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", ...)

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.

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 $σ$ 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.

Author

Tom Reynkens

Details

We need that $u \geq X_{n - k, n}$ , 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 $E (min (X - R)_{+}, L) = Π (R) - Π (R + L)$ where $Π (u) = E ((X - u)_{+}) = \int_{u}^{\infty} (1 - F (z)) d z$ is the premium of the excess-loss insurance with retention $u$ . When $L = \infty$ , the premium is equal to $Π (R)$ .

We estimate $Π$ by $\hat{Π} (u) = (k + 1) / (n + 1) \times {\hat{σ}}_{k} / (1 - {\hat{γ}}_{k}) \times (1 + {\hat{γ}}_{k} / {\hat{σ}}_{k} (u - X_{n - k, n}))^{1 - 1 / {\hat{γ}}_{k}},$ with ${\hat{γ}}_{k}$ and ${\hat{σ}}_{k}$ 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.

Examples

Run this code

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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