The Mathematical Derivation of Beta

So far, we have only explained a beta factor (b) by reference to a graphical relationship between the pricing or return of an individual security's risk and overall market risk. Let us now derive mathematical formulae for b by adapting our earlier notation and continuing with the equation numbering from previous Chapters. This ended with Equation (32) and began with Equation (33) in our Introduction the present one.

Suppose an individual was to place all their investment funds in all the financial securities that comprise the global stock market in proportion to the individual value of each constituent relative to the market's total value.

The market portfolio has a variance of VAR(m) and the covariance of an individual security j with the market average is COV(j,m). So, the relative risk (the security's beta) denoted by is given by our earlier equation:

Alternatively, we know from Chapter Two that given the relationship between the covariance and the linear correlation coefficient, the covariance term in Equation (33) can be rewritten as:

So, we can also define a theoretical value for beta as follows:

And simplifying, (allowing for the equation numbering in our Introduction to this Chapter):

If information on the variance or standard deviation and covariance or correlation coefficient is readily available, the calculation of beta is extremely straightforward using either equation. Ideally, we should determine b using forecast data (in order to appraise future investments). In its absence, however, we can derive an estimator using least-squares regression. This plots a security's historical periodic return against the corresponding return for the appropriate market index. For example, an ordinary share's return rt (common stock) is given by:

Obviously it needs to be adjusted for events such as bonus or rights issues and any capital reorganization-reconstruction. Fortunately, because of their ease of calculation, b estimators are published regularly by the financial services industry for stock exchange listings world-wide. A particularly fine example is the London Business School Risk Management Service (LBSRMS) that supplies details of equity betas, which are also geared up (leveraged) according to the firm's capital structure (more of which later in Chapter Seven).

Given the universal, freely available publication of beta factors, considerable empirical research on their behaviour has been undertaken over a long period of time. So much so, that as a measure of systematic risk they are now known to exhibit another extremely convenient property (which also explains their popularity within the investment community).

Although alpha risk varies considerably over time, numerous studies (beginning with Black, Jensen and Scholes in 1972) have continually shown that beta values are more stable. They move only slowly and display a near straight-line relationship with their returns. The longer the period analyzed, the better. The more data analyzed, the better. Thus, betas are invaluable for efficient portfolio selection. Investors can tailor a portfolio to their specific risk-return (utility) requirements, aiming to hold aggressive stocks with a b in excess of one while the market is rising, and less than one (defensive) when the market is falling.

Activity 2

Explain the investment implications of a beta factor of 1.15 and a beta factor that is less than the market portfolio

A beta of 1.15 implies that if the underlying market with a beta factor of one were to rise by 10 per cent, then the stock may be expected to rise by 11.5 per cent. Conversely, a security with a beta of less than one would not be as responsive to market movements. In this situation, smaller systemic risk would mean that investors would be satisfied with a return that is below the market average. The market portfolio has a beta of one precisely because the covariance of the market portfolio with itself is identical to the variance of the market portfolio. Needless to say, a risk-free investment has a beta of zero because its covariance with the market is zero.

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