combination.rule {CombinePortfolio} | R Documentation |

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

```
combination.rule(ret, gamma=1, superset=1:7, subset=NULL, detailed.output=FALSE,
RHO.grid.size= 100, Kmax.init= 500, tail.cut.exp= 20)
```

`ret` |
Matrix or data.frame of excess returns |

`gamma` |
Relative risk aversion parameter |

`superset` |
Vector of integers from 1,2,...,7. It gives the possible included target rules, |

`subset` |
Vector of integers of subset. It gives the target rules that must be included in the model, |

`detailed.output` |
If |

`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 |

`tail.cut.exp` |
See description of |

The target vectors are scaled so that their weights sum up to 1. Thus target rules are interpretable, i.e. 1 = tancency, 2 = GMV and 4 = naive (1/N). The function computes optimal portfolio weights given any combination rule of the riskfree asset and several target rule. These rules are called (and ordered) by and proportional to

`1 \equiv \widehat{\boldsymbol{\Sigma}}^{-1} \widehat{\boldsymbol{\mu}} `

`2 \equiv \widehat{\boldsymbol{\Sigma}}^{-1} \boldsymbol{1}`

`3 \equiv \widehat{\boldsymbol{\mu}} \ \ \ \ \ \ \ \ `

`4 \equiv \boldsymbol{1} \ \ \ `

`5 \equiv \widehat{\boldsymbol{S}}^{-2} \widehat{\boldsymbol{\mu}}`

`6 \equiv \widehat{\boldsymbol{S}}^{-2} \boldsymbol{1}`

`7 \equiv \widehat{\boldsymbol{S}}^{-1} \boldsymbol{1}`

where `\widehat{\boldsymbol{\mu}} `

and `\widehat{\boldsymbol{\Sigma}} `

are the Gaussian ML-estimators of the asset mean vector `\boldsymbol{\mu} `

and the covariance matrix `\boldsymbol{\Sigma} `

.
Moreover, we use the decomposition `\widehat{\boldsymbol{\Sigma}} = \widehat{\boldsymbol{S}}\widehat{\boldsymbol{R}}\widehat{\boldsymbol{S}} `

with
`\widehat{\boldsymbol{R}} `

as sample correlation matrix and `\widehat{\boldsymbol{S}} `

as diagonal matrix with the sample standard deviations on the diagonal.

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. The names of the combination rule are coded by their portfolio that is incorporated. If "'" is contained is the name `\theta^2 `

-adjusted estimation is used,
if "”" is contained is the name `\theta^2 `

-adjusted estimation is used. Hence e.g. "1'" represents the `\theta^2`

-adjusted 2-fund rule of Kan-Zhou(2007)
and "1”2" represents the `\psi^2 `

-adjusted 3-fund rule of Kan-Zhou(2007).

Florian Ziel

florian.ziel@uni-due.de

```
ret<- diff(log(EuStockMarkets))
combination.rule(ret) ## all 8-fund rule estimates
crule<- combination.rule(ret,gamma=5,detailed.output=TRUE)
crule$w["1'",] ## Adjusted Kan-Zhou(2007) 2-fund rule
crule$w["1''2",] ## Adjusted Kan-Zhou(2007) 3-fund rule
crule$w["124",] ## Combination rule: Tangency+GMV+naive 4-fund rule, plug-in estimator
crule$delta["124",] ## Combination weights
crule$V[,c(1,2,4)] ## Combination targets: Tangency, GMV and naive
## only models that can contain Tangency, GMV and naive, but must contain GMV
crule2<- combination.rule(ret, superset=c(1,2,4), subset=2, detailed.output=TRUE)
crule2$w # weights
crule2$delta # combination weights
crule2$V # target vectors
## case where T <= N - 4
ret2<- cbind(ret[1:10,], ret[11:20,], ret[21:30,]) ## (TxN) 10x12-matrix
combination.rule(ret2) ## only accessible solutions
```

[Package *CombinePortfolio* version 0.4 Index]