ExcessPareto | R Documentation |

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", ...)
```

`data` |
Vector of |

`gamma` |
Vector of |

`R` |
The retention level of the (re-)insurance. |

`L` |
The limit of the (re-)insurance, default is |

`endpoint` |
Endpoint for the truncated Pareto distribution. When |

`warnings` |
Logical indicating if warnings are displayed, default is |

`plot` |
Logical indicating if the estimates should be plotted as a function of |

`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 \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.

A list with following components:

`k` |
Vector of the values of the tail parameter |

`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.

`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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