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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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Index
a
adjustment coefficient
adverse development cover (ADC)
aggregate claim approximation
asymptotic
Bower’s gamma
Edgeworth
gamma
Gram‐Charlier
normal
normal‐power
orthonormal polynomials
shifted gamma
alternative risk transfer (ART)
annual aggregate deductible (AAD)
annual aggregate limit (AAL)
approximation
asymptotic
b
best estimate
bias
burning cost
c
capacity appetite limit (CAL) curve
capital market
captives
catastrophe (CAT) bonds
censoring
chain ladder
coefficient of extremal dependence
coefficient of tail dependence
coefficient of variation
combined ratio
compliance
Conditional Tail Expectation (CTE)
confidence interval
contingent convertible bonds (CoCo)
convolution
copula
cost of capital
Cramér‐Lundberg model
credibility
d
dependence
derivative plot
log‐normal
Pareto
Weibull
development time
deviance
explained
scaled
dispersion
index of
over‐
parameter
under‐
distribution
t
aggregate claim
asymptotic
beta
binomial
Burr
chi‐squared
Delaporte
empirical
Engen
equilibrium
Erlang
exponential
extended Pareto
extreme value
first Benktander
Fréchet
gamma
GB2
general inverse Gaussian
generalized Burr‐gamma
generalized Pareto
integrated tail,
see
equilibrium
inverse Gaussian
log‐gamma
log‐logistic
log‐normal
log‐Pearson
log‐
t
logarithmic
logistic
long‐tailed
long‐tailed beta
mixed Erlang
mixing
multivariate extreme value
multivariate generalized Pareto
multivariate mixed Erlang
negative binomial
normal
one‐sided Cauchy
one‐sided normal
one‐sided
t
Pareto
Paris‐Kestemont
phase‐type
Poisson
power‐ratio‐gamma
quasi‐log‐normal
second Benktander
shifted Pareto
shifted Poisson
Sichel
strict Pareto
sub‐exponential
transformed gamma
truncated Pareto
uniform
US‐Pareto
Wakeby
Weibull
dynamic financial analysis
e
EM algorithm
empirical mean excess values
equalization reserves
Esscher transform
estimator
Asmussen‐Kroese
bias‐reduced
control variate Monte Carlo
crude Monte Carlo
endpoint
extreme quantile
generalized Hill
Hill
Kaplan‐Meier
kernel
kernel weighted likelihood
large quantile
local polynomial ML
maximum likelihood
moment
Monte Carlo
penalized likelihood
quasi‐likelihood
quasi‐Monte Carlo
return period
scale
Turnbull
unbiased
weighted Monte Carlo
Weissman
excess‐of‐loss (XL)
adverse selection
attachment point
Cat XL
Cumulative
deductible
layer
moment calculations
multiline
payback period
per‐event
priority
rate on line
reinstatements
relative layer length
retention
unlimited
Expected Shortfall (ES)
experience rating
exponential family
exponential twisting
exposure curve
exposure rating
extremal coefficient
extreme value index (EVI)
f
Fast Fourier Transform
function
auto‐correlation
cumulant‐generating
likelihood ratio
link
mean excess
moment‐generating
partial auto‐correlation
probability generating
i
IBNER
IBNR
increased‐limits‐factors (ILS) curve
incurred loss
independence
industry loss warranty (ILW)
inflation
information criterion
insurance securitization
k
kurtosis
l
Laplace exponent
Laplace transform
large claims
limited liability
loss degree
loss development
loss portfolio transfer
m
max‐domain of attraction
mean squared error
memoryless property
mixtures
modelling
generalized additive
generalized linear
multivariate
non‐parametric
parametric
regression
Monte Carlo
conditional
control variates
crude
importance sampling
quasi‐Monte Carlo
moral hazard
o
objective function
open‐case estimate
see
RBNS
ordering
convex
stochastic dominance
stop‐loss
p
Panjer recursion
payback tariff
peaks‐over‐threshold (POT)
peaks‐over‐threshold, multivariate
premium
distortion principle
Esscher principle
expected value principle
expectile principle
exponential
mean value principle
properties
proportional
pure
risk‐adjusted principle
safety loading
standard deviation principle
stop‐loss
variance principle
zero utility
probable maximum loss (PML)
process
ARIMA
branching
claim counting
Cox
doubly stochastic Poisson
generalized Poisson‐Pascal
homogeneous Poisson
infinitely divisible
inhomogeneous Poisson
intensity
Lévy
log‐zero Poisson
mixed Poisson
multivariate Cox
nearly mixed Poisson
Neyman‐type A
Pólya
Pólya–Aeppli
Pascal
see
Pólya
Poisson
Poisson cluster
Poisson‐beta
Poisson‐gamma
Poisson‐inverse Gaussian
pure birth
renewal
SARIMA
shot noise
Sichel
Sparre Andersen
Yule
profit
proportional hazard transform
q
Q‐plot
exponential
generalized
log‐normal
normal
Pareto
Weibull
quasi‐Monte Carlo
r
rare event simulation
RBNS
regression fit
regular variation
reinstatements
reinsurance
administration cost
captives
cedent
change‐loss
collateralized
combining subportfolios
combining treaties
commission
deductible
Drop‐Down XL
dynamic proportional
dynamic reinsurance
dynamic XL
ECOMOR
excess‐of‐loss
facultative
financial pricing
finite risk
large claim
loss corridor
maximizing expected utility
mimizing ruin probability
minimizing retained variance
multi‐line
multi‐trigger
multi‐year
non‐proportional
obligatory
pools
proportional
quota‐share
reinsurance chains
reinsured amount
side‐cars
sliding scale commission
stop‐loss
structured
surplus
transaction cost
truncated stop‐loss
unearned premium reserve
reinsurance choice
expected utility
multi‐objective
objectives
regulatory ruin
RORAC criterion
ruin condition
security level condition
variance condition
reporting delay
reporting threshold
retrocession
return on risk‐adjusted capital (RORAC)
return period
risk
aversion coefficient
carriers
catastrophic
counterparty
foreign exchange
margin
market
measure
model
optimal sharing
Pareto‐optimal sharing
retention groups
risk margin
robustness
ruin probability
s
safety loading
self‐insurance
simulation see
Monte Carlo
skedasis function
skewness
slow variation
solvency capital
solvency capital requirement (SCR)
special‐purpose vehicle (SPV)
splicing
splines
spread loss treaties
square root rate on‐line (ROL)
stable tail dependence function
subordinator
gamma
inverse Gaussian
sum insured
t
tail quantile function
Tail‐VaR
tails of distribution
heavier than exponential (HTE)
lighter than exponential (LTE)
Pareto‐type
test
Kolmogorov–Smirnov
likelihood ratio
Shapiro–Wilk
upper‐truncation
theorem
Blackwell
Borch
central limit
thinning
truncation
u
ultimate loss
utility function
v
Value‐at‐Risk (VaR)
variance
variance reduction
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