[R-SIG-Finance] Returns used to compute the alpha and the beta

[Brian G. Peterson]

You will find all of the classic CAPM functions implemented in 
PerformanceAnalytics, including CAPM.alpha, CAPM.beta (and their bul/up 
and bear/down market counterparts), timing.ratio, Return.excess, 
CAPM.Riskpremium, and the SML and CML.  We have attempted to implement 
these using the definitions described in Sharpe's papers as well as 
generally agreed by other authors who have covered CAPM.  So all your 
functions, I hope, are already written, documented, and cited.

There is some disagreement in the literature and among practitioners 
regarding different aspects of CAPM that affect your inquiry.  One of 
them is the risk free rate.  In the original papers the risk free rate 
was a single (scalar numeric) number, while many more recent sources and 
practitioners prefer to use the risk free rate as a time series (for 
instance of returns on Treasury bills).  The function Return.excess 
allows either a scalar or time series representation. 

Classic CAPM alpha will not change based on the periodicity, as it 
measures the _portion_ of a set of returns that are not attributable to 
the benchmark return, and should be calculated with the highest regular 
periodicity available. Return.excess may calculate what you are 
referring to as "alpha" if by alpha you mean returns over a benchmark 
return, in which case you would first run Return.excess using the 
benchmark return as the parameter 'rf' and then cumulate your daily log 
returns to get a cumulative return over some other periodicity (annual 
in your query).

Returns and 'risk' may be annualized as a way to simplify comparison 
over longer time periods.  Although it requires a bit of estimating, 
such aggregation is popular because it offers a reference point for easy 
comparison. "Annualizing" the CAPM numbers and other related numbers 
such as the  Sharpe ratio you will find significant disagreement among 
different authors.  We have provided in our functions many of these 
different interpretations, including 'Return.annualized', 
'sd.annualized', and 'SharpeRatio.annualized' functions.

We have also provided many of the extensions of classic CAPM by Sharpe, 
Sortino, and others that are generally grouped into "Modern Portfolio 
Theory".  In addition to The Sortino ration and related semivariance and 
downside deviation measures, you will find active premium, information 
ration, and tracking error implemented in most of their commonly 
presented forms.

Hopefully my long response helps to answer your question.  If there are 
other questions that this raises, please ask them, and the list will 
attempt to provide a rational response.

Regards,

    - Brian

Benoit Schmid wrote:
> Good morning,
>
> I would like to know what people generally use when the compute
> the alpha and the beta (linear interpolation used for benchmark
> performance measure).
>
> From what I have seen, log return (log(V(i)/V(i-1))) and net return
> (V(i)/V(i-1) - 1) seems to be used.
> But, in general, what do people "prefer" to use?
> And why?
>
> I am working with daily values and I would like to provide annualized
> values.
> I have not found that many documentation on the web concerning
> that kind of annualized information.
>
>
> Intuitively the beta does not change, when it is annualized as it is a 
> ratio.
>
> Concerning the alpha, it is a return therefore I should use:
> 252*alphaDaily for logr and (alphaDaily + 1)252 for netr,
>
> Am I right?
>
> Thanks in advance for your answer.
>
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