ReIns (version 1.0.7)

ExcessGPD: Estimates for excess-loss premiums using GPD-MLE estimates

Description

Estimate premiums of excess-loss reinsurance with retention \(R\) and limit \(L\) using GPD-MLE estimates.

Usage

ExcessGPD(data, gamma, sigma, R, L = Inf, warnings = TRUE, plot = TRUE, add = FALSE, 
          main = "Estimates for premium of excess-loss insurance", ...)

Arguments

data

Vector of \(n\) observations.

gamma

Vector of \(n-1\) estimates for the EVI obtained from GPDmle.

sigma

Vector of \(n-1\) estimates for \(\sigma\) obtained from GPDmle.

R

The retention level of the (re-)insurance.

L

The limit of the (re-)insurance, default is Inf.

warnings

Logical indicating if warnings are displayed, default is TRUE.

plot

Logical indicating if the estimates should be plotted as a function of \(k\), default is FALSE.

add

Logical indicating if the estimates should be added to an existing plot, default is FALSE.

main

Title for the plot, default is "Estimates for premium of excess-loss insurance".

…

Additional arguments for the plot function, see plot for more details.

Value

A list with following components:

k

Vector of the values of the tail parameter \(k\).

premium

The corresponding estimates for the premium.

R

The retention level of the (re-)insurance.

L

The limit of the (re-)insurance.

Details

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.

References

Albrecher, H., Beirlant, J. and Teugels, J. (2017). Reinsurance: Actuarial and Statistical Aspects, Wiley, Chichester.

See Also

GPDmle, ExcessHill, ExcessEPD

Examples

# NOT RUN {
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))
# }