ExcessEPD function

Estimates for excess-loss premiums using EPD estimates

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", ...)

Arguments

• data: Vector of $n$ observations.
• gamma: Vector of $n-1$ estimates for the EVI, obtained from EPD.
• kappa: Vector of $n-1$ estimates for $\kappa$, obtained from EPD.
• tau: Vector of $n-1$ estimates for $\tau$, obtained from EPD.
• 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.

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.

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 $\Pi$ by

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.

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

EPD, ExcessHill, ExcessGPD

Examples

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))