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Activate the actuary menu. The option PML concerns the tarification with respect to the probable maximum loss. The next four options provide a first insight in the behavior of reserve processes within a finite time horizon. Note that the StatPascal program initres.xpl must be utilized for computing an estimate of the T-year initial reserve.

Reserve Process Path
Ruin Time Data
Ruin Probability
PML

Reserve Process Path

Let S(t) = X[1] + ... + X[N(t)] be the total claims process based on claim sizes X[i] and a claim number process N(t). With the option Reserve Process Path one can generate the path of a reserve process

U(t) = s + E(S(t)) + b(t) - S(t), t >= 0,

where Recollect that claim sizes X[i] and their arrival times T[i] may be jointly generated and visualized by Data... Generate Counting/Point Process... Marked Poisson Process....
Claim Number Process
Select Poisson claim number process or Polya-Lundberg claim number process and specify the corresponding parameters intensity lambda or alpha, sigma, respectively, and time horizon. Recollect that
E(S(t)) = E(X) * lambda * t and E(S(t)) = E(X) * alpha * sigma * t
for Poisson and Polya-Lundberg claim number processes.
Additional parameters
Specify initial reserve, safety exponent beta and the safety loading rho. The safety function is
b(t) = rho * E(X) * lambda * t**beta and b(t) = rho * E(X) * alpha * sigma * t**beta
for Poisson and Polya-Lundberg processes.
The claim arrival times and the pertaining reserve values are stored as Xtremes Multivariate Data. This data set is now the active one. The reserve process path can be generated interactively by means of the Visualizing button. Use the UpArrow-key to generate one data point, the PgUp-key to generate 20 data simultaneously. The generated values are displayed in a graphics window.
Select one of the following submenu items to specify the claim size distribution:

Exponential Claim Sizes

Select claim sizes X[1], X[2], X[3], ... distributed according to an Exponential (GP 0) distribution with location and scale parameters mu and sigma. One must specify these parameters in the Reserve Process Path dialog box.

Pareto Claim Sizes

Select claim sizes X[1], X[2], X[3], ... distributed according to a Pareto (GP 1) distribution with shape, location and scale parameters alpha, mu and sigma. One must specify these parameters in the Reserve Process Path dialog box.

GP Claim Sizes

Select claim sizes X[1], X[2], X[3], ... distributed according to a generalized Pareto (GP) distribution with shape, location and scale parameters gamma, mu and sigma. One must specify these parameters in the Reserve Process Path dialog box.

Ruin Time Data

Ruin occurs, when the reserve process path U(t) becomes negative. Ruin times are simulated within the framework as given in Reserve Process Path. Specify the filename of the data set and the Number of Simulations. The ruin times or, if no ruin occurs, the selected time horizon are stored as Xtremes Univariate Data.
Select one of the following submenu items to specify the claim size distribution:

Exponential Claim Sizes

Select claim sizes X[1], X[2], X[3], ... distributed according to an Exponential (GP 0) distribution with location and scale parameters mu and sigma. One must specify these parameters in the Ruin Time Data dialog box.

Pareto Claim Sizes

Select claim sizes X[1], X[2], X[3], ... distributed according to a Pareto (GP 1) distribution with shape, location and scale parameters alpha, mu and sigma. One must specify these parameters in the Ruin Time Data dialog box.

GP Claim Sizes

Select claim sizes X[1], X[2], X[3], ... distributed according to a generalized Pareto (GP) distribution with shape, location and scale parameters gamma, mu and sigma. One must specify these parameters in the Ruin Time Data dialog box.

Ruin Probability

Simulate ruin probabilities within a finite time horizon T. The simulation runs within the framework as given in Reserve Process Path. The probability of ruin up to time T is

psi(T,s) = P { tau(s) <= T }.

Applying the Ruin Probabilty option, one can calculate ruin probabilities for initial reserves which must be specified by smin, smax and steps. The resulting pairs (s, psi(T,s)) are stored as Xtremes Multivariate Data.

Simulation options
Specify the filename of data set and the Number of Simulations for which each of the ruin probabilities is simulated.
Select one of the following submenu items to specify the claim size distribution:

Exponential Claim Sizes

Select claim sizes X[1], X[2], X[3], ... distributed according to an Exponential (GP 0) distribution with location and scale parameters mu and sigma. One must specify these parameters in the Ruin Probability dialog box.

Pareto Claim Sizes

Select claim sizes X[1], X[2], X[3], ... distributed according to a Pareto (GP 1) distribution with shape, location and scale parameters alpha, mu and sigma. One must specify these parameters in the Ruin Probability dialog box.

GP Claim Sizes

Select claim sizes X[1], X[2], X[3], ... distributed according to a generalized Pareto (GP) distribution with shape, location and scale parameters gamma, mu and sigma. One must specify these parameters in the Ruin Probability dialog box.

PML

The tarification of policies with respect to the probable maximum loss (PML) is now on the agenda. First, read carefully Section 12.3 in Statistical Analysis. Our subsequent explanations are closely related to those given in the Demos 9.1 and 9.2.
The option Analyze Nearest Neighbors is applicable to a bivariate data set of type Xtremes Multivariate Data with PMLs in the first column and the pertaining claim sizes in the second column (e.g., select the columns "PML" and "Claim sizes" from im-pmlfi.dat by applying Data... Transform Data... Select Columns).
The option Analyze Segments is applicable to a data set of type Xtremes Multivariate Data with group numbers, lower and upper boundaries, PMLs, claim sizes and claim degrees in the first six columns (as given in im-pmlfi.dat).
If the given data set only consists of PMLs and claim sizes, then the options Transform to Segments and Add Degrees enable the user to prepare a data set for the analysis in PML groups. A first insight into the PML groups is gained by applying Functional Parameter Transformation.

Transfer to Segments

A bivariate data set of type Xtremes Multivariate Data, with PMLs and claim sizes in the two columns, is transformed to segments, i.e. to PML groups determined by priorities p[1], ... , p[k]. These priorities must be written into an edit field on the right-hand side of the Add Priorities dialog box. These values can also be entered by means of from, to, step or, respectively, Block Size. In the latter case, the priorities are given by PMLs. The new data set is stored to a file with "Number of PML group", "Lower bound", "Upper Bound", "PML" and "Claim sizes" in the first six columns. Claim degrees can be added by means of Add Degrees.

Add Degrees

Add claim degrees x[i,j]/z[i,j] to data sets as obtained by the option Transfer to Segments, where x[i,j] is a claim size belonging to PML group i and z[i,j] is the pertaining PML. The claim degrees are written into the sixth column.

Functional Parameter Transformation

For each of the PML groups, the priorities (boundaries), the center, the sample means and variances of the claim sizes and the sample means and variances of the claim degrees are stored in a file. This option is applicable to a data set as created by Add Degrees.

Analyze Segments

The option Analyze Segments is applicable to a data set of type Xtremes Multivariate Data with group numbers, lower and upper boundaries, PMLs, claim sizes and claim degrees in the first six columns (as already mentioned in PML).
First, choose Claim sizes or Claim degrees in the Analyze Segments dialog box. Pressing OK, the claim sizes or the claim degrees within each of the PML groups are written to files seg*.dat with * denoting the group number. Only values exceeding the threshold are dealt with. The threshold itself (likewise, the threshold divided by the upper group boundary) is included in the data set in the case of claim sizes (claim degrees).
Several estimators (Hill(GP1/GP2), MLE(GP), Moment(GP), MLE(GP0)) can either be applied to claim sizes or claim degrees within each PML group. If the Hill estimator is applied to claim degrees, then the data are internally transformed so that the Hill estimator becomes applicable within a beta model. Parameters of estimated beta distributions for the original claim degrees are finally stored in a data set.
For each of the PML groups, the group number, the center of the group, the sample mean, the parametric mean of the estimated GP distribution and the estimated shape, location and scale parameters are written to a text file with the default name para.dat. In addition, if one of the options Plot DF or Plot QF is activated, then parametric and nonparametric distribution or quantile functions are displayed in graphic windows for each of the PML groups (for which the parametric estimation was carried out successfully).

Analyze Nearest Neighbors

The option Analyze Nearest Neighbors is applicable to a bivariate data set of type Xtremes Multivariate Data with PMLs in the first column and the pertaining claim sizes in the second column (as already mentioned in PML). This option concerns the computations as carried out in Examples 12.3.3 and 12.3.4 in Statistical Analysis.
The estimation of the conditional expectation E(X|z) of claim sizes given the PML equal to z can be based on the claim sizes y[j], j = 1, ..., k, pertaining to the k PMLs z[j] closest to z. Then,

sigma * y[j]/k

is an empirical estimate of E(X|z).

Likewise, the estimation can be carried out within a GP model based on the nearest neighbors. The parametric mean is used as an estimate of E(X|z).

In the Analyze PML (Nearest Neighbors) dialog box, one must select either the MLE(GP) or an MLE of the scale parameter (for a fixed shape and location parameter) and the number of neighbors. The computations are carried out for each PML z. Then the PML, the empirical and parametric estimates of the conditional expectation E(X|z) and the parameters of the estimated GP distribution are written to a file.

© 2005 Xtremes Group · updated Jun 21, 2005