Estimate premiums of excess-loss reinsurance with retention \(R\) and limit \(L\) using GPD-MLE estimates.
ExcessGPD(data, gamma, sigma, R, L = Inf, warnings = TRUE, plot = TRUE, add = FALSE,
main = "Estimates for premium of excess-loss insurance", ...)
A list with following components:
Vector of the values of the tail parameter \(k\).
The corresponding estimates for the premium.
The retention level of the (re-)insurance.
The limit of the (re-)insurance.
Vector of \(n\) observations.
Vector of \(n-1\) estimates for the EVI obtained from GPDmle
.
Vector of \(n-1\) estimates for \(\sigma\) obtained from GPDmle
.
The retention level of the (re-)insurance.
The limit of the (re-)insurance, default is Inf
.
Logical indicating if warnings are displayed, default is TRUE
.
Logical indicating if the estimates should be plotted as a function of \(k\), default is FALSE
.
Logical indicating if the estimates should be added to an existing plot, default is FALSE
.
Title for the plot, default is "Estimates for premium of excess-loss insurance"
.
Additional arguments for the plot
function, see plot
for more details.
Tom Reynkens
We need that \(u \ge X_{n-k,n}\), the \((k+1)\)-th largest observation.
If this is not the case, we return NA
for the premium. A warning will be issued in
that case if warnings=TRUE
. One should then use global fits: ExcessSplice
.
The premium for the excess-loss insurance with retention \(R\) and limit \(L\) is given by $$E(\min{(X-R)_+, L}) = \Pi(R) - \Pi(R+L)$$ where \(\Pi(u)=E((X-u)_+)=\int_u^{\infty} (1-F(z)) dz\) is the premium of the excess-loss insurance with retention \(u\). When \(L=\infty\), the premium is equal to \(\Pi(R)\).
We estimate \(\Pi\) by $$ \hat{\Pi}(u) = (k+1)/(n+1) \times \hat{\sigma}_k/ (1-\hat{\gamma}_k) \times (1+\hat{\gamma}_k/\hat{\sigma}_k (u-X_{n-k,n}))^{1-1/\hat{\gamma}_k},$$ with \(\hat{\gamma}_k\) and \(\hat{\sigma}_k\) the estimates for the parameters of the GPD.
See Section 4.6 of Albrecher et al. (2017) for more details.
Albrecher, H., Beirlant, J. and Teugels, J. (2017). Reinsurance: Actuarial and Statistical Aspects, Wiley, Chichester.
GPDmle
, ExcessHill
, ExcessEPD
data(secura)
# GPDmle estimator
mle <- GPDmle(secura$size)
# Premium of excess-loss insurance with retention R
R <- 10^7
ExcessGPD(secura$size, gamma=mle$gamma, sigma=mle$sigma, R=R, ylim=c(0,2*10^4))
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