calculate single factor model (CAPM) beta

The single factor model or CAPM Beta is the beta of an asset to the variance and covariance of an initial portfolio. Used to determine diversification potential.

CAPM.beta(Ra, Rb, Rf = 0)

CAPM.beta.bull(Ra, Rb, Rf = 0)

CAPM.beta.bear(Ra, Rb, Rf = 0)

TimingRatio(Ra, Rb, Rf = 0)

Arguments

Ra	an xts, 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

Details

This function uses a linear intercept model to achieve the same results as the symbolic model used by BetaCoVariance $$\beta_{a,b}=\frac{CoV_{a,b}}{\sigma_{b}}=\frac{\sum((R_{a}-\bar{R_{a}})(R_{b}-\bar{R_{b}}))}{\sum(R_{b}-\bar{R_{b}})^{2}}$$

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 TimingRatio may 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.

While the classical CAPM has been almost completely discredited by the literature, it is an example of a simple single factor model, comparing an asset to any arbitrary benchmark.

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.alpha(managers[,1,drop=FALSE],
                        managers[,8,drop=FALSE],
                        Rf=.035/12)
#> [1] 0.005960609
    CAPM.alpha(managers[,1,drop=FALSE],
                        managers[,8,drop=FALSE],
                        Rf = managers[,10,drop=FALSE])
#> [1] 0.005774729
    CAPM.alpha(managers[,1:6],
                        managers[,8,drop=FALSE],
                        Rf=.035/12)
#>                        HAM1        HAM2        HAM3        HAM4        HAM5
#> Alpha: SP500 TR 0.005960609 0.009232784 0.006327417 0.004144392 0.001425414
#>                        HAM6
#> Alpha: SP500 TR 0.007244216
    CAPM.alpha(managers[,1:6],
                        managers[,8,drop=FALSE],
                        Rf = managers[,10,drop=FALSE])
#>                        HAM1        HAM2        HAM3        HAM4        HAM5
#> Alpha: SP500 TR 0.005774729 0.009092773 0.006216498 0.004029731 0.001733199
#>                        HAM6
#> Alpha: SP500 TR 0.007837454
    CAPM.alpha(managers[,1:6],
                        managers[,8:7,drop=FALSE],
                        Rf=.035/12)
#>                           HAM1        HAM2         HAM3         HAM4
#> Alpha: SP500 TR    0.005960609 0.009232784  0.006327417  0.004144392
#> Alpha: EDHEC LS EQ 0.003190734 0.001627709 -0.001716897 -0.003357739
#>                            HAM5        HAM6
#> Alpha: SP500 TR     0.001425414 0.007244216
#> Alpha: EDHEC LS EQ -0.002185688 0.004197041
    CAPM.alpha(managers[,1:6],
                        managers[,8:7,drop=FALSE],
                        Rf = managers[,10,drop=FALSE])
#>                           HAM1        HAM2         HAM3         HAM4
#> Alpha: SP500 TR    0.005774729 0.009092773  0.006216498  0.004029731
#> Alpha: EDHEC LS EQ 0.003134750 0.001732335 -0.001563904 -0.003406683
#>                            HAM5        HAM6
#> Alpha: SP500 TR     0.001733199 0.007837454
#> Alpha: EDHEC LS EQ -0.002255793 0.004212371
    CAPM.beta(managers[, "HAM2", drop=FALSE],
                        managers[, "SP500 TR", drop=FALSE],
                        Rf = managers[, "US 3m TR", drop=FALSE])
#> [1] 0.3383942
    CAPM.beta.bull(managers[, "HAM2", drop=FALSE],
                        managers[, "SP500 TR", drop=FALSE],
                        Rf = managers[, "US 3m TR", drop=FALSE])
#> [1] 0.5226596
    CAPM.beta.bear(managers[, "HAM2", drop=FALSE],
                        managers[, "SP500 TR", drop=FALSE],
                        Rf = managers[, "US 3m TR", drop=FALSE])
#> [1] 0.0698255
    TimingRatio(managers[, "HAM2", drop=FALSE],
                        managers[, "SP500 TR", drop=FALSE],
                        Rf = managers[, "US 3m TR", drop=FALSE])
#> [1] 7.485224
    chart.Regression(managers[, "HAM2", drop=FALSE],
                        managers[, "SP500 TR", drop=FALSE],
                        Rf = managers[, "US 3m TR", drop=FALSE],
                        fit="conditional",
                        main="Conditional Beta")