capm {TSEtools}

Capital assets pricing model including a risk-free asset.

Description

Compute the capital assets pricing model including a risk-free asset.

Usage

capm(x, Rf = 0.2/270, sh = FALSE, eRtn = NULL)

Arguments

x

a numeric matrix of random returns per unit of price within some holding period.

Rf

the return of the risk free, i.e. has variance 0.

sh

a logical indicating whether shortsales on the risky securities are allowed. Default is FALSE.

eRtn

a value of expected returen of portofilo. The mean of whole data defualt.

Details

Let \xi_1 , \ldots,\xi_n be random asset returns and w_1 , \ldots, w_n the portfolio weights. The expected returns are r_m = E\xi_m , m = 1, \ldots, n. In addition to these risky investments, there is a risk-free asset (a bond or bank account) available, which has return r_0. Denoting the weights of w_0 for the risk-free asset. The return of portfolio given by

R_p = w^t r

where, r = (r_1, \ldots, r_n)^t.

Risk is measure by a deviation functional \Sigma. It is a variance-covariance of asset returns. The risk-free component w_0 ignore in the objective. So, the standard deviation of portfolio is given by \sigma_p = w^t \Sigma w.

To obtain the optimum value of w_i, i = 1,\ldots, n, we solve the following model:

\min w^t \Sigma w,\;\;s.t:\;\; w^t r + w_0 r_0 > \mu \;\; and \;\;\sum w_i + w_0 = 1

Note that, the portfolio weights may be negative (selling short is allowed). Market portfolio is named MP where, the risk free weight w_0 is zero (see, the function of prtf()).

For any portfolio p,

E(R_p) = r_0 + \beta(p) (r_{MP} - r_0)

where, r_{MP} is return of market portfolio and \beta(p) is the beta coefficient of the portfolio p. It is given by \beta(p) = Cov( r_{MP}, r_p )/ SD(r_{MP}).

Value

wCAPM

weight of CAPM assets

wrF

weight of risk free assets

sd.capm

volatility of CAPM portfolio

rtn.capm

return of CAPM portfolio

beta

beta coefficient of portfolio

References

Pflug and Romisch (2007, ISBN: 9789812707406)

Examples

## Not run: 
x <- rnorm(500,0.05,0.02)
y <- rnorm(500,0.01,0.03)
z<-cbind(x, y)
colnames(z) <- c("prt1","prt2")

capm( z, sh = FALSE, Rf= 0.2/270, eRtn=0.02 )

## End(Not run)