Package 'FinCovRegularization'

Banding Opreator on Covariance Matrix

Description

Apply banding operator on a covariance matrix with a banding parameter.

Usage

banding(sigma, k = 0)

Arguments

sigma	a p*p covariance matrix
k	banding parameter

Value

a regularized covariance matrix after banding operation

References

"High-Dimensional Covariance Estimation" by Mohsen Pourahmadi

Examples

data(m.excess.c10sp9003)
cov.SAM <- cov(m.excess.c10sp9003)
banding(cov.SAM, 7)

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Select Tuning Parameter for Banding Covariance Matrix by CV

Description

Apply K-fold cross-validation for selecting tuning parameters for banding covariance matrix using grid search strategy

Usage

banding.cv(matrix, n.cv = 10, norm = "F", seed = 142857)

Arguments

matrix	a N*p matrix, N indicates sample size and p indicates the dimension
n.cv	times that cross-validation repeated, the default number is 10
norm	the norms used to measure the cross-validation errors, which can be the Frobenius norm "F" or the operator norm "O"
seed	random seed, the default value is 142857

Details

For cross-validation, this function split the sample randomly into two pieces of size n1 = n-n/log(n) and n2 = n/log(n), and repeat this k times

Value

An object of class "CovCv" containing the cross-validation's result for covariance matrix regularization, including:

regularization	regularization method, which is "Banding"
parameter.opt	selected optimal parameter by cross-validation
cv.error	the corresponding cross-validation errors
n.cv	times that cross-validation repeated
norm	the norm used to measure the cross-validation error
seed	random seed

References

"High-Dimensional Covariance Estimation" by Mohsen Pourahmadi

Examples

data(m.excess.c10sp9003)
retcov.cv <- banding.cv(m.excess.c10sp9003, n.cv = 10,
                        norm = "F", seed = 142857)
summary(retcov.cv)
plot(retcov.cv)
# Low dimension

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The Squared Frobenius Norm

Description

Calculate the squared Frobenius norm of a matrix

Usage

F.norm2(matrix)

Arguments

matrix

a matrix

Value

a scalar of the squared Frobenius norm

Examples

data(m.excess.c10sp9003)
cov.SAM <- cov(m.excess.c10sp9003)
F.norm2(cov.SAM)

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FinCovRegularization: Covariance Matrix Estimation and Regularization for Finance

Description

Estimation and regularization for covariance matrix of asset returns. For covariance matrix estimation, three major types of factor models are included: macroeconomic factor model, fundamental factor model and statistical factor model. For covariance matrix regularization, four regularized estimators are included: banding, tapering, hard-thresholding and soft-thresholding. The tuning parameters of these regularized estimators are selected via cross-validation.

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Covariance Matrix Estimation by Fundamental Factor Model

Description

Estimate covariance matrix by fitting a fundamental factor model using OLS or WLS regression

Usage

FundamentalFactor.Cov(assets, exposure, method = "WLS")

Arguments

assets	a N*p matrix of asset returns, N indicates sample size and p indicates the dimension of asset returns
exposure	a p*q matrix of exposure indicator for the fundamental factor model, p corresponds to the dimension of asset returns, q indicates the number of fundamental industries
method	a character, indicating regression method: "OLS" or "WLS"

Value

an estimated p*p covariance matrix

Examples

data(m.excess.c10sp9003)
assets <- m.excess.c10sp9003[,1:10]
Indicator <- matrix(0,10,3)
dimnames(Indicator) <- list(colnames(assets),c("Drug","Auto","Oil"))
Indicator[c("ABT","LLY","MRK","PFE"),"Drug"] <- 1
Indicator[c("F","GM"),"Auto"] <- 1
Indicator[c("BP","CVX","RD","XOM"),"Oil"] <- 1
FundamentalFactor.Cov(assets,exposure=Indicator,method="WLS")

---

Global Minimum Variance Portfolio

Description

Computing a global minimum variance portfolio weights from the estimated covariance matrix of return series.

Usage

GMVP(cov.mat, short = TRUE)

Arguments

cov.mat	an estimated p*p covariance matrix
short	logical flag, indicating whether shortsales on the risky assets are allowed

Value

a numerical vector containing the estimated portfolio weights

Examples

data(m.excess.c10sp9003)
assets <- m.excess.c10sp9003[,1:10]
GMVP(cov(assets), short=TRUE)
GMVP(cov(assets), short=FALSE)

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Hard-Thresholding Opreator on Covariance Matrix

Description

Apply hard-thresholding operator on a covariance matrix with a hard-thresholding parameter.

Usage

hard.thresholding(sigma, threshold = 0.5)

Arguments

sigma	a p*p covariance matrix
threshold	hard-thresholding parameter

Value

a regularized covariance matrix after hard-thresholding operation

References

"High-Dimensional Covariance Estimation" by Mohsen Pourahmadi

Examples

data(m.excess.c10sp9003)
cov.SAM <- cov(m.excess.c10sp9003)
hard.thresholding(cov.SAM, threshold = 0.001)

---

Independence opreator on Covariance Matrix

Description

Apply independence model on a covariance matrix.

Usage

Ind.Cov(sigma)

Arguments

sigma

a covariance matrix

Value

a regularized covariance matrix after applying independence model

Examples

data(m.excess.c10sp9003)
cov.SAM <- cov(m.excess.c10sp9003)
Ind.Cov(cov.SAM)

---

10 stock and S&P 500 excess returns

Description

A dataset containing monthly excess returns of 10 stocks and S$P 500 index return from January 1990 to December 2003

Usage

data(m.excess.c10sp9003)

Format

A matrix with 168 rows and 11 variables

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Covariance Matrix Estimation by Macroeconomic Factor Model

Description

Estimate covariance matrix by fitting a macroeconomic factor model using time series regression

Usage

MacroFactor.Cov(assets, factor)

Arguments

assets	a N*p matrix of asset returns, N indicates sample size and p indicates the dimension of asset returns
factor	a numerical vector of length N, or a N*q matrix of macroeconomic factor(s), q indicates the dimension of factors

Value

an estimated p*p covariance matrix

Examples

data(m.excess.c10sp9003)
assets <- m.excess.c10sp9003[,1:10]
factor <- m.excess.c10sp9003[,11]
MacroFactor.Cov(assets, factor)

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The Squared Operator Norm

Description

Calculate the squared Operator norm of a matrix

Usage

O.norm2(matrix)

Arguments

matrix

a matrix

Value

a scalar of the squared Operator norm

Examples

data(m.excess.c10sp9003)
cov.SAM <- cov(m.excess.c10sp9003)
O.norm2(cov.SAM)

---

Risk Parity Portfolio

Description

Computing a Risk Parity portfolio weights from the estimated covariance matrix of return series.

Usage

RiskParity(cov.mat)

Arguments

cov.mat

an estimated p*p covariance matrix

Value

a numerical vector containing the estimated portfolio weights

Examples

data(m.excess.c10sp9003)
assets <- m.excess.c10sp9003[,1:10]
RiskParity(cov(assets))

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Soft-Thresholding Opreator on Covariance Matrix

Description

Apply soft-thresholding operator on a covariance matrix with a soft-thresholding parameter.

Usage

soft.thresholding(sigma, threshold = 0.5)

Arguments

sigma	a covariance matrix
threshold	soft-thresholding parameter

Value

a regularized covariance matrix after soft-thresholding operation

References

"High-Dimensional Covariance Estimation" by Mohsen Pourahmadi

Examples

data(m.excess.c10sp9003)
cov.SAM <- cov(m.excess.c10sp9003)
soft.thresholding(cov.SAM, threshold = 0.001)

---

Covariance Matrix Estimation by Statistical Factor Model

Description

Estimate covariance matrix by fitting a statistical factor model using principle components analysis

Usage

StatFactor.Cov(assets, k = 0)

Arguments

assets	a matrix of asset returns
k	numbers of factors, if k = 0, automatically estimating by Kaiser method

Value

an estimated p*p covariance matrix

Examples

data(m.excess.c10sp9003)
assets <- m.excess.c10sp9003[,1:10]
StatFactor.Cov(assets, 3)

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Tapering Opreator on Covariance Matrix

Description

Apply tapering operator on a covariance matrix with tapering parameters.

Usage

tapering(sigma, l, h = 1/2)

Arguments

sigma	a p*p covariance matrix
l	tapering parameter
h	the ratio between taper l_h and parameter l

Value

a regularized covariance matrix after tapering operation

References

"High-Dimensional Covariance Estimation" by Mohsen Pourahmadi

Examples

data(m.excess.c10sp9003)
cov.SAM <- cov(m.excess.c10sp9003)
tapering(cov.SAM, l=7, h = 1/2)

---

Select Tuning Parameter for Tapering Covariance Matrix by CV

Description

Apply K-fold cross-validation for selecting tuning parameters for tapering covariance matrix using grid search strategy

Usage

tapering.cv(matrix, h = 1/2, n.cv = 10, norm = "F", seed = 142857)

Arguments

matrix	a N*p matrix, N indicates sample size and p indicates the dimension
h	the ratio between taper l_h and parameter l
n.cv	times that cross-validation repeated, the default number is 10
norm	the norms used to measure the cross-validation errors, which can be the Frobenius norm "F" or the operator norm "O"
seed	random seed, the default value is 142857

Details

For cross-validation, this function split the sample randomly into two pieces of size n1 = n-n/log(n) and n2 = n/log(n), and repeat this k times

Value

An object of class "CovCv" containing the cross-validation's result for covariance matrix regularization, including:

regularization	regularization method, which is "Tapering"
parameter.opt	selected optimal parameter by cross-validation
cv.error	the corresponding cross-validation errors
n.cv	times that cross-validation repeated
norm	the norm used to measure the cross-validation error
seed	random seed

References

"High-Dimensional Covariance Estimation" by Mohsen Pourahmadi

Examples

data(m.excess.c10sp9003)
retcov.cv <- tapering.cv(m.excess.c10sp9003, n.cv = 10,
                         norm = "F", seed = 142857)
summary(retcov.cv)
plot(retcov.cv)
# Low dimension

---

Select Tuning Parameter for Thresholding Covariance Matrix by CV

Description

Apply K-fold cross-validation for selecting tuning parameters for thresholding covariance matrix using grid search strategy

Usage

threshold.cv(matrix, method = "hard", thresh.len = 20, n.cv = 10,
  norm = "F", seed = 142857)

Arguments

matrix	a N*p matrix, N indicates sample size and p indicates the dimension
method	thresholding method, "hard" or "soft"
thresh.len	the number of thresholding values tested in cross-validation, the thresholding values will be a sequence of thresh.len equally spaced values from minimum threshold constant to largest covariance in sample covariance matrix
n.cv	times that cross-validation repeated, the default number is 10
norm	the norms used to measure the cross-validation errors, which can be the Frobenius norm "F" or the operator norm "O"
seed	random seed, the default value is 142857

Details

For cross-validation, this function split the sample randomly into two pieces of size n1 = n-n/log(n) and n2 = n/log(n), and repeat this k times

Value

An object of class "CovCv" containing the cross-validation's result for covariance matrix regularization, including:

regularization	regularization method, which is "Hard Thresholding" or "Soft Thresholding"
parameter.opt	selected optimal parameter by cross-validation
cv.error	the corresponding cross-validation errors
n.cv	times that cross-validation repeated
norm	the norm used to measure the cross-validation error
seed	random seed
threshold.grid	thresholding values tested in cross-validation

References

"High-Dimensional Covariance Estimation" by Mohsen Pourahmadi

Examples

data(m.excess.c10sp9003)
retcov.cv <- threshold.cv(m.excess.c10sp9003, method = "hard",
                          thresh.len = 20, n.cv = 10, norm = "F", seed = 142857)
summary(retcov.cv)
plot(retcov.cv)
# Low dimension

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