# Beta Calculations For 2 Assets

## Beta (Finance)

The beta coefficient, in terms of finance and investing, describes how the expected return of a stock or portfolio is correlated to the return of the financial market as a whole.
An asset with a beta of 0 means that its price is not at all correlated with the market; that asset is independent. A positive beta means that the asset generally follows the market. A negative beta shows that the asset inversely follows the market; the asset generally decreases in value if the market goes up.
Correlations are evident between companies within the same industry, or even within the same asset class (such as equities), as was demonstrated in the Wall Street crash of 1929. This correlated risk, measured by Beta, creates almost all of the risk in a diversified portfolio.
The beta coefficient is a key parameter in the capital asset pricing model (CAPM). It measures the part of the asset's statistical variance that cannot be mitigated by the diversification provided by the portfolio of many risky assets, because it is correlated with the return of the other assets that are in the portfolio. Beta can be estimated for individual companies using regression analysis against a stock market index.

Definition
The formula for the Beta of an asset within a portfolio is ,
where ra measures the rate of return of the asset, rp measures the rate of return of the portfolio of which the asset is a part and Cov(ra,rp) is the covariance between the rates of return. In the CAPM formulation, the portfolio is the market portfolio that contains all risky assets, and so the rp terms in the formula are replaced by rm, the rate of return of the market.
Beta is also referred to as financial elasticity or correlated relative volatility, and can be referred to as a measure of the sensitivity of the asset's returns to market returns, its non-diversifiable risk, its systematic risk or market risk. On an individual asset level, measuring beta can give clues to volatility and liquidity in the marketplace. On a portfolio level, measuring beta is thought to separate a manager's skill from his or her willingness to take risk.
The beta movement should be distinguished from the actual returns of the stocks. For example, a sector may be performing well and may have good prospects, but the fact that its movement does not correlate well with the broader market index may decrease its beta. However, it should not be taken as a reflection on the overall attractiveness or the loss of it for the sector, or stock as the case may be. Beta is a measure of risk and not to be confused with the attractiveness of the investment.
The beta coefficient was born out of linear regression analysis. It is linked to a regression analysis of the returns of a portfolio (such as a stock index) (x-axis) in a specific period versus the returns of an individual asset (y-axis) in a specific year. The regression line is then called the Security Characteristic Line (SCL).

α a is called the asset's alpha coefficient and βa is called the asset's beta coefficient. Both coefficients have an important role in Modern portfolio theory.
For an example, in a year where the broad market or benchmark index returns 25% above the risk free rate, suppose two managers gain 50% above the risk free rate. Since this higher return is theoretically possible merely by taking a leveraged position in the broad market to double the beta so it is exactly 2.0, we would expect a skilled portfolio manager to have built the outperforming portfolio with a beta somewhat less than 2, such that the excess return not explained by the beta is positive. If one of the managers' portfolios has an average beta of 3.0, and the other's has a beta of only 1.5, then the CAPM simply states that the extra return of the first manager is not sufficient to compensate us for that manager's risk, whereas the second manager has done more than expected given the risk. Whether investors can expect the second manager to duplicate that performance in future periods is of course a different question.

Beta volatility and correlation
β = (σ / σm)r
That is, beta is a combination of volatility and correlation. For example, if one stock has low volatility and high correlation, and the other stock has low correlation and high volatility, beta can decide which is more "risky".

In other words, beta sets a floor on volatility. For example, if market volatility is 10%, any stock (or fund) with a beta of 1 must have volatility at least 10%.

Investing
By definition, the market itself has an underlying beta of 1.0, and individual stocks are ranked according to how much they deviate from the macro market (for simplicity purposes, the S&P 500 is usually used as a proxy for the market as a whole). A stock that swings more than the market (i.e. more volatile) over time has a beta whose absolute value is above 1.0. If a stock moves less than the market, the absolute value of the stock's beta is less than 1.0.
More specifically, a stock that has a beta of 2 follows the market in an overall decline or growth, but does so by a factor of 2; meaning when the market has an overall decline of 3% a stock with a beta of 2 will fall 6%. Betas can also be negative, meaning the stock moves in the opposite direction of the market: a stock with a beta of -3 would decline 9% when the market goes up 3% and conversely would climb 9% if the market fell by 3%.
Higher-beta stocks mean greater volatility and are therefore considered to be riskier, but are in turn supposed to provide a potential for higher returns; low-beta stocks pose less risk but also lower returns. In the same way a stock's beta shows its relation to market shifts, it also is used as an indicator for required returns on investment (ROI). If the market with a beta of 1 has an expected return increase of 8%, a stock with a beta of 1.5 should increase return by 12%.
This expected return on equity, or equivalently, a firm's cost of equity, can be estimated using the Capital Asset Pricing Model (CAPM). According to the model, the expected return on equity is a function of a firm's equity beta (βE) which, in turn, is a function of both leverage and asset risk (βA):

where:
KE = firm's cost of equity
RF = risk-free rate (the rate of return on a "risk free investment", e.g. U.S. Treasury Bonds)
RM = return on the market portfolio
because:

and
Firm Value (V) = Debt Value (D) + Equity Value (E)
An indication of the systematic riskiness attaching to the returns on ordinary shares. It equates to the asset Beta for an ungeared firm, or is adjusted upwards to reflect the extra riskiness of shares in a geared firm., i.e. th Geared Beta.

Multiple Beta Model
The arbitrage pricing theory (APT) has multiple betas in its model. In contrast to the CAPM that has only one risk factor, namely the overall market, APT has multiple risk factors. Each risk factor has a corresponding beta indicating the responsiveness of the asset being priced to that risk factor.

Estimation of Beta
To estimate Beta, one needs a list of returns for the asset and returns for the index; these returns can be daily, weekly or any period. Next, a plot should be made, with the index returns on the x-axis and the asset returns on the y-axis, in order to check that there are no serious violations of the linear regression model assumptions. The slope of the fitted line from the linear least-squares calculation is the estimated Beta. The y-intercept is the alpha.
There is an inconsistency between how beta is interpreted and how it is calculated. The usual explanation is that it gives the asset volatility relative to the market volatility. If that were the case it should simply be the ratio of these volatilities. In fact, the standard estimation uses the slope of the least squares regression line - this gives a slope which is less than the volatility ratio. Specifically it gives the volatility ratio multiplied by the correlation of the plotted data. Tofallis, Chris, "Investment Volatility: A Critique of Standard Beta Estimation and a Simple Way Forward" . University of Hertfordshire Business School Working Paper No. 2004:3provides a discussion of this, together with a real example involving AT&T. The graph showing monthly returns from AT&T is visibly more volatile than the index and yet the standard estimate of beta for this is less than one.
Extreme and interesting cases
Beta has no upper or lower bound, and betas as large as 3 or 4 will occur with highly volatile stocks.
Beta can be zero. Some zero-beta securities are risk-free, such as treasury bonds and cash. However, simply because a beta is zero does NOT mean that it is risk free. A beta can be zero simply because the correlation between that item and the market is zero. An example would be betting on horse racing. The correlation with the market will be zero, but it is certainly not a risk free endeavor.
A negative beta simply means that the stock is inversely correlated with the market. Many gold-related stocks are beta-negative.
A negative beta might occur even when both the benchmark index and the stock under consideration have positive returns. It is possible that lower positive returns of the index coincide with higher positive returns of the stock, or vice versa. The slope of the regression line, i.e. the beta, in such a case will be negative.
Using beta as a measure of relative risk has its own limitations. Most analysis consider only the magnitude of beta. Beta is a statistical variable and should be considered with its statistical significance (R square value of the regression line). Higher R square value implies higher correlation and a stronger relationship between returns of the asset and benchmark index.

Since beta is a result of regression of one stock against the market where it is quoted, betas from different countries are not comparabale.
Staples stocks are thought to be less affected by cycles and usually have lower beta. Procter & Gamble is a classic example. They make soap. Other similar ones are Philip Morris (tobacco) and Anheuser-Busch (alcohol). Utility stocks are thought to be less cyclical and have lower beta as well, for similar reasons.
Foreign stocks may provide some diversification. World benchmarks such as S&P Global 100 have slightly lower betas than comparable US-only benchmarks such as S&P 100. However, this effect is not as good as it used to be; the various markets are now fairly correlated, especially the US and Western Europe.