Systemic risk contributions

marginal_expected_shortfall() is each component's mean over the simulations where the total is in its worst 1 - p; it equals allocate(x, distortion("tvar", p)) and adds up to the total's TVaR. esscher_allocation() is each component's mean under the Esscher transform of the total, E[X_j exp(h S)] / E[exp(h S)]; it adds up to esscher_premium(x, h). covar() is the total's VaR at level q over the simulations where one component is at or above its own VaR at p (Adrian and Brunnermeier's CoVaR, in the form of Girardi and Ergun): compare it with VaR(x, q) to see how much that component's bad years drag the portfolio.

Usage

marginal_expected_shortfall(x, p)

esscher_allocation(x, h)

covar(x, key, p, q)

Arguments

x

A predictive_distribution.

p

Level in [0, 1]: the total's tail for marginal_expected_shortfall(), the component's distress for covar().

h

Esscher parameter.

key

The component, a list with one entry per dimension.

q

Level of the total's VaR.

Value

marginal_expected_shortfall() and esscher_allocation(): the keys data frame of x with a contribution column. covar(): a single number.

Examples

pd <- predictive_distribution(
  matrix(c(1, 2, 3, 4, 0, 1, 5, 1), ncol = 2),
  data.frame(lob = c("a", "b"))
)
marginal_expected_shortfall(pd, 0.5)
#>   lob contribution
#> 1   a          3.5
#> 2   b          3.0
esscher_allocation(pd, 0.1)
#>   lob contribution
#> 1   a     2.697995
#> 2   b     2.231890
covar(pd, list(lob = "a"), 0.75, 0.5)
#> [1] 5