Estimate premiums of excess-loss reinsurance with retention \(R\) and limit \(L\) using EPD estimates.
ExcessEPD(data, gamma, kappa, tau, R, L = Inf, warnings = TRUE, plot = TRUE, add = FALSE,
main = "Estimates for premium of excess-loss insurance", ...)Vector of \(n\) observations.
Vector of \(n-1\) estimates for the EVI, obtained from EPD.
Vector of \(n-1\) estimates for \(\kappa\), obtained from EPD.
Vector of \(n-1\) estimates for \(\tau\), obtained from EPD.
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.
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.
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.
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 $$ \hat{\Pi}(u) = (k+1)/(n+1) \times (X_{n-k,n})^{1/\hat{\gamma}} \times ((1-\hat{\kappa}/\hat{\gamma})(1/\hat{\gamma}-1)^{-1}u^{1-1/\hat{\gamma}} + \hat{\kappa}/(\hat{\gamma}X_{n-k,n}^{\hat{\tau}})(1/\hat{\gamma}-\hat{\tau}-1)^{-1}u^{1+\hat{\tau}-1/\hat{\gamma}})$$ with \(\hat{\gamma}, \hat{\kappa}\) and \(\hat{\tau}\) the estimates for the parameters of the EPD.
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.
# NOT RUN {
data(secura)
# EPD estimator
epd <- EPD(secura$size)
# Premium of excess-loss insurance with retention R
R <- 10^7
ExcessEPD(secura$size, gamma=epd$gamma, kappa=epd$kappa, tau=epd$tau, R=R, ylim=c(0,2*10^4))
# }
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