be downloaded from here.

Links:

1.  Investopedia R-Squared Definition (http://www.investopedia.com/terms/r/r-squared.asp#axzz1cacIVWMK)

2.     Wikipedia Theory – Jensen’s Alpha (http://en.wikipedia.org/wiki/Jensen's_alpha)