combination.rule.restriction: Function for estimating portfolio weights of a restricted 8-fund rule

Description

This function computes optimal portfolio weights based on a restricted 8-fund rule.

Usage

combination.rule.restriction(ret,  HC, h0, rule, gamma=1, detailed.output=FALSE, 
		RHO.grid.size= 100, Kmax.init= 500, tail.cut.exp= 20)

Arguments

ret

Matrix or data.frame of excess returns

Scaled restriction matrix

Scaled restriction vector

rule

Vector of combination rule, subset of 1,2,... 7

gamma

Relative risk aversion parameter

detailed.output

If FALSE only the estimated portfolio weight vectors of the models are returned. If TRUE a list of the portfolio weight vectors, the combination weights, and the target rules is provided.

RHO.grid.size

Just for convergence issues, the larger the more time-consuming, but the higher the precision of the results, only relevant if one of 5, 6 or 7 rule is included.

Kmax.init

See description of RHO.grid.size

tail.cut.exp

See description of RHO.grid.size

Value

Returns matrix of estimated weights for possible combination rules. If detailed.output is TRUE TRUE a list of the portfolio weight vectors, the combination weights, and the target rules is provided.

Details

Note that only C=I is implemented. So HC = H.

Examples

Run this code

# NOT RUN {
##setting
ret<- diff(log(EuStockMarkets))
T<- dim(ret)[1]
N<- dim(ret)[2]
gamma<- 1
## Example Tu-Zhou(2011) on Markowitz portfolio
a1<- T/(T-N-2)
rule<- c(1,4) ## as. TZ on Tangency and naive  restriction index
HC<- array( c(c(gamma*a1,N ) ) , dim=c(length(rule), 1) )## C^{-1} H conditions...
h0<- c(1)
## plug-in estimator, theta^2-adjusted, psi^2-adjusted:
rcrule<-combination.rule.restriction(ret,rule=rule,HC=HC,h0=h0,gamma=gamma,detailed.output=TRUE)
rcrule

## compare with TZ:
we<- rep.int(1/N, N)
TT<- T
mu<- apply(ret, 2, mean)## exess return
Sigma<- cov(ret) * (TT-1)/TT
Sigma.inv<- solve(Sigma)
sharpe.squared<- as.numeric( tcrossprod(crossprod(mu, Sigma.inv),mu) )	
Sigma.inv.unb<- Sigma.inv * (TT-N-2)/TT
w.Markowitz<- 1/gamma * crossprod(Sigma.inv.unb, mu) ##
weSigmawe<- as.numeric( tcrossprod(crossprod(we, Sigma),we) )	 
wemu<- crossprod(we,mu)
pi1<- as.numeric( weSigmawe - 2/gamma * wemu + 1/gamma^2 *sharpe.squared )
bb<- (TT-2)*(TT-N-2)/( (TT-N-1)*(TT-N-4) ) ##c1 in tu-zhou
pi2<- (bb-1) * sharpe.squared /gamma^2 + bb/gamma^2 * N/TT
pi3<- 0
delta.TZ.Markowitz<- (pi1 - pi3)/(pi1 + pi2 - 2*pi3)
w.TZ.Markowitz<- (1- delta.TZ.Markowitz)* we + delta.TZ.Markowitz * w.Markowitz
w.TZ.Markowitz	
rcrule$w["r:14",]

## adjusted Tu-Zhou on Markowitz
ibeta<- function(x,a,b) pbeta(x,a,b) * beta(a,b) ## incomplete beta
sharpe.squared.adj<- ((TT-N-2)*sharpe.squared - N)/TT + 2*(sharpe.squared^(N/2)*
	(1+ sharpe.squared)^(-(TT-2)/2))/TT/ibeta(sharpe.squared/(1+sharpe.squared),N/2,(TT-N)/2)
pi1.adj<- as.numeric( weSigmawe - 2/gamma * wemu + 1/gamma^2 *sharpe.squared.adj )
pi2.adj<- (bb-1) * sharpe.squared.adj /gamma^2 + bb/gamma^2 * N/TT
delta.TZ.Markowitz.adj<- (pi1.adj - pi3)/(pi1.adj + pi2.adj - 2*pi3)
w.TZ.Markowitz.adj<- (1- delta.TZ.Markowitz.adj)* we + delta.TZ.Markowitz.adj * w.Markowitz
w.TZ.Markowitz.adj
rcrule$w["r:1'4",]


## Example Tu-Zhou(2011) on Kan-Zhou(2007) 3-fund
cd<- combination.rule(ret, detailed.output=TRUE)[[2]]["1''2",1:2] ## KZ3fund combination weights
rule<- c(1,2,4) ## as. TZ on KZ3fund  restriction index
HC<- array( c(c(gamma,0, N*cd[1] ), c(0, gamma, N*cd[2] )) , dim=c(length(rule), 2) )
h0<- c(cd[1]/N, cd[2]/N)
combination.rule.restriction(ret, rule=rule, HC=HC, h0=h0) 

# }