Estimate premiums of excess-loss reinsurance with retention \(R\) and limit \(L\) using a (truncated) Pareto model.
ExcessPareto(data, gamma, R, L = Inf, endpoint = Inf, warnings = TRUE, plot = TRUE,
add = FALSE, main = "Estimates for premium of excess-loss insurance", ...)
ExcessHill(data, gamma, R, L = Inf, endpoint = Inf, warnings = TRUE, plot = TRUE,
add = FALSE, main = "Estimates for premium of excess-loss insurance", ...)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.
Vector of \(n\) observations.
Vector of \(n-1\) estimates for the EVI, obtained from Hill or trHill.
The retention level of the (re-)insurance.
The limit of the (re-)insurance, default is Inf.
Endpoint for the truncated Pareto distribution. When Inf, the default, the ordinary Pareto model is used.
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.
Tom Reynkens
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 \(\Pi\) (for the untruncated Pareto distribution) by $$ \hat{\Pi}(u) = (k+1)/(n+1) / (1/H_{k,n}-1) \times (X_{n-k,n}^{1/H_{k,n}} u^{1-1/H_{k,n}}),$$ with \(H_{k,n}\) the Hill estimator.
The ExcessHill function is the same function but with a different name for compatibility with old versions of the package.
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.
Hill, ExcessEPD, ExcessGPD, ExcessSplice
data(secura)
# Hill estimator
H <- Hill(secura$size)
# Premium of excess-loss insurance with retention R
R <- 10^7
ExcessPareto(secura$size, H$gamma, R=R)
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