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qrmtools (version 0.0-17)

risk_measures: Risk Measures

Description

Computing risk measures.

Usage

## Value-at-risk
VaR_np(x, level, names = FALSE, type = 1, ...)
VaR_t(level, loc = 0, scale = 1, df = Inf)
VaR_t01(level, df = Inf)
VaR_GPD(level, shape, scale)
VaR_Par(level, shape, scale = 1)
VaR_GPDtail(level, threshold, p.exceed, shape, scale)

## Expected shortfall ES_np(x, level, method = c(">", ">="), verbose = FALSE, ...) ES_t(level, loc = 0, scale = 1, df = Inf) ES_t01(level, df = Inf) ES_GPD(level, shape, scale) ES_Par(level, shape, scale = 1) ES_GPDtail(level, threshold, p.exceed, shape, scale)

## Range value-at-risk RVaR_np(x, level, ...)

## Multivariate geometric value-at-risk and expectiles gVaR(x, level, start = colMeans(x), method = if(length(level) == 1) "Brent" else "Nelder-Mead", ...) gEX(x, level, start = colMeans(x), method = if(length(level) == 1) "Brent" else "Nelder-Mead", ...)

Value

VaR_np(), ES_np(), RVaR_np() estimate value-at-risk, expected shortfall and range value-at-risk non-parametrically. For expected shortfall, if method = ">="

(method = ">", the default), losses greater than or equal to (strictly greater than) the nonparametric value-at-risk estimate are averaged; in the former case, there might be no such loss, in which case NaN is returned. For range value-at-risk, losses greater than the nonparametric VaR estimate at level

level[1] and less than or equal to the nonparametric VaR estimate at level level[2] are averaged.

VaR_t(), ES_t() compute value-at-risk and expected shortfall for the \(t\) (or normal) distribution. VaR_t01(),

ES_t01() compute value-at-risk and expected shortfall for the standardized \(t\) (or normal) distribution, so scaled \(t\)

distributions to have mean 0 and variance 1; note that they require a degrees of freedom parameter greater than 2.

VaR_GPD(), ES_GPD() compute value-at-risk and expected shortfall for the generalized Pareto distribution (GPD).

VaR_Par(), ES_Par() compute value-at-risk and expected shortfall for the Pareto distribution.

gVaR(), gEX() compute the multivariate geometric value-at-risk and expectiles suggested by Chaudhuri (1996) and Herrmann et al. (2018), respectively.

Arguments

x

gVaR(), gEX():

matrix of (rowwise) multivariate losses.

VaR_np(), ES_np(), RVaR_np():

if x is a matrix then rowSums() is applied first (so value-at-risk and expected shortfall of the sum is computed).

otherwise:

vector of losses.

level

RVaR_np():

vector of length 1 or 2 giving the lower and upper confidence level; if of length 1, it is interpreted as the lower confidence level and the upper one is taken to be 1.

gVaR(), gEX():

vector or matrix of (rowwise) confidence levels \(\alpha\) (all in \([0,1]\)).

otherwise:

confidence level \(\alpha\in[0,1]\).

names

see ?quantile.

type

see ?quantile.

loc

location parameter \(\mu\).

shape

VaR_GPD(), ES_GPD():

GPD shape parameter \(\xi\), a real number.

VaR_Par(), ES_Par():

Pareto shape parameter \(\theta\), a positive number.

scale

VaR_t(), ES_t():

\(t\) scale parameter \(\sigma\), a positive number.

VaR_GPD(), ES_GPD():

GPD scale parameter \(\beta\), a positive number.

VaR_Par(), ES_Par():

Pareto scale parameter \(\kappa\), a positive number.

df

degrees of freedom, a positive number; choose df = Inf for the normal distribution. For the standardized \(t\) distributions, df has to be greater than 2.

threshold

threhold \(u\) (used to estimate the exceedance probability based on the data x).

p.exceed

exceedance probability; typically mean(x > threshold) for x being the data modeled with the peaks-over-threshold (POT) method.

start

vector of initial values for the underlying optim().

method

ES_np():

character string indicating the method for computing expected shortfall.

gVaR(), gEX():

the optimization method passed to the underlying optim().

verbose

logical indicating whether verbose output is given (in case the mean is computed over (too) few observations).

...

VaR_np():

additional arguments passed to the underlying quantile().

ES_np(), RVaR_np():

additional arguments passed to the underlying VaR_np().

gVaR(), gEX():

additional arguments passed to the underlying optim().

Author

Marius Hofert

Details

The distribution function of the Pareto distribution is given by $$F(x) = 1-(\kappa/(\kappa+x))^{\theta},\ x\ge 0,$$ where \(\theta > 0\), \(\kappa > 0\).

References

McNeil, A. J., Frey, R. and Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques, Tools. Princeton University Press.

Chaudhuri, P. (1996). On a geometric notion of quantiles for multivariate data. Journal of the American Statistical Assosiation 91(434), 862--872.

Herrmann, K., Hofert, M. and Mailhot, M. (2018). Multivariate geometric expectiles. Scandinavian Actuarial Journal, 2018(7), 629--659.

Examples

Run this code
### 1 Univariate measures ######################################################

## Generate some losses and (non-parametrically) estimate VaR_alpha and ES_alpha
set.seed(271)
L <- rlnorm(1000, meanlog = -1, sdlog = 2) # L ~ LN(mu, sig^2)
## Note: - meanlog = mean(log(L)) = mu, sdlog = sd(log(L)) = sig
##       - E(L) = exp(mu + (sig^2)/2), var(L) = (exp(sig^2)-1)*exp(2*mu + sig^2)
##         To obtain a sample with E(L) = a and var(L) = b, use:
##         mu = log(a)-log(1+b/a^2)/2 and sig = sqrt(log(1+b/a^2))
VaR_np(L, level = 0.99)
ES_np(L,  level = 0.99)

## Example 2.16 in McNeil, Frey, Embrechts (2015)
V <- 10000 # value of the portfolio today
sig <- 0.2/sqrt(250) # daily volatility (annualized volatility of 20%)
nu <- 4 # degrees of freedom for the t distribution
alpha <- seq(0.001, 0.999, length.out = 256) # confidence levels
VaRnorm <- VaR_t(alpha, scale = V*sig, df = Inf)
VaRt4 <- VaR_t(alpha, scale = V*sig*sqrt((nu-2)/nu), df = nu)
ESnorm <- ES_t(alpha, scale = V*sig, df = Inf)
ESt4 <- ES_t(alpha, scale = V*sig*sqrt((nu-2)/nu), df = nu)
ran <- range(VaRnorm, VaRt4, ESnorm, ESt4)
plot(alpha, VaRnorm, type = "l", ylim = ran, xlab = expression(alpha), ylab = "")
lines(alpha, VaRt4, col = "royalblue3")
lines(alpha, ESnorm, col = "darkorange2")
lines(alpha, ESt4, col = "maroon3")
legend("bottomright", bty = "n", lty = rep(1,4), col = c("black",
       "royalblue3", "darkorange3", "maroon3"),
       legend = c(expression(VaR[alpha]~~"for normal model"),
                  expression(VaR[alpha]~~"for "*t[4]*" model"),
                  expression(ES[alpha]~~"for normal model"),
                  expression(ES[alpha]~~"for "*t[4]*" model")))


### 2 Multivariate measures ####################################################

## Setup
library(copula)
n <- 1e4 # MC sample size
nu <- 3 # degrees of freedom
th <- iTau(tCopula(df = nu), tau = 0.5) # correlation parameter
cop <- tCopula(param = th, df = nu) # t copula
set.seed(271) # for reproducibility
U <- rCopula(n, cop = cop) # copula sample
theta <- c(2.5, 4) # marginal Pareto parameters
stopifnot(theta > 2) # need finite 2nd moments
X <- sapply(1:2, function(j) qPar(U[,j], shape = theta[j])) # generate X
N <- 17 # number of angles (rather small here because of run time)
phi <- seq(0, 2*pi, length.out = N) # angles
r <- 0.98 # radius
alpha <- r * cbind(alpha1 = cos(phi), alpha2 = sin(phi)) # vector of confidence levels

## Compute geometric value-at-risk
system.time(res <- gVaR(X, level = alpha))
gvar <- t(sapply(seq_len(nrow(alpha)), function(i) {
    x <- res[[i]]
    if(x[["convergence"]] != 0) # 0 = 'converged'
        warning("No convergence for alpha = (", alpha[i,1], ", ", alpha[i,2],
                ") (row ", i, ")")
    x[["par"]]
})) # (N, 2)-matrix

## Compute geometric expectiles
system.time(res <- gEX(X, level = alpha))
gex <- t(sapply(seq_len(nrow(alpha)), function(i) {
    x <- res[[i]]
    if(x[["convergence"]] != 0) # 0 = 'converged'
        warning("No convergence for alpha = (", alpha[i,1], ", ", alpha[i,2],
                ") (row ", i, ")")
    x[["par"]]
})) # (N, 2)-matrix

## Plot geometric VaR and geometric expectiles
plot(gvar, type = "b", xlab = "Component 1 of geometric VaRs and expectiles",
     ylab = "Component 2 of geometric VaRs and expectiles",
     main = "Multivariate geometric VaRs and expectiles")
lines(gex, type = "b", col = "royalblue3")
legend("bottomleft", lty = 1, bty = "n", col = c("black", "royalblue3"),
       legend = c("geom. VaR", "geom. expectile"))
lab <- substitute("MC sample size n ="~n.*","~t[nu.]~"copula with Par("*th1*
                  ") and Par("*th2*") margins",
                  list(n. = n, nu. = nu, th1 = theta[1], th2 = theta[2]))
mtext(lab, side = 4, line = 1, adj = 0)

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