PerformanceAnalytics (version 0.9.7.1)

CAPM.beta: calculate CAPM beta

Description

CAPM Beta is the beta of an asset to the variance and covariance of an initial portfolio. Used to determine diversification potential. also called "systematic beta" by several papers.

This function uses a linear intercept model to achieve the same results as the symbolic model used by BetaCoVariance

Usage

CAPM.beta(Ra, Rb, rf = 0, digits = 4)
CAPM.beta.bull(Ra, Rb, rf = 0, digits = 4)
CAPM.beta.bear(Ra, Rb, rf = 0, digits = 4)
timing.ratio(Ra, Rb, rf = 0, digits = 4)

Arguments

Ra

a vector, matrix, data frame, timeSeries or zoo object of asset returns

Rb

return vector of the benchmark asset

rf

risk free rate, in same period as your returns

digits

number of digits to round results to

Value

• systematic beta of an asset to the index, perhaps conditioned on positive or negative returns.

Details

$$\beta_{a,b}=\frac{CoV_{a,b}}{\sigma_{a}}=\frac{\sum((R_{a}-\bar{R_{a}})(R_{b}-\bar{R_{b}}))}{\sum(R_{a}-\bar{R_{a}})^{2}}$$

Ruppert(2004) reports that this equation will give the estimated slope of the linear regression of $R_{a}$ on $R_{b}$ and that this slope can be used to determine the risk premium or excess expected return (see Eq. 7.9 and 7.10, p. 230-231).

Two other functions apply the same notion of best fit to positive and negative market returns, separately. The CAPM.beta.bull is a regression for only positive market returns, which can be used to understand the behavior of the asset or portfolio in positive or 'bull' markets. Alternatively, CAPM.beta.bear provides the calculation on negative market returns.

The timing.ratio can help assess whether the manager is a good timer of asset allocation decisions. The ratio, which is calculated as $$TimingRatio =\frac{\beta^{+}}{\beta^{-}}$$ is best when greater than one in a rising market and less than one in a falling market.

References

Sharpe, W.F. Capital Asset Prices: A theory of market equilibrium under conditions of risk. Journal of finance, vol 19, 1964, 425-442. Ruppert, David. Statistics and Finance, an Introduction. Springer. 2004. Bacon, Carl. Practical portfolio performance measurement and attribution. Wiley. 2004.

See Also

BetaCoVariance CAPM.alpha CAPM.utils

Examples

data(managers)
CAPM.beta(managers[, "HAM2", drop=FALSE], managers[, "SP500.TR", drop=FALSE], rf = managers[, "US.3m.TR", drop=FALSE])
CAPM.beta.bull(managers[, "HAM2", drop=FALSE], managers[, "SP500.TR", drop=FALSE], rf = managers[, "US.3m.TR", drop=FALSE])
CAPM.beta.bear(managers[, "HAM2", drop=FALSE], managers[, "SP500.TR", drop=FALSE], rf = managers[, "US.3m.TR", drop=FALSE])
timing.ratio(managers[, "HAM2", drop=FALSE], managers[, "SP500.TR", drop=FALSE], rf = managers[, "US.3m.TR", drop=FALSE])
chart.Regression(managers[, "HAM2", drop=FALSE], managers[, "SP500.TR", drop=FALSE], Rf = managers[, "US.3m.TR", drop=FALSE], fit="conditional", main="Conditional Beta")