Hidden Risks in Casualty (Re)insurance
Casualty Actuaries in Reinsurance (CARe) 2007
David R. Clark, Vice President
Munich Reinsurance America, Inc.
Agenda
Overview of Inflation Risk
3
A Model for Inflation
5
Including Inflation Variability in the Collective Risk Model
10
Including Inflation Variability in Portfolio Management
15
2
Hidden Risks in Casualty (Re)insurance
Introduction
1) Parameter Risk

Recognize uncertainty in pricing parameters
2) Adverse Selection

Which companies supply their own development patterns?

Which companies need to purchase this type of coverage?
3) Systematic Risks

Inflation  We will only cover this item

Market Cycle

Other coverage changes (e.g., WC reforms)
4) Mega-Risks (the “next asbestos”)
3
Hidden Risks in Casualty (Re)insurance
Introduction
Key Ideas:

Estimate variability due to inflation using time-series analysis*

Use this variability as a “mixing” distribution

For aggregate ground-up losses

To adjust contagion for excess frequency
*Based on Variance and Covariance Due to Inflation, in the Fall 2006 Reserves Call
Paper Program
4
Hidden Risks in Casualty (Re)insurance
Inflation Model
There are two components to the inflation variability model:
1) An expected pattern of future loss payments
 Typically on a quarterly or annual basis
2) The inflation index covering the same time horizon as the payout pattern
 Ideally this should be derived from insurance statistics
 Practically, it may be based on economic statistics (e.g., the most
relevant CPI)
5
Hidden Risks in Casualty (Re)insurance
Inflation Model
Inflation Variability for Sample Loss Payout
25.0%
200
180
160
140
15.0%
120
100
20.4%
10.0%
80
Inflation Index
Percent of Reserve Paid by Year
20.0%
18.4%
15.3%
60
11.5%
5.0%
40
8.6%
6.5%
4.8%
20
3.6%
2.7%
2.0%
0.0%
0
12
24
36
48
60
72
Payment Year
84
96
108
120
6
Hidden Risks in Casualty (Re)insurance
Inflation Model
For the inflation model, we will use a mean-reverting random walk.
X t 1 
Projected
Inflation Rate
for Time t+1
X t  r    (1  r )  et 1
Projected
Inflation Rate
for Time t
Long-Run
Average
Inflation Rate
Normal Error
Term N 0, 


Note: Xt will actually be log(1+inflation rate at time t) rather than the inflation rate
directly.
7
Hidden Risks in Casualty (Re)insurance
Inflation Model
 The idea of the mean-reverting random walk is that the inflation rate for
one year is dependent on the inflation rate in the immediately prior year,
but with a tendency back towards some long-run average rate.
 This is sometimes illustrated by a drunk whose every step could be in a
different direction, but who has at least a vague idea of how to stagger
back home.
8
Hidden Risks in Casualty (Re)insurance
Including Inflation in Collective Risk Model
Given a model of inflation and an expected payout pattern for losses, we can
estimate the variability of the loss amount.
This inflation variability is implemented as a “mixing distribution that puts
variability around the scale of the severity curve.
For the next step, we need to consider the Collective Risk Model, which
treats the number of claims (N) and the amount of each individual claim
(Xi) as random variables.
9
Hidden Risks in Casualty (Re)insurance
Including Inflation in Collective Risk Model
Variance formula for the Collective Risk Model:
Var Z   E N  Var  X   Var N   E  X 
2
This includes two key independence assumptions:
1) Frequency and severity are independent
2) Each loss (severity) is independent
If we also include the correlation between losses, then this formula is
expanded for a covariance term:
 

Var Z   E N  Var  X   E N 2  E N    Var  X   Var N   E  X 
2
covariance
10
Hidden Risks in Casualty (Re)insurance
Including Inflation in Collective Risk Model
For an excess layer, the change in loss cost is mostly due to
frequency rather than to layer severity. That will give us a way to
approximate the impact of inflation by changing just the frequency.
Layer:
500,000
excess of
Untrended
100
Trended
100
Pareto B
Pareto Q
Overall Severity
125,000
1.55
227,273
135,000
1.55
245,455
8.0%
Layer Counts
Layer Severity
Layer Loss Cost
8.3
313,899
2,590,513
9.1
315,687
2,864,008
9.9%
0.6%
10.6%
Total # of Claims
Numbers for illustration only.
500,000
Trend %
11
Hidden Risks in Casualty (Re)insurance
Including Inflation in Collective Risk Model
Variance Formulas for Ground-up (GU) and Excess (XS)
Layer Counts:
VarNGU   ENGU   c  ENGU 
2
VarN XS   EN XS   c  EN XS 
2
VarN XS   E p  NGU   c  E p  NGU 
2
c = “contagion” : constant for all layers
p = 1-F(AttPt) = probability of hitting the excess layer
12
Hidden Risks in Casualty (Re)insurance
Including Inflation in Collective Risk Model
To include variance from inflation in the excess (XS) layer
counts, we simply add a term to the “contagion”:
Var N XS 
 Var  p 
2
 E  p  N GU   c 
  E  p  N GU 
2
p 

c = “contagion”
p = 1-F(AttPt) = probability of hitting the excess layer
Var(p) = variance in the probability “p” due to inflation
13
Hidden Risks in Casualty (Re)insurance
Including Inflation in Collective Risk Model
Importance of Contract Terms:

Annual Aggregate Deductibles (AADs) are very sensitive to inflation

Limited Reinstatements can help avoid extreme fluctuations

Some protection comes from policy limits on the underlying subject
business

Indexation Clauses are another protection:

Goal is to share inflation risk equitably with the ceding company –
inflation risk is tempered but not eliminated

Potential “basis risk” in choosing a base index
14
Hidden Risks in Casualty (Re)insurance
Including Inflation in Portfolio Management
Including Inflation Variability in Portfolio Management:

The impact of inflation on an individual treaty may be small compared to
other sources of variability.

Major significance is the systematic impact: inflation will affect much of
the portfolio simultaneously. It is not a fully diversifiable risk.

The next graph illustrates that as the portfolio grows, the systematic risk
begins to dominate the variability.
15
Hidden Risks in Casualty (Re)insurance
Including Inflation in Portfolio Management
Systematic Risk dominates larger portfolios:
Systematic
Variance
Random
Variance
E[n] = .50
E[n] = 5
E[n] = 50
E[n] = 500
Numbers for illustration only.
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Hidden Risks in Casualty (Re)insurance
Conclusions:

It is possible to estimate the variability from inflation

The biggest impact from this variability is the implied correlation between
losses and between treaties – it is a SYSTEMATIC effect

One practical approach is the use of mixing parameters, such as the
“contagion“ in the Negative Binomial frequency distribution
17
Thank you very much for your attention.
David R. Clark, Vice President
Munich Reinsurance America, Inc.
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