sandbox/valueAtRiskFunctions.r
In cloudcell/PerformanceAnalytics: Econometric Tools for Performance and Risk Analysis
## valueAtRiskFunctions.r
##
## author: Eric Zivot
## created: December 6, 2008
## updated: October 1, 2009
##
## purpose: functions for computing value-at-risk
##
## Functions:
## normalVaR
## normalPortfolioVaR
## normalMarginalVaR
## normalComponentVaR
## normalIncrementalVaR
## normalVaRreport
## modifiedVaR
## modifiedPortfolioVaR
## modifiedMarginalVaR needs to be finished anc checked
## modifiedComponentVaR needs to be finished anc checked
## modifiedIncrementalVaR
## modifiedVaRreport needs to be finished anc checked
## bootstrapVaR
## bootstrapPortfolioVaR
## bootstrapIncrementalVaR
## bootstrapMarginalVaR
## bootstrapComponentVaR
## bootstrapVaRreport
##
## Comments:
## 1. Cornish-Fisher (modified) VaR functions are in the package PerformanceAnalytics
## 2. Standard errors for VaR calculations can be computed using the bootstrap
##
## References:
## 1. Dowd, K. (2002). Measuring Market Risk, John Wiley and Sons
## 2. Jorian, P. (2007). Value at Risk, Third Edition, McGraw Hill
## 3. Hallerback (2003). "Decomposing Portfolio Value-at-Risk: A General Analysis",
## The Journal of Risk 5/2.
## 4. Boudt, C., B. Peterson and C. Croux (2008). "Estimation and Decomposition of Downside
## Risk for Portfolios with Non-Normal Returns," Journal of Risk.
## 5. Yamai and Yoshiba (2002). "Comparative Analyses of Expected Shortfall and
## Value-at-Risk: Their Estimation Error, Decomposition, and Optimization",
## Bank of Japan.
## 6. Gourieroux, C., P. Laurent and O. Scaliet (2000). "Sensitivity Analysis of Values at
## Risk", Journal of Empirical Finance.
##
## Normal VaR functions for portfolio VaR report
##
normalVaR <- function(mu, sigma, tail.prob = 0.01) {
## compute normal VaR for collection of assets given mean and sd vector
## inputs:
## mu n x 1 vector of expected returns
## sigma n x 1 vector of standard deviations
## tail.prob scalar tail probability
## output:
## VaR n x 1 vector of left tail return quantiles returned as a positive number
## References:
## Jorian (2007) pg 111.
mu = as.matrix(mu)
sigma = as.matrix(sigma)
if ( nrow(mu) != nrow(sigma) )
stop("mu and sigma must have same number of elements")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
VaR = mu + sigma*qnorm(tail.prob)
return(-VaR)
}
normalPortfolioVaR <- function(mu, Sigma, w, tail.prob = 0.01) {
## compute normal portfolio VaR given portfolio weights, mean vector and
## covariance matrix
## inputs:
## mu n x 1 vector of expected returns
## Sigma n x n return covariance matrix
## w n x 1 vector of portfolio weights
## tail.prob scalar tail probability
## output:
## pVaR scalar left tail return quantile of normal portfolio distn returned
## as a positive number
## References:
## Jorian (2007) pg 162
mu = as.matrix(mu)
Sigma = as.matrix(Sigma)
w = as.matrix(w)
if ( nrow(mu) != nrow(Sigma) )
stop("mu and Sigma must have same number of rows")
if ( nrow(mu) != nrow(w) )
stop("mu and w must have same number of elements")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
pVaR = crossprod(w, mu) + sqrt( t(w) %*% Sigma %*% w ) * qnorm(tail.prob)
return(as.numeric(-pVaR))
}
normalMarginalVaR <- function(mu, Sigma, w, tail.prob = 0.01) {
## compute normal marginal VaR for collection of assets in a portfolio given
## mean vector and covariance matrix.
## marginal VaR is defined as the derivative of VaR wrt portfolio weight
## inputs:
## mu n x 1 vector of expected returns
## Sigma n x n return covariance matrix
## w n x 1 vector of portfolio weights
## tail.prob scalar tail probability
## output:
## mVaR n x 1 vector of left tail return quantiles returned as a positive number
## References:
## Jorian (2007) pg. 167
mu = as.matrix(mu)
Sigma = as.matrix(Sigma)
w = as.matrix(w)
if ( nrow(mu) != nrow(Sigma) )
stop("mu and Sigma must have same number of rows")
if ( nrow(mu) != nrow(w) )
stop("mu and w must have same number of elements")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
sigma.p = as.numeric(sqrt( t(w) %*% Sigma %*% w ))
mVaR = mu + Sigma %*% w * qnorm(tail.prob) / sigma.p
return(-mVaR)
}
normalComponentVaR <- function(mu, Sigma, w, tail.prob = 0.01) {
## compute normal component VaR for collection of assets in a portfolio given
## mean vector and covariance matrix
## component VaR is defined as the asset specific contribution to portfolio VaR
## such that portfolio VaR is equal to the sum of the asset specific contributions.
## By Euler's theorem component VaR = marginal VaR * portfolio weight
## inputs:
## mu n x 1 vector of expected returns
## Sigma n x n return covariance matrix
## w n x 1 vector of portfolio weights
## tail.prob scalar tail probability
## output:
## cVaR n x 1 vector of component VaR quantities
## References:
## Jorian (2007) pg. 173
mu = as.matrix(mu)
Sigma = as.matrix(Sigma)
w = as.matrix(w)
if ( nrow(mu) != nrow(Sigma) )
stop("mu and Sigma must have same number of rows")
if ( nrow(mu) != nrow(w) )
stop("mu and w must have same number of elements")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
mVaR = normalMarginalVaR(mu, Sigma, w, tail.prob)
cVaR = mVaR * w
return(cVaR)
}
normalIncrementalVaR <- function(mu, Sigma, w, tail.prob = 0.01) {
## compute normal incremental VaR given portfolio weights, mean vector and
## covariance matrix
## Incremental VaR is defined as the change in portfolio VaR that occurs
## when an asset is removed from the portfolio
## inputs:
## mu n x 1 vector of expected returns
## Sigma n x n return covariance matrix
## w n x 1 vector of portfolio weights
## tail.prob scalar tail probability
## output:
## iVaR n x 1 vector of incremental VaR values
## References:
## Jorian (2007) pg. 168
mu = as.matrix(mu)
Sigma = as.matrix(Sigma)
w = as.matrix(w)
if ( nrow(mu) != nrow(Sigma) )
stop("mu and Sigma must have same number of rows")
if ( nrow(mu) != nrow(w) )
stop("mu and w must have same number of elements")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
n.w = nrow(mu)
## portfolio VaR with all assets
pVaR = crossprod(w, mu) + sqrt( t(w) %*% Sigma %*% w ) * qnorm(tail.prob)
temp.w = w
iVaR = matrix(0, n.w, 1)
for (i in 1:n.w) {
## set weight for asset i to zero and renormalize
temp.w[i,1] = 0
temp.w = temp.w/sum(temp.w)
pVaR.new = crossprod(temp.w, mu) + sqrt( t(temp.w) %*% Sigma %*% temp.w ) * qnorm(tail.prob)
iVaR[i,1] = pVaR.new - pVaR
## reset weight
temp.w = w
}
return(-iVaR)
}
normalVaRreport <- function(mu, Sigma, w, nav, nav.p, fundName, fundStrategy,
tail.prob = 0.01) {
## compute normal VaR report for collection of assets in a portfolio given
## mean vector and covariance matrix. Report format follows that of Excel VaR report
## inputs:
## mu n x 1 vector of expected returns
## Sigma n x n return covariance matrix
## w n x 1 vector of portfolio weights
## nav n x 1 vector of net asset values in each fund
## vav.p scalar, net asset value of portfolio
## fundName n x 1 vector of fund names
## fundStrategy n x 1 vector of fund strategies
## tail.prob scalar tail probability
## output:
## VaRreport.df dataframe with n rows and the following columns
## Strategy character strategy group abbreviation
## Net.Asset.Value net asset value
## Allocation asset weights
## Mean mean return
## Std.Dev standard deviation of return
## Asset.Var asset specific VaR values
## cVaR asset specific component VaR values
## cVaR.dollar asset specific component VaR values multiplied by net asset value
## pcVaR asset specific percent contribution to VaR values
## iVaR asset specific incremental VaR values
## iVaR.dollar asset specific incremental VaR values multiplied by net asset value
## mVaR asset specific marginal VaR values
## mVaR.dollar asset specific marginal VaR values multiplied by net asset value
##
## To-do: Add information for cash position.
mu = as.matrix(mu)
Sigma = as.matrix(Sigma)
w = as.matrix(w)
nav = as.matrix(nav)
if ( nrow(mu) != nrow(Sigma) )
stop("mu and Sigma must have same number of rows")
if ( nrow(mu) != nrow(w) )
stop("mu and w must have same number of elements")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
asset.VaR = normalVaR(mu, sqrt(diag(Sigma)), tail.prob)
portfolio.VaR = normalPortfolioVaR(mu, Sigma, w, tail.prob)
marginal.VaR = normalMarginalVaR(mu, Sigma, w, tail.prob)
component.VaR = normalComponentVaR(mu, Sigma, w, tail.prob)
incremental.VaR = normalIncrementalVaR(mu, Sigma, w, tail.prob)
VaRreport.df = data.frame(Strategy = fundStrategy,
Net.Asset.Value = nav,
Allocation = w,
Mean = mu,
Std.Dev = sqrt(diag(Sigma)),
Asset.VaR = asset.VaR,
cVaR = component.VaR,
cVaR.dollar = component.VaR*nav.p,
pcVaR = component.VaR/portfolio.VaR,
iVaR = incremental.VaR,
iVaR.dollar = incremental.VaR*nav.p,
mVaR = marginal.VaR,
mVaR.dollar = marginal.VaR*nav.p)
rownames(VaRreport.df) = fundName
return(VaRreport.df)
}
##
## modifed (Cornish-Fisher) VaR functions. These functions make use
## of several functions in the PerformanceAnalytics package
##
modifiedVaR <- function(mu, sigma, skew, kurt, tail.prob = 0.01) {
## compute modified (Cornish-Fisher) VaR for collection of assets given mean and
## sd vectors.
## inputs:
## mu n x 1 vector of expected returns
## sigma n x 1 vector of standard deviations
## skew n x 1 vector of skewness values
## kurt n x 1 vector of kurtosis values
## tail.prob scalar tail probability
## output:
## mVaR n x 1 vector of left tail return quantiles returned as a positive number
## References:
## Jorian (2007) Value At Risk, pg. 273
## Boudt, Peterson and Croux (2008), "Estimation and Decomposition of Downside
## Risk for Portfolios with Non-Normal Returns," Journal of Risk.
mu = as.matrix(mu)
sigma = as.matrix(sigma)
if ( nrow(mu) != nrow(sigma) )
stop("mu and sigma must have same number of elements")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
zc = qnorm(tail.prob)
Zcf = zc + (((zc^2 - 1) * skew)/6) + (((zc^3 - 3 * zc) *
kurt)/24) - ((((2 * zc^3) - 5 * zc) * skew^2)/36)
mVaR = mu + sigma*Zcf
return(-mVaR)
}
modifiedPortfolioVaR <- function(mu, Sigma, M3, M4, w, tail.prob = 0.01) {
## compute modified (Cornish-Fisher) portfolio VaR given portfolio weights, mean vector,
## covariance matrix, co-skewness matrix and co-kurtosis matrix
## inputs:
## mu n x 1 vector of expected returns
## Sigma n x n return covariance matrix
## M3 n x n^2 co-skewness matrix
## M4 n x m^3 co-kurtosis matrix
## w n x 1 vector of portfolio weights
## tail.prob scalar tail probability
## output:
## pMVaR scalar left tail return quantile of portfolio distn returned
## as a positive number
## References:
## 1. Calculations follow Boudt et al (2008) "Estimation and Decomposition of Downside
## Risk for Portfolios with Non-Normal Returns," Journal of Risk, and make use
## of the function mVaR.MM in the PerformanceAnalytics package
require(PerformanceAnalytics)
p = 1 - tail.prob
mu = as.matrix(mu)
Sigma = as.matrix(Sigma)
w = as.matrix(w)
if ( nrow(mu) != nrow(Sigma) )
stop("mu and Sigma must have same number of rows")
if ( nrow(mu) != nrow(w) )
stop("mu and w must have same number of elements")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
pMVaR = mVaR.MM(w, mu, Sigma, M3, M4, p)
return(pMVaR)
}
modifiedMarginalVaR <- function(mu, Sigma, M3, M4, w, tail.prob = 0.01) {
## compute Cornish-Fisher (modified) marginal VaR for collection of assets in a
## portfolio given mean vector, covariance matrix, co-skewness matrix and co-kurtosis
## matrix.
## marginal VaR is defined as the derivative of VaR wrt portfolio weight
## inputs:
## mu n x 1 vector of expected returns
## Sigma n x n return covariance matrix
## w n x 1 vector of portfolio weights
## tail.prob scalar tail probability
## output:
## mVaR n x 1 vector of left tail return quantiles returned as a positive number
## References:
## Boudt, Peterson and Croux (2008), "Estimation and Decomposition of Downside
## Risk for Portfolios with Non-Normal Returns," Journal of Risk.
mu = as.matrix(mu)
Sigma = as.matrix(Sigma)
w = as.matrix(w)
if ( nrow(mu) != nrow(Sigma) )
stop("mu and Sigma must have same number of rows")
if ( nrow(mu) != nrow(w) )
stop("mu and w must have same number of elements")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
sigma.p = as.numeric(sqrt( t(w) %*% Sigma %*% w ))
mVaR = mu + Sigma %*% w * qnorm(tail.prob) / sigma.p
return(-mVaR)
}
modifiedComponentVaR <- function(mu, Sigma, M3, M4, w, tail.prob = 0.01) {
## compute Cornish-Fisher (modified) component VaR for collection of assets in
## a portfolio given mean vector, covariance matrix, co-skewness matrix and co-kurtosis
## matrix.
## component VaR is defined as the asset specific contribution to portfolio VaR
## such that portfolio VaR is equal to the sum of the asset specific contributions.
## By Euler's theorem component VaR = marginal VaR * portfolio weight
## inputs:
## mu n x 1 vector of expected returns
## Sigma n x n return covariance matrix
## w n x 1 vector of portfolio weights
## tail.prob scalar tail probability
## output:
## cVaR n x 1 vector of component VaR quantities
## References:
## Boudt, Peterson and Croux (2008), "Estimation and Decomposition of Downside
## Risk for Portfolios with Non-Normal Returns," Journal of Risk.
mu = as.matrix(mu)
Sigma = as.matrix(Sigma)
w = as.matrix(w)
if ( nrow(mu) != nrow(Sigma) )
stop("mu and Sigma must have same number of rows")
if ( nrow(mu) != nrow(w) )
stop("mu and w must have same number of elements")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
mVaR = modifiedMarginalVaR(mu, Sigma, M3, M4, w, tail.prob)
cVaR = mVaR * w
return(cVaR)
}
modifiedIncrementalVaR <- function(mu, Sigma, M3, M4, w, tail.prob = 0.01) {
## compute modified incremental VaR given portfolio weights, mean vector,
## covariance matrix, co-skewness matrix and co-kurtosis matrix
## Incremental VaR is defined as the change in portfolio VaR that occurs
## when an asset is removed from the portfolio
## inputs:
## mu n x 1 vector of expected returns
## Sigma n x n return covariance matrix
## w n x 1 vector of portfolio weights
## tail.prob scalar tail probability
## output:
## iVaR n x 1 vector of incremental VaR values
## References:
## Jorian (2007) pg. 168
mu = as.matrix(mu)
Sigma = as.matrix(Sigma)
w = as.matrix(w)
if ( nrow(mu) != nrow(Sigma) )
stop("mu and Sigma must have same number of rows")
if ( nrow(mu) != nrow(w) )
stop("mu and w must have same number of elements")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
n.w = nrow(mu)
## portfolio VaR with all assets
pVaR = modifiedPortfolioVaR(mu, Sigma, M3, M4, w, tail.prob)
temp.w = w
iVaR = matrix(0, n.w, 1)
for (i in 1:n.w) {
## set weight for asset i to zero and renormalize
temp.w[i,1] = 0
temp.w = temp.w/sum(temp.w)
pVaR.new = modifiedPortfolioVaR(mu, Sigma, M3, M4, w, tail.prob)
iVaR[i,1] = pVaR.new - pVaR
## reset weight
temp.w = w
}
return(iVaR)
}
modifiedVaRreport <- function(mu, Sigma, skew, kurt, M3, M4, w, nav, nav.p, fundName, fundStrategy,
tail.prob = 0.01) {
## compute Cornish-Fisher (modified) VaR report for collection of assets in a portfolio given
## mean vector, covariance matrix, co-skewness matrix and co-kurtosis matrix.
## Report format follows that of Excel VaR report
## inputs:
## mu n x 1 vector of expected returns
## Sigma n x n return covariance matrix
## skew n x 1 vector of skewness values
## kurt n x 1 vector of kurtosis values
## M3 n x n^2 co-skewness matrix
## M4 n x m^3 co-kurtosis matrix
## w n x 1 vector of portfolio weights
## nav n x 1 vector of net asset values in each fund
## vav.p scalar, net asset value of portfolio
## fundName n x 1 vector of fund names
## fundStrategy n x 1 vector of fund strategies
## tail.prob scalar tail probability
## output:
## VaRreport.df dataframe with n rows and the following columns
## Strategy character strategy group abbreviation
## Net.Asset.Value net asset value
## Allocation asset weights
## Mean mean return
## Std.Dev standard deviation of return
## Asset.Var asset specific VaR values
## cVaR asset specific component VaR values
## cVaR.dollar asset specific component VaR values multiplied by net asset value
## pcVaR asset specific percent contribution to VaR values
## iVaR asset specific incremental VaR values
## iVaR.dollar asset specific incremental VaR values multiplied by net asset value
## mVaR asset specific marginal VaR values
## mVaR.dollar asset specific marginal VaR values multiplied by net asset value
##
## To-do: Add information for cash position.
## extract asset skewness and excess kurtosis values from co-skewness and co-kurtosis
## matrices
## skew = M3[]
## kurt = M4[]
## where each VaR component has been multiplied by -1
mu = as.matrix(mu)
Sigma = as.matrix(Sigma)
w = as.matrix(w)
nav = as.matrix(nav)
if ( nrow(mu) != nrow(Sigma) )
stop("mu and Sigma must have same number of rows")
if ( nrow(mu) != nrow(w) )
stop("mu and w must have same number of elements")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
asset.VaR = modifiedVaR(mu, sqrt(diag(Sigma)), skew, kurt, tail.prob)
portfolio.VaR = modifiedPortfolioVaR(mu, Sigma, M3, M4, w, tail.prob)
marginal.VaR = modifiedMarginalVaR(mu, Sigma, M3, M4, w, tail.prob)
component.VaR = marginal.VaR*w
incremental.VaR = modifiedIncrementalVaR(mu, Sigma, M3, M4, w, tail.prob)
VaRreport.df = data.frame(Strategy = fundStrategy,
Net.Asset.Value = nav,
Allocation = w,
Mean = mu,
Std.Dev = sqrt(diag(Sigma)),
Asset.VaR = asset.VaR,
cVaR = component.VaR,
cVaR.dollar = component.VaR*nav.p,
pcVaR = component.VaR/portfolio.VaR,
iVaR = incremental.VaR,
iVaR.dollar = incremental.VaR*nav.p,
mVaR = marginal.VaR,
mVaR.dollar = marginal.VaR*nav.p)
rownames(VaRreport.df) = fundName
return(VaRreport.df)
}
##
## To-do: Add functions for plotting risk budgeting information
##
##
## VaR functions for use with simulated (bootstrapped) return data
##
bootstrapVaR <- function(bootData, tail.prob = 0.01,
method=c("HS", "CornishFisher")) {
## Compute VaR given bootstrap data VaR is computed either as the sample quantile or as
## an estimated quantile using the Cornish-Fisher expansion
## inputs:
## bootData B x n matrix of bootstrap returns on n assets in portfolio
## tail.prob scalar tail probability
## method character, method for computing VaR. Valid choices are "HS" for
## historical simulation (empirical quantile); "CornishFisher" for
## modified VaR based on Cornish-Fisher quantile estimate. Cornish-Fisher
## computation is done with the VaR.CornishFisher in the PerformanceAnalytics
## package
## output:
## VaR n x 1 matrix of VaR values for each asset
## References:
require(PerformanceAnalytics)
method = method[1]
bootData = as.matrix(bootData)
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
if (method == "HS") {
## use empirical quantile
VaR = -as.matrix(apply(bootData, 2, FUN=quantile, prob=tail.prob))
} else {
## use Cornish-Fisher expansion
VaR = t(as.matrix(VaR.CornishFisher(bootData, p=(1-tail.prob))))
}
return(VaR)
}
bootstrapPortfolioVaR <- function(bootData, w, tail.prob = 0.01,
method=c("HS", "CornishFisher")) {
## Compute portfolio VaR given bootstrap data and portfolio weights. VaR is
## computed either as the sample quantile or as an estimated quantile using the
## Cornish-Fisher expansion
## inputs:
## bootData B x n matrix of bootstrap returns on n assets in portfolio
## w n x 1 vector of portfolio weights
## tail.prob scalar tail probability
## method character, method for computing VaR. Valid choices are "HS" for
## historical simulation (empirical quantile); "CornishFisher" for
## modified VaR based on Cornish-Fisher quantile estimate. Cornish-Fisher
## computation is done with the VaR.CornishFisher in the PerformanceAnalytics
## package
## output:
## pVaR scalar, portfolio VaR
## References:
require(PerformanceAnalytics)
method = method[1]
bootData = as.matrix(bootData)
w = as.matrix(w)
if ( ncol(bootData) != nrow(w) )
stop("Columns of bootData and rows of w do not match")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
r.p = bootData %*% w
if (method == "HS") {
## use empirical quantile
pVaR = -quantile(r.p, prob=tail.prob)
} else {
## use Cornish-Fisher expansion
pVaR = VaR.CornishFisher(r.p, p=(1-tail.prob))
}
return(pVaR)
}
bootstrapIncrementalVaR <- function(bootData, w, tail.prob = 0.01,
method=c("HS", "CornishFisher")) {
## Compute incremental VaR given bootstrap data and portfolio weights.
## Incremental VaR is defined as the change in portfolio VaR that occurs
## when an asset is removed from the portfolio and allocation is spread equally
## among remaining assets. VaR is computed either as the sample quantile or as
## an estimated quantile using the Cornish-Fisher expansion
## inputs:
## bootData B x n matrix of bootstrap returns on n assets in portfolio
## w n x 1 vector of portfolio weights
## tail.prob scalar tail probability
## method character, method for computing VaR. Valid choices are "HS" for
## historical simulation (empirical quantile); "CornishFisher" for
## modified VaR based on Cornish-Fisher quantile estimate. Cornish-Fisher
## computation is done with the VaR.CornishFisher in the PerformanceAnalytics
## package
## output:
## iVaR n x 1 matrix of incremental VaR values for each asset
## References:
## Jorian (2007) pg. 168
##
require(PerformanceAnalytics)
method = method[1]
bootData = as.matrix(bootData)
w = as.matrix(w)
if ( ncol(bootData) != nrow(w) )
stop("Columns of bootData and rows of w do not match")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
n.w = nrow(w)
## portfolio VaR with all assets
r.p = bootData %*% w
if (method == "HS") {
## use empirical quantile
pVaR = -quantile(r.p, prob=tail.prob)
temp.w = w
iVaR = matrix(0, n.w, 1)
for (i in 1:n.w) {
## set weight for asset i to zero and renormalize
temp.w[i,1] = 0
temp.w = temp.w/sum(temp.w)
temp.r.p = bootData %*% temp.w
pVaR.new = -quantile(temp.r.p, prob=tail.prob)
iVaR[i,1] = pVaR.new - pVaR
## reset weight
temp.w = w
}
} else {
## use Cornish-Fisher VaR
pVaR = VaR.CornishFisher(r.p, p=(1-tail.prob))
temp.w = w
iVaR = matrix(0, n.w, 1)
for (i in 1:n.w) {
## set weight for asset i to zero and renormalize
temp.w[i,1] = 0
temp.w = temp.w/sum(temp.w)
temp.r.p = bootData %*% temp.w
pVaR.new = VaR.CornishFisher(temp.r.p, p=(1-tail.prob))
iVaR[i,1] = pVaR.new - pVaR
## reset weight
temp.w = w
}
}
return(iVaR)
}
bootstrapMarginalVaR <- function(bootData, w, delta.w = 0.001, h=NULL, tail.prob = 0.01,
method=c("derivative", "average"),
VaR.method=c("HS", "CornishFisher")) {
## Compute marginal VaR given bootstrap data and portfolio weights.
## Marginal VaR is computed either as the numerical derivative of VaR wrt portfolio weight or
## as the expected fund return given portfolio return is equal to portfolio VaR
## VaR is compute either as the sample quantile or as an estimated quantile
## using the Cornish-Fisher expansion
## inputs:
## bootData B x n matrix of bootstrap returns on assets in portfolio.
## w n x 1 vector of portfolio weights
## delta.w scalar, change in portfolio weight for computing numerical derivative
## h integer, number of obvs on each side of VaR. Default is h=round(sqrt(B)/2)
## tail.prob scalar tail probability
## method character, method for computing marginal VaR. Valid choices are
## "derivative" for numerical computation of the derivative of portfolio
## VaR wrt fund portfolio weight; "average" for approximating E[Ri | Rp=VaR]
## VaR.method character, method for computing VaR. Valid choices are "HS" for
## historical simulation (empirical quantile); "CornishFisher" for
## modified VaR based on Cornish-Fisher quantile estimate. Cornish-Fisher
## computation is done with the VaR.CornishFisher in the PerformanceAnalytics
## package
## output:
## mVaR n x 1 matrix of marginal VaR values for each fund
## References:
## 1. Hallerback (2003), "Decomposing Portfolio Value-at-Risk: A General Analysis",
## The Journal of Risk 5/2.
## 2. Yamai and Yoshiba (2002). "Comparative Analyses of Expected Shortfall and
## Value-at-Risk: Their Estimation Error, Decomposition, and Optimization
## Bank of Japan.
require(PerformanceAnalytics)
method = method[1]
VaR.method = VaR.method[1]
bootData = as.matrix(bootData)
w = as.matrix(w)
if ( ncol(bootData) != nrow(w) )
stop("Columns of bootData and rows of w do not match")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
# determine number of obvs to average around VaR
if (is.null(h)) {
h = round(sqrt(nrow(bootData)))
} else h = round(h)
n.w = nrow(w)
## portfolio VaR with all assets
r.p = bootData %*% w
if (VaR.method == "HS") {
pVaR = -quantile(r.p, prob=tail.prob)
} else {
pVaR = VaR.CornishFisher(r.p, p=(1-tail.prob))
}
##
## compute marginal VaR
##
if (method=="derivative") {
## compute marginal var as derivative wrt portfolio weight
temp.w = w
mVaR = matrix(0, n.w, 1)
for (i in 1:n.w) {
## increment weight for asset i by delta.w
temp.w[i,1] = w[i,1] + delta.w
temp.r.p = bootData %*% temp.w
if (VaR.method == "HS") {
pVaR.new = -quantile(temp.r.p, prob=tail.prob)
} else {
pVaR.new = VaR.CornishFisher(temp.r.p, p=(1-tail.prob))
}
mVaR[i,1] = (pVaR.new - pVaR)/delta.w
## reset weight
temp.w = w
}
} else {
## compute marginal VaR as expected value of fund return given portfolio
## return is equal to portfolio VaR
r.p.sort = sort(r.p)
idx.lower = which(r.p.sort <= -pVaR)
idx.upper = which(r.p.sort > -pVaR)
r.p.vals = c(r.p.sort[tail(idx.lower,n=h)], r.p.sort[head(idx.upper,n=h)])
idx = which(r.p %in% r.p.vals)
mVaR = -as.matrix(colMeans(bootData[idx,]))
}
## compute correction factor so that sum of weighted marginal VaR adds to portfolio VaR
cf = as.numeric( pVaR / sum(mVaR*w) )
return(cf*mVaR)
}
bootstrapComponentVaR <- function(bootData, w, delta.w = 0.001, h=NULL, tail.prob = 0.01,
method=c("derivative", "average"),
VaR.method=c("HS", "CornishFisher")) {
## Compute component VaR given bootstrap data and portfolio weights.
## Marginal VaR is computed either as the derivative of VaR wrt portfolio weight or
## as the expected fund return given portfolio return is equal to portfolio VaR
## VaR is compute either as the sample quantile or as an estimated quantile
## using the Cornish-Fisher expansion
## inputs:
## bootData B x n matrix of bootstrap returns on assets in portfolio.
## w n x 1 vector of portfolio weights
## delta.w scalar, change in portfolio weight for computing numerical derivative
## h integer, number of obvs on each side of VaR. Default is h=round(sqrt(B)/2)
## tail.prob scalar tail probability
## method character, method for computing marginal VaR. Valid choices are
## "derivative" for numerical computation of the derivative of portfolio
## VaR wrt fund portfolio weight; "average" for approximating E[Ri | Rp=VaR]
## VaR.method character, method for computing VaR. Valid choices are "HS" for
## historical simulation (empirical quantile); "CornishFisher" for
## modified VaR based on Cornish-Fisher quantile estimate. Cornish-Fisher
## computation is done with the VaR.CornishFisher in the PerformanceAnalytics
## package
## output:
## cVaR n x 1 matrix of component VaR values for each fund
## References:
## 1. Hallerback (2003), "Decomposing Portfolio Value-at-Risk: A General Analysis",
## The Journal of Risk 5/2.
## 2. Yamai and Yoshiba (2002). "Comparative Analyses of Expected Shortfall and
## Value-at-Risk: Their Estimation Error, Decomposition, and Optimization
## Bank of Japan.
require(PerformanceAnalytics)
method = method[1]
VaR.method = VaR.method[1]
bootData = as.matrix(bootData)
w = as.matrix(w)
# determine number of obvs to average around VaR
if (is.null(h)) {
h = round(sqrt(nrow(bootData)))
} else h = round(h)
if ( ncol(bootData) != nrow(w) )
stop("Columns of bootData and rows of w do not match")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
mVaR = bootstrapMarginalVaR(bootData, w, delta.w, h, tail.prob,
method, VaR.method)
cVaR = mVaR * w
return(cVaR)
}
bootstrapVaRreport <- function(bootData, w, delta.w = 0.001, h=NULL, tail.prob = 0.01,
method=c("derivative", "average"),
VaR.method=c("HS", "CornishFisher"),
nav, nav.p, fundName, fundStrategy) {
## compute VaR report for collection of assets in a portfolio given
## simulated (bootstrapped) return data
## Report format follows that of Excel VaR report
## inputs:
## bootData B x n matrix of bootstrap returns on assets in portfolio.
## w n x 1 vector of portfolio weights
## delta.w scalar, change in portfolio weight for computing numerical derivative
## h integer, number of obvs on each side of VaR. Default is h=round(sqrt(B)/2)
## tail.prob scalar tail probability
## method character, method for computing marginal VaR. Valid choices are
## "derivative" for numerical computation of the derivative of portfolio
## VaR wrt fund portfolio weight; "average" for approximating E[Ri | Rp=VaR]
## VaR.method character, method for computing VaR. Valid choices are "HS" for
## historical simulation (empirical quantile); "CornishFisher" for
## modified VaR based on Cornish-Fisher quantile estimate. Cornish-Fisher
## computation is done with the VaR.CornishFisher in the PerformanceAnalytics
## package
## nav n x 1 vector of net asset values in each fund
## nav.p scalar, net asset value of portfolio
## fundName n x 1 vector of fund names
## fundStrategy n x 1 vector of fund strategies
## output:
## VaRreport.df dataframe with n rows and the following columns
## Strategy character strategy group abbreviation
## Net.Asset.Value net asset value
## Allocation asset weights
## Mean mean return
## Std.Dev standard deviation of return
## Asset.Var asset specific VaR values
## cVaR asset specific component VaR values
## cVaR.dollar asset specific component VaR values multiplied by net asset value
## pcVaR asset specific percent contribution to VaR values
## iVaR asset specific incremental VaR values
## iVaR.dollar asset specific incremental VaR values multiplied by net asset value
## mVaR asset specific marginal VaR values
## mVaR.dollar asset specific marginal VaR values multiplied by net asset value
##
## To-do: Add information for cash position.
require(PerformanceAnalytics)
method = method[1]
VaR.method = VaR.method[1]
bootData = as.matrix(bootData)
w = as.matrix(w)
nav = as.matrix(nav)
if ( ncol(bootData) != nrow(w) )
stop("Columns of bootData and rows of w do not match")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
# determine number of obvs to average around VaR
if (is.null(h)) {
h = round(sqrt(nrow(bootData)))
} else h = round(h)
asset.VaR = bootstrapVaR(bootData, tail.prob, VaR.method)
portfolio.VaR = bootstrapPortfolioVaR(bootData, w, tail.prob, VaR.method)
marginal.VaR = bootstrapMarginalVaR(bootData, w, delta.w, h, tail.prob = 0.01,
method, VaR.method)
component.VaR = marginal.VaR*w
incremental.VaR = bootstrapIncrementalVaR(bootData, w, tail.prob, method)
VaRreport.df = data.frame(Strategy = fundStrategy,
Net.Asset.Value = nav,
Allocation = w,
Mean = colMeans(bootData),
Std.Dev = apply(bootData, 2, sd),
Asset.VaR = asset.VaR,
cVaR = component.VaR,
cVaR.dollar = component.VaR*nav.p,
pcVaR = component.VaR/portfolio.VaR,
iVaR = incremental.VaR,
iVaR.dollar = incremental.VaR*nav.p,
mVaR = marginal.VaR,
mVaR.dollar = marginal.VaR*nav.p)
rownames(VaRreport.df) = fundName
return(VaRreport.df)
}
###############################################################################
# R (http://r-project.org/) Econometrics for Performance and Risk Analysis
#
# This file originally written and contributed by Eric Zivot, 2010
# package Copyright (c) 2004-2014 Peter Carl and Brian G. Peterson
#
# This R package is distributed under the terms of the GNU Public License (GPL)
# for full details see the file COPYING
#
# $Id: valueAtRiskFunctions.r 3523 2014-09-04 10:06:12Z braverock $
#
###############################################################################
cloudcell/PerformanceAnalytics documentation built on May 13, 2019, 8:01 p.m.
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
## valueAtRiskFunctions.r
##
## author: Eric Zivot
## created: December 6, 2008
## updated: October 1, 2009
##
## purpose: functions for computing value-at-risk
##
## Functions:
## normalVaR
## normalPortfolioVaR
## normalMarginalVaR
## normalComponentVaR
## normalIncrementalVaR
## normalVaRreport
## modifiedVaR
## modifiedPortfolioVaR
## modifiedMarginalVaR needs to be finished anc checked
## modifiedComponentVaR needs to be finished anc checked
## modifiedIncrementalVaR
## modifiedVaRreport needs to be finished anc checked
## bootstrapVaR
## bootstrapPortfolioVaR
## bootstrapIncrementalVaR
## bootstrapMarginalVaR
## bootstrapComponentVaR
## bootstrapVaRreport
##
## Comments:
## 1. Cornish-Fisher (modified) VaR functions are in the package PerformanceAnalytics
## 2. Standard errors for VaR calculations can be computed using the bootstrap
##
## References:
## 1. Dowd, K. (2002). Measuring Market Risk, John Wiley and Sons
## 2. Jorian, P. (2007). Value at Risk, Third Edition, McGraw Hill
## 3. Hallerback (2003). "Decomposing Portfolio Value-at-Risk: A General Analysis",
## The Journal of Risk 5/2.
## 4. Boudt, C., B. Peterson and C. Croux (2008). "Estimation and Decomposition of Downside
## Risk for Portfolios with Non-Normal Returns," Journal of Risk.
## 5. Yamai and Yoshiba (2002). "Comparative Analyses of Expected Shortfall and
## Value-at-Risk: Their Estimation Error, Decomposition, and Optimization",
## Bank of Japan.
## 6. Gourieroux, C., P. Laurent and O. Scaliet (2000). "Sensitivity Analysis of Values at
## Risk", Journal of Empirical Finance.
##
## Normal VaR functions for portfolio VaR report
##
normalVaR <- function(mu, sigma, tail.prob = 0.01) {
## compute normal VaR for collection of assets given mean and sd vector
## inputs:
## mu n x 1 vector of expected returns
## sigma n x 1 vector of standard deviations
## tail.prob scalar tail probability
## output:
## VaR n x 1 vector of left tail return quantiles returned as a positive number
## References:
## Jorian (2007) pg 111.
mu = as.matrix(mu)
sigma = as.matrix(sigma)
if ( nrow(mu) != nrow(sigma) )
stop("mu and sigma must have same number of elements")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
VaR = mu + sigma*qnorm(tail.prob)
return(-VaR)
}
normalPortfolioVaR <- function(mu, Sigma, w, tail.prob = 0.01) {
## compute normal portfolio VaR given portfolio weights, mean vector and
## covariance matrix
## inputs:
## mu n x 1 vector of expected returns
## Sigma n x n return covariance matrix
## w n x 1 vector of portfolio weights
## tail.prob scalar tail probability
## output:
## pVaR scalar left tail return quantile of normal portfolio distn returned
## as a positive number
## References:
## Jorian (2007) pg 162
mu = as.matrix(mu)
Sigma = as.matrix(Sigma)
w = as.matrix(w)
if ( nrow(mu) != nrow(Sigma) )
stop("mu and Sigma must have same number of rows")
if ( nrow(mu) != nrow(w) )
stop("mu and w must have same number of elements")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
pVaR = crossprod(w, mu) + sqrt( t(w) %*% Sigma %*% w ) * qnorm(tail.prob)
return(as.numeric(-pVaR))
}
normalMarginalVaR <- function(mu, Sigma, w, tail.prob = 0.01) {
## compute normal marginal VaR for collection of assets in a portfolio given
## mean vector and covariance matrix.
## marginal VaR is defined as the derivative of VaR wrt portfolio weight
## inputs:
## mu n x 1 vector of expected returns
## Sigma n x n return covariance matrix
## w n x 1 vector of portfolio weights
## tail.prob scalar tail probability
## output:
## mVaR n x 1 vector of left tail return quantiles returned as a positive number
## References:
## Jorian (2007) pg. 167
mu = as.matrix(mu)
Sigma = as.matrix(Sigma)
w = as.matrix(w)
if ( nrow(mu) != nrow(Sigma) )
stop("mu and Sigma must have same number of rows")
if ( nrow(mu) != nrow(w) )
stop("mu and w must have same number of elements")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
sigma.p = as.numeric(sqrt( t(w) %*% Sigma %*% w ))
mVaR = mu + Sigma %*% w * qnorm(tail.prob) / sigma.p
return(-mVaR)
}
normalComponentVaR <- function(mu, Sigma, w, tail.prob = 0.01) {
## compute normal component VaR for collection of assets in a portfolio given
## mean vector and covariance matrix
## component VaR is defined as the asset specific contribution to portfolio VaR
## such that portfolio VaR is equal to the sum of the asset specific contributions.
## By Euler's theorem component VaR = marginal VaR * portfolio weight
## inputs:
## mu n x 1 vector of expected returns
## Sigma n x n return covariance matrix
## w n x 1 vector of portfolio weights
## tail.prob scalar tail probability
## output:
## cVaR n x 1 vector of component VaR quantities
## References:
## Jorian (2007) pg. 173
mu = as.matrix(mu)
Sigma = as.matrix(Sigma)
w = as.matrix(w)
if ( nrow(mu) != nrow(Sigma) )
stop("mu and Sigma must have same number of rows")
if ( nrow(mu) != nrow(w) )
stop("mu and w must have same number of elements")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
mVaR = normalMarginalVaR(mu, Sigma, w, tail.prob)
cVaR = mVaR * w
return(cVaR)
}
normalIncrementalVaR <- function(mu, Sigma, w, tail.prob = 0.01) {
## compute normal incremental VaR given portfolio weights, mean vector and
## covariance matrix
## Incremental VaR is defined as the change in portfolio VaR that occurs
## when an asset is removed from the portfolio
## inputs:
## mu n x 1 vector of expected returns
## Sigma n x n return covariance matrix
## w n x 1 vector of portfolio weights
## tail.prob scalar tail probability
## output:
## iVaR n x 1 vector of incremental VaR values
## References:
## Jorian (2007) pg. 168
mu = as.matrix(mu)
Sigma = as.matrix(Sigma)
w = as.matrix(w)
if ( nrow(mu) != nrow(Sigma) )
stop("mu and Sigma must have same number of rows")
if ( nrow(mu) != nrow(w) )
stop("mu and w must have same number of elements")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
n.w = nrow(mu)
## portfolio VaR with all assets
pVaR = crossprod(w, mu) + sqrt( t(w) %*% Sigma %*% w ) * qnorm(tail.prob)
temp.w = w
iVaR = matrix(0, n.w, 1)
for (i in 1:n.w) {
## set weight for asset i to zero and renormalize
temp.w[i,1] = 0
temp.w = temp.w/sum(temp.w)
pVaR.new = crossprod(temp.w, mu) + sqrt( t(temp.w) %*% Sigma %*% temp.w ) * qnorm(tail.prob)
iVaR[i,1] = pVaR.new - pVaR
## reset weight
temp.w = w
}
return(-iVaR)
}
normalVaRreport <- function(mu, Sigma, w, nav, nav.p, fundName, fundStrategy,
tail.prob = 0.01) {
## compute normal VaR report for collection of assets in a portfolio given
## mean vector and covariance matrix. Report format follows that of Excel VaR report
## inputs:
## mu n x 1 vector of expected returns
## Sigma n x n return covariance matrix
## w n x 1 vector of portfolio weights
## nav n x 1 vector of net asset values in each fund
## vav.p scalar, net asset value of portfolio
## fundName n x 1 vector of fund names
## fundStrategy n x 1 vector of fund strategies
## tail.prob scalar tail probability
## output:
## VaRreport.df dataframe with n rows and the following columns
## Strategy character strategy group abbreviation
## Net.Asset.Value net asset value
## Allocation asset weights
## Mean mean return
## Std.Dev standard deviation of return
## Asset.Var asset specific VaR values
## cVaR asset specific component VaR values
## cVaR.dollar asset specific component VaR values multiplied by net asset value
## pcVaR asset specific percent contribution to VaR values
## iVaR asset specific incremental VaR values
## iVaR.dollar asset specific incremental VaR values multiplied by net asset value
## mVaR asset specific marginal VaR values
## mVaR.dollar asset specific marginal VaR values multiplied by net asset value
##
## To-do: Add information for cash position.
mu = as.matrix(mu)
Sigma = as.matrix(Sigma)
w = as.matrix(w)
nav = as.matrix(nav)
if ( nrow(mu) != nrow(Sigma) )
stop("mu and Sigma must have same number of rows")
if ( nrow(mu) != nrow(w) )
stop("mu and w must have same number of elements")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
asset.VaR = normalVaR(mu, sqrt(diag(Sigma)), tail.prob)
portfolio.VaR = normalPortfolioVaR(mu, Sigma, w, tail.prob)
marginal.VaR = normalMarginalVaR(mu, Sigma, w, tail.prob)
component.VaR = normalComponentVaR(mu, Sigma, w, tail.prob)
incremental.VaR = normalIncrementalVaR(mu, Sigma, w, tail.prob)
VaRreport.df = data.frame(Strategy = fundStrategy,
Net.Asset.Value = nav,
Allocation = w,
Mean = mu,
Std.Dev = sqrt(diag(Sigma)),
Asset.VaR = asset.VaR,
cVaR = component.VaR,
cVaR.dollar = component.VaR*nav.p,
pcVaR = component.VaR/portfolio.VaR,
iVaR = incremental.VaR,
iVaR.dollar = incremental.VaR*nav.p,
mVaR = marginal.VaR,
mVaR.dollar = marginal.VaR*nav.p)
rownames(VaRreport.df) = fundName
return(VaRreport.df)
}
##
## modifed (Cornish-Fisher) VaR functions. These functions make use
## of several functions in the PerformanceAnalytics package
##
modifiedVaR <- function(mu, sigma, skew, kurt, tail.prob = 0.01) {
## compute modified (Cornish-Fisher) VaR for collection of assets given mean and
## sd vectors.
## inputs:
## mu n x 1 vector of expected returns
## sigma n x 1 vector of standard deviations
## skew n x 1 vector of skewness values
## kurt n x 1 vector of kurtosis values
## tail.prob scalar tail probability
## output:
## mVaR n x 1 vector of left tail return quantiles returned as a positive number
## References:
## Jorian (2007) Value At Risk, pg. 273
## Boudt, Peterson and Croux (2008), "Estimation and Decomposition of Downside
## Risk for Portfolios with Non-Normal Returns," Journal of Risk.
mu = as.matrix(mu)
sigma = as.matrix(sigma)
if ( nrow(mu) != nrow(sigma) )
stop("mu and sigma must have same number of elements")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
zc = qnorm(tail.prob)
Zcf = zc + (((zc^2 - 1) * skew)/6) + (((zc^3 - 3 * zc) *
kurt)/24) - ((((2 * zc^3) - 5 * zc) * skew^2)/36)
mVaR = mu + sigma*Zcf
return(-mVaR)
}
modifiedPortfolioVaR <- function(mu, Sigma, M3, M4, w, tail.prob = 0.01) {
## compute modified (Cornish-Fisher) portfolio VaR given portfolio weights, mean vector,
## covariance matrix, co-skewness matrix and co-kurtosis matrix
## inputs:
## mu n x 1 vector of expected returns
## Sigma n x n return covariance matrix
## M3 n x n^2 co-skewness matrix
## M4 n x m^3 co-kurtosis matrix
## w n x 1 vector of portfolio weights
## tail.prob scalar tail probability
## output:
## pMVaR scalar left tail return quantile of portfolio distn returned
## as a positive number
## References:
## 1. Calculations follow Boudt et al (2008) "Estimation and Decomposition of Downside
## Risk for Portfolios with Non-Normal Returns," Journal of Risk, and make use
## of the function mVaR.MM in the PerformanceAnalytics package
require(PerformanceAnalytics)
p = 1 - tail.prob
mu = as.matrix(mu)
Sigma = as.matrix(Sigma)
w = as.matrix(w)
if ( nrow(mu) != nrow(Sigma) )
stop("mu and Sigma must have same number of rows")
if ( nrow(mu) != nrow(w) )
stop("mu and w must have same number of elements")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
pMVaR = mVaR.MM(w, mu, Sigma, M3, M4, p)
return(pMVaR)
}
modifiedMarginalVaR <- function(mu, Sigma, M3, M4, w, tail.prob = 0.01) {
## compute Cornish-Fisher (modified) marginal VaR for collection of assets in a
## portfolio given mean vector, covariance matrix, co-skewness matrix and co-kurtosis
## matrix.
## marginal VaR is defined as the derivative of VaR wrt portfolio weight
## inputs:
## mu n x 1 vector of expected returns
## Sigma n x n return covariance matrix
## w n x 1 vector of portfolio weights
## tail.prob scalar tail probability
## output:
## mVaR n x 1 vector of left tail return quantiles returned as a positive number
## References:
## Boudt, Peterson and Croux (2008), "Estimation and Decomposition of Downside
## Risk for Portfolios with Non-Normal Returns," Journal of Risk.
mu = as.matrix(mu)
Sigma = as.matrix(Sigma)
w = as.matrix(w)
if ( nrow(mu) != nrow(Sigma) )
stop("mu and Sigma must have same number of rows")
if ( nrow(mu) != nrow(w) )
stop("mu and w must have same number of elements")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
sigma.p = as.numeric(sqrt( t(w) %*% Sigma %*% w ))
mVaR = mu + Sigma %*% w * qnorm(tail.prob) / sigma.p
return(-mVaR)
}
modifiedComponentVaR <- function(mu, Sigma, M3, M4, w, tail.prob = 0.01) {
## compute Cornish-Fisher (modified) component VaR for collection of assets in
## a portfolio given mean vector, covariance matrix, co-skewness matrix and co-kurtosis
## matrix.
## component VaR is defined as the asset specific contribution to portfolio VaR
## such that portfolio VaR is equal to the sum of the asset specific contributions.
## By Euler's theorem component VaR = marginal VaR * portfolio weight
## inputs:
## mu n x 1 vector of expected returns
## Sigma n x n return covariance matrix
## w n x 1 vector of portfolio weights
## tail.prob scalar tail probability
## output:
## cVaR n x 1 vector of component VaR quantities
## References:
## Boudt, Peterson and Croux (2008), "Estimation and Decomposition of Downside
## Risk for Portfolios with Non-Normal Returns," Journal of Risk.
mu = as.matrix(mu)
Sigma = as.matrix(Sigma)
w = as.matrix(w)
if ( nrow(mu) != nrow(Sigma) )
stop("mu and Sigma must have same number of rows")
if ( nrow(mu) != nrow(w) )
stop("mu and w must have same number of elements")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
mVaR = modifiedMarginalVaR(mu, Sigma, M3, M4, w, tail.prob)
cVaR = mVaR * w
return(cVaR)
}
modifiedIncrementalVaR <- function(mu, Sigma, M3, M4, w, tail.prob = 0.01) {
## compute modified incremental VaR given portfolio weights, mean vector,
## covariance matrix, co-skewness matrix and co-kurtosis matrix
## Incremental VaR is defined as the change in portfolio VaR that occurs
## when an asset is removed from the portfolio
## inputs:
## mu n x 1 vector of expected returns
## Sigma n x n return covariance matrix
## w n x 1 vector of portfolio weights
## tail.prob scalar tail probability
## output:
## iVaR n x 1 vector of incremental VaR values
## References:
## Jorian (2007) pg. 168
mu = as.matrix(mu)
Sigma = as.matrix(Sigma)
w = as.matrix(w)
if ( nrow(mu) != nrow(Sigma) )
stop("mu and Sigma must have same number of rows")
if ( nrow(mu) != nrow(w) )
stop("mu and w must have same number of elements")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
n.w = nrow(mu)
## portfolio VaR with all assets
pVaR = modifiedPortfolioVaR(mu, Sigma, M3, M4, w, tail.prob)
temp.w = w
iVaR = matrix(0, n.w, 1)
for (i in 1:n.w) {
## set weight for asset i to zero and renormalize
temp.w[i,1] = 0
temp.w = temp.w/sum(temp.w)
pVaR.new = modifiedPortfolioVaR(mu, Sigma, M3, M4, w, tail.prob)
iVaR[i,1] = pVaR.new - pVaR
## reset weight
temp.w = w
}
return(iVaR)
}
modifiedVaRreport <- function(mu, Sigma, skew, kurt, M3, M4, w, nav, nav.p, fundName, fundStrategy,
tail.prob = 0.01) {
## compute Cornish-Fisher (modified) VaR report for collection of assets in a portfolio given
## mean vector, covariance matrix, co-skewness matrix and co-kurtosis matrix.
## Report format follows that of Excel VaR report
## inputs:
## mu n x 1 vector of expected returns
## Sigma n x n return covariance matrix
## skew n x 1 vector of skewness values
## kurt n x 1 vector of kurtosis values
## M3 n x n^2 co-skewness matrix
## M4 n x m^3 co-kurtosis matrix
## w n x 1 vector of portfolio weights
## nav n x 1 vector of net asset values in each fund
## vav.p scalar, net asset value of portfolio
## fundName n x 1 vector of fund names
## fundStrategy n x 1 vector of fund strategies
## tail.prob scalar tail probability
## output:
## VaRreport.df dataframe with n rows and the following columns
## Strategy character strategy group abbreviation
## Net.Asset.Value net asset value
## Allocation asset weights
## Mean mean return
## Std.Dev standard deviation of return
## Asset.Var asset specific VaR values
## cVaR asset specific component VaR values
## cVaR.dollar asset specific component VaR values multiplied by net asset value
## pcVaR asset specific percent contribution to VaR values
## iVaR asset specific incremental VaR values
## iVaR.dollar asset specific incremental VaR values multiplied by net asset value
## mVaR asset specific marginal VaR values
## mVaR.dollar asset specific marginal VaR values multiplied by net asset value
##
## To-do: Add information for cash position.
## extract asset skewness and excess kurtosis values from co-skewness and co-kurtosis
## matrices
## skew = M3[]
## kurt = M4[]
## where each VaR component has been multiplied by -1
mu = as.matrix(mu)
Sigma = as.matrix(Sigma)
w = as.matrix(w)
nav = as.matrix(nav)
if ( nrow(mu) != nrow(Sigma) )
stop("mu and Sigma must have same number of rows")
if ( nrow(mu) != nrow(w) )
stop("mu and w must have same number of elements")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
asset.VaR = modifiedVaR(mu, sqrt(diag(Sigma)), skew, kurt, tail.prob)
portfolio.VaR = modifiedPortfolioVaR(mu, Sigma, M3, M4, w, tail.prob)
marginal.VaR = modifiedMarginalVaR(mu, Sigma, M3, M4, w, tail.prob)
component.VaR = marginal.VaR*w
incremental.VaR = modifiedIncrementalVaR(mu, Sigma, M3, M4, w, tail.prob)
VaRreport.df = data.frame(Strategy = fundStrategy,
Net.Asset.Value = nav,
Allocation = w,
Mean = mu,
Std.Dev = sqrt(diag(Sigma)),
Asset.VaR = asset.VaR,
cVaR = component.VaR,
cVaR.dollar = component.VaR*nav.p,
pcVaR = component.VaR/portfolio.VaR,
iVaR = incremental.VaR,
iVaR.dollar = incremental.VaR*nav.p,
mVaR = marginal.VaR,
mVaR.dollar = marginal.VaR*nav.p)
rownames(VaRreport.df) = fundName
return(VaRreport.df)
}
##
## To-do: Add functions for plotting risk budgeting information
##
##
## VaR functions for use with simulated (bootstrapped) return data
##
bootstrapVaR <- function(bootData, tail.prob = 0.01,
method=c("HS", "CornishFisher")) {
## Compute VaR given bootstrap data VaR is computed either as the sample quantile or as
## an estimated quantile using the Cornish-Fisher expansion
## inputs:
## bootData B x n matrix of bootstrap returns on n assets in portfolio
## tail.prob scalar tail probability
## method character, method for computing VaR. Valid choices are "HS" for
## historical simulation (empirical quantile); "CornishFisher" for
## modified VaR based on Cornish-Fisher quantile estimate. Cornish-Fisher
## computation is done with the VaR.CornishFisher in the PerformanceAnalytics
## package
## output:
## VaR n x 1 matrix of VaR values for each asset
## References:
require(PerformanceAnalytics)
method = method[1]
bootData = as.matrix(bootData)
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
if (method == "HS") {
## use empirical quantile
VaR = -as.matrix(apply(bootData, 2, FUN=quantile, prob=tail.prob))
} else {
## use Cornish-Fisher expansion
VaR = t(as.matrix(VaR.CornishFisher(bootData, p=(1-tail.prob))))
}
return(VaR)
}
bootstrapPortfolioVaR <- function(bootData, w, tail.prob = 0.01,
method=c("HS", "CornishFisher")) {
## Compute portfolio VaR given bootstrap data and portfolio weights. VaR is
## computed either as the sample quantile or as an estimated quantile using the
## Cornish-Fisher expansion
## inputs:
## bootData B x n matrix of bootstrap returns on n assets in portfolio
## w n x 1 vector of portfolio weights
## tail.prob scalar tail probability
## method character, method for computing VaR. Valid choices are "HS" for
## historical simulation (empirical quantile); "CornishFisher" for
## modified VaR based on Cornish-Fisher quantile estimate. Cornish-Fisher
## computation is done with the VaR.CornishFisher in the PerformanceAnalytics
## package
## output:
## pVaR scalar, portfolio VaR
## References:
require(PerformanceAnalytics)
method = method[1]
bootData = as.matrix(bootData)
w = as.matrix(w)
if ( ncol(bootData) != nrow(w) )
stop("Columns of bootData and rows of w do not match")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
r.p = bootData %*% w
if (method == "HS") {
## use empirical quantile
pVaR = -quantile(r.p, prob=tail.prob)
} else {
## use Cornish-Fisher expansion
pVaR = VaR.CornishFisher(r.p, p=(1-tail.prob))
}
return(pVaR)
}
bootstrapIncrementalVaR <- function(bootData, w, tail.prob = 0.01,
method=c("HS", "CornishFisher")) {
## Compute incremental VaR given bootstrap data and portfolio weights.
## Incremental VaR is defined as the change in portfolio VaR that occurs
## when an asset is removed from the portfolio and allocation is spread equally
## among remaining assets. VaR is computed either as the sample quantile or as
## an estimated quantile using the Cornish-Fisher expansion
## inputs:
## bootData B x n matrix of bootstrap returns on n assets in portfolio
## w n x 1 vector of portfolio weights
## tail.prob scalar tail probability
## method character, method for computing VaR. Valid choices are "HS" for
## historical simulation (empirical quantile); "CornishFisher" for
## modified VaR based on Cornish-Fisher quantile estimate. Cornish-Fisher
## computation is done with the VaR.CornishFisher in the PerformanceAnalytics
## package
## output:
## iVaR n x 1 matrix of incremental VaR values for each asset
## References:
## Jorian (2007) pg. 168
##
require(PerformanceAnalytics)
method = method[1]
bootData = as.matrix(bootData)
w = as.matrix(w)
if ( ncol(bootData) != nrow(w) )
stop("Columns of bootData and rows of w do not match")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
n.w = nrow(w)
## portfolio VaR with all assets
r.p = bootData %*% w
if (method == "HS") {
## use empirical quantile
pVaR = -quantile(r.p, prob=tail.prob)
temp.w = w
iVaR = matrix(0, n.w, 1)
for (i in 1:n.w) {
## set weight for asset i to zero and renormalize
temp.w[i,1] = 0
temp.w = temp.w/sum(temp.w)
temp.r.p = bootData %*% temp.w
pVaR.new = -quantile(temp.r.p, prob=tail.prob)
iVaR[i,1] = pVaR.new - pVaR
## reset weight
temp.w = w
}
} else {
## use Cornish-Fisher VaR
pVaR = VaR.CornishFisher(r.p, p=(1-tail.prob))
temp.w = w
iVaR = matrix(0, n.w, 1)
for (i in 1:n.w) {
## set weight for asset i to zero and renormalize
temp.w[i,1] = 0
temp.w = temp.w/sum(temp.w)
temp.r.p = bootData %*% temp.w
pVaR.new = VaR.CornishFisher(temp.r.p, p=(1-tail.prob))
iVaR[i,1] = pVaR.new - pVaR
## reset weight
temp.w = w
}
}
return(iVaR)
}
bootstrapMarginalVaR <- function(bootData, w, delta.w = 0.001, h=NULL, tail.prob = 0.01,
method=c("derivative", "average"),
VaR.method=c("HS", "CornishFisher")) {
## Compute marginal VaR given bootstrap data and portfolio weights.
## Marginal VaR is computed either as the numerical derivative of VaR wrt portfolio weight or
## as the expected fund return given portfolio return is equal to portfolio VaR
## VaR is compute either as the sample quantile or as an estimated quantile
## using the Cornish-Fisher expansion
## inputs:
## bootData B x n matrix of bootstrap returns on assets in portfolio.
## w n x 1 vector of portfolio weights
## delta.w scalar, change in portfolio weight for computing numerical derivative
## h integer, number of obvs on each side of VaR. Default is h=round(sqrt(B)/2)
## tail.prob scalar tail probability
## method character, method for computing marginal VaR. Valid choices are
## "derivative" for numerical computation of the derivative of portfolio
## VaR wrt fund portfolio weight; "average" for approximating E[Ri | Rp=VaR]
## VaR.method character, method for computing VaR. Valid choices are "HS" for
## historical simulation (empirical quantile); "CornishFisher" for
## modified VaR based on Cornish-Fisher quantile estimate. Cornish-Fisher
## computation is done with the VaR.CornishFisher in the PerformanceAnalytics
## package
## output:
## mVaR n x 1 matrix of marginal VaR values for each fund
## References:
## 1. Hallerback (2003), "Decomposing Portfolio Value-at-Risk: A General Analysis",
## The Journal of Risk 5/2.
## 2. Yamai and Yoshiba (2002). "Comparative Analyses of Expected Shortfall and
## Value-at-Risk: Their Estimation Error, Decomposition, and Optimization
## Bank of Japan.
require(PerformanceAnalytics)
method = method[1]
VaR.method = VaR.method[1]
bootData = as.matrix(bootData)
w = as.matrix(w)
if ( ncol(bootData) != nrow(w) )
stop("Columns of bootData and rows of w do not match")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
# determine number of obvs to average around VaR
if (is.null(h)) {
h = round(sqrt(nrow(bootData)))
} else h = round(h)
n.w = nrow(w)
## portfolio VaR with all assets
r.p = bootData %*% w
if (VaR.method == "HS") {
pVaR = -quantile(r.p, prob=tail.prob)
} else {
pVaR = VaR.CornishFisher(r.p, p=(1-tail.prob))
}
##
## compute marginal VaR
##
if (method=="derivative") {
## compute marginal var as derivative wrt portfolio weight
temp.w = w
mVaR = matrix(0, n.w, 1)
for (i in 1:n.w) {
## increment weight for asset i by delta.w
temp.w[i,1] = w[i,1] + delta.w
temp.r.p = bootData %*% temp.w
if (VaR.method == "HS") {
pVaR.new = -quantile(temp.r.p, prob=tail.prob)
} else {
pVaR.new = VaR.CornishFisher(temp.r.p, p=(1-tail.prob))
}
mVaR[i,1] = (pVaR.new - pVaR)/delta.w
## reset weight
temp.w = w
}
} else {
## compute marginal VaR as expected value of fund return given portfolio
## return is equal to portfolio VaR
r.p.sort = sort(r.p)
idx.lower = which(r.p.sort <= -pVaR)
idx.upper = which(r.p.sort > -pVaR)
r.p.vals = c(r.p.sort[tail(idx.lower,n=h)], r.p.sort[head(idx.upper,n=h)])
idx = which(r.p %in% r.p.vals)
mVaR = -as.matrix(colMeans(bootData[idx,]))
}
## compute correction factor so that sum of weighted marginal VaR adds to portfolio VaR
cf = as.numeric( pVaR / sum(mVaR*w) )
return(cf*mVaR)
}
bootstrapComponentVaR <- function(bootData, w, delta.w = 0.001, h=NULL, tail.prob = 0.01,
method=c("derivative", "average"),
VaR.method=c("HS", "CornishFisher")) {
## Compute component VaR given bootstrap data and portfolio weights.
## Marginal VaR is computed either as the derivative of VaR wrt portfolio weight or
## as the expected fund return given portfolio return is equal to portfolio VaR
## VaR is compute either as the sample quantile or as an estimated quantile
## using the Cornish-Fisher expansion
## inputs:
## bootData B x n matrix of bootstrap returns on assets in portfolio.
## w n x 1 vector of portfolio weights
## delta.w scalar, change in portfolio weight for computing numerical derivative
## h integer, number of obvs on each side of VaR. Default is h=round(sqrt(B)/2)
## tail.prob scalar tail probability
## method character, method for computing marginal VaR. Valid choices are
## "derivative" for numerical computation of the derivative of portfolio
## VaR wrt fund portfolio weight; "average" for approximating E[Ri | Rp=VaR]
## VaR.method character, method for computing VaR. Valid choices are "HS" for
## historical simulation (empirical quantile); "CornishFisher" for
## modified VaR based on Cornish-Fisher quantile estimate. Cornish-Fisher
## computation is done with the VaR.CornishFisher in the PerformanceAnalytics
## package
## output:
## cVaR n x 1 matrix of component VaR values for each fund
## References:
## 1. Hallerback (2003), "Decomposing Portfolio Value-at-Risk: A General Analysis",
## The Journal of Risk 5/2.
## 2. Yamai and Yoshiba (2002). "Comparative Analyses of Expected Shortfall and
## Value-at-Risk: Their Estimation Error, Decomposition, and Optimization
## Bank of Japan.
require(PerformanceAnalytics)
method = method[1]
VaR.method = VaR.method[1]
bootData = as.matrix(bootData)
w = as.matrix(w)
# determine number of obvs to average around VaR
if (is.null(h)) {
h = round(sqrt(nrow(bootData)))
} else h = round(h)
if ( ncol(bootData) != nrow(w) )
stop("Columns of bootData and rows of w do not match")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
mVaR = bootstrapMarginalVaR(bootData, w, delta.w, h, tail.prob,
method, VaR.method)
cVaR = mVaR * w
return(cVaR)
}
bootstrapVaRreport <- function(bootData, w, delta.w = 0.001, h=NULL, tail.prob = 0.01,
method=c("derivative", "average"),
VaR.method=c("HS", "CornishFisher"),
nav, nav.p, fundName, fundStrategy) {
## compute VaR report for collection of assets in a portfolio given
## simulated (bootstrapped) return data
## Report format follows that of Excel VaR report
## inputs:
## bootData B x n matrix of bootstrap returns on assets in portfolio.
## w n x 1 vector of portfolio weights
## delta.w scalar, change in portfolio weight for computing numerical derivative
## h integer, number of obvs on each side of VaR. Default is h=round(sqrt(B)/2)
## tail.prob scalar tail probability
## method character, method for computing marginal VaR. Valid choices are
## "derivative" for numerical computation of the derivative of portfolio
## VaR wrt fund portfolio weight; "average" for approximating E[Ri | Rp=VaR]
## VaR.method character, method for computing VaR. Valid choices are "HS" for
## historical simulation (empirical quantile); "CornishFisher" for
## modified VaR based on Cornish-Fisher quantile estimate. Cornish-Fisher
## computation is done with the VaR.CornishFisher in the PerformanceAnalytics
## package
## nav n x 1 vector of net asset values in each fund
## nav.p scalar, net asset value of portfolio
## fundName n x 1 vector of fund names
## fundStrategy n x 1 vector of fund strategies
## output:
## VaRreport.df dataframe with n rows and the following columns
## Strategy character strategy group abbreviation
## Net.Asset.Value net asset value
## Allocation asset weights
## Mean mean return
## Std.Dev standard deviation of return
## Asset.Var asset specific VaR values
## cVaR asset specific component VaR values
## cVaR.dollar asset specific component VaR values multiplied by net asset value
## pcVaR asset specific percent contribution to VaR values
## iVaR asset specific incremental VaR values
## iVaR.dollar asset specific incremental VaR values multiplied by net asset value
## mVaR asset specific marginal VaR values
## mVaR.dollar asset specific marginal VaR values multiplied by net asset value
##
## To-do: Add information for cash position.
require(PerformanceAnalytics)
method = method[1]
VaR.method = VaR.method[1]
bootData = as.matrix(bootData)
w = as.matrix(w)
nav = as.matrix(nav)
if ( ncol(bootData) != nrow(w) )
stop("Columns of bootData and rows of w do not match")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
# determine number of obvs to average around VaR
if (is.null(h)) {
h = round(sqrt(nrow(bootData)))
} else h = round(h)
asset.VaR = bootstrapVaR(bootData, tail.prob, VaR.method)
portfolio.VaR = bootstrapPortfolioVaR(bootData, w, tail.prob, VaR.method)
marginal.VaR = bootstrapMarginalVaR(bootData, w, delta.w, h, tail.prob = 0.01,
method, VaR.method)
component.VaR = marginal.VaR*w
incremental.VaR = bootstrapIncrementalVaR(bootData, w, tail.prob, method)
VaRreport.df = data.frame(Strategy = fundStrategy,
Net.Asset.Value = nav,
Allocation = w,
Mean = colMeans(bootData),
Std.Dev = apply(bootData, 2, sd),
Asset.VaR = asset.VaR,
cVaR = component.VaR,
cVaR.dollar = component.VaR*nav.p,
pcVaR = component.VaR/portfolio.VaR,
iVaR = incremental.VaR,
iVaR.dollar = incremental.VaR*nav.p,
mVaR = marginal.VaR,
mVaR.dollar = marginal.VaR*nav.p)
rownames(VaRreport.df) = fundName
return(VaRreport.df)
}
###############################################################################
# R (http://r-project.org/) Econometrics for Performance and Risk Analysis
#
# This file originally written and contributed by Eric Zivot, 2010
# package Copyright (c) 2004-2014 Peter Carl and Brian G. Peterson
#
# This R package is distributed under the terms of the GNU Public License (GPL)
# for full details see the file COPYING
#
# $Id: valueAtRiskFunctions.r 3523 2014-09-04 10:06:12Z braverock $
#
###############################################################################
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
Embedding an R snippet on your website
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.