library(qrmtools) library(copula) library(sn) # for skew-normal distribution library(RColorBrewer) # for Dark2 color palette col <- brewer.pal(8, name = "Dark2")[c(7, 3, 5, 4, 6)] # some colors
In this vignette, we briefly present some examples on geometric value-at-risk (VaR) and geometric expectiles; see Chaudhuri (1996, "On a geometric notion of quantiles for multivariate data") and Herrmann et al. (2017, "Multivariate geometric expectiles"). Geometric VaR and geometric expectile are defined by \begin{align} \mathrm{VaR}{\mathbf{\alpha}}(\mathbf{X})&=\mathrm{argmin}{\mathbf{c}\in\mathbb{R}^d}\ \mathbb{E}(\Phi_{\mathbf{\alpha}}(\mathbf{X}-\mathbf{c}))\quad\text{for}\quad\Phi_{\mathbf{\alpha}}(\mathbf{t})=(\lVert\mathbf{t}\rVert_2+\langle\mathbf{\alpha},\mathbf{t}\rangle)/2,\ \mathrm{e}{\mathbf{\alpha}}(\mathbf{X})&=\mathrm{argmin}{\mathbf{c}\in\mathbb{R}^d}\ \mathbb{E}(\Lambda_{\mathbf{\alpha}}(\mathbf{X}-\mathbf{c}))\quad\text{for}\quad\Lambda_{\mathbf{\alpha}}(\mathbf{t})=\lVert\mathbf{t}\rVert_2(\lVert\mathbf{t}\rVert_2+\langle\mathbf{\alpha},\mathbf{t}\rangle)/2, \end{align} respectively.
Consider the following two sets of confidence levels.
## Setup lphi <- 33 # number of angles phi <- seq(0, 2*pi, length.out = lphi) # equidistant angles circle <- cbind(cos(phi), sin(phi)) # unit circle evaluated at angles a1 <- 0.98 * circle # first set of alphas a2 <- cbind(0.98 * circle[,1], 0.9 * circle[,2]) # other set of alphas
Here is a graphical representation. We also identify two specific points in these sets of confidence levels.
## Plot par(pty = "s") plot(circle, type = "l", lty = 2, lwd = 2, col = gray(0.5), xlab = expression(alpha[1]), ylab = expression(alpha[2]), xlim = c(-1, 1), ylim = c(-1, 1)) # gray circle abline(v = 0, h = 0, lty = 5) # alpha_1 = 0 and alpha_2 = 0 lines(a1[,1], a1[,2], lwd = 2, col = col[1]) # first set of alphas lines(a2[,1], a2[,2], lwd = 2, col = col[2]) # other set of alphas p1 <- 0.98 * c(cos(3/4*pi), sin(3/4*pi)) # location of "1" p2 <- c(0.98 * cos(1/4*pi), 0.90 * sin(1/4*pi)) # location of "2" arrows(0, 0, p1[1], p1[2], lwd = 2, col = col[1]) # arrow to "1" arrows(0, 0, p2[1], p2[2], lwd = 2, col = col[2]) # arrow to "2" points(p1[1], p1[2], pch = "1", cex = 1.2) # "1" points(p2[1], p2[2], pch = "2", cex = 1.2) # "2" points(0, 0, pch = 19, cex = 1.2) # filled dot
Let us generate an example data set based on which we compute geometric VaRs and geometric expectiles (see below) for the two sets of confidence levels. To keep run time small, note that we use rather small sample sizes here (and below).
## Generate a copula sample set.seed(42) # for reproducibility n <- 2e3 # sample size cop <- gumbelCopula(iTau(gumbelCopula(), tau = 0.5)) # Gumbel copula U <- rCopula(n, copula = cop) # sample ## Map to skew-normal and t_4 margins xi <- -1; om <- 1; al <- 2 # parameters for skew-normal margin nu <- 4 # parameter for t margin X <- cbind(qsn(U[,1], xi = xi, omega = om, alpha = al), qt(U[,2], df = nu)) # map to skew-normal and t_4 margins
For getting an intuition for the appearing shapes, we also evaluate the density of $\mathbf{X}$ on a grid.
## Evaluate the joint density (according to Sklar's Theorem) x <- seq(-4, 4, length.out = 64) y <- seq(-4, 4, length.out = 64) dH <- function(x, y) dCopula(cbind(psn(x, xi = xi, omega = om, alpha = al), pt(y, df = nu)), copula = cop) * dsn(x, xi = xi, omega = om, alpha = al) * dt(y, df = nu) h <- outer(x, y, FUN = dH)
Now we can compute geometric VaRs and geometric expectiles for both sets of
confidence levels; note that gVaR()
and gEX()
are matricized in level
.
## Compute geometric VaR and expectile for both sets of indices alpha gVaR.a1 <- matrix(vapply(gVaR(X, level = a1), `[[`, numeric(2), "par"), ncol = 2, byrow = TRUE) gEX.a1 <- matrix(vapply(gEX (X, level = a1), `[[`, numeric(2), "par"), ncol = 2, byrow = TRUE) gVaR.a2 <- matrix(vapply(gVaR(X, level = a2), `[[`, numeric(2), "par"), ncol = 2, byrow = TRUE) gEX.a2 <- matrix(vapply(gEX (X, level = a2), `[[`, numeric(2), "par"), ncol = 2, byrow = TRUE)
We can also compute the geometric VaR and geometric expectile at the
specific points p1
and p2
chosen above.
## Compute geometric VaR and expectile for the points p1 and p2 gVaR.p1 <- gVaR(X, level = p1)$par gEX.p1 <- gEX(X, level = p1)$par gVaR.p2 <- gVaR(X, level = p2)$par gEX.p2 <- gEX(X, level = p2)$par
Finally, let us plot the geometric risk measures.
## Plot par(pty = "s") ran <- range(gVaR.a1, gVaR.a2, gEX.a1, gEX.a2, gVaR.p1, gVaR.p2, gEX.p1, gEX.p2) # determine plotting range plot(NA, type = "l", xlim = ran, ylim = ran, xlab = "Component 1 of geometric VaRs and expectiles", ylab = "Component 2 of geometric VaRs and expectiles") # set up plot region abline(v = 0, h = 0, lty = 5) contour(x, y, h, nlevels = 12, col = gray(0.7), add = TRUE) lines(gVaR.a1, lwd = 2, col = col[1], lty = 2) # geometric VaRs for alpha = a1 lines(gVaR.a2, lwd = 2, col = col[2], lty = 2) # geometric VaRs for alpha = a2 lines(gEX.a1, lwd = 2, col = col[1]) # geometric expectiles for alpha = a1 lines(gEX.a2, lwd = 2, col = col[2]) # geometric expectiles for alpha = a2 points(rbind(gVaR.p1), pch = "1", cex = 1.2) # geometric VaR for alpha = p1 points(rbind(gVaR.p2), pch = "2", cex = 1.2) # geometric VaR for alpha = p2 points(rbind(gEX.p1), pch = "1", cex = 1.2) # geometric expectiles for alpha = p1 points(rbind(gEX.p2), pch = "2", cex = 1.2) # geometric expectiles for alpha = p2 points(rbind(colMeans(X)), pch = 19, cex = 1.2) # filled dot
Revisit the above example and let us focus on geometric expectiles. Since the functional underlying this risk measure is typically evaluated by Monte Carlo, let us generated various samples (instead of just one) and see what kind of ``variation'' we obtain. Note that in this case, we choose a significantly smaller sample size and also only a small number of bootstrap replications in order to keep run time comparably small in this vignette.
## Bootstrap B <- 16 # bootstrap replications n <- 250 # sample size U <- rCopula(B * n, copula = cop) # sample copula B * n times X <- cbind(qsn(U[,1], xi = xi, omega = om, alpha = al), qt(U[,2], df = nu)) # map to skew-normal and t_4 margins res <- lapply(1:B, function(b) { # iterate over 1:B matrix(vapply(gEX(X[250 * (b-1) + (1:250),], level = a1), `[[`, numeric(2), "par"), ncol = 2, byrow = TRUE) # (33, 2)-matrix })
Here are the various geometric expectile curves (each based on the first set of confidence levels considered before).
## Plot par(pty = "s") ran <- range(gEX.a1, res) # determine plotting range plot(NA, type = "l", xlim = ran, ylim = ran, xlab = "Component 1 of geometric expectiles", ylab = "Component 2 of geometric expectiles") # set up plot region abline(v = 0, h = 0, lty = 5) contour(x, y, h, nlevels = 12, col = gray(0.7), add = TRUE) for(b in 1:B) lines(res[[b]], col = adjustcolor(col[1], alpha.f = 10/B)) # bootstrap geom. expectiles lines(gEX.a1, col = col[1]) # "true" geometric expectiles points(rbind(colMeans(X)), pch = 19, cex = 1.2) # filled dot
In this example, let us fix a confidence-level direction.
## Determine alphas u <- c(1, 1) / sqrt(2) # direction n. <- 64 mag <- tail(head(seq(-1, 1, length.out = n.), n = -1), n = -1) # magnitude a <- matrix(mag * rep(u, each = n. - 2), ncol = 2) # alpha
Now compute geometric VaRs and geometric expectiles for these confidence levels.
## Compute geometric VaRs and expectiles for the alphas gVaR.a <- matrix(vapply(gVaR(X, level = a), `[[`, numeric(2), "par"), ncol = 2, byrow = TRUE) gEX.a <- matrix(vapply(gEX (X, level = a), `[[`, numeric(2), "par"), ncol = 2, byrow = TRUE)
Here is a graphical representation.
## Plot of margins of geometric VaRs and expectiles ## Note: bold() does not respect Greek letters yran <- range(gVaR.a, gEX.a) # determine plotting range xlab <- expression("r for"~bold(alpha)~"= r"*bold(u)) # x-axis label plot(mag, gVaR.a[,1], type = "l", lty = 2, ylim = yran, lwd = 2, col = col[1], xlab = xlab, ylab = "Marginal risk measure") # 1st-margin geometric VaRs lines(mag, gVaR.a[,2], lty = 2, lwd = 2, col = col[2]) # 2nd-margin geometric VaRs lines(mag, gEX.a[,1], lwd = 2, col = col[1]) # 1st-margin geometric expectiles lines(mag, gEX.a[,2], lwd = 2, col = col[2]) # 2nd-margin geometric expectiles legend("topleft", bty = "n", lty = c(2, 2, 1, 1), lwd = 2, col = col[c(1, 2, 1, 2)], legend = c(expression("1st component of"~VaR[bold(alpha)](bold(X))), expression("2nd component of"~VaR[bold(alpha)](bold(X))), expression("1st component of"~e[bold(alpha)](bold(X))), expression("2nd component of"~e[bold(alpha)](bold(X)))))
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