Let's get back to discussing beta. In my previous article, I pointed out to frequent misuses of the financial statistic known as beta, which can lead to wrong investment decisions. Now, I will try to give a right interpretation of beta from its mathematical formulation.
Formally, the beta of a stock measures the linear dependence of the stock's return to the return of the market in proportion to the stock to market volatility ratio. This is mathematically expressed as follows:
Beta((NYSE:A)) = Corr(R_{A}, R_{M}) x std(R_{A})/std(R_{M}) (*)
where Beta is the beta of stock A, Corr(R_{A}, R_{M}) measures the correlation of the stock's return R_{A} and the market's return R_{M}_{,} and std(R_{A})/std(R_{M}) is the stock to market volatility ratio (the quotient of the respective standard deviations of the stock and of the market). Here the market stands for a market index, such as S&P 500, to which the stock belongs to.
A first approach to understanding the values of beta.
From equation (*) one can produce the following table containing interpretations of the possible range of values of a stock's beta in terms of the movements of the market:
Value of Beta | Effect on correlation and volatility ratio |
Interpretation |
---|---|---|
Beta(A)<0 |
Corr(R_{A}, R_{M})<0 std(R_{A})/std(R_{M})>0 |
Stock moves in opposite direction to the movement of the market |
Beta(A)=0 | Corr(R_{A}, R_{M})=0 | Movements of the stock and the market are uncorrelated |
0<Beta(A) ≤1 |
Corr(R_{A}, R_{M})>0 0 < std(R_{A})/std(R_{M}) ≤ 1/Corr(R_{A}, R_{M}) |
Stock moves in the same direction as the market, volatility of stock can be less or more than volatility of market |
Beta(A)>1 |
Corr(R_{A}, R_{M})>0 std(R_{A})/std(R_{M})> 1/Corr(R_{A}, R_{M}) >1 |
Stock moves in the same direction as the market but with greater volatility |
Note that the above interpretations differ from those found on Wikipedia, Investopedia, and other popular websites of public knowledge, so beware.
Now, let's look at another way of estimating beta. Many investors, including myself, consult their stock's statistics at websites dedicated to financial markets such as Yahoo or Google Finance. There one finds the value of beta posted for every stock. How is this number calculated?
The beta in Yahoo measures the stock's monthly risk premium relative to the monthly market premium, where the market in this case is in fact the S&P 500 Index. This beta is the quotient of the monthly expected excess rate of return of the stock, denoted E(R_{A}) - r , and the monthly expected excess rate of return of the market, denoted E(R_{M}) - r , and computed over three years (or 36 months). So,
Beta = (E(R_{A})-r) / (E(R_{M})-r) (**)
Here r is a risk-free interest rate which may be fixed (for simplicity) or variable; for example, r is the current T-bill rate.
The above formulation is saying that the expected return of the stock A is linearly related to the expected excess return of the market (i.e., the S&P 500); that is:
E(R_{A}) = r + Beta(E(R_{M})-r)
This formulation of beta comes from a model of price formation known as the capital asset pricing model (CAPM), which further says that beta can be estimated from data, regardless of the risk-free interest rate, as
Beta = Cov(R_{A,} R_{M}) / Var(R_{M})
where Cov(R_{A,} R_{M}) is the covariance of the stock A and the market M, and Var(R_{M}) is the variance of the market. Whether you believe in such linear relationship between stocks and the market is a separate discussion, but this is what underlies the calculation of this number beta as found in the above mentioned webs of financial services. This last equation links with equation (*) since
Beta = Cov(R_{A,} R_{M}) / Var(R_{M}) = Corr(R_{A}, R_{M}) x std(R_{A})/std(R_{M})
Thus, equations (*) and (**) are two equivalent forms of estimating beta. And the combination of both equations give us sharper interpretations of the values of beta.
In the following section, I will discuss interpretations of beta and give some examples.
Extending the interpretation of the values of beta.
Consider first the case of beta< 0 . This value is possible only if the correlation of the stock's and the market's return is negative (i.e., Corr(R_{A}, R_{M}) < 0), since the standard deviation quotient, std(R_{A})/std(R_{M}), is always positive. The immediate implication of this negative sign of beta is that the stock's return runs contrary to the market's return, but not much can be said about the relative risk of the stock with respect to the risk of the market, unless you compute the quotient Beta/Corr(R_{A}, R_{M}), as I suggested in my previous article, to know exactly the value of this relative risk.
However, regardless of the result of this extra calculation, we can infer from Equation (**) that a beta < 0 implies that the expected rate of return of the stock should be less than the risk-free interest rate, that is, E(R_{A}) < r, since under normal market conditions (that is to say, not in the middle of a market crisis) the market premium should be positive, thus leaving only the term (E(R_{A}) - r) in (**) to account for the negative sign of beta. Thus, this would be the kind of stock running contrary to the market, and leaving no benefit in the mean term.
It is very rare to come across with stocks with significantly negative beta, since common stocks almost always are in positive correlation with the members of their own sector, but for what have been said above if you run across a stock with negative beta, it will certainly be a bad investment. I must confess that I have not seen a negative beta posted at Yahoo, so these are indeed a rarity.
A beta = 0, or very close to 0 (since in practice it is very difficult to have a stock with beta exactly equal to 0), means that the stock and the market are (almost) uncorrelated. In this case, from Equation (**) we can conclude that E(R_{A}) = r, or that the risk premium is zero. The explanation for this conclusion is that the risk of a stock which is uncorrelated with an efficient portfolio gets neutralized by the risk compounded from the different positions of the portfolio, and hence, one should not expect greater benefits than those that could be obtained at the risk free rate.
A recent example of a stock with an almost zero beta is Verizon (NYSE:VZ), which by May 2014 had a beta of 0.04. From what have been said, this implies that VZ runs uncorrelated with the S&P 500 (and, indeed, by the given date it reports a 52-week change of -5.48%, while the S&P 500 had a 52 week change of 13.14%). This stock then would be a choice for neutralizing risk in our portfolio, assuming possible benefits not better than a bond (although this particular stock pays an attractive dividend of 4.3%).
A different situation is presented by beta > 1. This implies that the stock and the market are positively correlated; but since Corr(R_{A}, R_{M}) ≤ 1 always, the risk of the stock is greater than the risk of the market. This extra risk might have its compensation since, from Equation (**), it follows that E(R_{A}) > E(R_{M}) - i.e., the stock could beat the market.
Positive and greater than 1 beta is often the case for blue chips like General Motors (NYSE:GM) and General Electric (NYSE:GE). And there are stocks with positively enormous values of beta, like Zevotek, Inc. (OTCPK:ZVTK) with a beta = 605, as of may 2014, a real roller-coaster.
If beta is ≤ 1 but positive, we conclude a similar relation on the correlation of the stock and the market as in the previous case, but not much can be said about their relative risk. In my previous article, I showed how Apple (NASDAQ:AAPL) having a beta of 0.76 by July 2013 had a standard deviation that was almost double that of the S&P 500. One can also find stocks with beta < 1 and relative volatility < 1 (not much excitement here).
On the other hand, a beta ≤ 1 implies from Equation (**) that E(R_{A})≤ E(R_{M}), so the stock should not be expected to do better than the market, which was the case of AAPL in that second quarter of 2013.
I hope this analysis helps you in forming correct investment decisions based on beta.
Disclosure: I have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.