Calculate single factor model alpha and beta coefficients

The single factor model is the beta of an asset to the variance and covariance of an initial portfolio. Used to determine diversification potential. "Alpha" purports to be a measure of a manager's skill by measuring the portion of the managers returns that are not attributable to "Beta", or the portion of performance attributable to a benchmark.

SFM.coefficients(
  Ra,
  Rb,
  Rf = 0,
  subset = TRUE,
  ...,
  method = "Robust",
  family = "mopt",
  digits = 3,
  benchmarkCols = T,
  Model = F,
  warning = T
)

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

subset

a logical vector representing the set of observations to be used in regression

...

Other parameters like max.it or bb specific to lmrobdetMM regression. Another interesting parameter is round.by.alphas, and round.by.betas which can be set to round off the returned values, otherwise the values

method

(Optional): string representing linear regression model, "LS" for Least Squares and "Robust" for robust. Defaults to "LS

family

(Optional): If method == "Robust": This is a string specifying the name of the family of loss function to be used (current valid options are "bisquare", "opt" and "mopt"). Incomplete entries will be matched to the current valid options. Defaults to "mopt". Else: the parameter is ignored

digits

(Optional): Number of digits to round the results to. Defaults to 3.

benchmarkCols

(Optional): Boolean to show the benchmarks as columns. Defaults to TRUE.

Model

(Optional): Boolean to return the fitted model. Defaults to FALSE

warning

(Optional): Boolean to show warnings or not. Defaults to TRUE.

Details

This function is designed to be used a a wrapper for SFM's regression models. Using this, one can easily fit different types of linear models like Ordinary Least Squares or Robust Estimators like Huber.

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

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).

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 SFM.alpha CAPM.utils SFM.beta

Author

Dhairya Jain

Examples

if(requireNamespace("RobStatTM", quietly = TRUE)) { # CRAN requires conditional execution
  data(managers)
  SFM.coefficients(managers[,1], managers[,8]) 
  SFM.coefficients(managers[,1:6], managers[,8:9], Rf = managers[,10])
   SFM.coefficients(managers[,1:6], managers[,8:9], 
           Rf=.035/12, method="Robust", 
           family="mopt", bb=0.25, 
           max.it=200)
} # CRAN requires conditional execution
#>      Alpha : SP500 TR Alpha : US 10Y TR Beta : SP500 TR Beta : US 10Y TR
#> HAM1            0.006             0.010           0.357           -0.238
#> HAM2            0.005             0.005           0.290           -0.126
#> HAM3            0.005             0.007           0.599           -0.203
#> HAM4            0.003             0.010           0.709           -0.247
#> HAM5            0.004             0.002           0.296            0.056
#> HAM6            0.007             0.009           0.333           -0.339