Beta (finance)

The market beta ${\displaystyle \beta _{i}}$ of an asset ${\displaystyle i}$, observed on ${\displaystyle t}$ occasions, is defined by (and best obtained via) a linear regression of the rate of return ${\displaystyle r_{i,t}}$ of asset ${\displaystyle i}$ on the rate of return ${\displaystyle r_{m,t}}$ of the (typically value-weighted) stock-market index ${\displaystyle m}$:

${\displaystyle r_{i,t}=\alpha _{i}+\beta _{i}\cdot r_{m,t}+\varepsilon _{t},}$

where ${\displaystyle \varepsilon _{t}}$ is an unbiased error term whose squared error should be minimized. The coefficient ${\displaystyle \alpha _{i}}$ is often referred to as the alpha.

The ordinary least squares solution is

${\displaystyle \beta _{i}={\frac {\operatorname {Cov} (r_{i},r_{m})}{\operatorname {Var} (r_{m})}},}$

where ${\displaystyle \operatorname {Cov} }$ and ${\displaystyle \operatorname {Var} }$ are the covariance and variance operators. Betas with respect to different market indexes are not comparable.

By using the relationship between standard deviation and variance, ${\displaystyle \sigma \equiv {\sqrt {\operatorname {Var} (r)}}}$ and the definition of correlation ${\displaystyle \rho _{a,b}\equiv {\frac {\operatorname {Cov} (r_{a},r_{b})}{\sqrt {\operatorname {Var} (r_{a})\operatorname {Var} (r_{b})}}}}$, market beta can also be written as

${\displaystyle \beta _{i}=\rho _{i,m}{\frac {\sigma _{i}}{\sigma _{m}}}}$,

where ${\displaystyle \rho _{i,m}}$ is the correlation of the two returns, and ${\displaystyle \sigma _{i}}$, ${\displaystyle \sigma _{m}}$ are the respective volatilities. This equation shows that the idiosyncratic risk (${\displaystyle \sigma _{i}}$) is related to but often very different to market beta. If the idiosyncratic risk is 0 (i.e., the stock returns do not move), so is the market-beta. The reverse is not the case: A coin toss bet has a zero beta but not zero risk.

Attempts have been made to estimate the three ingredient components separately, but this has not led to better estimates of market-betas.

Suppose an investor has all his money in the market ${\displaystyle m}$ and wishes to move a small amount into asset class ${\displaystyle i}$. The new portfolio is defined by

${\displaystyle r_{p}=(1-\delta )r_{m}+\delta r_{i}.}$

The variance can be computed as

${\displaystyle \operatorname {Var} (r_{p})=(1-\delta )^{2}\operatorname {Var} (r_{m})+2\delta (1-\delta )\operatorname {Cov} (r_{m},r_{i})+\delta ^{2}\operatorname {Var} (r_{i}).}$

For small values of ${\displaystyle \delta }$, the terms in ${\displaystyle \delta ^{2}}$ can be ignored,

${\displaystyle \operatorname {Var} (r_{p})\approx (1-2\delta )\operatorname {Var} (r_{m})+2\delta \operatorname {Cov} (r_{m},r_{i}).}$

Using the definition of ${\displaystyle \beta _{i}=\operatorname {Cov} (r_{m},r_{i})/\operatorname {Var} (r_{m}),}$ this is

${\displaystyle \operatorname {Var} (r_{p})/\operatorname {Var} (r_{m})\approx 1+2\delta (\beta _{i}-1).}$

This suggests that an asset with ${\displaystyle \beta }$ greater than 1 increases the portfolio variance, while an asset with ${\displaystyle \beta }$ less than 1 decreases it if added in a small amount.

Market-beta can be weighted, averaged, added, etc. That is, if a portfolio consists of 80% asset A and 20% asset B, then the beta of the portfolio is 80% times the beta of asset A and 20% times the beta of asset B.

${\displaystyle r_{p}=w_{a}\cdot r_{a}+w_{b}\cdot r_{b}\Rightarrow \beta _{p,m}=w_{a}\cdot \beta _{a,m}+w_{b}\cdot \beta _{b,m}.}$

Other beta estimators reflect the tendency of betas (like rates of return) for regression toward the mean, induced not only by measurement error but also by underlying changes in the true beta and/or historical randomness. (Intuitively, one would not suggest a company with high return [e.g., a drug discovery] last year also to have as high a return next year.) Such estimators include the Blume/Bloomberg beta[3] (used prominently on many financial websites), the Vasicek beta,[4] the Scholes-Williams beta,[5] and the Dimson beta.[6]

• The Blume beta estimates the future beta as 2/3 times the historical OLS beta plus 1/3 times the number 1. A version based on monthly rates of return is widely distributed by Capital IQ and quoted on all financial websites. It predicts future market-beta poorly .
• The Vasicek beta varies the weight between the historical OLS beta and the number 1 (or the average market beta if the portfolio is not value-weighted) by the volatility of the stock and the heterogeneity of betas in the overall market. It can be viewed either as an optimal Bayesian estimator or a random-effects estimator under the (violated) assumption that the underlying market-beta does not move. It is modestly difficult to implement. It performs modestly better than the OLS beta.
• The Scholes-Williams and Dimson betas are estimators that account for infrequent trading causing non-synchronously quoted prices. They are rarely useful when stock prices are quoted at day's end and easily available to analysts (as they are in the US), because they incur an efficiency loss when trades are reasonably synchronous. However, they can be very useful in cases in which frequent trades are not observed (e.g., as in private equity) or in markets with rare trading activity.

These estimators attempt to uncover the instant prevailing market-beta. When long-term market-betas are required, further regression toward the mean over long horizons should be considered.