Author: Marius Hofert, Klaus Herrmann and Mélina Mailhot
Date: 2025-12-10

# Geometric Risk Measures

#### Marius Hofert, Klaus Herrmann and Mélina Mailhot

#### 2025-12-10

    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.

## 1 Geometric VaR and expectile for two sets of confidence levels

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

## 2 Bootstrapped geometric expectiles

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

## 3 Comparison of geometric VaR and expectile for a given direction

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)))))