ExcessPareto function

Estimates for excess-loss premiums using a Pareto model

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", ...)

Arguments

• data: Vector of $n$ observations.
• gamma: Vector of $n-1$ estimates for the EVI, obtained from Hill or trHill.
• R: The retention level of the (re-)insurance.
• L: The limit of the (re-)insurance, default is Inf.
• endpoint: Endpoint for the truncated Pareto distribution. When Inf, the default, the ordinary Pareto model is used.
• 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.

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

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

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.

Returns

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.

References

Albrecher, H., Beirlant, J. and Teugels, J. (2017). Reinsurance: Actuarial and Statistical Aspects, Wiley, Chichester.

Author(s)

Tom Reynkens

See Also

Hill, ExcessEPD, ExcessGPD, ExcessSplice

Examples

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)