Estimates for excess-loss premiums using GPD-MLE estimates
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", ...)data |
Vector of n observations. |
gamma |
Vector of n-1 estimates for the EVI obtained from |
sigma |
Vector of n-1 estimates for σ obtained from |
R |
The retention level of the (re-)insurance. |
L |
The limit of the (re-)insurance, default is |
warnings |
Logical indicating if warnings are displayed, default is |
plot |
Logical indicating if the estimates should be plotted as a function of k, default is |
add |
Logical indicating if the estimates should be added to an existing plot, default is |
main |
Title for the plot, default is |
... |
Additional arguments for the |
We need that u ≥ 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^{∞} (1-F(z)) dz is the premium of the excess-loss insurance with retention u. When L=∞, 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.
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. |
Tom Reynkens
Albrecher, H., Beirlant, J. and Teugels, J. (2017). Reinsurance: Actuarial and Statistical Aspects, Wiley, Chichester.
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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