Pareto Package Vignette
Ulrich Riegel
2020-07-09
Introduction
The (European) Pareto distribution is probably the most popular distribution for modeling large losses in reinsurance pricing. There are good reasons for this popularity, which are discussed in detail in Fackler (2013). We recommend Philbrick (1985) and Schmutz et.al. (1998) for an impression of how the (European) Pareto distribution is applied in practice.
In cases where the Pareto distribution is not flexible enough, pricing actuaries sometimes use piecewise Pareto distributions. For instance, a Pareto alpha of 1.5 is used to model claim sizes between USD 1M and USD 5M and an alpha of 2.5 is used above USD 5M. A particularly useful and non-trivial application of the piecewise Pareto distribution is that it can be used to match a tower of expected layer losses with a layer independent collective loss model. Details are described in Riegel (2018), who also provides a matching algorithm that works for an arbitrary number of reinsurance layers.
The package provides a tool kit for the Pareto, the piecewise Pareto and the generalized Pareto distribution, which is useful for pricing of reinsurance treaties. In particular, the package provides the matching algorithm for layer losses.
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:
## [1] 0.0000000 0.7500000 0.8888889 0.9375000 0.9600000 0.9722222 0.9795918
## [8] 0.9843750 0.9876543 0.9900000
## [1] 0.000000e+00 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
The package also provides the quantile function:
## [1] 1000.000 1054.093 1118.034 1195.229 1290.994 1414.214 1581.139 1825.742
## [9] 2236.068 3162.278 Inf
Simulation:
## [1] 1474.666 5650.373 1773.300 1205.131 1766.101 1587.097 1665.768 1259.295
## [9] 2580.145 1431.201 1188.809 1078.929 1068.653 1061.356 1911.431 1164.972
## [17] 2950.296 1511.268 1000.528 1096.830
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
## [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
## [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 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:
## [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:
## [1] 2
Check:
## [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).
## [1] 1.086263
Frequency @ 10:
## [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?
## [1] 0.0625
Vice versa:
## [1] 2.5
Maximum likelihood estimation of the parameter alpha
For \(i=1,\dots,n\) let \(X_i\sim \text{Pareto}(t_i,\alpha)\) be Pareto distributed observations. Then we have the ML estimator \[
\hat{\alpha}^{ML}=\frac{n}{\sum_{i=1}^n\log(X_i/t_i)}.
\] Example:
Pareto distributed losses with a reporting threshold of \(t=100\) and \(\alpha = 2\):
## [1] 2.106583
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
## [1] 0.0000000 0.7500000 0.8333333 0.9296875 0.9991894 0.9999789 0.9999990
## [8] 0.9999999 1.0000000 1.0000000
Density
## [1] 0.000000e+00 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
Simulation
## [1] 1162.864 1199.878 1032.403 1070.109 4105.392 1444.605 1683.931 1122.714
## [9] 4008.938 1376.195 1289.021 1229.029 1132.429 1153.457 1117.178 1048.721
## [17] 1007.457 1008.221 1500.681 1356.458
Layer mean
## [1] 826.6969
Layer variance
## [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 \(\mathbf{t}:=(100,\,200,\, 300)\) and \(\boldsymbol\alpha:=(1,\, 2,\, 3)\):
## [1] 0.9877786 1.9711218 3.0946606
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 truncationtruncation_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:
| 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:
##
## 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:
## [1] 100 90 50 40 100
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 distributiondispersion = 1: Poisson distributiondispersion > 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:
##
## 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:
## [1] 2475.811
## [1] 2676.332
## [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:
## [,1] [,2] [,3]
## [1,] 1266.640 NaN NaN
## [2,] NaN NaN NaN
## [3,] 5991.073 1304.936 NaN
## [4,] 1282.121 4005.112 1175.979
## [5,] 2367.291 NaN NaN
## [6,] 1248.373 NaN NaN
## [7,] 3064.119 2888.808 1251.142
## [8,] 1732.366 1811.926 NaN
## [9,] 2211.160 NaN NaN
## [10,] 1081.810 1086.841 2319.059
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:
## [1] 0.0000000 0.5555556 0.7500000 0.8400000 0.8888889 0.9183673 0.9375000
## [8] 0.9506173 0.9600000 0.9669421
## [1] 0.000000e+00 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
The package also provides the quantile function:
## [1] 1000.000 1108.185 1236.068 1390.457 1581.989 1828.427 2162.278 2651.484
## [9] 3472.136 5324.555 Inf
Simulation:
## [1] 1772.913 1478.241 8354.756 2111.723 3162.379 32850.393 5597.288
## [8] 1766.094 1079.832 1016.452 4039.234 1015.216 5659.635 7340.540
## [15] 1387.975 2592.041 1066.031 1354.714 1731.588 3555.537
Layer mean:
## [1] 484.8485
Layer variance:
## [1] 908942.5
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:
##
## 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:
## [1] 2484.33
## [1] 2756.15
## [1] 7596365
## [1] 2.0000000 2.0000000 0.8509022 0.1614435 0.0000000 0.0000000
## [,1] [,2] [,3] [,4]
## [1,] 2620.326 1422.076 3499.159 2495.885
## [2,] 1546.914 NaN NaN NaN
## [3,] 1377.625 1447.268 1704.737 NaN
## [4,] 1597.050 1088.183 2490.309 NaN
## [5,] 4847.857 NaN NaN NaN
## [6,] 1241.716 NaN NaN NaN
## [7,] 1864.495 4397.882 1270.887 NaN
## [8,] 1325.962 NaN NaN NaN
## [9,] 2099.793 2724.719 NaN NaN
## [10,] 5570.219 1007.457 NaN NaN
References
Fackler, M. (2013) Reinventing Pareto: Fits for both small and large losses. ASTIN Colloquium Den Haag
Johnson, N.L., and Kotz, S. (1970) Continuous Univariate Distributions-I. Houghton Mifflin Co
Philbrick, S.W. (1985) A Practical Guide to the Single Parameter Pareto Distribution. PCAS LXXII: 44–84
Riegel, U. (2008) Generalizations of common ILF models. Bl"{a}tter der DGVFM 29: 45–71
Riegel, U. (2018) Matching tower information with piecewise Pareto. European Actuarial Journal 8(2): 437–460
Schmutz, M., and Doerr, R.R. (1998) Das Pareto-Modell in der Sach-Rueckversicherung. Formeln und Anwendungen. Swiss Re Publications, Zuerich