Compute and Plot Alpha Proxy

Description

Computes (and possibly generates a contour plot of) the alpha proxy – a measure of the effect that volatility and correlation have on the utility of the investor. As the name might suggest, it is in the same units as alpha (that is, expected returns).

Usage

alpha.proxy(weight = 0.2, vol.man = 0.2, vol.bench = 0.2, 
	vol.other = 0.2, cor.man = 0.2, cor.bench = 0.2, 
	plot.it = TRUE, transpose = FALSE, ...)

Arguments

Value

a vector of the alpha proxies (in basis points) if less than 2 of the first 6 arguments have length more than 1.

Otherwise it is a list (returned invisibly if plot.it is TRUE) with the following components:

Side Effects

if plot.it is TRUE, then a contour plot is created.

An error occurs if more than 2 of the first 6 arguments have length greater than 1.

Details

The first 6 arguments are the variables that determine the alpha proxy. The investor is faced with the decision of hiring the manager for some portion (the weight) of the portfolio rather than using a replication of the benchmark. If the alpha proxy is positive, then the volatility and correlation of the manager's portfolio is improving whatever outperformance the manager may have (or is offsetting the losses).

Revision

This help was last revised 2010 January 05.

References

Burns, Patrick (2003). "Portfolio Sharpening". Working Paper, Burns Statistics http://www.burns-stat.com/.

See Also

partial.rainbow.

Examples

# return vector of alpha proxies
alpha.proxy(weigh=.05, vol.man=.17, cor.man=seq(0, .2, len=21))

# create a contour plot
alpha.proxy(weigh=.05, vol.man=seq(.15, .25, len=20), 
	cor.man=seq(0, .2, len=21))

# commands used to create figures in the paper
alpha.proxy(vol.man=seq(.15, .25, len=50), weight=seq(.01, .7, leng=60), 
        color.palette=partial.rainbow(start=0, end=.32))

alpha.proxy(cor.man=seq(0, .3, len=50), weight=seq(.01, .7, leng=60), 
        color.palette=partial.rainbow(start=.07))

alpha.proxy(cor.man=seq(0, .3, len=50), vol.man=seq(.15, .25, leng=60), 
        color.palette=partial.rainbow(start=0))

Estimate Variance Matrix via Statistical Factors

Description

Creates a variance matrix based on the principal components of the variables that have no missing values.

Usage

factor.model.stat(x, weights = seq(0.5, 1.5, length.out = nobs), 
	output = "full", center = TRUE, frac.var = 0.5, iter.max = 1, 
	nfac.miss = 1, full.min = 20, reg.min = 40, sd.min = 20, 
	quan.sd = 0.9, tol = 0.001, zero.load = FALSE, 
	range.factors = c(0, Inf), constant.returns.okay = FALSE, 
	specific.floor = 0.1, floor.type = "quantile", verbose=2)

Arguments

Value

if output is "full", then a variance matrix with dimensions equal to the number of columns in the input x. This has two additional attributes: number.of.factors that says how many factors are used in the model, and timestamp that gives the date and time that the object was created.

if output is "systematic", then a matrix with dimensions equal to the number of columns in the input x that contains the systematic portion of the variance matrix.

if output is "specific", then a diagonal matrix with dimensions equal to the number of columns in the input x that contains the specific variance portion of the variance matrix. The full variance matrix is the sum of the systematic and specific matrices.

If output is "factor", then an object of class "statfacmodBurSt" which is a list with components:

Details

Observations that are missing on all variables are deleted. Then a principal components factor model is estimated with the variables that have complete data.

For variables that have missing values, the standard deviation is estimated when there are enough obeservations otherwise a given quantile of the standard deviations of the other assets is used as the estimate. The loadings for these variables are set to be either the average loading for the variables with no missing data, or zero. The loadings for the most important factors are modified by performing a regression with the non-missing data for each variable (if there is enough data to do the regression).

The treatment of variables with missing values can be quite important. You may well benefit from specializing how missing values are handled to your particular problem. To do this, set the output to "factor" – then you can modify the loadings (and per force the uniquenesses), and the standard deviations to fit your situation. This may include taking sectors and countries into account, for example.

The default settings for missing value treatment are suitable for creating a variance matrix for long-only portfolio optimization – high volatility and average correlation. Take note that the proper treatment of missing values is HIGHLY dependent on the use to which the variance matrix is to be put.

OBSERVATION WEIGHTS. Time weights are quite helpful for estimating variances from returns. The default weighting seems to perform reasonably well over a range of situations.

FACTOR MODEL TO FULL MODEL. This class of object has a method for fitted which returns the variance matrix corresponding to the factor model representation.

Warning

The default value for weights assumes that the last row is the most recent observation and the first observation is the most ancient observation.

Research Issues

The method of handling missing values used in the function has not been well studied. It seems not to be the worst approach, but undoubtedly can be improved.

The default method of boosting the result away from singularity is completely unstudied. For optimization it is wise to move away from singularity, just how to do that best seems like a research question.

Revision

This help was last revised 2014 March 09.

Author(s)

Burns Statistics

See Also

fitted.statfacmodBurSt, var.shrink.eqcor, cov.wt, slideWeight.

Examples

## Not run: 
varian1 <- factor.model.stat(retmat)

varfac <- factor.model.stat(retmat, nfac=0, zero=TRUE, output="fact")

varian2 <- fitted(varfac) # get matrix from factor model

varian3 <- factor.model.stat(retmat, nfac=rep(c(5,3,1), c(20,40,1)))

## End(Not run)

Variance Matrix From Statistical Factor Model

Description

Takes a statistical factor model object and produces the variance matrix that it represents.

Usage

## S3 method for class 'statfacmodBurSt'
fitted(object, output = "full", ...)

Arguments

Value

a numeric matrix of the variance represented by the factor model, or the systematic or specific variance portion of the model.

This has two additional attributes:

Revision

This help was last revised 2012 February 12.

Author(s)

Burns Statistics

See Also

factor.model.stat.

Examples

## Not run: 
varfac <- factor.model.stat(retmat, zero=TRUE, output="factor")
# perhaps modify loadings and uniquenesses for missing values
varian2 <- fitted(varfac) # get variance matrix from factor model

specif <- fitted(varfac, output="specific") # diagonal matrix

## End(Not run)

Create Palette Function for Part of Rainbow

Description

Returns a function suitable as the color.palette argument to filled.contour that contains a specified portion of the rainbow.

Usage

partial.rainbow(start = 0, end = 0.35)

Arguments

Value

a function similar to rainbow but with the start and end arguments (possibly) changed.

Details

This function was made to facilitate the construction of contour plots in the alpha.proxy function, but is of general use.

Revision

This help was last revised 2010 January 05.

References

Burns, Patrick (2003). "Portfolio Sharpening". Working Paper, Burns Statistics http://www.burns-stat.com/.

See Also

rainbow, filled.contour, alpha.proxy.

Examples

ap1 <- alpha.proxy(cor.man=seq(0, .3, len=50), 
	vol.man=seq(.15, .25, leng=60), plot=FALSE)
filled.contour(ap1, color.palette=partial.rainbow(start=.05, end=.3))

Generate Time Weights Flexiibly

Description

Returns a vector of weights between 1 and 0 (inclusive). The general form is: some recent weights are 1, some early weights are 0, and weights between these vary linearly. It is allowable for no points to be specified as 0 or 1.

Usage

slideWeight(n, fractions = c(0, 1), observations = NULL, 
	locations = NULL)

Arguments

Value

a numeric vector of length n with values within the range of 0 and 1 (inclusive). The values are non-decreasing – meaning put more weight on more recent observations.

Details

The arguments fractions, observations and locations are all to control where the ends of the weights strictly between 0 and 1 are. Only one of them is used. locations takes precedence. fractions is used if neither of the other two are given.

This is suitable to give as the weights argument to var.shrink.eqcor and factor.model.stat.

Revision

This help was last revised 2014 March 09.

See Also

var.shrink.eqcor, factor.model.stat.

Examples

# examples assume number of observations is 200

# all weights either 0 or 1
rollwin50 <- slideWeight(200, observations=c(50,50))

# 50 with full weight, 100 more with partial weight
swght50.150 <- slideWeight(200, observations=c(50, 150))

# approximately default weights of var.shrink.eqcor and factor.model.stat
lindec3 <- slideWeight(200, fractions=c(-1/2, 3/2))

Combine matrices into 3D array

Description

Returns a three-dimensional array given one or more matrices and instructions on how to combine them.

Usage

threeDarr(..., rep = 1, union = TRUE, slicenames = NULL)

Arguments

Value

an array with the first two dimensions being the union or intersection of the first two dimensions of the input matrices, and third dimension equal to the number of input matrices times the number of replications.

Details

The full name of arguments rep, union and slicenames must be given (no abbreviations) because they come after the three-dots construct.

Revision

This help was last revised 2012 January 22.

See Also

More general functionality of this sort can be found in the abind package.

Examples

## Not run: 
multiple.var <- threeDarr(var1, var2, slicenames=c("standardVar", "crashVar"))

reparr <- threeDarr(matrix1, rep=3)

## End(Not run)

Expand a Variance Matrix to Include a Benchmark

Description

Takes a variance matrix (or 3-dimensional array) and a vector of weights for the benchmark. An object like the input is returned which includes a new asset representing the benchmark.

Usage

var.add.benchmark(variance, benchmark.weights, name = "benchmark",
	sum.to.one = TRUE)

Arguments

Value

a matrix or array similar to the input variance, but including an additional asset.

Details

The (absolute value of) the weights ideally sum to one. You can give it weights that do not sum to one (perhaps they sum to 100 or the market cap of the benchmark), but you will get a warning that the weights are being adjusted.

If your "benchmark" is something that doesn't sum to 1 (such as portfolio weights minus benchmark weights), then set the sum.to.one argument to FALSE.

Revision

This help was last revised 2012 January 20.

See Also

var.relative.benchmark, var.shrink.eqcor, factor.model.stat, threeDarr.

Examples

## Not run: 
varian.ben <- var.add.benchmark(varian, ftse.constituents, "ftse100")

## End(Not run)

var.orig <- array(c(400, 32, 24, 32, 64, 9.6, 24, 9.6, 144), c(3,3),
	list(c("equities", "bonds", "commodities"), 
	c("equities", "bonds", "commodities")))
var.aa <- var.add.benchmark(var.orig, c(equities=.6, bonds=.4), "e60b40")

Transform a Variance Matrix to be Relative to a Benchmark

Description

Returns a matrix (or 3-dimensional array) with dimensions one smaller than the input. The returned variance is relative to the benchmark.

Usage

var.relative.benchmark(variance, benchmark)

Arguments

Value

a matrix or array similar to the input variance, but with one less asset and containing variances that are relative to that asset.

There is a call attribute which gives the command that created the object (and hence the benchmark to which the object is relative).

Revision

This help was last revised 2012 January 20.

See Also

var.add.benchmark, var.shrink.eqcor, factor.model.stat, threeDarr.

Examples

var.orig <- array(c(400, 32, 24, 32, 64, 9.6, 24, 9.6, 144), c(3,3),
	list(c("equities", "bonds", "commodities"), 
	c("equities", "bonds", "commodities")))
var.aa <- var.add.benchmark(var.orig, c(equities=.6, bonds=.4), "e60b40")

var.rel <- var.relative.benchmark(var.aa, "e60b40")

Ledoit-Wolf Shrinkage Variance Estimate

Description

Returns a variance matrix that shrinks towards the equal correlation matrix – a Ledoit-Wolf estimate.

Usage

var.shrink.eqcor(x, weights = seq(0.5, 1.5, length = nt), shrink = NULL, 
	center = TRUE, vol.shrink = 0, sd.min = 20, quan.sd = 0.9, 
	tol = 1e-4, compatible = FALSE, verbose=2)

Arguments

Value

an estimate of the variance matrix of x. The sample variance is shrunk towards equal correlation.

This has two additional attributes:

Details

Time weights are quite helpful for estimating variances from returns. The default weighting seems to perform reasonably well over a range of situations. However, time weighting was not studied for this estimator.

Warning

The default value for weights assumes that the last row is the most recent observation and the first observation is the most ancient observation.

Research Issues

The method of handling missing values used in the function has not been studied (at all).

The method of boosting the result away from singularity is completely unstudied. For optimization it is wise to move away from singularity, just how to do that best seems like a research question. The method used boosts the smallest eigenvalues, it might be better to increase the diagonal.

Revision

This help was last revised 2014 March 09.

Author(s)

Burns Statistics

References

Olivier Ledoit and Michael Wolf (2004) "Honey, I shrunk the sample covariance matrix". The Journal of Portfolio Management, volume 30, number 4.

See Also

factor.model.stat, cov.wt, slideWeight.

Examples

## Not run: 
var1 <- var.shrink.eqcor(return.matrix)

var.unweighted <- var.shrink.eqcor(return.matrix, weights=1)

## End(Not run)