A Reinsurer’s Perspective on Capital and Property Cat Pricing
Ron Wilkins, FCAS, MAAA
Vice President and Corporate Actuarial Manager
March 20, 2012
The following presentation is for general information, education and discussion purposes only, in connection with the Casualty Actuarial Society
RPM
Any views or opinions expressed, whether oral or in writing are those of the speaker alone. They do not constitute legal or
July 8,Seminar.
2010
professional advice; and do not necessarily reflect, in whole or in part, any corporate position, opinion or view of Partner Re ,
or its affiliates, or a corporate endorsement, position or preference with respect to any issue or area covered in the
presentation.
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3
Outline
 Risk Management Culture
 Attributed Capital and Pricing
 Managing Catastrophe Risk
 Property Portfolio Model
4
PartnerRe's Risk Management Culture
 Transparent risk management communication both internally and
externally
 Risk assumption and risk management at core of the company’s
value proposition
 Embedded in strategy and stated goals of the company: To satisfy
client needs and provide unquestioned ability to pay claims, while
providing attractive risk adjusted returns to shareholders
 Appreciation of risk, capital, returns, correlations, processes, limits
and controls embedded throughout organization
5
Risk Appetite
UNACCEPTABLE AT
ANY PRICE
Return
Beyond our appetite
or capacity to absorb
downside of risk
Potential creation
of value over time
for assuming
risks
Capital/Value @ Risk
Potential loss of value
6
Attributed Capital and Pricing
7
Framework for Capital Attribution
 Board and CEO risk appetite: expressed as tolerance
for loss from specific large risks to “unlikely” and
“remote” events as a percent of economic capital
 Limits and solvency thresholds
 Models – tail risk metrics, catastrophe and non-Cat
 Uncertainty – additional loads
 Capital – attributed to the product line, diversification is
considered
 Pricing – capital based on a mixture of the treaty’s
product line and its individual features
Required Capital
Capital Coverage =
board appetite
Estimate of known unmodeled loss
Economic Capital
Minimum
Required
Economic
Capital
Modeled Loss
Curve
1-in-100
1-in-250
Return Period
9
Attributed Capital: Serves Multiple Purposes
 Common view of risk across the organization achieved
through attributing capital to every treaty and
investment class
 Attributed capital forms the basis for
 Determining deployed capital
 Profitability measurement for pricing purposes
 Hindsight performance measurement and annual incentive
 Principle and process of attributing capital
 Each business unit has control over tactical capital deployment
so diversification between classes within unit considered with
the exception of catastrophe risk
 Iterative process during plan and dynamic process during
pricing
10
Tactical Capital Deployment
EXPECTED
RETURN
Hard Market
Transitional
Market
A
B
Soft Market
LOWER
RETURN
TO BUY
DOWN RISK
C
RISK
RISK REDUCTION
A = Hard Market Risk and Returns
B = Transitional Market Returns; but high risk
C = Soft Market; reduced Risk in period of narrow spreads
11
Attributed Capital: Profitability Measurement
 Attritional loss distributions and Cat event loss tables
are inputs into the Group Capital Model
 Capital charges are calibrated from model output





Additional loads reflect risk appetite and unknowns
Leverage ratios calibrated to product line level
Underwriting risk – capital to premium ratio
Reserving risk – capital to reserve ratio
Selected capital will depart from product line leverage based on
features of each individual treaty.
 Hypothetical Example – Relatively volatile transaction
 capital to premium (product line)
 capital to premium (transaction)
 capital to premium (selected)
= 75%
= 95%
= 85%
12
Managing Catastrophe Risk
13
Managing Catastrophe Risk - Introduction
 Manage the maximum foreseeable loss from any one
catastrophe event
 Manage the annual aggregate for multiple events
 Monitor and control limits for each exposure zone
 The company’s approach combines




Vendor models
Proprietary modeling
Qualitative underwriting
Maintaining a geographically diversified book of business
Managing Catastrophe Risk - Pricing Approach
Pricing is based on a balance of information from:
 Proprietary and vendor
models’ output
 Actuarial techniques
 Loss history – can be used to
calibrate vendor models
 Statistical models for poorly
modeled or non-modeled perils
 Cat risk is encapsulated in an
event loss table
Balancing quantitative
and qualitative analyses
15
Managing Catastrophe Risk - Event Loss Tables
Event Number
Mean Loss
Frequency
Description
102001
102002
102003
102004
102006
102007
102008
102009
102010
102012
102013
102014
102015
102016
102017
102018
102019
102022
102023
102024
102025
102026
102027
…
39,233,238.24
56,159,115.56
34,896,366.84
45,305,203.88
44,888,362.68
42,268,580.35
46,426,644.46
41,640,576.96
37,393,727.13
37,398,656.75
38,435,115.50
21,711,491.12
23,800,957.37
24,824,763.83
25,429,748.48
19,905,788.97
19,838,504.55
19,776,686.34
19,702,744.07
20,038,266.45
20,008,866.45
18,486,003.06
8,668,954.98
…
0.001628%
0.003015%
0.002702%
0.002741%
0.002330%
0.003178%
0.002980%
0.002169%
0.002324%
0.001895%
0.001794%
0.002704%
0.002177%
0.002150%
0.002923%
0.003125%
0.001785%
0.001693%
0.002338%
0.003079%
0.001952%
0.002508%
0.002520%
…
Class 2 Hurr FL
100,000,000
Class 4 Hurr NY
Class 3 Hurr MA
Class 5 Hurr FL
80,000,000
…
60,000,000
40,000,000
20,000,000
0
200
400
600
800
1000
 Use table to compute remote return period
losses
 Some major United States peril zones:
•South East Hurricane
•North East Hurricane
•Southern California Earthquake
•Northern California Earthquake
16
Managing Catastrophe Risk - Secondary Perils
 Earthquakes, hurricanes, and terrorist attacks are
catastrophic perils for which vendor-licensed and inhouse Cat models play a crucial role.
 Secondary perils are defined as all non Earthquake,
non Hurricane, and non Terrorism perils:
 Tornado, Hail, Convective storms
 Flood
 Freeze, Winter Storms
 Experienced based approach:




Burning costs
Actuarial methods using loss distributions
Monte-Carlo simulation of loss data
Market share / market loss analyses
17
Combining Attritional and Cat losses
 Attritional portfolio simulation model output:
 Rectangle of simulated loss outcomes
 One row for each simulated year
 One column for each product line
 Cat risk, event loss table:
 One row for each event in the model’s event set.
 One column for each cat-exposure product line.
 Combine attritional and Cat assuming independence
 Modeled output is used to create metrics and exhibits:
 Risk versus return profile of lines, business units, and the Group
 Diversification ratios
18
Combining Attritional and Cat losses
Attritional: Fire and other non-Cat perils
Natural Catastrophes:
19
Property Portfolio Model
20
Property Portfolio Model - Framework
 Begin with individual reinsurance treaties
 Combine to the product line level (using a copula)
 Combine multiple lines into a portfolio (using a copula)
 Property is combined with other lines when calculating the
Group’s economic capital.
 This results in a model of the attritional portfolio
 Catastrophe risk is modeled in event loss tables
 Combine attritional and catastrophe distributions by
assuming independence
 This results in a model of the overall portfolio, attritional
plus catastrophe. Property is a component of this model
(represented as a separate column or set of columns).
21
Property Portfolio Model - Attritional
 For attritional risk, we use a copula model. Not for Cat.
 Construct a product line loss distribution by combining
the attritional distributions of individual treaties
 Select correlations between product lines
 Simulate nominal losses from the distributions,
considering correlations among the product lines.
 Convert nominal losses into financial losses:
 Premium and expenses are treated as fixed
 Discount factors are based on the risk-free yield curve and the
duration of the product line
 Financial Loss = PV(Loss) + PV(Expenses) – PV(Premium)
22
Property Portfolio Model - Attritional Loss Distribution
 Discrete model of how annual non-Cat losses are
spread over the range of possible outcomes.
 Selecting the loss distribution for a single treaty:
 Estimate the probability that the layer will be loss free and the
probability of a full limit loss.
 Benchmark curves are based on product line and layer. If the
experience is extensive consider historical experience as a
basis for an estimate of annual variability.
 Larger subject base implies smaller relative volatility
 Consider features that significantly modify ultimate cash flows:
 Annual Aggregate Deductible
 Reinstatements
 Loss Ratio Cap
 Sliding scale commission
23
Copula Model Background
 A copula is defined as the joint cumulative distribution
function of k uniform random variables.
 Key idea: The correlation relationship is generated
separately from the marginal densities.
Gaussian copula
80% correlation
24
Copula Model Background – The Gaussian Copula
 Correlation Matrix
 k is the number of units in the model. The matrix is k by k.
 Correlation coefficients represent the closeness of outcomes
across units in the model.
 Positive semi-definite - This means that the correlations are
consistent with each other.
 The units could be treaties, product lines, or business units
which combine multiple product lines.
 More flexible than closed form multivariate models:
 Correlation assumptions are kept separate from the individual
loss distributions
 Some multivariate models only allow the user to adjust one
parameter for correlation.
25
Copula Model Background – Hypothetical Inputs
 Correlation Matrix example – between product lines:
Prop - Natl
Prop - Regl
Auto
Prop - Natl
100%
Prop - Regl
75%
100%
Auto
20%
20%
100%
Crop
25%
25%
5%
Crop
100%
 Loss distribution example
 Discrete form
 More points in the tail
 Represents a product line,
such as Property National
cumulative
probability
10%
20%
30%
40%
50%
60%
70%
80%
90%
95%
96%
97%
98%
99%
loss
ratio
28%
32%
35%
46%
58%
63%
70%
88%
115%
210%
280%
360%
450%
720%
26
Copula Model Parameters - Correlation Matrix Key Input / Assumption
 Consider correlations between loss ratios (not losses)
 Rate adequacy in different lines is subject to a market cycle
 When selecting correlations across lines consider:
 Geography: Regional versus national
 Perils: for example, drought is a crucial driver of crop insurance
results but probably irrelevant for casualty
 Macroeconomic forces: Impact premium and losses in several
lines at once
 Inflationary trends
 Tort environment trends
 Unemployment rate and GDP growth
27
Copula Model Parameters - Correlation Matrix Key Input / Assumption
 When selecting correlations across treaties consider:
 If an insurer buys a tower of several excess of loss layers, then
those layers would probably be highly correlated
 If two reinsurance treaties cover national property E&S insurers
who participate in towers on the same large buildings, that
would suggest higher correlation
 Density of risks: can a single fire destroy multiple buildings?
 Some risks that are lumped into the attritional distribution are
truly more catastrophic in nature, such as winter storm damage.
 Techniques for selecting correlation coefficients:
 Estimate using historical company or industry data.
 Expert opinion
 Stress testing – important in practice
28
Layering
20,000,000 20,000,000
$20M xs $20M: Fourth excess layer . This is shared, 50%
each, by two competing reinsurers,
Overall, the primary insurer
cedes $39M xs $1M.
10,000,000
$10M xs $10M: Third excess layer ceded to reinsurers.
5,000,000
$5M xs $5M: Second excess layer ceded to reinsurers.
4,000,000
$4M xs $1M: First excess layer ceded to reinsurers.
1,000,000
First million of each occurrence is retained by primary insurer.
The primary insurer has decided to retain the first million of each occurrence and cede the rest.
Above one million, the risk is ceded to reinsurers, split into four excess of loss layers.
It is common in practice for a single layer to be shared by several reinsurers.
Can also represent coverage for a large building, layered among Excess & Surplus insurers.
29
Copula Model Parameters - Additional Thoughts
 Gaussian is a common starting point, but modelers
should consider whether it provides a heavy enough tail
 t-copula has stronger tail correlation
 Degrees of freedom parameter – influences the degree of tail
correlation
 Like the Gaussian in that the modeler specifies the entire matrix
 Copula selection could be thought of holistically, in the
context of related assumptions:
 Correlation coefficient selection
 Loss distributions
 Handling of treaty features such as loss ratio caps
30
Goals of the Portfolio Risk-Return Model
 Monitor Reinsurance Portfolio
 Exhibits discussed at Quarterly senior management meeting
 Analysis covers the risk-versus-return position, how it has
recently changed, and where it is moving.
 Evaluate Strategic Decisions
 Typical question: “How would the risk and return change if we
significantly added volume to a particular product line?”
 Acquisitions, product line growth, and market segmentation
present strategic questions.
 Parameter assumptions are sensitivity tested.
31
Diversification Analysis: 1-in-100 Year TVaR
Financial Losses ($ MM) – Hypothetical Example
In-force 2009
In-force 2010
100
250
300
200
50
Stand Alone
(700)
490
300
200
300
350
400
80
30
100
200
200
150
300
Diversified BU
(450)
Diversified US
(300)
Stand Alone
(800)
200
40
100
150
Diversified BU
(450)
Diversified US
(310)
Diversification Benefit
BU1
BU2
BU3
Diversification Ratios
Business Unit
(1) Within BU %
(2) Across BU %
(3) Total %
Business Unit
(1) Within BU %
(2) Across BU %
(3) Total %
BU1
BU2
BU3
Overall
50
33
33
36
20
33
17
21
70
67
50
57
BU1
BU2
BU3
Overall
60
33
33
44
20
33
17
18
80
67
50
61
Stand Alone: Each product line is treated as its own business. Add TVaR across lines.
Div BU: Diversification credit is given between lines within a BU, but not between BUs.
Div US: Full diversification credit is given.
In this hypothetical example, diversification decreased from 2009 to 2010, due to a
reduction in diversification in BU1.
32
Return Period For Fixed Dollar Losses – Hypothetical Example
(By Business Unit and Overall, Financial Losses)
In-force 2010
Business Unit
25 MM
Premium Return period
in-force
in years
Market loss
50 MM
Return period
in years
Market loss
BU1
BU2
200
300
20
10
250
250
200
15
500
500
BU3
Overall
500
1000
5
5
250
250
15
10
500
500
In-force 2009
BU1
BU2
200
300
17
8
250
250
180
12
500
500
BU3
Overall
500
1000
4
4
250
250
12
8
500
500
Perspective: Look at a fixed meaningful amount of loss to the firm (such as $50 MM) and
ask how often such a full-year loss is expected.
The U.S. return period for $50 MM increased from eight to ten years.
Lengthening return periods of adverse outcomes correspond to decreasing firm risk.
33
Event Set Analysis – Integrates Cat and Non-Cat Risk
Hypothetical example centered around 100-year return period
In-force 2010 – millions USD
Quantile
98.95
98.96
98.97
98.98
98.99
99.00
99.01
99.02
99.03
99.04
99.05
Agriculture
PV Profits
PV Tech Ratio
5
(10)
(5)
20
(20)
(10)
(20)
(10)
(10)
10
0
90
100
100
80
120
110
120
110
110
80
100
Casualty
PV Profits PV Tech Ratio
(200)
(130)
(70)
(200)
(130)
(25)
(200)
0
(150)
(200)
0
140
135
110
140
135
100
130
90
130
137
100
Property
PV Profits PV Tech Ratio
1
(40)
(130)
(20)
(30)
(160)
30
(250)
(50)
(10)
(200)
All Others
PV Profits PV Tech Ratio
90
110
140
100
100
150
80
160
110
100
160
(76)
(91)
(66)
(72)
(92)
(78)
(83)
(14)
(64)
(75)
(75)
90
100
90
90
120
90
95
85
87
90
90
PV Profits
(270)
(271)
(271)
(272)
(272)
(273)
(273)
(274)
(274)
(275)
(275)
Total
PV Tech Ratio
120
121
121
122
122
123
123
124
124
125
125
In-force 2009 – millions USD
98.95
98.96
98.97
98.98
98.99
99.00
99.01
99.02
99.03
99.04
99.05
(40)
10
(20)
(10)
30
(10)
5
(80)
10
(90)
0
110
90
100
100
80
100
90
125
90
135
90
0
50
70
0
(100)
(100)
(70)
(50)
(130)
(20)
(180)
90
80
80
90
120
120
110
100
125
100
150
(200)
(300)
(300)
(200)
(200)
(150)
(150)
(100)
(100)
(120)
(80)
160
190
190
160
160
140
140
120
120
125
110
(60)
(61)
(51)
(92)
(32)
(43)
(88)
(74)
(84)
(75)
(45)
90
90
85
100
75
75
100
95
100
95
75
Event set perspective: examine specific tail events to identify what drove them.
In 2010 the 1-in-100 year event was property driven, but in 2009 it was a combined
property and casualty event.
(300)
(301)
(301)
(302)
(302)
(303)
(303)
(304)
(304)
(305)
(305)
125
126
126
127
127
128
128
129
129
130
130
34
Return versus Risk: Hypothetical Example
In-force 2010 vs. In-force 2009
ROP Excl OH (%)
25
20
BU3
Overall - 2010
15
BU2
Overall - 2009
10
BU1 - 2010
5
BU1 - 2009
0
0
10
20
30
40
50
60
70
TVaR of Financial Losses At 99% Level Divided By PV Losses
Finance perspective: plot return on the vertical axis and risk on the horizontal axis.
Movements to the Northwest are unambiguously good: lower risk and higher return
BU1 reduced risk and improved returns, which drove a portfolio-wide improvement.
The other two business units were stable.
35
Conclusion
 A simulation model of economic capital can be a key
component of a reinsurer’s capital attribution approach
 Property Cat risk needs to be integrated into the
portfolio model for purposes of:
 Tracking key risk, return, and diversification metrics
 Capital attribution
 Capital deployment can be an iterative process:
 Transaction-level analysis is incorporated into the selected Cat
event tables and attritional loss distributions
 Transactions and lines are combined to the unit and Group
levels in the portfolio model
 Management risk appetite is built into pricing leverage ratios
 Leverage ratios are key inputs to transaction pricing