CAS RATEMAKING AND PRODUCT MANAGEMENT
SEMINAR- MARCH 2012
RR-2: RISK LOAD/COST OF CAPITAL: HOW
REINSURERS CONSIDER THESE IN
REINSURANCE RATES FOR PROPERTY CAT
COVERS
Ira Robbin, PhD

 The Casualty Actuarial Society is committed to adhering strictly to the letter and spirit of the antitrust laws. Seminars conducted under the auspices of the CAS are designed solely to provide a forum for the expression of various points of view on topics described in the programs or agendas for such meetings.
 Under no circumstances shall CAS seminars be used as a means for competing companies or firms to reach any understanding – expressed or implied – that restricts competition or in any way impairs the ability of members to exercise independent business judgment regarding matters affecting competition.
 It is the responsibility of all seminar participants to be aware of antitrust regulations, to prevent any written or verbal discussions that appear to violate these laws, and to adhere in every respect to the CAS antitrust compliance policy.
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




No statements of the Endurance corporate position or the position of any prior employers will be made or should be inferred.
No liability whatsoever is assumed for any damages, either direct or indirect, that may be attributed to use of the methods discussed in this presentation.
Writing CAT covers is risky – results may be catastrophic to your bottom line.
Examples are for illustrative purposes only. Do not use the results from any example in real-world applications.
There may be a quiz at the end – so pay attention!
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







CAT Pricing Process Fundamentals
 Event Loss Table
 Random Trials
CAT Context
Pricing Overview
Basic Equations
Required Capital Paradigms
Order Dependence and Reference Portfolios
Risk Measures
 Properties
 Take your pick
 Ranking definitions of Var and TVaR
Conclusions
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4

Event
Rank Peril Region
3
4
5
1
2
8
9
6
7
10
.
.
.
1998 HU
EQ
.
.
.
1999 HU
2000 HU
EQ
EQ
HU
EQ
HU
EQ CA
EQ PNW
HU FLA
EQ PNW
CA
CA
FLA
CA
LA
AK
.
.
.
NC
FL
SC
Annual
Prob
0.021%
0.040%
0.080%
0.070%
0.045%
0.055%
0.006%
0.150%
0.010%
0.025%
.
.
.
2.000%
4.000%
3.000%
Treaty A
Loss
300
0
0
900
0
700
0
0
0
0
.
.
.
0
0
0
Treaty B
Loss
.
.
.
1,200
1,000
0
400
0
0
400
550
0
0
0
0
0
Treaty C
Loss
.
.
.
5,500
2
2
0
3,000
0
2,100
0
0
700
500
100
900
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
Total
Portfolio
Loss
125,000
100,000
90,000
80,000
75,000
70,000
60,000
50,000
50,000
40,000
.
.
.
3
2
1
5

k
Event Rank
1
2
5
6
3
4
7
8
9
10
.
.
.
1998
1999
2000
EQ
.
.
.
HU
HU
HU
Peril
EQ
EQ
HU
EQ
HU
EQ
EQ
HU
EQ
AK
.
.
.
NC
FL
SC
Region
CA
CA
FLA
CA
LA
CA
PNW
FLA
PNW p(k)
Annual Prob
0.021%
0.040%
0.080%
0.070%
0.045%
0.055%
0.006%
0.150%
0.010%
.
.
.
0.025%
2.000%
4.000%
3.000%
EP(k)
Exceeding
Probability
0.021%
0.061%
0.141%
0.211%
0.256%
0.311%
0.317%
0.466%
0.476%
0.501%
24.000%
27.040%
29.229%
Portfolio
Event Loss
125,000
100,000
90,000
80,000
75,000
70,000
60,000
50,000
50,000
.
.
.
40,000
3
2
1
EP(k) = Prob(Largest Loss in a Year will equal or exceed the kth largest loss in the event table)
(

1)

( )
 
( ))

(

1)
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9
10
.
.
.
9998
9999
10000
7
8
5
6
Trial
Year Event 1 Event 2 Event 3
3
4
1
2
40,000
2,100
-
5,500
-
3,500
-
27,550 -
-
-
450
700
1,250
8,750
75
400
900
-
45
50
25
-
70,000
-
15
.
-
3,500
.
-
45
.
25
550
650
-
7,750
-
-
-
-
…
-
-
Largest Event over the Year
40,000
3,500
0
27,550
.
.
.
700
1,250
8,750
70,000
0
3,500
25
7,750
650
Total Annual
Loss
40,000
6,050
0
33,050
.
.
.
1,150
2,175
8,750
70,120
0
3,560
25
8,300
650
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


Trial Year Rank
1
2
3
6
7
4
5
8
9
.
.
10
.
99
100
101
.
.
.
9998
9999
10000
100/10000 = 1.0%
100 year return period
AEP PML =36,675
OEP PML= 21,000
PML = Probable Maximum Loss
AEP = Annual Exceeding Probability
OEP = Occurrence Exceeding Probability
Largest Event Total Annual Loss
125,000 175,000
125,000
125,000
170,000
165,000
100,000
100,000
100,000
90,000
137,500
135,000
130,000
125,000
90,000
90,000
90,000
115,000
110,000
110,000
.
.
.
.
.
.
21,250
21,000
21,000
.
.
.
-
-
-
.
.
.
37,500
36,675
35,950
0
0
0
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




CAT Pricing is part of the process of writing CAT business, but not the only part.
Pricing models give indications – the market sets the price.
Risk Management sets limits on PMLs and
TIV/Limit Aggregations by peril/zone .
 Compliance monitoring essential
Business bunched –lots of 1/1s. Quotes go out with authorized shares. By the time an account is bound, the portfolio has changed.
Selection problem is constrained optimization:
Reinsurers looks to get most profitable portfolio with smallest risk. No one prices that way.
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


Emerald City Pricing: Don’t look at that man behind the curtain
 Different vendor models
 Some adopt new versions –others wait.
 Differences in data quality
 Loading factors
Non-modeled CAT events (Thai flood): Not always priced
 Ostrich Excuse - “It was not in the model”
 Hiding-in-Plain-Sight Swan - May not show up on risk management radar – obvious after the fact.
Pricing Method Flavors: Different ways of translating model stats into indicated prices.
 Can’t we just all agree?
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



P= E[X]+ RL(X)
P = Indicated premium prior to expense loading
X = CAT Loss
RL(X) = Risk Load
RL(X) = r target
*C(X)
C(X) = Required Capital
RORAC Approach
 Universally used in actual CAT Treaty pricing
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


Standalone: C(X) = r
(X) , where is r
(X) is a risk measure.
Incremental: Let T be the existing portfolio
C(X|T) = r
(T+X) r
(T)
,
Real Allocation
C(X|T) = A(X,T) * r
(T+X)
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



 Order Dependence – Pricing depends on the order in which accounts are priced (Mango)
 Universe A : Zoe’s CAT Treaty is priced first at $100 then Jessica’s CAT Treaty is priced next at $150
 Universe B: Jessica’s CAT Treaty is priced first at $100 then Zoe’s CAT Treaty is priced next at $150
A major problem for Incremental methods
A small problem for Allocation methods
Not a problem for Standalone
Reference Portfolio Cure
 Portfolio fixed over a given period
 How often should it be updated??
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 A risk measure, r
, is a monotonic function that maps a real-valued random variable, X, to a non-negative number, r
(X), such that:

1.
2.
Risk Measure Basic Properties
Non-negative: r
(X) ≥ 0
Monotonic Premium: If X
1
E[X
1
]+ r
(X
1
)

E[X
2
]+ r
(X
2
)

X
2
, then
 A risk measure is pure if it maps constants to zero: r
(c) = 0
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1.
2.
3.
Scalable: r
( l
X) = lr
(X)
Translation Invariant: r
(X+ a
) = r
(X)
Subadditive: r
(X
1
+ X
2
) ≤ r
(X
1
) + r
( X
2
)
 Some academicians refuse to refer to a function as a risk measure unless it is coherent
 Most academicians uses reverse signs ( X represents the value of assets instead of CAT losses)
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4.
5.
6.
1.
2.
3.
7.
8.
9.
m) 2 ]
Semivariance: Var + (X): E[(Xm) 2
| X≥ m
]*Prob(X ≥ m
)
Standard Deviation: s
= Var ½ (X)
Semi Standard Deviation: s 
= Var

½ (X)
Value at Risk: for 0< < 1 , VaR( ) = sup{x| F(x)≤ }
Tail Value at Risk: TVaR( ) = conditional mean for all x values associated with the tail, 1- , of probability
Excess Tail Value at Risk: XTVaR( ) = TVaR( ) m
Distortion Risk Measure: (Wang) E*[X] = E[X*] where
F*(x) = g(F(X)) for g a distortion function
Excess Distortion Risk Measure: E*[X] –E[X]
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18


Let X
1
≥ X
2 …
≥ X n
Suppose k = (1 - )n be an ordering of n trials of X
VaR
TVaR

  X k
 k
1 j k 

1
X j


Note TVaR is not necessarily equal to the Conditional
Tail Expectation (CTE) when the data is discrete.
CTE( ) = E[X|X>VaR( ) ]
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Statistic
Trials
Pct
Rank
Loss Data by Trial
6
7
4
5
8
Trial
1
2
3
9
10
A
8.00
0.00
0.00
0.00
1.00
2.00
7.00
2.00
4.00
4.00
Value
10
50%
5
Results
Mean
VaR
TVaR
CTE
Ref A+Ref
12.00
37.00
36.00
20.00
37.00
36.00
35.00
33.00
17.00
16.00
33.00
27.00
14.00
35.00
34.00
19.00
23.00
35.00
31.00
18.00
Ordered Loss Data
Rank
1
2
3
9
10
6
7
4
5
8
A
8.00
7.00
4.00
4.00
2.00
2.00
1.00
0.00
0.00
0.00
A
2.80
2.00
5.00
5.75
Ref A+Ref
26.00
28.80
33.00
34.80
36.00
34.00
35.40
35.75
Ref A+Ref
37.00
36.00
35.00
37.00
36.00
35.00
33.00
33.00
27.00
17.00
16.00
14.00
12.00
35.00
34.00
31.00
23.00
20.00
19.00
18.00
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Statistic
Trials
Pct
Rank
Value
10
50%
5
Results
Mean
VaR
TVaR
A
2.80
2.00
5.00
Ref A+Ref
26.00
33.00
34.80
28.80
37.00
39.40
Loss Data by Trial
5
6
3
4
Trial
1
2
9
10
7
8
A
0.00
0.00
8.00
7.00
4.00
2.00
0.00
4.00
2.00
1.00
Ref A+Ref
12.00
37.00
12.00
37.00
36.00
35.00
33.00
17.00
44.00
42.00
37.00
19.00
16.00
33.00
27.00
14.00
16.00
37.00
29.00
15.00
Ordered Loss Data
Rank
1
2
5
6
3
4
9
10
7
8
A
8.00
7.00
4.00
4.00
2.00
2.00
1.00
0.00
0.00
0.00
Ref A+Ref
37.00
36.00
44.00
42.00
35.00
33.00
33.00
27.00
37.00
37.00
37.00
29.00
17.00
16.00
14.00
12.00
19.00
16.00
15.00
12.00
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1.
Stand-alone Risk Measure as Allocation Base
2.

Marginal Risk Measure as Allocation Base
Adjusted for Order Dependence (Mango)
3.
4.
5.
Game theory –(LeMaire) Allocation of Portfolio
Consolidation Benefit
Co-Measures – (Kreps)
Percentile Allocation (Bodoff)
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21

98
99
100
101
102
10,000
Rank
1
VaR
Percentage
Portfolio
Loss
Risk A
Loss
99.02%
99.01%
99.00%
98.99%
98.98%
$422
$408
$405
$395
$390
$6
$0
$20
$0
$4
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



Target return on required capital is the basis for reinsurer pricing indications.
Debate is over required capital
A profusion of methods and approaches
Tail focus and portfolio dependence are key areas of disagreement.
 Some of the most popular methods fail to satisfy one or more basic properties
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