Pareto Package Vignette

Pareto distribution

Definition: Let $t>0$ and $\alpha>0$. The Pareto distribution $\text{Pareto}(t,\alpha)$ is defined by the distribution function \[ F_{t,\alpha}(x):=\begin{cases} 0 & \text{ for $x\le t$} \\ \displaystyle 1-\left(\frac{t}{x}\right)^{\alpha} & \text{ for $x>t$.} \end{cases} \] This version of the Pareto distribution is also known as Pareto type I, European Pareto or single-parameter Pareto.

Distribution function and density

The functions pPareto and dPareto provide the distribution function and the density function of the Pareto distribution:

library(Pareto)
x <- c(1:10) * 1000
pPareto(x, 1000, 2)

##  [1] 0.0000000 0.7500000 0.8888889 0.9375000 0.9600000 0.9722222 0.9795918
##  [8] 0.9843750 0.9876543 0.9900000

plot(pPareto(1:5000, 1000, 2), xlab = "x", ylab = "CDF(x)")

dPareto(x, 1000, 2)

##  [1] 2.000000e-03 2.500000e-04 7.407407e-05 3.125000e-05 1.600000e-05
##  [6] 9.259259e-06 5.830904e-06 3.906250e-06 2.743484e-06 2.000000e-06

plot(dPareto(1:5000, 1000, 2), xlab = "x", ylab = "PDF(x)")

The package also provides the quantile function:

qPareto(0:10 / 10, 1000, 2)

##  [1] 1000.000 1054.093 1118.034 1195.229 1290.994 1414.214 1581.139 1825.742
##  [9] 2236.068 3162.278      Inf

Simulation:

rPareto(20, 1000, 2)

##  [1] 19558.051  4001.775  1094.137  1097.425  1188.666  5579.462  1604.935
##  [8]  1303.746  1108.651  1049.821  1258.915  1126.669  1276.232  2347.034
## [15]  2367.378  1165.916  1542.368  1556.208  1268.018  1134.410

Layer mean:

Let $X\sim \text{Pareto}(t,\alpha)$ and $a, c\ge 0$. Then \[ E(\min[c,\max(X-a,0)]) = \int_a^{c+a}(1-F_{t,\alpha}(x))\, dx =: I_{t,\alpha}^{\text{$c$ xs $a$}} \] is the layer mean of $c$ xs $a$, i.e. the expected loss to the layer given a single loss $X$.

Example: $t=500$, $\alpha = 2$, Layer 4000 xs 1000

Pareto_Layer_Mean(4000, 1000, 2, t = 500)

## [1] 200

Layer variance:

Let $X\sim \text{Pareto}(t,\alpha)$ and $a, c\ge 0$. Then the variance of the layer loss $\min[c,\max(X-a,0)]$ can be calculated with the function Pareto_Layer_Var.

Example: $t=500$, $\alpha = 2$, Layer 4000 xs 1000

Pareto_Layer_Var(4000, 1000, 2, t = 500)

## [1] 364719

Why is the Pareto distribution so popular?

Lemma:

Let $X \sim \text{Pareto}(t,\alpha )$. Then $cX \sim \text{Pareto}(ct,\alpha )$ for all $c>0$.
Let $X \sim \text{Pareto}(t_{1} ,\alpha )$. For $t_2 > t_1$ we then have $X|(X>t_2 ) \sim \text{Pareto}(t_2 ,\alpha )$

Consequences:

The Pareto alpha is invariant wrt scaling (which implies that $\alpha$ does not depend on currencies and inflation)
For Pareto distributed data the Pareto alpha does not depend on the reporting threshold
For layers and thresholds above $t$ the ratio between expected layer losses and/or excess frequencies depends only on $\alpha$ (and not on $t$)

Pareto extrapolation

Consider two layers $c_i$ xs $a_i$ and a $\text{Pareto}(t,\alpha)$ distributed severity with sufficiently small $t$. What is the expected loss of $c_2$ xs $a_2$ given the expected loss of $c_1$ xs $a_1$?

Example: Assume $\alpha = 2$ and the expected loss of 4000 xs 1000 is 500. Calculate the expected loss of the layer 5000 xs 5000.

Pareto_Extrapolation(4000, 1000, 5000, 5000, 2) * 500

## [1] 62.5

Pareto_Extrapolation(4000, 1000, 5000, 5000, 2, ExpLoss_1 = 500)

## [1] 62.5

Pareto alpha between two layers:

Given the expected losses of two layers, there is typically a unique Pareto alpha $\alpha$ which is consistent with the ratio of the expected layer losses.

Example: Expected loss of 4000 xs 1000 is 500. Expected loss of 5000 xs 5000 is 62.5. Alpha between the two layers:

Pareto_Find_Alpha_btw_Layers(4000, 1000, 500, 5000, 5000, 62.5)

## [1] 2

Check: see previous example

Pareto alpha between a frequency and layer:

Given the expected excess frequency at a threshold and the expected loss of a layer, then there is typically a unique Pareto alpha $\alpha$ which is consistent with this data.

Example: Expected frequency in excess of 500 is 2.5. Expected loss of 4000 xs 1000 is 500. Alpha between the frequency and the layer:

Pareto_Find_Alpha_btw_FQ_Layer(500, 2.5, 4000, 1000, 500)

## [1] 2

Check:

Pareto_Layer_Mean(4000, 1000, 2, t = 500) * 2.5

## [1] 500

Matching the expected losses of two layers:

Given the expected losses of two layers, we can use these techniques to obtain a Poisson-Pareto model which matches the expected loss of both layers.

Example: Expected loss of 30 xs 10 is 26.66 (Burning Cost). Expected loss of 60 xs 40 is 15.95 (Exposure model).

Pareto_Find_Alpha_btw_Layers(30, 10, 26.66, 60, 40, 15.95)

## [1] 1.086263

Frequency @ 10:

26.66 / Pareto_Layer_Mean(30, 10, 1.086263)

## [1] 2.040392

A collective model $\sum_{n=1}^NX_n$ with $X_N\sim \text{Pareto}(10, 1.09)$ and $N\sim \text{Poisson}(2.04)$ matches both expected layer losses.

Frequency extrapolation and alpha between frequencies:

Given the frequency $f_1$ in excess of $t_1$ the frequency $f_2$ in excess of $t_2$ can directly be calculated as follows: \[ f_2 = f_1 \cdot \left(\frac{t_1}{t_2}\right)^\alpha \] Vice versa, we can calculate the Pareto alpha, if the two excess frequencies $f_1$ and $f_2$ are given: \[ \alpha = \frac{\log(f_2/f_1)}{\log(t_1/t_2)}. \]

Example:

Expected frequency excess 1000 is 2. What is the expected frequency excess 4000 if we have a Pareto alpha of 2.5?

t_1 <- 1000
f_1 <- 2
t_2 <- 4000
(f_2 <- f_1 * (t_1 / t_2)^2.5)

## [1] 0.0625

Vice versa:

Pareto_Find_Alpha_btw_FQs(t_1, f_1, t_2, f_2)

## [1] 2.5

Maximum likelihood estimation of the parameter alpha

For $i=1,\dots,n$ let $X_i\sim \text{Pareto}(t,\alpha)$ be Pareto distributed observations. Then we have the ML estimator \[ \hat{\alpha}^{ML}=\frac{n}{\sum_{i=1}^n\log(X_i/t)}. \] Example:

Pareto distributed losses with a reporting threshold of $t=1000$ and $\alpha = 2$:

losses <- rPareto(1000, t = 1000, alpha = 2)
Pareto_ML_Estimator_Alpha(losses, t = 1000)

## [1] 1.974864

In reinsurance, sometimes large loss data from different sources are used for severity fits. Then the losses are typically only available in excess of certain reporting thresholds which may vary by data source. Assume that two data sources each contain 5000 losses in excess of 1000, which are Pareto distributed with an alpha of 2 but from data source 2 we only know the losses exceeding a reporting threshold of 3000. If we apply the standard ML estimator with a threshold of 1000, then we obtain an alpha which is too low, since we ignore that the loss data is not complete in excess of 1000:

losses_1 <- rPareto(5000, t = 1000, alpha = 2)
losses_2 <- rPareto(5000, t = 1000, alpha = 2)
reported <- losses_2 > 3000
losses_2 <- losses_2[reported]
losses <- c(losses_1, losses_2)
Pareto_ML_Estimator_Alpha(losses, t = 1000)

## [1] 1.673069

In the function Pareto_ML_Estimator_Alpha the user can define reporting threshold for each loss in order to handle this situation:

reporting_thresholds_1 <- rep(1000, length(losses_1))
reporting_thresholds_2 <- rep(3000, length(losses_2))
reporting_thresholds <- c(reporting_thresholds_1, reporting_thresholds_2)
Pareto_ML_Estimator_Alpha(losses, t = 1000, reporting_thresholds = reporting_thresholds)

## [1] 2.028599

Now, assume that the underlying policies have limits of 5000 or 10000 and that a loss is censored if it exceeds the respective limit. If the underlying losses are Pareto distributed before they are censored then ML estimation leads to a too large value for alpha:

limits <- sample(c(5000, 10000), length(losses), replace = T)
censored <- losses > limits
losses[censored] <- limits[censored]
reported <- losses > reporting_thresholds
losses <- losses[reported]
reporting_thresholds <- reporting_thresholds[reported]
Pareto_ML_Estimator_Alpha(losses, t = 1000, reporting_thresholds = reporting_thresholds)

## [1] 2.135917

In order to deal with this situation the function allows to specify for each loss if it is censored or not:

Pareto_ML_Estimator_Alpha(losses, t = 1000, reporting_thresholds = reporting_thresholds, 
                          is.censored = censored)

## [1] 2.033894

Truncation

Let $X\sim \text{Pareto}(t,\alpha)$ and $T>t$. Then $X|(X<T)$ has a truncated Pareto distribution. The Pareto functions mentioned above are also available for the truncated Pareto distribution.

Piecewise Pareto distribution

Definition: Let $\mathbf{t}:=(t_1,\dots,t_n)$ be a vector of thresholds with $0<t_1<\dots<t_n<t_{n+1}:=+\infty$ and let $\boldsymbol\alpha:=(\alpha_1,\dots,\alpha_n)$ be a vector of Pareto alphas with $\alpha_i\ge 0$ and $\alpha_n>0$. The piecewise Pareto distribution} $\text{PPareto}(\mathbf{t},\boldsymbol\alpha)$ is defined by the distribution function \[ F_{\mathbf{t},\boldsymbol\alpha}(x):=\begin{cases} 0 & \text{ for $x<t_1$} \\ \displaystyle 1-\left(\frac{t_{k}}{x}\right)^{\alpha_k}\prod_{i=1}^{k-1}\left(\frac{t_i}{t_{i+1}}\right)^{\alpha_i} & \text{ for $x\in [t_k,t_{k+1}).$} \end{cases} \]

The family of piecewise Pareto distributions is very flexible:

Proposition: The set of Piecewise Pareto distributions is dense in the space of all positive-valued distributions (with respect to the Lévy metric).

This means that we can approximate any positive valued distribution as good as we want with piecewise Pareto. A very good approximation typically comes at the cost of many Pareto pieces. Piecewise Pareto is often a good alternative to a discrete distribution, since it is much better to handle!

The Pareto package also provides functions for the piecewise Pareto distribution. For instance:

Distribution function

x <- c(1:10) * 1000
t <- c(1000, 2000, 3000, 4000)
alpha <- c(2, 1, 3, 20)
pPiecewisePareto(x, t, alpha)

##  [1] 0.0000000 0.7500000 0.8333333 0.9296875 0.9991894 0.9999789 0.9999990
##  [8] 0.9999999 1.0000000 1.0000000

plot(pPiecewisePareto(1:5000, t, alpha), xlab = "x", ylab = "CDF(x)")

Density

dPiecewisePareto(x, t, alpha)

##  [1] 2.000000e-03 1.250000e-04 1.666667e-04 3.515625e-04 3.242592e-06
##  [6] 7.048328e-08 2.768239e-09 1.676381e-10 1.413089e-11 1.546188e-12

plot(dPiecewisePareto(1:5000, t, alpha), xlab = "x", ylab = "PDF(x)")

Simulation

rPiecewisePareto(20, t, alpha)

##  [1] 3951.284 1261.649 1707.848 1337.723 1966.081 1089.231 3082.267 1095.236
##  [9] 1142.188 1017.210 1015.655 1422.136 3108.096 1168.951 4036.404 1484.274
## [17] 1119.853 1415.625 1102.989 1047.392

Layer mean

PiecewisePareto_Layer_Mean(4000, 1000, t, alpha)

## [1] 826.6969

Layer variance

PiecewisePareto_Layer_Var(4000, 1000, t, alpha)

## [1] 922221.2

Maximum likelihood estimation of the alphas

Let $\mathbf{t}:=(t_1,\dots,t_n)$ be a vector of thresholds and let $\boldsymbol\alpha:=(\alpha_1,\dots,\alpha_n)$ be a vector of Pareto alphas. For $i=1,\dots,n$ let $X_i\sim \text{PPareto}(\mathbf{t},\boldsymbol\alpha)$. If the vector $\mathbf{t}$ is known, then the parameter vector $\boldsymbol\alpha$ can be estimated with maximum likelihood.

Example:

Piecewise Pareto distributed losses with $\mathbf{t}:=(1000,\,2000,\, 3000)$ and $\boldsymbol\alpha:=(1,\, 2,\, 3)$:

losses <- rPiecewisePareto(10000, t = c(1000, 2000, 3000), alpha = c(1, 2, 3))
PiecewisePareto_ML_Estimator_Alpha(losses, c(1000, 2000, 3000))

## [1] 1.006808 1.902564 2.954470

Reporting thresholds and censoring of losses can be taken into account as described for the function Pareto_ML_Estimator_Alpha.

losses_1 <- rPiecewisePareto(5000, t = c(1000, 2000, 3000), alpha = c(1, 2, 3))
losses_2 <- rPiecewisePareto(5000, t = c(1000, 2000, 3000), alpha = c(1, 2, 3))
reported <- losses_2 > 3000
losses_2 <- losses_2[reported]
losses <- c(losses_1, losses_2)
PiecewisePareto_ML_Estimator_Alpha(losses, c(1000, 2000, 3000))

## [1] 0.7820443 1.2085217 2.9213366

reporting_thresholds_1 <- rep(1000, length(losses_1))
reporting_thresholds_2 <- rep(3000, length(losses_2))
reporting_thresholds <- c(reporting_thresholds_1, reporting_thresholds_2)
PiecewisePareto_ML_Estimator_Alpha(losses, c(1000, 2000, 3000), 
                                   reporting_thresholds = reporting_thresholds)

## [1] 1.023555 1.996905 2.921337

limits <- sample(c(2500, 5000, 10000), length(losses), replace = T)
censored <- losses > limits
losses[censored] <- limits[censored]
reported <- losses > reporting_thresholds
losses <- losses[reported]
reporting_thresholds <- reporting_thresholds[reported]
censored <- censored[reported]
PiecewisePareto_ML_Estimator_Alpha(losses, c(1000, 2000, 3000), 
                                   reporting_thresholds = reporting_thresholds)

## [1] 1.023555 2.860323 3.323436

PiecewisePareto_ML_Estimator_Alpha(losses, c(1000, 2000, 3000), 
                                   reporting_thresholds = reporting_thresholds, 
                                   is.censored = censored)

## [1] 1.023555 1.982386 2.937254

Truncation

The package also provides truncated versions of the piecewise Pareto distribution. There are two options available:

truncation_type = 'lp': Below the largest threshold $t_n$, the distribution function equals the distribution of the piecewise Pareto distribution without truncation. The last Pareto piece, however, is truncated at truncation
truncation_type = 'wd': The whole piecewise Pareto distribution is truncated at `truncation’

Matching a tower of layer losses

The Pareto distribution can be used to build a collective model which matches the expected loss of two layers. We can use piecewise Pareto if we want to match the expected loss of more than two layers.

Consider a sequence of attachment points $0 < a_1 <\dots < a_n<a_{n+1}:=+\infty$. Let $c_i:=a_{i+1}-a_i$ and let $e_i$ be the expected loss of the layer $c_i$ xs $a_i$. Moreover, let $f_1$ be the expected frequency in excess of $a_1$.

The following matching algorithm uses one Pareto piece per layer and is straight forward:

Calculate the Pareto alpha $\alpha_1$ between the excess frequency $f_1$ and the layer $c_1$ xs $a_1$
Calculate the frequency $f_2$ in excess of $a_2$: $f_2:=(a_1/a_2)^{\alpha_1}\cdot f_1$
Calculate the Pareto alpha $\alpha_2$ between the excess frequency $f_2$ and the layer $c_2$ xs $a_2$
Calculate the frequency $f_3$ in excess of $a_3$: $f_3:=(a_2/a_3)^{\alpha_2}\cdot f_3$
$\dots$
Use a collective model $\sum_{n=1}^NX_n$ with $E(N)=f_1$ and $X_n\sim\text{PPareto}(\mathbf{t},\boldsymbol\alpha)$.

This approach always works for three layers, but it often does not work if we have three or more layers. For instance, Riegel (2018) shows that it does not work for the following example:

$i$	Cover $c_i$	Att. Pt. $a_i$	Exp. Loss $e_i$	Rate on Line $e_i/c_i$
1	500	1000	100	0.20
2	500	1500	90	0.18
3	500	2000	50	0.10
4	500	2500	40	0.08

The Pareto package provides a more complex matching approach that uses two Pareto pieces per layer. Riegel (2018) shows that this approach works for an arbitrary number of layers with consistent expected losses.

Example:

attachment_points <- c(1000, 1500, 2000, 2500, 3000)
exp_losses <- c(100, 90, 50, 40, 100)
fit <- PiecewisePareto_Match_Layer_Losses(attachment_points, exp_losses)
fit

## 
## Panjer & Piecewise Pareto model
## 
## Collective model with a Poisson distribution for the claim count and a Piecewise Pareto distributed severity.
## 
## Poisson Distribution:
## Expected Frequency:   0.2136971
## 
## Piecewise Pareto Distribution:
## Thresholds:         1000   1500   1932.059   2000   2147.531   2500   2847.756   3000
## Alphas:              0.3091209   0.1753613   9.685189   3.538534   0.817398   0.7663698   5.086828   2.845488
## The distribution is not truncated.
## 
## Status:               0
## Comments:             OK

The function PiecewisePareto_Match_Layer_Losses returns a PPP_Model object (PPP stands for Panjer & Piecewise Pareto) which contains the information required to specify a collective model with a Panjer distributed claim count and a piecewise Pareto distributed severity. The results can be checked using the attributes FQ, t and alpha of the object:

c(PiecewisePareto_Layer_Mean(500, 1000, fit$t, fit$alpha) * fit$FQ,
  PiecewisePareto_Layer_Mean(500, 1500, fit$t, fit$alpha) * fit$FQ,
  PiecewisePareto_Layer_Mean(500, 2000, fit$t, fit$alpha) * fit$FQ,
  PiecewisePareto_Layer_Mean(500, 2500, fit$t, fit$alpha) * fit$FQ,
  PiecewisePareto_Layer_Mean(Inf, 3000, fit$t, fit$alpha) * fit$FQ)

## [1] 100  90  50  40 100

There are, however, functions which can directly use PPP_Models:

covers <- c(diff(attachment_points), Inf)
Layer_Mean(fit, covers, attachment_points)

## [1] 100  90  50  40 100

Matching reference information

The function PiecewisePareto_Match_Layer_Losses can be used to match the expected losses of a complete tower of layers. If we want to match the expected losses of some reference layers which do not form a complete tower then it is more convenient to use the function Fit_References. Also excess frequencies can be provided as reference information. The function can be seen as a user interface for PiecewisePareto_Match_Layer_Losses:

  covers <- c(1000, 1000, 1000)
  att_points <- c(1000, 2000, 5000)
  exp_losses <- c(100, 50, 10)
  thresholds <- c(4000, 10000)
  fqs <- c(0.04, 0.005)
  fit <- Fit_References(covers, att_points, exp_losses, thresholds, fqs)
  Layer_Mean(fit, covers, att_points)

## [1] 100  50  10

  Excess_Frequency(fit, thresholds)

## [1] 0.040 0.005

If the package lpSolve is installed then the funcion Fit_References can handle ovelapping layers.

Interpolation of PML curves

The function Fit_PML_Curve can be used fit a PPP_Model that reproduces and interpolates the information provided in the PML curve. A PML curve is a table containing return periods and the corresponding loss amounts:

$i$	Return Period $r_i$	Amount $x_i$
1	1	1000
2	5	4000
3	10	7000
4	20	10000
5	50	13000
6	100	14000

The information contained in such a PML curve can be used to create a PPP_Model that has the expected excess frequency $1/r_i$ at $x_i$.

Example:

return_periods <- c(1, 5, 10, 20, 50, 100)
amounts <- c(1000, 4000, 7000, 10000, 13000, 14000)
fit <- Fit_PML_Curve(return_periods, amounts)
1 / Excess_Frequency(fit, amounts)

## [1]   1   5  10  20  50 100

PPP_Models (Panjer & Piecewise Pareto Models)

A PPP_Model object contains the information required to specify a collective model with a Panjer distributed claim count and a piecewise Pareto distributed severity.

Claim count distribution: The Panjer class contains the binomial distribution, the Poisson distribution and the negative binomial distribution. The distribution of the claim count $N$ is specified by the expected frequency $E(N)$ (attribute FQ of the object) and the dispersion $D(N):=Var(N)/E(N)$ (attribute dispersion of the object). We have the following cases:

dispersion < 1: binomial distribution
dispersion = 1: Poisson distribution
dispersion > 1: negative binomial distribution.

Severity distribution: The piecewise Pareto distribution is specified by the vectors t, alpha, truncation and truncation_type.

The function PiecewisePareto_Match_Layer_Losses returns PPP_Model object. Such an object can also be directly created using the constructor function:

PPPM <- PPP_Model(FQ = 2, t = c(1000, 2000), alpha = c(1, 2), 
                  truncation = 10000, truncation_type = "wd", dispersion = 1.5)
PPPM

## 
## Panjer & Piecewise Pareto model
## 
## Collective model with a Negative Binomial distribution for the claim count and a Piecewise Pareto distributed severity.
## 
## Negative Binomial Distribution:
## Expected Frequency:   2
## Dispersion:           1.5 (i.e. contagion = 0.25)
## 
## Piecewise Pareto Distribution:
## Thresholds:         1000   2000
## Alphas:              1   2
## Truncation:           10000
## Truncation Type:      'wd'
## 
## Status:               0
## Comments:             OK

Expected Loss, Standard Deviation and Variance for Reinsurance Layers

A PPP_Model can directly be used to calculate the expected loss, the standard deviation or the variance of a reinsurance layer: function:

PPPM <- PPP_Model(FQ = 2, t = c(1000, 2000), alpha = c(1, 2), 
                  truncation = 10000, truncation_type = "wd", dispersion = 1.5)
Layer_Mean(PPPM, 4000, 1000)

## [1] 2475.811

Layer_Sd(PPPM, 4000, 1000)

## [1] 2676.332

Layer_Var(PPPM, 4000, 1000)

## [1] 7162754

Expected Excess Frequency

A PPP_Model can directly be used to calculate the expected frequency in excess of a threshold:

PPPM <- PPP_Model(FQ = 2, t = c(1000, 2000), alpha = c(1, 2), 
                  truncation = 10000, truncation_type = "wd", dispersion = 1.5)
thresholds <- c(0, 1000, 2000, 5000, 10000, Inf)
Excess_Frequency(PPPM, thresholds)

## [1] 2.0000000 2.0000000 0.9795918 0.1224490 0.0000000 0.0000000

Simulation of Losses

A PPP_Model can directly be used to simulate losses with the corresponding collective model:

PPPM <- PPP_Model(FQ = 2, t = c(1000, 2000), alpha = c(1, 2), 
                  truncation = 10000, truncation_type = "wd", dispersion = 1.5)
Simulate_Losses(PPPM, 10)

##           [,1]     [,2]     [,3]     [,4]     [,5]     [,6]
##  [1,] 2216.330 1317.064 1178.209 5334.729      NaN      NaN
##  [2,] 1114.320 1661.739 1125.267      NaN      NaN      NaN
##  [3,] 3271.311      NaN      NaN      NaN      NaN      NaN
##  [4,] 2152.098 2165.751 2004.136      NaN      NaN      NaN
##  [5,] 1201.509 2433.928      NaN      NaN      NaN      NaN
##  [6,]      NaN      NaN      NaN      NaN      NaN      NaN
##  [7,] 1029.217 2128.913 1514.048      NaN      NaN      NaN
##  [8,] 1453.590 1187.804      NaN      NaN      NaN      NaN
##  [9,] 4007.314 1709.672 2489.485 1334.261 2260.757 1008.616
## [10,] 1318.774      NaN      NaN      NaN      NaN      NaN

The function Simulate_Losses returns a matrix where each row contains the losses from one simulation.

Note that for a given expected frequency FQ not every dispersion dispersion < 1 is possible for the binomial distribution. In this case a binomial distribution with the smallest dispersion larger than or equal to dispersion is used for the simulation.

Generalized Pareto Distribution

Definition: Let $t>0$ and $\alpha_\text{ini}, \alpha_\text{tail}>0$. The generalized Pareto distribution $\text{GenPareto}(t,\alpha_\text{ini}, \alpha_\text{tail})$ is defined by the distribution function \[ F_{t,\alpha_\text{ini}, \alpha_\text{tail}}(x):=\begin{cases} 0 & \text{ for $x\le t$} \\ \displaystyle 1-\left(1+\frac{\alpha_\text{ini}}{\alpha_\text{tail}} \left(\frac{x}{t}-1\right)\right)^{-\alpha_\text{tail}} & \text{ for $x>t$.} \end{cases} \] We do not the standard parameterization from extreme value theory but the parameterization from Riegel (2008) which is useful in a reinsurance context.

Distribution function and density

The functions pGenPareto and dGenPareto provide the distribution function and the density function of the Pareto distribution:

x <- c(1:10) * 1000
pGenPareto(x, t = 1000, alpha_ini = 1, alpha_tail = 2)

##  [1] 0.0000000 0.5555556 0.7500000 0.8400000 0.8888889 0.9183673 0.9375000
##  [8] 0.9506173 0.9600000 0.9669421

plot(pGenPareto(1:5000, 1000, 1, 2), xlab = "x", ylab = "CDF(x)")

dGenPareto(x, t = 1000, alpha_ini = 1, alpha_tail = 2)

##  [1] 1.000000e-03 2.962963e-04 1.250000e-04 6.400000e-05 3.703704e-05
##  [6] 2.332362e-05 1.562500e-05 1.097394e-05 8.000000e-06 6.010518e-06

plot(dGenPareto(1:5000, 1000, 1, 2), xlab = "x", ylab = "PDF(x)")

The package also provides the quantile function:

qGenPareto(0:10 / 10, 1000, 1, 2)

##  [1] 1000.000 1108.185 1236.068 1390.457 1581.989 1828.427 2162.278 2651.484
##  [9] 3472.136 5324.555      Inf

Simulation:

rGenPareto(20, 1000, 1, 2)

##  [1]  1224.324  5502.200  1076.812  2063.607  3035.002  1792.905  1628.436
##  [8]  1758.421  1467.709 13372.238  3854.364  1426.958  1006.056  3105.047
## [15]  5112.588  1197.770  1279.133  1981.269  3609.662  1486.011

Layer mean:

GenPareto_Layer_Mean(4000, 1000, t = 500, alpha_ini = 1, alpha_tail = 2)

## [1] 484.8485

Layer variance:

GenPareto_Layer_Var(4000, 1000, t = 500, alpha_ini = 1, alpha_tail = 2)

## [1] 908942.5

Maximum likelihood estimation of the alpha_ini and alpha_tail

Let $t>0$ and $\alpha_\text{ini}, \alpha_\text{tail}>0$ and let $X_i\sim \text{GenPareto}(t,\alpha_\text{ini}, \alpha_\text{tail})$. For known $t$ the parameters $\alpha_\text{ini}, \alpha_\text{tail}$ can be estimated with maximum likelihood.

Example:

Generalized Pareto distributed losses with $t:=1000$ and $\alpha_\text{ini}=1$, $\alpha_\text{tail}=2$:

losses <- rGenPareto(10000, t = 1000, alpha_ini = 1, alpha_tail = 2)
GenPareto_ML_Estimator_Alpha(losses, 1000)

## [1] 1.00598 1.94554

Reporting thresholds and censoring of losses can be taken into account as described for the function Pareto_ML_Estimator_Alpha.

losses_1 <- rGenPareto(5000, t = 1000, alpha_ini = 1, alpha_tail = 2)
losses_2 <- rGenPareto(5000, t = 1000, alpha_ini = 1, alpha_tail = 2)
reported <- losses_2 > 3000
losses_2 <- losses_2[reported]
losses <- c(losses_1, losses_2)
GenPareto_ML_Estimator_Alpha(losses, 1000)

## [1] 0.6369308 2.3083172

reporting_thresholds_1 <- rep(1000, length(losses_1))
reporting_thresholds_2 <- rep(3000, length(losses_2))
reporting_thresholds <- c(reporting_thresholds_1, reporting_thresholds_2)
GenPareto_ML_Estimator_Alpha(losses, 1000, 
                             reporting_thresholds = reporting_thresholds)

## [1] 0.9757415 2.0675578

limits <- sample(c(2500, 5000, 10000), length(losses), replace = T)
censored <- losses > limits
losses[censored] <- limits[censored]
reported <- losses > reporting_thresholds
losses <- losses[reported]
reporting_thresholds <- reporting_thresholds[reported]
censored <- censored[reported]
GenPareto_ML_Estimator_Alpha(losses, 1000, 
                             reporting_thresholds = reporting_thresholds)

## [1] 0.9065316 6.0750798

GenPareto_ML_Estimator_Alpha(losses, 1000, 
                             reporting_thresholds = reporting_thresholds, 
                             is.censored = censored)

## [1] 0.9803169 2.0641422

Truncation

Let $X\sim \text{GenPareto}(t, \alpha_\text{ini}, \alpha_\text{tail})$ and $T>t$. Then $X|(X<T)$ has a truncated generalized Pareto distribution. The Pareto functions mentioned above are also available for the truncated generalized Pareto distribution.

PGP_Models (Panjer & Generalized Pareto Models)

A PGP_Model object contains the information required to specify a collective model with a Panjer distributed claim count and a generalized Pareto distributed severity.

Claim count distribution: Like in a PPP_Model the claim count distribution from the Panjer class is specified by the expected frequency $E(N)$ (attribute FQ of the object) and the dispersion $D(N):=Var(N)/E(N)$ (attribute dispersion of the object).

Severity distribution: The generalized Pareto distribution is specified by the parameters t, alpha_ini, alpha_tail and truncation.

A PPP_Model object can be created using the constructor function:

PGPM <- PGP_Model(FQ = 2, t = 1000, alpha_ini = 1, alpha_tail = 2, 
                  truncation = 10000, dispersion = 1.5)
PGPM

## 
## Panjer & Generalized Pareto model
## 
## Collective model with a Negative Binomial distribution for the claim count and a generalized Pareto distributed severity.
## 
## Negative Binomial Distribution:
## Expected Frequency:   2
## Dispersion:           1.5 (i.e. contagion = 0.25)
## Generalized Pareto Distribution:
## Threshold:            1000
## alpha_ini:            1
## alpha_tail:           2
## Truncation:           10000
## 
## Status:               0
## Comments:             OK

Methods for PGP_Models

For PGP_Models the same methods are available as for PPP_Models:

PGPM <- PGP_Model(FQ = 2, t = 1000, alpha_ini = 1, alpha_tail = 2, 
                  truncation = 10000, dispersion = 1.5)
Layer_Mean(PGPM, 4000, 1000)

## [1] 2484.33

Layer_Sd(PGPM, 4000, 1000)

## [1] 2756.15

Layer_Var(PGPM, 4000, 1000)

## [1] 7596365

thresholds <- c(0, 1000, 2000, 5000, 10000, Inf)
Excess_Frequency(PGPM, thresholds)

## [1] 2.0000000 2.0000000 0.8509022 0.1614435 0.0000000 0.0000000

Simulate_Losses(PGPM, 10)

##           [,1]     [,2]     [,3]     [,4]     [,5]
##  [1,] 1296.060      NaN      NaN      NaN      NaN
##  [2,] 1029.742 1867.765 1151.948 1052.723 3017.628
##  [3,] 7729.719 1251.795      NaN      NaN      NaN
##  [4,] 1142.286 2139.064      NaN      NaN      NaN
##  [5,] 2261.024 1976.714 1692.425      NaN      NaN
##  [6,] 1625.904 1356.037      NaN      NaN      NaN
##  [7,] 5898.024      NaN      NaN      NaN      NaN
##  [8,] 1346.330 2344.134      NaN      NaN      NaN
##  [9,] 1031.435      NaN      NaN      NaN      NaN
## [10,] 1608.139 2163.278 3220.170 1184.772 2108.927

Pareto Package Vignette

Ulrich Riegel

2023-04-08

Introduction

Pareto distribution

Distribution function and density

Simulation:

Layer mean:

Layer variance:

Why is the Pareto distribution so popular?

Pareto extrapolation

Pareto alpha between two layers:

Pareto alpha between a frequency and layer:

Matching the expected losses of two layers:

Frequency extrapolation and alpha between frequencies:

Maximum likelihood estimation of the parameter alpha

Truncation

Piecewise Pareto distribution

Distribution function

Density

Simulation

Layer mean

Layer variance

Maximum likelihood estimation of the alphas

Truncation

Matching a tower of layer losses

Matching reference information

Interpolation of PML curves

PPP_Models (Panjer & Piecewise Pareto Models)

Expected Loss, Standard Deviation and Variance for Reinsurance Layers

Expected Excess Frequency

Simulation of Losses

Generalized Pareto Distribution

Distribution function and density

Simulation:

Layer mean:

Layer variance:

Maximum likelihood estimation of the alpha_ini and alpha_tail

Truncation

PGP_Models (Panjer & Generalized Pareto Models)

Methods for PGP_Models

References