VaR_ES_bounds_rearrange: Worst and Best Value-at-Risk and Best Expected Shortfall for...
In qrmtools: Tools for Quantitative Risk Management
VaR_ES_bounds_rearrange R Documentation
Worst and Best Value-at-Risk and Best Expected Shortfall
for Given Marginals via Rearrangements
Description
Compute the worst and best Value-at-Risk (VaR) and the best expected
shortfall (ES) for given marginal distributions via rearrangements.
Usage
## Workhorses
## Column rearrangements
rearrange(X, tol = 0, tol.type = c("relative", "absolute"),
n.lookback = ncol(X), max.ra = Inf,
method = c("worst.VaR", "best.VaR", "best.ES"),
sample = TRUE, is.sorted = FALSE, trace = FALSE, ...)
## Block rearrangements
block_rearrange(X, tol = 0, tol.type = c("absolute", "relative"),
n.lookback = ncol(X), max.ra = Inf,
method = c("worst.VaR", "best.VaR", "best.ES"),
sample = TRUE, trace = FALSE, ...)
## User interfaces
## Rearrangement Algorithm
RA(level, qF, N, abstol = 0, n.lookback = length(qF), max.ra = Inf,
method = c("worst.VaR", "best.VaR", "best.ES"), sample = TRUE)
## Adaptive Rearrangement Algorithm
ARA(level, qF, N.exp = seq(8, 19, by = 1), reltol = c(0, 0.01),
n.lookback = length(qF), max.ra = 10*length(qF),
method = c("worst.VaR", "best.VaR", "best.ES"),
sample = TRUE)
## Adaptive Block Rearrangement Algorithm
ABRA(level, qF, N.exp = seq(8, 19, by = 1), absreltol = c(0, 0.01),
n.lookback = NULL, max.ra = Inf,
method = c("worst.VaR", "best.VaR", "best.ES"),
sample = TRUE)
Arguments
X
(N, d)-matrix of quantiles (to be
rearranged). If is.sorted it is assumed that the columns of
X are sorted in increasing order.
tol
(absolute or relative) tolerance to determine (the
individual) convergence. This should normally be a number
greater than or equal to 0, but rearrange() also allows
for tol = NULL which means that columns are rearranged
until each column is oppositely ordered to the sum of all other
columns.
tol.type
character string indicating the
type of convergence tolerance function to be used ("relative"
for relative tolerance and "absolute" for absolute tolerance).
n.lookback
number of rearrangements to look back for deciding
about numerical convergence. Use this option with care.
max.ra
maximal number of (considered) column rearrangements
of the underlying matrix of quantiles (can be set to Inf).
method
character string indicating whether bounds
for the worst/best VaR or the best ES should be computed.
These bounds are termed \underline{s}_N and \overline{s}_N
in the literature (and below) and are theoretically not guaranteed
bounds of worst/best VaR or best ES;
however, they are treated as such in practice and are typically in
line with results from VaR_bounds_hom() in the homogeneous
case, for example.
sample
logical indicating whether each column of
the two underlying matrices of quantiles (see Step 3 of the Rearrangement
Algorithm in Embrechts et al. (2013))
are randomly permuted before the rearrangements begin. This typically has
quite a positive effect on run time (as most of the time is spent
(oppositely) ordering columns (for rearrange()) or blocks
(for block_rearrange())).
is.sorted
logical indicating whether the columns of
X are sorted in increasing order.
trace
logical indicating whether the underlying
matrix is printed after each rearrangement step. See
vignette("VaR_bounds", package = "qrmtools") for how to interpret
the output.
level
confidence level \alpha for VaR and ES (e.g., 0.99).
qF
d-list containing the marginal quantile functions.
N
number of discretization points.
abstol
absolute convergence tolerance \epsilon
to determine the individual convergence, i.e., the change in the computed
minimal row sums (for method = "worst.VaR") or maximal
row sums (for method = "best.VaR") or expected shortfalls (for
method = "best.ES") for the lower bound \underline{s}_N
and the upper bound \overline{s}_N. abstol is typically
\ge0; it can also be NULL, see tol
above.
N.exp
exponents of the number of discretization points
(a vector) over which the algorithm iterates to find
the smallest number of discretization points for which the desired
accuracy (specified by abstol and reltol) is attained;
for each number of discretization points, at most max.ra-many
column rearrangements are of the underlying matrix of quantiles
are considered.
reltol
vector of length two containing the
individual (first component; used to determine convergence of the
minimal row sums (for method = "worst.VaR") or maximal row
sums (for method = "best.VaR") or expected shortfalls (for
method = "best.ES") for \underline{s}_N and
\overline{s}_N) and the joint (second component; relative
tolerance between the computed \underline{s}_N and
\overline{s}_N with respect to \overline{s}_N) relative
convergence tolerances. reltol can also be of length one
in which case it denotes the joint relative tolerance; the
individual relative tolerance is taken as NULL
(see tol above) in this case.
absreltol
vector of length two containing the
individual (first component; used to determine convergence of the
minimal row sums (for method = "worst.VaR") or maximal row
sums (for method = "best.VaR") or expected shortfalls (for
method = "best.ES") for \underline{s}_N and
\overline{s}_N) absolute and the joint (second component; relative
tolerance between the computed \underline{s}_N and
\overline{s}_N with respect to \overline{s}_N) relative
convergence tolerances. absreltol can also be of length one
in which case it denotes the joint relative tolerance; the
individual absolute tolerance is taken as 0 in this case.
...
additional arguments passed to the underlying
optimization function. Currently, this is only used if
method = "best.ES" in which case the required confidence
level \alpha must be provided as argument level.
Details
rearrange() is an auxiliary
function (workhorse). It is called by RA() and ARA().
After a column rearrangement of X, the tolerance between the
minimal row sum (for the worst VaR) or maximal row sum (for the best
VaR) or expected shortfall (obtained from the row sums; for the best
ES) after this rearrangement and the one of n.lookback
rearrangement steps before is computed and convergence determined.
For performance reasons, no input checking is done for
rearrange() and it can change in future versions to (futher)
improve run time. Overall it should only be used by experts.
block_rearrange(), the workhorse underlying ABRA(),
is similar to rearrange() in that it
checks whether convergence has occurred after every rearrangement by
comparing the change to the row sum variance from n.lookback
rearrangement steps back. block_rearrange() differs from
rearrange in the following ways. First, instead of single columns,
whole (randomly chosen) blocks (two at a time) are chosen and
oppositely ordered. Since some of the ideas for improving the speed of
rearrange() do not carry over to block_rearrange(), the
latter should in general not be as fast as the former.
Second, instead of using minimal or maximal row
sums or expected shortfall to determine numerical convergence,
block_rearrange() uses the variance of the vector of row sums
to determine numerical convergence. By default, it targets a variance
of 0 (which is also why the default tol.type is "absolute").
For the Rearrangement Algorithm RA(), convergence of
\underline{s}_N and \overline{s}_N is determined if the
minimal row sum (for the worst VaR) or maximal row sum (for the best
VaR) or expected shortfall (obtained from the row sums; for the best ES)
satisfies the specified abstol (so \le\epsilon)
after at most max.ra-many column rearrangements. This is different
from Embrechts et al. (2013) who use <\epsilon and
only check for convergence after an iteration through all
columns of the underlying matrix of quantiles has been completed.
For the Adaptive Rearrangement Algorithm ARA()
and the Adaptive Block Rearrangement Algorithm ABRA(),
convergence of \underline{s}_N and \overline{s}_N
is determined if, after at most max.ra-many column
rearrangements, the (the individual relative tolerance)
reltol[1] is satisfied and the
relative (joint) tolerance between both bounds is at most reltol[2].
Note that RA(), ARA() and ABRA() need to evalute the
0-quantile (for the lower bound for the best VaR) and
the 1-quantile (for the upper bound for the
worst VaR). As the algorithms, due to performance reasons, can only
handle finite values, the 0-quantile and the 1-quantile need to be
adjusted if infinite. Instead of the 0-quantile,
the \alpha/(2N)-quantile is
computed and instead of the 1-quantile the
\alpha+(1-\alpha)(1-1/(2N))-quantile
is computed for such margins (if the 0-quantile or the 1-quantile is
finite, no adjustment is made).
rearrange(), block_rearrange(), RA(), ARA()
and ABRA() compute \underline{s}_N and
\overline{s}_N which are, from a practical
point of view, treated as bounds for the worst (i.e., largest) or the
best (i.e., smallest) VaR or the best (i.e., smallest ES), but which are
not known to be such bounds from a theoretical point of view; see also above.
Calling them “bounds” for worst/best VaR or best ES is thus
theoretically not correct (unless proven) but “practical”.
The literature thus speaks of (\underline{s}_N, \overline{s}_N) as
the rearrangement gap.
Value
rearrange() and block_rearrange() return a
list containing
bound:computed \underline{s}_N
or \overline{s}_N.
tol:reached tolerance (i.e., the (absolute or
relative) change of the minimal row sum (for
method = "worst.VaR") or maximal row sum
(for method = "best.VaR") or expected shortfall (for
method = "best.ES") after the last rearrangement).
converged:logical indicating whether
the desired (absolute or relative) tolerance tol has been
reached.
opt.row.sums:vector containing the
computed optima (minima for method = "worst.VaR"; maxima
for method = "best.VaR"; expected shortfalls for
method = "best.ES") for the row sums after each (considered)
rearrangement.
X.rearranged:(N, d)-matrix
containing the rearranged X.
X.rearranged.opt.row:vector containing
the row of X.rearranged which leads to the final optimal
sum. If there is more than one such row, the columnwise averaged
row is returned.
RA() returns a list containing
bounds:bivariate vector containing the computed
\underline{s}_N and \overline{s}_N (the so-called
rearrangement range) which are typically treated as bounds for
worst/best VaR or best ES; see also above.
rel.ra.gap:reached relative tolerance (also known as
relative rearrangement gap) between
\underline{s}_N and \overline{s}_N computed with
respect to \overline{s}_N.
ind.abs.tol:bivariate vector containing
the reached individual absolute tolerances (i.e., the absolute change
of the minimal row sums
(for method = "worst.VaR") or maximal row sums
(for method = "best.VaR") or expected shortfalls
(for mehtod = "best.ES") for computing \underline{s}_N
and \overline{s}_N;
see also tol returned by rearrange() above).
converged:bivariate logical vector
indicating convergence of the computed \underline{s}_N and
\overline{s}_N (i.e., whether the desired tolerances were
reached).
num.ra:bivariate vector containing the number
of column rearrangments of the underlying matrices
of quantiles for
\underline{s}_N and \overline{s}_N.
opt.row.sums:list of length two containing
the computed optima (minima for method = "worst.VaR"; maxima
for method = "best.VaR"; expected shortfalls for
method = "best.ES") for the row sums after each
(considered) column rearrangement for the computed
\underline{s}_N and \overline{s}_N; see also
rearrange().
X:initially constructed (N, d)-matrices
of quantiles for computing
\underline{s}_N and \overline{s}_N.
X.rearranged:rearranged matrices X for
\underline{s}_N and \overline{s}_N.
X.rearranged.opt.row:rows corresponding to optimal
row sum (see X.rearranged.opt.row as returned by
rearrange()) for \underline{s}_N and \overline{s}_N.
ARA() and ABRA() return a list containing
bounds:see RA().
rel.ra.gap:see RA().
tol:trivariate vector containing
the reached individual (relative for ARA(); absolute for
ABRA()) tolerances and the reached joint
relative tolerance (computed with respect to \overline{s}_N).
converged:trivariate logical
vector indicating individual convergence of the computed
\underline{s}_N (first entry) and \overline{s}_N
(second entry) and indicating joint convergence of the two bounds
according to the attained joint relative tolerance (third entry).
N.used:actual N used for computing
the (final) \underline{s}_N and \overline{s}_N.
num.ra:see RA(); computed for N.used.
opt.row.sums:see RA(); computed for N.used.
X:see RA(); computed for N.used.
X.rearranged:see RA(); computed for N.used.
X.rearranged.opt.row:see RA(); computed for N.used.
Author(s)
Marius Hofert
References
Embrechts, P., Puccetti, G., Rüschendorf, L.,
Wang, R. and Beleraj, A. (2014). An Academic Response to Basel
3.5. Risks 2(1), 25–48.
Embrechts, P., Puccetti, G. and Rüschendorf, L. (2013).
Model uncertainty and VaR aggregation. Journal of Banking &
Finance 37, 2750–2764.
McNeil, A. J., Frey, R. and Embrechts, P. (2015).
Quantitative Risk Management: Concepts, Techniques, Tools.
Princeton University Press.
Hofert, M., Memartoluie, A., Saunders, D. and Wirjanto, T. (2017).
Improved Algorithms for Computing Worst
Value-at-Risk. Statistics & Risk Modeling
or, for an earlier version, https://arxiv.org/abs/1505.02281.
Bernard, C., Rüschendorf, L. and Vanduffel,
S. (2013). Value-at-Risk bounds with variance constraints.
See https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2342068.
Bernard, C. and McLeish, D. (2014). Algorithms for Finding Copulas
Minimizing Convex Functions of Sums.
See https://arxiv.org/abs/1502.02130v3.
See Also
VaR_bounds_hom() for an “analytical” approach for
computing best and worst Value-at-Risk in the homogeneous casse.
vignette("VaR_bounds", package = "qrmtools")
for more example calls, numerical challenges
encoutered and a comparison of the different methods for computing
the worst (i.e., largest) Value-at-Risk.
Examples
### 1 Reproducing selected examples of McNeil et al. (2015; Table 8.1) #########
## Setup
alpha <- 0.95
d <- 8
theta <- 3
qF <- rep(list(function(p) qPar(p, shape = theta)), d)
## Worst VaR
N <- 5e4
set.seed(271)
system.time(RA.worst.VaR <- RA(alpha, qF = qF, N = N, method = "worst.VaR"))
RA.worst.VaR$bounds
stopifnot(RA.worst.VaR$converged,
all.equal(RA.worst.VaR$bounds[["low"]],
RA.worst.VaR$bounds[["up"]], tol = 1e-4))
## Best VaR
N <- 5e4
set.seed(271)
system.time(RA.best.VaR <- RA(alpha, qF = qF, N = N, method = "best.VaR"))
RA.best.VaR$bounds
stopifnot(RA.best.VaR$converged,
all.equal(RA.best.VaR$bounds[["low"]],
RA.best.VaR$bounds[["up"]], tol = 1e-4))
## Best ES
N <- 5e4 # actually, we need a (much larger) N here (but that's time consuming)
set.seed(271)
system.time(RA.best.ES <- RA(alpha, qF = qF, N = N, method = "best.ES"))
RA.best.ES$bounds
stopifnot(RA.best.ES$converged,
all.equal(RA.best.ES$bounds[["low"]],
RA.best.ES$bounds[["up"]], tol = 5e-1))
### 2 More Pareto examples (d = 2, d = 8; hom./inhom. case; explicit/RA/ARA) ###
alpha <- 0.99 # VaR confidence level
th <- 2 # Pareto parameter theta
qF <- function(p, theta = th) qPar(p, shape = theta) # Pareto quantile function
pF <- function(q, theta = th) pPar(q, shape = theta) # Pareto distribution function
### 2.1 The case d = 2 #########################################################
d <- 2 # dimension
## ``Analytical''
VaRbounds <- VaR_bounds_hom(alpha, d = d, qF = qF) # (best VaR, worst VaR)
## Adaptive Rearrangement Algorithm (ARA)
set.seed(271) # set seed (for reproducibility)
ARAbest <- ARA(alpha, qF = rep(list(qF), d), method = "best.VaR")
ARAworst <- ARA(alpha, qF = rep(list(qF), d))
## Rearrangement Algorithm (RA) with N as in ARA()
RAbest <- RA(alpha, qF = rep(list(qF), d), N = ARAbest$N.used, method = "best.VaR")
RAworst <- RA(alpha, qF = rep(list(qF), d), N = ARAworst$N.used)
## Compare
stopifnot(all.equal(c(ARAbest$bounds[1], ARAbest$bounds[2],
RAbest$bounds[1], RAbest$bounds[2]),
rep(VaRbounds[1], 4), tolerance = 0.004, check.names = FALSE))
stopifnot(all.equal(c(ARAworst$bounds[1], ARAworst$bounds[2],
RAworst$bounds[1], RAworst$bounds[2]),
rep(VaRbounds[2], 4), tolerance = 0.003, check.names = FALSE))
### 2.2 The case d = 8 #########################################################
d <- 8 # dimension
## ``Analytical''
I <- crude_VaR_bounds(alpha, qF = qF, d = d) # crude bound
VaR.W <- VaR_bounds_hom(alpha, d = d, method = "Wang", qF = qF)
VaR.W.Par <- VaR_bounds_hom(alpha, d = d, method = "Wang.Par", shape = th)
VaR.dual <- VaR_bounds_hom(alpha, d = d, method = "dual", interval = I, pF = pF)
## Adaptive Rearrangement Algorithm (ARA) (with different relative tolerances)
set.seed(271) # set seed (for reproducibility)
ARAbest <- ARA(alpha, qF = rep(list(qF), d), reltol = c(0.001, 0.01), method = "best.VaR")
ARAworst <- ARA(alpha, qF = rep(list(qF), d), reltol = c(0.001, 0.01))
## Rearrangement Algorithm (RA) with N as in ARA and abstol (roughly) chosen as in ARA
RAbest <- RA(alpha, qF = rep(list(qF), d), N = ARAbest$N.used,
abstol = mean(tail(abs(diff(ARAbest$opt.row.sums$low)), n = 1),
tail(abs(diff(ARAbest$opt.row.sums$up)), n = 1)),
method = "best.VaR")
RAworst <- RA(alpha, qF = rep(list(qF), d), N = ARAworst$N.used,
abstol = mean(tail(abs(diff(ARAworst$opt.row.sums$low)), n = 1),
tail(abs(diff(ARAworst$opt.row.sums$up)), n = 1)))
## Compare
stopifnot(all.equal(c(VaR.W[1], ARAbest$bounds, RAbest$bounds),
rep(VaR.W.Par[1],5), tolerance = 0.004, check.names = FALSE))
stopifnot(all.equal(c(VaR.W[2], VaR.dual[2], ARAworst$bounds, RAworst$bounds),
rep(VaR.W.Par[2],6), tolerance = 0.003, check.names = FALSE))
## Using (some of) the additional results computed by (A)RA()
xlim <- c(1, max(sapply(RAworst$opt.row.sums, length)))
ylim <- range(RAworst$opt.row.sums)
plot(RAworst$opt.row.sums[[2]], type = "l", xlim = xlim, ylim = ylim,
xlab = "Number or rearranged columns",
ylab = paste0("Minimal row sum per rearranged column"),
main = substitute("Worst VaR minimal row sums ("*alpha==a.*","~d==d.*" and Par("*
th.*"))", list(a. = alpha, d. = d, th. = th)))
lines(1:length(RAworst$opt.row.sums[[1]]), RAworst$opt.row.sums[[1]], col = "royalblue3")
legend("bottomright", bty = "n", lty = rep(1,2),
col = c("black", "royalblue3"), legend = c("upper bound", "lower bound"))
## => One should use ARA() instead of RA()
### 3 "Reproducing" examples from Embrechts et al. (2013) ######################
### 3.1 "Reproducing" Table 1 (but seed and eps are unknown) ###################
## Left-hand side of Table 1
N <- 50
d <- 3
qPar <- rep(list(qF), d)
p <- alpha + (1-alpha)*(0:(N-1))/N # for 'worst' (= largest) VaR
X <- sapply(qPar, function(qF) qF(p))
cbind(X, rowSums(X))
## Right-hand side of Table 1
set.seed(271)
res <- RA(alpha, qF = qPar, N = N)
row.sum <- rowSums(res$X.rearranged$low)
cbind(res$X.rearranged$low, row.sum)[order(row.sum),]
### 3.2 "Reproducing" Table 3 for alpha = 0.99 #################################
## Note: The seed for obtaining the exact results as in Table 3 is unknown
N <- 2e4 # we use a smaller N here to save run time
eps <- 0.1 # absolute tolerance
xi <- c(1.19, 1.17, 1.01, 1.39, 1.23, 1.22, 0.85, 0.98)
beta <- c(774, 254, 233, 412, 107, 243, 314, 124)
qF.lst <- lapply(1:8, function(j){ function(p) qGPD(p, shape = xi[j], scale = beta[j])})
set.seed(271)
res.best <- RA(0.99, qF = qF.lst, N = N, abstol = eps, method = "best.VaR")
print(format(res.best$bounds, scientific = TRUE), quote = FALSE) # close to first value of 1st row
res.worst <- RA(0.99, qF = qF.lst, N = N, abstol = eps)
print(format(res.worst$bounds, scientific = TRUE), quote = FALSE) # close to last value of 1st row
### 4 Further checks ###########################################################
## Calling the workhorses directly
set.seed(271)
ra <- rearrange(X)
bra <- block_rearrange(X)
stopifnot(ra$converged, bra$converged,
all.equal(ra$bound, bra$bound, tolerance = 6e-3))
## Checking ABRA against ARA
set.seed(271)
ara <- ARA (alpha, qF = qPar)
abra <- ABRA(alpha, qF = qPar)
stopifnot(ara$converged, abra$converged,
all.equal(ara$bound[["low"]], abra$bound[["low"]], tolerance = 2e-3),
all.equal(ara$bound[["up"]], abra$bound[["up"]], tolerance = 6e-3))
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| VaR_ES_bounds_rearrange | R Documentation |
Worst and Best Value-at-Risk and Best Expected Shortfall for Given Marginals via Rearrangements
Description
Compute the worst and best Value-at-Risk (VaR) and the best expected shortfall (ES) for given marginal distributions via rearrangements.
Usage
## Workhorses
## Column rearrangements
rearrange(X, tol = 0, tol.type = c("relative", "absolute"),
n.lookback = ncol(X), max.ra = Inf,
method = c("worst.VaR", "best.VaR", "best.ES"),
sample = TRUE, is.sorted = FALSE, trace = FALSE, ...)
## Block rearrangements
block_rearrange(X, tol = 0, tol.type = c("absolute", "relative"),
n.lookback = ncol(X), max.ra = Inf,
method = c("worst.VaR", "best.VaR", "best.ES"),
sample = TRUE, trace = FALSE, ...)
## User interfaces
## Rearrangement Algorithm
RA(level, qF, N, abstol = 0, n.lookback = length(qF), max.ra = Inf,
method = c("worst.VaR", "best.VaR", "best.ES"), sample = TRUE)
## Adaptive Rearrangement Algorithm
ARA(level, qF, N.exp = seq(8, 19, by = 1), reltol = c(0, 0.01),
n.lookback = length(qF), max.ra = 10*length(qF),
method = c("worst.VaR", "best.VaR", "best.ES"),
sample = TRUE)
## Adaptive Block Rearrangement Algorithm
ABRA(level, qF, N.exp = seq(8, 19, by = 1), absreltol = c(0, 0.01),
n.lookback = NULL, max.ra = Inf,
method = c("worst.VaR", "best.VaR", "best.ES"),
sample = TRUE)
Arguments
X |
( |
tol |
(absolute or relative) tolerance to determine (the
individual) convergence. This should normally be a number
greater than or equal to 0, but |
tol.type |
|
n.lookback |
number of rearrangements to look back for deciding about numerical convergence. Use this option with care. |
max.ra |
maximal number of (considered) column rearrangements
of the underlying matrix of quantiles (can be set to |
method |
|
sample |
|
is.sorted |
|
trace |
|
level |
confidence level |
qF |
|
N |
number of discretization points. |
abstol |
absolute convergence tolerance |
N.exp |
exponents of the number of discretization points
(a |
reltol |
|
absreltol |
|
... |
additional arguments passed to the underlying
optimization function. Currently, this is only used if
|
Details
rearrange() is an auxiliary
function (workhorse). It is called by RA() and ARA().
After a column rearrangement of X, the tolerance between the
minimal row sum (for the worst VaR) or maximal row sum (for the best
VaR) or expected shortfall (obtained from the row sums; for the best
ES) after this rearrangement and the one of n.lookback
rearrangement steps before is computed and convergence determined.
For performance reasons, no input checking is done for
rearrange() and it can change in future versions to (futher)
improve run time. Overall it should only be used by experts.
block_rearrange(), the workhorse underlying ABRA(),
is similar to rearrange() in that it
checks whether convergence has occurred after every rearrangement by
comparing the change to the row sum variance from n.lookback
rearrangement steps back. block_rearrange() differs from
rearrange in the following ways. First, instead of single columns,
whole (randomly chosen) blocks (two at a time) are chosen and
oppositely ordered. Since some of the ideas for improving the speed of
rearrange() do not carry over to block_rearrange(), the
latter should in general not be as fast as the former.
Second, instead of using minimal or maximal row
sums or expected shortfall to determine numerical convergence,
block_rearrange() uses the variance of the vector of row sums
to determine numerical convergence. By default, it targets a variance
of 0 (which is also why the default tol.type is "absolute").
For the Rearrangement Algorithm RA(), convergence of
\underline{s}_N and \overline{s}_N is determined if the
minimal row sum (for the worst VaR) or maximal row sum (for the best
VaR) or expected shortfall (obtained from the row sums; for the best ES)
satisfies the specified abstol (so \le\epsilon)
after at most max.ra-many column rearrangements. This is different
from Embrechts et al. (2013) who use <\epsilon and
only check for convergence after an iteration through all
columns of the underlying matrix of quantiles has been completed.
For the Adaptive Rearrangement Algorithm ARA()
and the Adaptive Block Rearrangement Algorithm ABRA(),
convergence of \underline{s}_N and \overline{s}_N
is determined if, after at most max.ra-many column
rearrangements, the (the individual relative tolerance)
reltol[1] is satisfied and the
relative (joint) tolerance between both bounds is at most reltol[2].
Note that RA(), ARA() and ABRA() need to evalute the
0-quantile (for the lower bound for the best VaR) and
the 1-quantile (for the upper bound for the
worst VaR). As the algorithms, due to performance reasons, can only
handle finite values, the 0-quantile and the 1-quantile need to be
adjusted if infinite. Instead of the 0-quantile,
the \alpha/(2N)-quantile is
computed and instead of the 1-quantile the
\alpha+(1-\alpha)(1-1/(2N))-quantile
is computed for such margins (if the 0-quantile or the 1-quantile is
finite, no adjustment is made).
rearrange(), block_rearrange(), RA(), ARA()
and ABRA() compute \underline{s}_N and
\overline{s}_N which are, from a practical
point of view, treated as bounds for the worst (i.e., largest) or the
best (i.e., smallest) VaR or the best (i.e., smallest ES), but which are
not known to be such bounds from a theoretical point of view; see also above.
Calling them “bounds” for worst/best VaR or best ES is thus
theoretically not correct (unless proven) but “practical”.
The literature thus speaks of (\underline{s}_N, \overline{s}_N) as
the rearrangement gap.
Value
rearrange() and block_rearrange() return a
list containing
bound:computed
\underline{s}_Nor\overline{s}_N.tol:reached tolerance (i.e., the (absolute or relative) change of the minimal row sum (for
method = "worst.VaR") or maximal row sum (formethod = "best.VaR") or expected shortfall (formethod = "best.ES") after the last rearrangement).converged:logicalindicating whether the desired (absolute or relative) tolerancetolhas been reached.opt.row.sums:vectorcontaining the computed optima (minima formethod = "worst.VaR"; maxima formethod = "best.VaR"; expected shortfalls formethod = "best.ES") for the row sums after each (considered) rearrangement.X.rearranged:(
N,d)-matrixcontaining the rearrangedX.X.rearranged.opt.row:vectorcontaining the row ofX.rearrangedwhich leads to the final optimal sum. If there is more than one such row, the columnwise averaged row is returned.
RA() returns a list containing
bounds:bivariate vector containing the computed
\underline{s}_Nand\overline{s}_N(the so-called rearrangement range) which are typically treated as bounds for worst/best VaR or best ES; see also above.rel.ra.gap:reached relative tolerance (also known as relative rearrangement gap) between
\underline{s}_Nand\overline{s}_Ncomputed with respect to\overline{s}_N.ind.abs.tol:bivariate
vectorcontaining the reached individual absolute tolerances (i.e., the absolute change of the minimal row sums (formethod = "worst.VaR") or maximal row sums (formethod = "best.VaR") or expected shortfalls (formehtod = "best.ES") for computing\underline{s}_Nand\overline{s}_N; see alsotolreturned byrearrange()above).converged:bivariate
logicalvector indicating convergence of the computed\underline{s}_Nand\overline{s}_N(i.e., whether the desired tolerances were reached).num.ra:bivariate vector containing the number of column rearrangments of the underlying matrices of quantiles for
\underline{s}_Nand\overline{s}_N.opt.row.sums:listof length two containing the computed optima (minima formethod = "worst.VaR"; maxima formethod = "best.VaR"; expected shortfalls formethod = "best.ES") for the row sums after each (considered) column rearrangement for the computed\underline{s}_Nand\overline{s}_N; see alsorearrange().X:initially constructed (
N,d)-matrices of quantiles for computing\underline{s}_Nand\overline{s}_N.X.rearranged:rearranged matrices
Xfor\underline{s}_Nand\overline{s}_N.X.rearranged.opt.row:rows corresponding to optimal row sum (see
X.rearranged.opt.rowas returned byrearrange()) for\underline{s}_Nand\overline{s}_N.
ARA() and ABRA() return a list containing
bounds:see
RA().rel.ra.gap:see
RA().tol:trivariate
vectorcontaining the reached individual (relative forARA(); absolute forABRA()) tolerances and the reached joint relative tolerance (computed with respect to\overline{s}_N).converged:trivariate
logicalvectorindicating individual convergence of the computed\underline{s}_N(first entry) and\overline{s}_N(second entry) and indicating joint convergence of the two bounds according to the attained joint relative tolerance (third entry).N.used:actual
Nused for computing the (final)\underline{s}_Nand\overline{s}_N.num.ra:see
RA(); computed forN.used.opt.row.sums:see
RA(); computed forN.used.X:see
RA(); computed forN.used.X.rearranged:see
RA(); computed forN.used.X.rearranged.opt.row:see
RA(); computed forN.used.
Author(s)
Marius Hofert
References
Embrechts, P., Puccetti, G., Rüschendorf, L., Wang, R. and Beleraj, A. (2014). An Academic Response to Basel 3.5. Risks 2(1), 25–48.
Embrechts, P., Puccetti, G. and Rüschendorf, L. (2013). Model uncertainty and VaR aggregation. Journal of Banking & Finance 37, 2750–2764.
McNeil, A. J., Frey, R. and Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques, Tools. Princeton University Press.
Hofert, M., Memartoluie, A., Saunders, D. and Wirjanto, T. (2017). Improved Algorithms for Computing Worst Value-at-Risk. Statistics & Risk Modeling or, for an earlier version, https://arxiv.org/abs/1505.02281.
Bernard, C., Rüschendorf, L. and Vanduffel, S. (2013). Value-at-Risk bounds with variance constraints. See https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2342068.
Bernard, C. and McLeish, D. (2014). Algorithms for Finding Copulas Minimizing Convex Functions of Sums. See https://arxiv.org/abs/1502.02130v3.
See Also
VaR_bounds_hom() for an “analytical” approach for
computing best and worst Value-at-Risk in the homogeneous casse.
vignette("VaR_bounds", package = "qrmtools")
for more example calls, numerical challenges
encoutered and a comparison of the different methods for computing
the worst (i.e., largest) Value-at-Risk.
Examples
### 1 Reproducing selected examples of McNeil et al. (2015; Table 8.1) #########
## Setup
alpha <- 0.95
d <- 8
theta <- 3
qF <- rep(list(function(p) qPar(p, shape = theta)), d)
## Worst VaR
N <- 5e4
set.seed(271)
system.time(RA.worst.VaR <- RA(alpha, qF = qF, N = N, method = "worst.VaR"))
RA.worst.VaR$bounds
stopifnot(RA.worst.VaR$converged,
all.equal(RA.worst.VaR$bounds[["low"]],
RA.worst.VaR$bounds[["up"]], tol = 1e-4))
## Best VaR
N <- 5e4
set.seed(271)
system.time(RA.best.VaR <- RA(alpha, qF = qF, N = N, method = "best.VaR"))
RA.best.VaR$bounds
stopifnot(RA.best.VaR$converged,
all.equal(RA.best.VaR$bounds[["low"]],
RA.best.VaR$bounds[["up"]], tol = 1e-4))
## Best ES
N <- 5e4 # actually, we need a (much larger) N here (but that's time consuming)
set.seed(271)
system.time(RA.best.ES <- RA(alpha, qF = qF, N = N, method = "best.ES"))
RA.best.ES$bounds
stopifnot(RA.best.ES$converged,
all.equal(RA.best.ES$bounds[["low"]],
RA.best.ES$bounds[["up"]], tol = 5e-1))
### 2 More Pareto examples (d = 2, d = 8; hom./inhom. case; explicit/RA/ARA) ###
alpha <- 0.99 # VaR confidence level
th <- 2 # Pareto parameter theta
qF <- function(p, theta = th) qPar(p, shape = theta) # Pareto quantile function
pF <- function(q, theta = th) pPar(q, shape = theta) # Pareto distribution function
### 2.1 The case d = 2 #########################################################
d <- 2 # dimension
## ``Analytical''
VaRbounds <- VaR_bounds_hom(alpha, d = d, qF = qF) # (best VaR, worst VaR)
## Adaptive Rearrangement Algorithm (ARA)
set.seed(271) # set seed (for reproducibility)
ARAbest <- ARA(alpha, qF = rep(list(qF), d), method = "best.VaR")
ARAworst <- ARA(alpha, qF = rep(list(qF), d))
## Rearrangement Algorithm (RA) with N as in ARA()
RAbest <- RA(alpha, qF = rep(list(qF), d), N = ARAbest$N.used, method = "best.VaR")
RAworst <- RA(alpha, qF = rep(list(qF), d), N = ARAworst$N.used)
## Compare
stopifnot(all.equal(c(ARAbest$bounds[1], ARAbest$bounds[2],
RAbest$bounds[1], RAbest$bounds[2]),
rep(VaRbounds[1], 4), tolerance = 0.004, check.names = FALSE))
stopifnot(all.equal(c(ARAworst$bounds[1], ARAworst$bounds[2],
RAworst$bounds[1], RAworst$bounds[2]),
rep(VaRbounds[2], 4), tolerance = 0.003, check.names = FALSE))
### 2.2 The case d = 8 #########################################################
d <- 8 # dimension
## ``Analytical''
I <- crude_VaR_bounds(alpha, qF = qF, d = d) # crude bound
VaR.W <- VaR_bounds_hom(alpha, d = d, method = "Wang", qF = qF)
VaR.W.Par <- VaR_bounds_hom(alpha, d = d, method = "Wang.Par", shape = th)
VaR.dual <- VaR_bounds_hom(alpha, d = d, method = "dual", interval = I, pF = pF)
## Adaptive Rearrangement Algorithm (ARA) (with different relative tolerances)
set.seed(271) # set seed (for reproducibility)
ARAbest <- ARA(alpha, qF = rep(list(qF), d), reltol = c(0.001, 0.01), method = "best.VaR")
ARAworst <- ARA(alpha, qF = rep(list(qF), d), reltol = c(0.001, 0.01))
## Rearrangement Algorithm (RA) with N as in ARA and abstol (roughly) chosen as in ARA
RAbest <- RA(alpha, qF = rep(list(qF), d), N = ARAbest$N.used,
abstol = mean(tail(abs(diff(ARAbest$opt.row.sums$low)), n = 1),
tail(abs(diff(ARAbest$opt.row.sums$up)), n = 1)),
method = "best.VaR")
RAworst <- RA(alpha, qF = rep(list(qF), d), N = ARAworst$N.used,
abstol = mean(tail(abs(diff(ARAworst$opt.row.sums$low)), n = 1),
tail(abs(diff(ARAworst$opt.row.sums$up)), n = 1)))
## Compare
stopifnot(all.equal(c(VaR.W[1], ARAbest$bounds, RAbest$bounds),
rep(VaR.W.Par[1],5), tolerance = 0.004, check.names = FALSE))
stopifnot(all.equal(c(VaR.W[2], VaR.dual[2], ARAworst$bounds, RAworst$bounds),
rep(VaR.W.Par[2],6), tolerance = 0.003, check.names = FALSE))
## Using (some of) the additional results computed by (A)RA()
xlim <- c(1, max(sapply(RAworst$opt.row.sums, length)))
ylim <- range(RAworst$opt.row.sums)
plot(RAworst$opt.row.sums[[2]], type = "l", xlim = xlim, ylim = ylim,
xlab = "Number or rearranged columns",
ylab = paste0("Minimal row sum per rearranged column"),
main = substitute("Worst VaR minimal row sums ("*alpha==a.*","~d==d.*" and Par("*
th.*"))", list(a. = alpha, d. = d, th. = th)))
lines(1:length(RAworst$opt.row.sums[[1]]), RAworst$opt.row.sums[[1]], col = "royalblue3")
legend("bottomright", bty = "n", lty = rep(1,2),
col = c("black", "royalblue3"), legend = c("upper bound", "lower bound"))
## => One should use ARA() instead of RA()
### 3 "Reproducing" examples from Embrechts et al. (2013) ######################
### 3.1 "Reproducing" Table 1 (but seed and eps are unknown) ###################
## Left-hand side of Table 1
N <- 50
d <- 3
qPar <- rep(list(qF), d)
p <- alpha + (1-alpha)*(0:(N-1))/N # for 'worst' (= largest) VaR
X <- sapply(qPar, function(qF) qF(p))
cbind(X, rowSums(X))
## Right-hand side of Table 1
set.seed(271)
res <- RA(alpha, qF = qPar, N = N)
row.sum <- rowSums(res$X.rearranged$low)
cbind(res$X.rearranged$low, row.sum)[order(row.sum),]
### 3.2 "Reproducing" Table 3 for alpha = 0.99 #################################
## Note: The seed for obtaining the exact results as in Table 3 is unknown
N <- 2e4 # we use a smaller N here to save run time
eps <- 0.1 # absolute tolerance
xi <- c(1.19, 1.17, 1.01, 1.39, 1.23, 1.22, 0.85, 0.98)
beta <- c(774, 254, 233, 412, 107, 243, 314, 124)
qF.lst <- lapply(1:8, function(j){ function(p) qGPD(p, shape = xi[j], scale = beta[j])})
set.seed(271)
res.best <- RA(0.99, qF = qF.lst, N = N, abstol = eps, method = "best.VaR")
print(format(res.best$bounds, scientific = TRUE), quote = FALSE) # close to first value of 1st row
res.worst <- RA(0.99, qF = qF.lst, N = N, abstol = eps)
print(format(res.worst$bounds, scientific = TRUE), quote = FALSE) # close to last value of 1st row
### 4 Further checks ###########################################################
## Calling the workhorses directly
set.seed(271)
ra <- rearrange(X)
bra <- block_rearrange(X)
stopifnot(ra$converged, bra$converged,
all.equal(ra$bound, bra$bound, tolerance = 6e-3))
## Checking ABRA against ARA
set.seed(271)
ara <- ARA (alpha, qF = qPar)
abra <- ABRA(alpha, qF = qPar)
stopifnot(ara$converged, abra$converged,
all.equal(ara$bound[["low"]], abra$bound[["low"]], tolerance = 2e-3),
all.equal(ara$bound[["up"]], abra$bound[["up"]], tolerance = 6e-3))
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