fPortfolio (version 240.10068)

AssetsModelling: Modelling of Multivariate Asset Sets

Description

A collection and description of functions which generate multivariate artificial data sets of assets, which fit the parameters to a multivariate normal, skew normal, or (skew) Student-t distribution and which compute some benchmark statistics. In addition a function is provided which allows for the selection and clustering of individual assets from portfolios using hierarchical and k-means clustering approaches. The functions are: ll{ assetsSim Simulates a data set of assets, assetsSelect Asset Selection from Portfolios, assetsFit Fits the parameter of a data set of assets, assetsStats Computes benchmark statistics of asset sets, assetsMeanCov Computes mean and covariance matri, assetsTest Test for multivariate Normal distribution, print S3 print method for an object of class 'fASSETS', plot S3 Plot method for an object of class 'fASSETS", summary S3 summary method for an object of class 'fASSETS'. }

Usage

assetsSim(n, dim = 2, model = list(mu = rep(0, dim), Omega = diag(dim), 
    alpha = rep(0, dim), df = Inf), assetNames = NULL) 
assetsSelect(x, method = c("hclust", "kmeans"), 
    kmeans.centers = 5, kmeans.maxiter = 10, doplot = TRUE, ...)
    
assetsFit(x, method = c("st", "snorm", "norm"), title = NULL, 
    description = NULL, fixed.df = NA, ...)
assetsMeanCov(x, method = c("cov", "mve", "mcd", "nnve", "shrink", "bagged"), 
    check = TRUE, force = TRUE, baggedR = 100, ...)
assetsStats(x)
assetsTest(x, method = c("shapiro", "energy"), Replicates = 100, 
    title = NULL, description = NULL)
## S3 method for class 'fASSETS':
print(x, \dots)
## S3 method for class 'fASSETS':
plot(x, which = "ask", \dots)
## S3 method for class 'fASSETS':
summary(object, which = "all", \dots)

Arguments

assetNames

[assetsSim] - a vector of character strings of length dim allowing for modifying the names of the individual assets.

baggedR

[assetsMeanCov] - an integer value, the number of bootstrap replicates, by default 100. This value is only used if method="bagged".

check

[assetsMeanCov] - a logical flag. Should the covariance matrix be tested to be positive definite? By default TRUE.

description

[assetsFit] - a character string, assigning a brief description to an "fASSETS" object.

doplot

[assetsSelect] - a logical, should a plot be displayed?

fixed.df

[assetsFit] - either NA, the default, or a numeric value assigning the number of degrees of freedom to the model. In the case that fixed.df=NA the value of df will be included in the optimiz

force

[assetsMeanCov] - a logical flag. Should the covariance matrix be forced to be positive definite? By default TRUE.

kmeans.centers

[assetsSelect] - either the number of clusters or a set of initial cluster centers. If the first, a random set of rows in x are chosen as the initial centers.

kmeans.maxiter

[assetsSelect] - the maximum number of iterations allowed.

method

[assetsFit] - a character string, which type of distribution should be fitted? method="st" denotes a multivariate skew-Student-t distribution, method="snorm" a multivariate skew-Normal distribution, and <

model

[assetsSim] - a list of model parameters: mu a vector of mean values, one for each asset series, Omega the covariance matrix of assets, alpha the skewness vector, and df the number of degrees of fre

n, dim

[assetsSim] - integer values giving the number of data records to be simulated, and the dimension of the assets set.

object

[summary] - An object of class fASSETS.

Replicates

[assetsTest] - an integer value, the number of bootstrap replicates, by default 100. This value is only used if method="energy".

title

[assetsFit] - a character string, assigning a title to an "fASSETS" object.

which

which of the five plots should be displayed? which can be either a character string, "all" (displays all plots) or "ask" (interactively asks which one to display), or a vector of 5 logical

x

[assetsFit][assetsStats][assetsMeanVar] - a numeric matrix of returns or any other rectangular object like a data.frame or a multivariate time series object which can be transformed by the function as.matrix to an object of

...

optional arguments to be passed.

Value

• assetsFit returns a S4 object class of class "fASSETS", with the following slots:
• @call
  the matched function call.
• @data
  the input data in form of a data.frame.
• @description
  allows for a brief project description.
• @fit
  the results as a list returned from the underlying fitting function.
• @method
  the selected method to fit the distribution, one of "norm", "snorm", "st".
• @model
  the model parameters describing the fitted parameters in form of a list, model=list(mu, Omega, alpha, df.
• @title
  a title string.
• The @fit slot is a list with the following compontents: (Note, not all are documented here).
• @fit$dp
  a list containing the direct parameters beta, Omega, alpha. Here, beta is a matrix of regression coefficients with dim(beta)=c(nrow(X), ncol(y)), Omega is a covariance matrix of order dim, alpha is a vector of shape parameters of length dim.
• @fit$se
  a list containing the components beta, alpha, info. Here, beta and alpha are the standard errors for the corresponding point estimates; info is the observed information matrix for the working parameter, as explained below.
• fit@optim
  the list returned by the optimizer optim; see the documentation of this function for explanation of its components.
• Note that the @fit$model slot can be used as input to the function assetsSim for simulating a similar portfolio of assets compared with the original portfolio data, usually market assets. assetsMeanCov returns a list with two entries named mu and Sigma{Sigma}. The first denotes the vector of assets means, and the second the covariance matrix. Note, that the output of this function can be used as data input for the portfolio functions to compute the efficient frontier. assetsSelect if method="hclust" was selected then the function returns a S3 object of class "hclust", otherwise if method="kmeans" was selected then the function returns an obkject of class list. For details we refer to the help pages of hclust and kmeans. assetsSim returns a matrix, the artifical data records represent the assets of the portfolio. Row names and column names are not created, they have to be added afterwards. assetsStats returns a data frame with the following entries per column and asset: Records - number of records (length of time series), paMean - annualized (pa, per annum) Mean of Returns, paAve - annualized Average of Returns, paVola - annualized Volatility (standard Deviation), paSkew - Skewness of Returns, paKurt - Kurtosis of Returns, maxDD - maximum Drawdown, TUW - Time under Water, mMaxLoss - Monthly maximum Loss, mVaR - Monthly 99 mModVaR - Monthly 99 mSharpe - Monthly Sharpe Ratio, mModSharpe - Monthly Modified Sharpe Ratio, and skPrice - Skewness/Kurtosis Price. assetsTest returns an object of class fHTEST.

Details

Data sets of assets x can be expressed as multivariate 'timeSeries' objects, as 'data.frame' objects, or any other rectangular object which can be transformed into an object of class 'matrix'. Parameter Estimation: The function assetsFit for the parameter estimation and assetsSim for the simulation of assets sets use code based on functions from the contributed packages "mtvnorm" and "sn". The required functionality for fitting data to a multivariate Normal, skew-Normal, or skew-Student-t is available from builtin functions, so it is not necessary to load the packages "mtvnorm" and "sn". Assets Mean and Covariance: The function assetsMeanCov computes the mean vector and covariance matrix of an assets set. For the covariance matrix one can select from three choicdes: The standard covariance computation through R's base function cov and a shrinked and bagged version for the covariance. The latter two choices implement the covariance computation from the functions cov.shrink() and cov.bagged() which are part of the contributed R package corpcov. Assets Statistics: The function assetsStats implements benchmark formulas and statistics as reported in the help page of the hedge fund software from www.AlternativeSoft.com. The computed statistics are listed in the 'Value' section below. Note, that the functions were written for monthly recorded data sets. Be aware of this when you use or generate asset sets on different time scales, then you have to scale them properly. Assets Selection: The function assetsSelect calls the functions hclust or kmeans from R's "stats" package. hclust performs a hierarchical cluster analysis on the set of dissimilarities hclust(dist(t(x))) and kmeans performs a k-means clustering on the data matrix itself. Assets Tests: The function assetsTest performs two tests for multivariate Normality of an assets Set.

References

Azzalini A. (1985); A Class of Distributions Which Includes the Normal Ones, Scandinavian Journal of Statistics 12, 171--178.

Azzalini A. (1986); Further Results on a Class of Distributions Which Includes the Normal Ones, Statistica 46, 199--208.

Azzalini A., Dalla Valle A. (1996); The Multivariate Skew-normal Distribution, Biometrika 83, 715--726.

Azzalini A., Capitanio A. (1999); Statistical Applications of the Multivariate Skew-normal Distribution, Journal Roy. Statist. Soc. B61, 579--602.

Azzalini A., Capitanio A. (2003); Distributions Generated by Perturbation of Symmetry with Emphasis on a Multivariate Skew-t Distribution, Journal Roy. Statist. Soc. B65, 367--389. Breiman L. (1996); Bagging Predictors, Machine Learning 24, 123--140. Genz A., Bretz F. (1999); Numerical Computation of Multivariate t-Probabilities with Application to Power Calculation of Multiple Contrasts, Journal of Statistical Computation and Simulation 63, 361--378.

Genz A. (1992); Numerical Computation of Multivariate Normal Probabilities, Journal of Computational and Graphical Statistics 1, 141--149. Genz A. (1993); Comparison of Methods for the Computation of Multivariate Normal Probabilities, Computing Science and Statistics 25, 400--405. Hothorn T., Bretz F., Genz A. (2001); On Multivariate t and Gauss Probabilities in R, R News 1/2, 27--29. Ledoit O., Wolf. M. (2003); ImprovedEestimation of the Covariance Matrix of Stock Returns with an Application to Portfolio Selection, Journal of Empirical Finance 10, 503--621.

Rizzo M.L. (2002); A New Rotation Invariant Goodness-of-Fit Test, PhD dissertation, Bowling Green State University.

Schaefer J., Strimmer K. (2005); A Shrinkage Approach to Large-Scale Covariance Estimation and Implications for Functional Genomics, Statist. Appl. Genet. Mol. Biol. 4, 32. Szekely G.J., Rizzo, M.L. (2005); A New Test for Multivariate Normality, Journal of Multivariate Analysis 93, 58--80. Szekely G.J. (1989); Potential and Kinetic Energy in Statistics, Lecture Notes, Budapest Institute of Technology, TechnicalUniversity.

See Also

MultivariateDistribution, hclust and kmeans.

Examples

## SOURCE("fPortfolio.101A-AssetsModelling")

## berndtInvest -
   xmpPortfolio("\nStart: Load monthly data set of returns > ")
   data(berndtInvest)
   # Exclude Date, Market and Interest Rate columns from data frame,
   # then multiply by 100 for percentual returns ...
   berndtAssets = berndtInvest[, -c(1, 11, 18)]
   rownames(berndtAssets) = berndtInvest[, 1]
   head(berndtAssets)
    
## assetsSelect -
   xmpPortfolio("\nNext: Select 4 most dissimilar assets from hclust > ")
   clustered = assetsSelect(berndtAssets, doplot = FALSE)
   myAssets = berndtAssets[, c(clustered$order[1:4])]
   colnames(myAssets)
   # Scatter and time series plot:
   par(mfrow = c(2, 1), cex = 0.7)
   plot(clustered)  
   myPrices = apply(myAssets, 2, cumsum)
   ts.plot(myPrices, main = "Selected Assets", 
     xlab = "Months starting 1978", ylab = "Price", col = 1:4)
   legend(0, 3, legend = colnames(myAssets), pch = "----", col = 1:4, cex = 1)
   
## assetsStats -
   if (require(fBasics)) assetsStats(myAssets)
   
## assetsSim -
   xmpPortfolio("\nNext: Fit a Skew Student-t > ")
   fit = assetsFit(myAssets)
   # Show Model Slot:
   fit @model
   # Simulate set with same properties:
   set.seed(1953)
   simAssets = assetsSim(n = 120, dim = 4, model = fit@model)
   head(simAssets)
   simPrices = apply(simAssets, 2, cumsum)
   ts.plot(simPrices, main = "Simulated Assets", 
     xlab = "Number of Months", ylab = "Simulated Price", col = 1:4)
   legend(0, 3, legend = colnames(simAssets), pch = "----", col = 1:4, cex = 1)
   
## plot -
   xmpPortfolio("\nNext: Show Simulated Assets Plots > ")
   if (require(fExtremes)) {
     # Show Scatterplot:
     par(mfrow = c(1, 1), cex = 0.7)
     plot(fit, which = c(TRUE, FALSE, FALSE, FALSE, FALSE))
     # Show  QQ and PP Plots:
     par(mfrow = c(2, 2), cex = 0.7)
     plot(fit, which = !c(TRUE, FALSE, FALSE, FALSE, FALSE))
   }