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 $\sigma$ 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.

Author

Tom Reynkens

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.

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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