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by Jozef L. Teugels, Jan Beirlant, Hansjöerg Albrecher
Reinsurance
Cover
Title Page
Preface
1 Introduction
1.1 What is Reinsurance?
1.2 Why Reinsurance?
1.3 Reinsurance Data
1.4 Notes and Bibliography
2 Reinsurance Forms and their Properties
2.1 Quota‐share Reinsurance
2.2 Surplus Reinsurance
2.3 Excess‐of‐loss Reinsurance
2.4 Stop‐loss Reinsurance
2.5 Large Claim Reinsurance
2.6 Combinations of Reinsurance Forms and Global Protections
2.7 Facultative Contracts
2.8 Notes and Bibliography
3 Models for Claim Sizes
3.1 Tails of Distributions
3.2 Large Claims
3.3 Common Claim Size Distributions
3.4 Mean Excess Analysis
3.5 Full Models: Splicing
3.6 Multivariate Modelling of Large Claims
4 Statistics for Claim Sizes
4.1 Heavy or Light Tails: QQ‐ and Derivative Plots
4.2 Large Claims Modelling through Extreme Value Analysis
4.3 Global Fits: Splicing, Upper‐truncation and Interval Censoring
4.4 Incorporating Covariate Information
4.5 Multivariate Analysis of Claim Distributions
4.6 Estimation of Other Tail Characteristics
4.7 Further Case Studies
4.8 Notes and Bibliography
5 Models for Claim Counts
5.1 General Treatment
5.2 The Poisson Process and its Extensions
5.3 Other Claim Number Processes
5.4 Discrete Claim Counts
5.5 Statistics of Claim Counts
5.6 Claim Numbers under Reinsurance
5.7 Notes and Bibliography
6 Total Claim Amount
6.1 General Formulas for Aggregating Independent Risks
6.2 Classical Approximations for the Total Claim Size
6.3 Panjer Recursion
6.4 Fast Fourier Transform
6.5 Total Claim Amount under Reinsurance
6.6 Numerical Illustrations
6.7 Aggregation for Dependent Risks
6.8 Notes and Bibliography
7 Reinsurance Pricing
7.1 Classical Principles of Premium Calculation
7.2 Solvency Considerations
7.3 Pricing Proportional Reinsurance
7.4 Pricing Non‐proportional Reinsurance
7.5 The Aggregate Risk Margin
7.6 Leading and Following Reinsurers
7.7 Notes and Bibliography
8 Choice of Reinsurance
8.1 Decision Criteria
8.2 Classical Optimality Results
8.3 Solvency Constraints and Cost of Capital
8.4 Minimizing Other Risk Measures
8.5 Combining Reinsurance Treaties
8.6 Reinsurance Chains
8.7 Dynamic Reinsurance
8.8 Beyond Piecewise Linear Contracts
8.9 Notes and Bibliography
9 Simulation
9.1 The Monte Carlo Method
9.2 Variance Reduction Techniques
9.3 Quasi‐Monte Carlo Techniques
9.4 Notes and Bibliography
10 Further Topics
10.1 More on Large Claim Reinsurance
10.2 Alternative Risk Transfer
10.3 Reinsurance and Finance
10.4 Catastrophic Risk
References
Index
End User License Agreement
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Reinsurance: Actuarial and Statistical Aspects
Table of Contents
Cover
Title Page
Preface
1 Introduction
1.1 What is Reinsurance?
1.2 Why Reinsurance?
1.3 Reinsurance Data
1.4 Notes and Bibliography
2 Reinsurance Forms and their Properties
2.1 Quota‐share Reinsurance
2.2 Surplus Reinsurance
2.3 Excess‐of‐loss Reinsurance
2.4 Stop‐loss Reinsurance
2.5 Large Claim Reinsurance
2.6 Combinations of Reinsurance Forms and Global Protections
2.7 Facultative Contracts
2.8 Notes and Bibliography
3 Models for Claim Sizes
3.1 Tails of Distributions
3.2 Large Claims
3.3 Common Claim Size Distributions
3.4 Mean Excess Analysis
3.5 Full Models: Splicing
3.6 Multivariate Modelling of Large Claims
4 Statistics for Claim Sizes
4.1 Heavy or Light Tails: QQ‐ and Derivative Plots
4.2 Large Claims Modelling through Extreme Value Analysis
4.3 Global Fits: Splicing, Upper‐truncation and Interval Censoring
4.4 Incorporating Covariate Information
4.5 Multivariate Analysis of Claim Distributions
4.6 Estimation of Other Tail Characteristics
4.7 Further Case Studies
4.8 Notes and Bibliography
5 Models for Claim Counts
5.1 General Treatment
5.2 The Poisson Process and its Extensions
5.3 Other Claim Number Processes
5.4 Discrete Claim Counts
5.5 Statistics of Claim Counts
5.6 Claim Numbers under Reinsurance
5.7 Notes and Bibliography
6 Total Claim Amount
6.1 General Formulas for Aggregating Independent Risks
6.2 Classical Approximations for the Total Claim Size
6.3 Panjer Recursion
6.4 Fast Fourier Transform
6.5 Total Claim Amount under Reinsurance
6.6 Numerical Illustrations
6.7 Aggregation for Dependent Risks
6.8 Notes and Bibliography
7 Reinsurance Pricing
7.1 Classical Principles of Premium Calculation
7.2 Solvency Considerations
7.3 Pricing Proportional Reinsurance
7.4 Pricing Non‐proportional Reinsurance
7.5 The Aggregate Risk Margin
7.6 Leading and Following Reinsurers
7.7 Notes and Bibliography
8 Choice of Reinsurance
8.1 Decision Criteria
8.2 Classical Optimality Results
8.3 Solvency Constraints and Cost of Capital
8.4 Minimizing Other Risk Measures
8.5 Combining Reinsurance Treaties
8.6 Reinsurance Chains
8.7 Dynamic Reinsurance
8.8 Beyond Piecewise Linear Contracts
8.9 Notes and Bibliography
9 Simulation
9.1 The Monte Carlo Method
9.2 Variance Reduction Techniques
9.3 Quasi‐Monte Carlo Techniques
9.4 Notes and Bibliography
10 Further Topics
10.1 More on Large Claim Reinsurance
10.2 Alternative Risk Transfer
10.3 Reinsurance and Finance
10.4 Catastrophic Risk
References
Index
End User License Agreement
List of Tables
Chapter 01
Table 1.1 Global premium volume 2015 (in US$ billions).
Table 1.2 Company A: characteristics of the claims from Figure 1.4
Table 1.3 Company A: observed number of claims per accident year and per number of development years in 2010 (
DY
2010
)
Chapter 03
Table 3.1
Chapter 04
Table 4.1
Chapter 05
Table 5.1
Table 5.2
Chapter 06
Table 6.1 MTPL data for Company A: Key figures for models of
X
,
N
365
and
S
365
using Poisson and Cox models with inverse Gaussian subordinator
Table 6.2 Dutch fire insurance claim data: Key figures for models of
X
,
N
365
and
S
365
using a Poisson model (
λ
= 0.28) and a Cox model with inverse Gaussian subordinator (cf. Figure 5.8).
List of Illustrations
Chapter 01
Figure 1.1 Claim development scheme.
Figure 1.2 Indexed reporting thresholds of Companies A and B.
Figure 1.3 Company A: histogram of reporting delays.
Figure 1.4 MTPL data: development pattern of four particular claims.
Figure 1.5 Company A: incurred losses (top); Kaplan–Meier estimator for the distribution function of the number of development years (bottom).
Figure 1.6 Company B: incurred losses (top); Kaplan–Meier estimator for the distribution function of the number of development years (bottom).
Figure 1.7 Ultimate versus incurred losses with least squares regression fit for the open claims of Company A (top) and Company B (bottom).
Figure 1.8 Company A: observed cluster sizes of the claim number process.
Figure 1.9 Dutch fire insurance claims: log‐claims as a function of time for Dutch fire insurance (top); observed cluster sizes of the claim number process (bottom).
Figure 1.10 Normalized loss data against wind index
W
for Vienna (top) and Upper Austria (bottom); original scale (left) and log‐scale (right).
Figure 1.11 Flood risk: aggregate annual losses (in log scale) by percentage of building value for Germany and the UK.
Figure 1.12 Induced (dark points) earthquakes in the northern part of the Netherlands with magnitudes larger than 1.5.
Figure 1.13 Danish fire insurance data: scatterplot matrix on the log‐scale.
Figure 1.14 Danish fire insurance data: cluster arrivals, selecting the claim dates where each component is addressed.
Chapter 02
Figure 2.1 Proportionality factor of the reinsurer as a function of insured sum.
Figure 2.2 Comparison of reinsured claim amounts for some combinations of claim sizes
X
i
and corresponding insured sums
Q
i
for QS reinsurance with
a
= 0.3 (left), surplus reinsurance (middle), and
L
xs
M
reinsurance (right).
Figure 2.3 Graphical interpretation for the splitting of the expected claim size between insurer and reinsurer.
Figure 2.4 100 xs 100 treaty with one reinstatement and claims
X
1
= 150,
X
2
= 175,
X
3
= 225, and
X
4
= 150.
Chapter 04
Figure 4.1 Dutch fire insurance data: exponential QQ‐plot and mean excess plot (
x
n
−
k
,
n
,
e
k
,
n
) (top); Pareto QQ‐plot and Hill plot
(second line); log‐normal QQ‐plot and derivative plot
(third line); Weibull QQ‐plot and derivative plot
(bottom). For each QQ‐plot the regression line through
X
n
−99,
n
, …,
X
n
,
n
is plotted.
Figure 4.2 MTPL data for Company A, ultimate data values at evaluation: exponential QQ‐plot and mean excess plot (
x
n
−
k
,
n
,
e
k
,
n
) (top); Pareto QQ‐plot and Hill plot
(second line); log‐normal QQ‐plot and derivative plot
(third line); Weibull QQ‐plot and derivative plot
(bottom).
Figure 4.3 MTPL data for Company B, ultimate data values at evaluation: exponential QQ‐plot and mean excess plot (
x
n
−
k
,
n
,
e
k
,
n
) (top); Pareto QQ‐plot and Hill plot
(second line); log‐normal QQ‐plot and derivative plot
(third line); Weibull QQ‐plot and derivative plot
(bottom).
Figure 4.4 Pareto QQ‐plot and Hill plot (
k
,
H
k
,
n
): Dutch fire insurance data with 95% confidence intervals for each
k
(top); MTPL data for Company A, ultimate values (middle); MTPL data for Company B, ultimate values (bottom).
Figure 4.5 Hill estimates and bias‐reduced versions, regression approach as a function of
k
(left) and log
k
(right), and EPD approach (middle) as a function of
k
: Dutch fire insurance data (top); MTPL data for Company A, ultimate values (middle); MTPL data for Company B, ultimate values (bottom).
Figure 4.6 Quantile estimates
and
(left) and log‐return periods
and
(right), as a function of
k
: Dutch fire insurance data (
x
= 200 million, top); MTPL data for Company A, ultimate values (
x
= 4 million, middle); MTPL data for Company B, ultimate values (
x
= 6 million, bottom).
Figure 4.7 Scale estimates
Â
k
,
n
and
as a function of
k
: Dutch fire insurance data (top); MTPL data for Company A, ultimate values (middle); MTPL data for Company B, ultimate values (bottom).
Figure 4.8 Generalized QQ‐plot (middle) and estimators of EVI
,
,
as a function of
k
(left) and log
k
(right): Dutch fire insurance data (first and second line); MTPL data for Company A, ultimate values (third and fourth line); MTPL data for Company B, ultimate values (bottom two lines).
Figure 4.9 Large quantile estimators
,
,
(left) and log‐return period estimators
,
,
(right) as a function of
k
: Dutch fire insurance data (
x
= 200 million, top); MTPL data for Company A, ultimate values (
x
= 4 million, middle); MTPL data for Company B, ultimate values (
x
= 6 million, bottom).
Figure 4.10 MTPL data for Company A, ultimate estimates:
and
H
k
,
n
as a function of
k
(top left), P‐value of
T
B
,
k
as a function of
k
(top right),
and
as a function of
k
(middle left), estimates of endpoint
(middle right),
as a function of
k
(bottom left), truncated Pareto fit to Pareto QQ‐plot based on top
k
= 250 observations (bottom right).
Figure 4.11 MTPL data for Company A ultimates: fit of spliced model with a mixed Erlang and a Pareto component with thresholds
t
indicated on mean excess plot (top left); empirical and model survival function (top right); PP plot of empirical survival function against splicing model RTF (bottom left); idem with
transformation (bottom right).
Figure 4.12 The different classes of censored observations.
Figure 4.13 Dutch fire claim data: fit of spliced model mixed Erlang and Pareto with threshold
t
indicated on the mean excess plot (top left); empirical and model survival function (top right); PP plot of empirical survival function against splicing model RTF (bottom left); idem with
transformation (bottom right).
Figure 4.14 Dutch fire claim data: fit of spliced model with a mixed Erlang and two Pareto components with thresholds
t
1
and
t
2
as indicated on the mean excess plot (top left); empirical and model survival function (top right); PP plot of empirical survival function against splicing model RTF (bottom left); idem with
transformation (bottom right).
Figure 4.15 MTPL data for Company A: percentage of closed claims with incurred value being a correct upper bound for final payment as a function of the number of development years (DY) (top); boxplots of
R
i
,
d
for every development year
d
and factor
f
d
used in the interval censoring approach (bottom).
Figure 4.16 MTPL data for Company A: fit of spliced mixed Erlang and Pareto models with interval censoring based on upper bounds
I
i
,
d
,
i
= 1, …, 596,
d
= 5, …, 16, for non‐closed claims: mean excess plot based on (4.3.44) (top left); Hill plot based on (4.3.45) (top right); empirical and model survival function (middle left); PP plot of empirical survival function against splicing model RTF (middle right); idem with
transformation (bottom).
Figure 4.17 MTPL data for Company A (1995–2009): fit of spliced mixed Erlang and Pareto models with interval censoring based on upper bounds
Ĩ
i
,
d
,
i
= 1, …, 849,
d
= 1, …, 15 for non‐closed claims: mean excess plot based on (4.3.44) (top left); Hill plot based on (4.3.45) (top right); empirical and model survival function (middle); PP plot of
survival function, empirical against splicing mode (bottom left); size of confidence intervals using interval censoring with upper bounds
I
i
,
d
and
Ĩ
i
,
d
(bottom right).
Figure 4.18 MTPL data for Company B: mean excess plot based on (4.3.44) (top left) and Hill plot based on (4.3.45)(top right) for interval censored data based on accidents from 1990 to 2005; fit of spliced model mixed Erlang and Pareto models: empirical and model survival function (middle left); PP plot of empirical survival function against splicing model RTF (middle right) left); idem with
transformation (bottom).
Figure 4.19 Hill plots adapted for interval censoring
with upper bounds
I
i
,
d
and
Ĩ
i
,
d
, and Hill estimates based on random right censoring
without upper bounds. The vertical line indicates the splicing threshold
t
used above.
Figure 4.20 Austrian storm claim data: plot of
for Upper Austria and Vienna area (top);
(middle left) and residual QQ‐plot (middle right) for Upper Austria;
(bottom left) with and without outlier and residual QQ‐plot (bottom right) for Vienna.
Figure 4.21 MTPL data for Company A: time plots of cumulative payments
Z
i
as a function of
nDY
e
(top); Pareto QQ‐plots (middle) and Hill estimates (bottom) adapted for right censoring at development years
nDY
= 3, 8, 13.
Figure 4.22 MTPL data for Company A: fit of splicing model at development years
nDY
= 3, 8, 13; PP plot of empirical survival function against splicing model RTF at
nDY
= 8 (top); idem with
transformation (bottom).
Figure 4.23 Danish fire insurance data, building and contents: Hill and bias‐reduced Hill plots, building (top left) and contents (top right), plot of
against
k
(middle left) and
Ĉ
(
u
,
u
) against
u
∈ (0, 1) (middle right), plot of
(bottom left) and cumulative distribution function of the fitted bivariate splicing model (bottom right).
Figure 4.24 MTPL data for Company A: XL pure premium Π(
u
) based on ME‐Pa fit taking interval censoring into account. Comparison with the result when the upper bounds are ignored (right censoring) and when the premium is based on the ultimates.
Figure 4.25 Austrian storm claim data: XL pure premium Π(exp(0.001
w
)) for Upper Austria based based on a GPD regression fit with the wind index
W
as covariate.
Figure 4.26 UK flood loss data: mean excess plot (
x
n
−
k
,
n
,
e
k
,
n
) (top left); Hill plot
(top right); log‐normal derivative plot
(second line left); Weibull derivative plot
(second line right);
γ
estimates (third line left); P‐values of
T
B
test (third line right); endpoint estimates
(bottom left); Pareto QQ‐plot with truncated Pareto fit (full line) and Pareto fit (dashed line) (bottom right).
Figure 4.27 Flood loss data Germany: mean excess plot (
x
n
−
k
,
n
,
e
k
,
n
) (top left); Hill plot
(top right); log‐normal derivative plot
(second line left); Weibull derivative plot
(second line right);
γ
estimates (third line left); P‐values of
T
B
test (third line right); endpoint estimates
(bottom left); Pareto QQ‐plot with truncated Pareto fit (full line), and Pareto fit (dashed line) (bottom right).
Figure 4.28 Earthquake magnitude data from Groningen area. Exponential QQ‐plot based on magnitudes (top left); estimates of
γ
(top right);
T
B
P‐value plot (middle left);
estimates (middle right); Pareto QQ‐plot of energy values with truncated Pareto fit (bottom right); estimates of maximum magnitude
(bottom right).
Chapter 05
Figure 5.1 10 simulated Poisson paths on [0, 100]: homogeneous Poisson
Ñ
1
(top) and discrete mixed Poisson with
λ
1
= 0.5,
λ
2
= 1.5, and
p
= 0.5 (bottom).
Figure 5.2 Simulated path on [0, 100] with mean value and mean value +/‐ 2 standard deviation curves: homogeneous Poisson
Ñ
1
(top); inhomogeneous Poisson with
(middle); doubly stochastic Poisson directed by a gamma subordinator (
α
= 2,
β
= 2) (bottom).
Figure 5.3 Dutch fire insurance data, claim counts: time plot of the yearly occurrences (top); non‐homogeneous Poisson and negative binomial fits using GAM on
μ
i
, together with horizontal line from Poisson fit with no regression
μ
i
=
μ
(bottom).
Figure 5.4 MTPL data for Company A, claim counts: time plot of the yearly occurrences (top); non‐homogeneous Poisson and negative binomial fits using GAM on
μ
i
, together with horizontal line from Poisson fit with no regression
μ
i
=
μ
(middle); Poisson and negative binomial fits with exposure as offset (bottom).
Figure 5.5 MTPL data for Company B, claim counts: time plot of the yearly occurrences (top); non‐homogeneous Poisson and negative binomial fits using GAM on
μ
i
, together with horizontal line from Poisson fit with no regression
μ
i
=
μ
(bottom).
Figure 5.6 Dutch fire insurance data: distribution of time points (5.5.30) (top left); moving average estimates of intensity function (with
m
= 50) (top right); estimated time transformation (middle left); moving average estimates of intensity function after time transformation (middle right); exponential QQ‐plot (5.5.31) based on
μ
‐transformed waiting times (bottom left); boxplots of monthly mean waiting times after time transformation (bottom right).
Figure 5.7 Dutch fire insurance data. Model (5.5.34) fit: observed and simulated intensities (top left); normal QQ‐plot of residuals (top right); auto‐correlation (second line left); partial auto‐correlation (second line right). Model (5.5.35) fit: observed and simulated intensities (third line left); normal QQ‐plot of residuals (third line right); auto‐correlation (bottom left); partial auto‐correlation (bottom right).
Figure 5.8 Dutch fire insurance data. Simulated confidence intervals and observed cumulative counting data for fitted Cox models directed by a gamma Lévy subordinator (top left) and an inverse Gaussian Lévy subordinator (top right); mean values of locally fitted Cox gamma models using 30 days moving windows (middle left) and exponential QQ‐plot based on waiting times between clusters (middle right); cluster sizes of a simulated claim number process following the Cox model with the IG fitted parameters (bottom).
Figure 5.9 MTPL data for Company A: distribution of time points (5.5.30) (top left); moving average estimates of intensity function (with
m
= 50) (top right); estimated time transformation (middle left); moving average estimates of intensity function after time transformation (middle right); exponential QQ‐plot (5.5.31) based on
μ
‐transformed waiting times (bottom left); boxplots of monthly mean waiting times after time transformation (bottom right).
Figure 5.10 MTPL data for Company A: Simulated confidence intervals and observed cumulative counting data for fitted Cox models directed by a gamma Lévy subordinator (top left) and an inverse Gaussian Lévy subordinator (top right); mean values of locally fitted Cox gamma models using 30 days moving windows (second line) and exponential QQ‐plot based on waiting times between clusters (third line); cluster sizes of a simulated claim number process following the Cox model with the IG fitted parameters (bottom).
Chapter 06
Figure 6.1 Discretizing the claim size distribution with Δ = 1.
Figure 6.2 Dutch fire insurance claim data: Panjer bounds (and equally FFT bounds) for
F
S
based on interval splicing loss model and Poisson counting model:
M
= 2
14
and Δ = 250 000 (top);
M
= 2
17
and Δ = 25 000 (bottom).
Figure 6.3 Dutch fire insurance claim data: application of (6.2.5), (6.2.18) and (6.2.19), and FFT as approximations of
F
S
based on interval splicing loss model and Poisson counting model (top); restricted graph with cumulative probabilities larger than 0.99 (bottom).
Figure 6.4 Compound negative binomial model with gamma‐distributed claim sizes: normal approximation (6.2.5), Gram–Charlier approximation (6.2.8), asymptotic approximation (6.2.17), and FFT as approximations of
F
S
(top); restricted graph with cumulative probabilities larger than 0.95 (bottom).
Chapter 07
Figure 7.1 Sample path of a Cramér–Lundberg process
C
(
t
).
Chapter 09
Figure 9.1 Dutch fire insurance data: confidence intervals for the simulation of
for the compound Poisson case of Figure 6.2 as a function of
n
for crude MC and the Asmussen–Kroese estimator.
Figure 9.2 Estimation of VaR
0.995
(
X
) for a gamma(3.3,0.9)‐distributed
X
as a function of
n
for crude MC and an importance sampling estimator.
Figure 9.3 Confidence intervals for the simulation of
, where
X
i
are gamma(3.3,2) distributed, as a function of
n
for crude MC and the importance sampling estimator (9.2.4).
Figure 9.4 Dutch fire insurance data: confidence intervals for the simulation of
for the compound Poisson case of Figure 6.2 as a function of
n
for crude MC (top left), Asmussen–Kroese estimator
Z
SL
(
u
) (top right), and its control variate improvement
(bottom).
Figure 9.5 Confidence intervals for the simulation of
for 1 −
F
X
(
x
) = (1 +
x
)
−1.5
and
λ
= 10 as a function of
n
for
Z
SL
(
u
) and its control variate improvement
.
Figure 9.6 Two‐dimensional sequence of 250 pseudorandom numbers (left), Halton numbers with
b
1
= 2 and
b
2
= 3 (middle) and Sobol numbers (right).
Figure 9.7 Dutch fire insurance data: QMC simulation of
for the compound Poisson case of Figure 6.2 as a function of
n
for the Asmussen–Kroese estimator.
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