R 2
Richard Rosengarten
May 18, 2015

• Collective Risk Model (CRM) for multiple lines of business with correlation.
• Well-Trodden Ground:
Wang
Meyers and Collaborators
Mildenhall
Homer-Rosengarten
Many Others
• Correlation: By common shock method as found in several of the references above – with a twist.
• Along the way point out some underappreciated aspects of CRM.
• Actually parameterizing simulation method consistent with the model .
Parameterizing Collective Risk Models 2

• Requirements:
Efficient as to runtime.
Efficient as to parameterization – relativity low number of parameters,
Simulate small and large losses – and reflect the appropriate dependency. Generate individual large losses and small losses in the aggregate.
Reflect correlation between lines/years.
-
Consisten t with underlying CRM.
Parameterizing Collective Risk Models 3

• CV: For any random variable 𝑌 , the coefficient of variation , or CV is
𝝂 𝑌 = 𝑉𝑎𝑟 𝑌 𝐸 𝑌
• CV is unit-less, makes for nice formulas.
•
Collective Risk Model ,
𝑍 = 𝑋
1
+ ⋯ +𝑋
𝑁
, 𝑋 𝑖 𝑖𝑖𝑑, 𝑋, 𝑁 independent.
•
Where 𝑍 = aggregate losses, 𝑋 = severity, and the random variable 𝑁 is the claim count, or “frequency”
•
Independence of 𝑋, 𝑁 could be violated by inhomogeneous data.
• Large/Small Losses – Threshold 𝑇 such that (severity) losses ≥ 𝑇 are “large”, losses <
𝑇 are “small
Parameterizing Collective Risk Models 4

• Induced CRMs
𝑍
𝐿
= 𝑋
1,𝐿
+ ⋯ +𝑋
𝑁,𝐿
, 𝑍
𝑆
= 𝑋
1,𝑆
+ ⋯ +𝑋
𝑁,𝑆
• Contagion Parameter 𝑐 = 𝑐
𝐿
= 𝑐
𝑆
– Set 𝑐 = 𝝂 2 𝑁 − . Then 𝑐 is invariant in the sense
(follows from independence if 𝑋, 𝑁 )
• Assume 𝑐 > 0 (positive contagion).
• Moments of CRM:
-
𝐸 𝑍 = 𝐸 𝑁 𝐸 𝑋 𝝂 𝑍 = 𝝂 2 𝑋 + 1 𝐸 𝑁 + 𝑐
• It follows that 𝝂 𝑍 → 𝑐 as 𝐸 𝑁 → ∞
Parameterizing Collective Risk Models 5

• Correlation: 𝝆 𝑍
𝑆
, 𝑍
𝐿
= 𝝂 𝑍
𝑆 𝝂 𝑍
𝐿
(common shock based on identical mixing distributions)
• Total Variation:
𝐸 2 𝑍 𝝂 2 𝑍 − 𝑐 = 𝐸 2 𝑍
𝐿 𝝂 2 𝑍
𝐿
− 𝑐 + 𝐸 2 𝑍
𝑆 𝝂 2 𝑍
𝑆
− 𝑐 .
Parameterizing Collective Risk Models 6

• Interval for 𝝂 𝒁 : 𝑐 + 𝐸2 𝑍𝐿
𝐸2 𝑍 𝝂 2 𝑍
𝐿
−𝑐
≤ 𝝂 𝑍 ≤ 𝑐 + 𝐸2 𝑍𝐿
𝐸2 𝑍 𝝂 2 𝑍
𝐿
−𝑐
+ 𝑇
𝐸 𝑍
1−
𝐸 𝑍𝐿
𝐸 𝑍
(*) 𝑐 ≤ 𝝂 𝑍
𝑆
≤ 𝑐 +
𝑇
𝐸 𝑍
𝑆
•
Inequality is sharp.
•
Proof : Dividing the total variation equation by 𝐸 2 𝑍 immediately gives the left-hand inequality in (*).
To prove the right-hand side, must show that
𝐸
2
𝐸 2
𝑍
𝑆
𝑍 𝝂 2 𝑍
𝑆
− 𝑐 ≤
𝑇
𝐸 𝑍
1 − 𝐸 𝑍𝐿
𝐸 𝑍
, which reduces to 𝝂 2 𝑍
𝑆
𝑇
− 𝑐 ≤
𝐸 𝑍
𝑆
Parameterizing Collective Risk Models 7

We use the following:
Fact : If 𝑌 is a non-negative random variable with support on 0, 𝑇 ,then
𝑉𝑎𝑟 𝑌 ≤ 𝐸 𝑌 𝑇 − 𝐸 𝑌 .
Proof of Fact:
𝑇
2
≥ 𝐸
4
𝑇
2
− 𝑌 gives the result.
2
=
𝑇
2
− 𝑇𝐸 𝑌 + 𝐸 𝑌 2
4
=
𝑇
2
− 𝑇𝐸 𝑌 + 𝐸 2
4
𝑌 + 𝑉𝑎𝑟 𝑌 , which
Using fact: 𝜈 2 𝑍
𝑆
− 𝑐 =
𝐸 𝑋
𝑆
2
𝐸 𝑁
𝑆
𝐸 2 𝑋
𝑆
1
=
𝐸 𝑁
𝑆
1 +
𝑉𝑎𝑟 𝑋
𝐸 2 𝑋
𝑆
𝑆
≤
1
𝐸 𝑁
𝑆
1 +
𝐸 𝑋
𝑆
𝑇 − 𝐸 𝑋
𝐸 2 𝑋
𝑆
𝑆
=
𝐸 𝑁
𝑆
𝑇
𝐸 𝑋
𝑆
=
𝑇
𝐸 𝑍
𝑆
, as required
Parameterizing Collective Risk Models 8

For sharpness note that if we hold 𝐸 𝑍
𝑆 so that 𝝂 𝑍 → 𝑐 + 𝐸2 𝑍𝐿
𝐸2 𝑍 𝝂 2 𝑍
𝐿
−𝑐 fixed while letting 𝐸 𝑁
𝑆
→ ∞ , then 𝝂 2 𝑍
𝑆
→ 𝑐
, which is the left-hand side of (*). Furthermore if we
, take 𝑋
𝑆 to be a 2-point distribution with masses at 𝑋
𝑆
= 0 and 𝑋
𝑆
= 𝑇 (with probability 𝑝 =
𝐸 𝑍
𝑆
𝐸 𝑁
𝑆
𝑇
), then equality holds for the right-hand side of (*).
Parameterizing Collective Risk Models 9

• We now assume that the claim count r.v
𝑁 is of mixed Poisson type, meaning
𝑁~𝑃𝑜𝑖𝑠𝑠𝑜𝑛 𝐸 𝑁 𝐺 , where 𝐺 is a r.v with mean 1 .
• To draw from 𝑁 :
1. Draw 𝑔 from 𝐺 .
2. Draw from 𝑃𝑜𝑖𝑠𝑠𝑜𝑛 𝐸 𝑁 𝑔 .
• 𝑉𝑎𝑟 𝐺 = 𝑐 . Will use the notation 𝐺 𝑐
• 𝑁
𝐿
, 𝑁
𝑆 are also mixed Poisson with the same mixing distribution 𝐺 .
• Example: 𝐺~𝑔𝑎𝑚𝑚𝑎 . Then 𝑁~𝑁𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝐵𝑖𝑛𝑜𝑚𝑖𝑎𝑙 .
𝐷
• Fact (“Severity is Irrelevant”): 𝑍 𝐸 𝑍 → 𝐺 𝑎𝑠 𝐸 𝑁 → ∞
Parameterizing Collective Risk Models 10

• Ref:Homer-Rosengarten (2011), Meyers-Klinker-LaLonde (2003)
• Full Info CAD (Have 𝑁, 𝑋 )
Draw from 𝑁 (i.e. draw from 𝐺 and then from 𝑃𝑜𝑖𝑠𝑠𝑜𝑛 𝐸 𝑁 𝐺 )
Draw 𝑁
𝐿 from 𝐵𝑖𝑛 𝑁, 𝑞 , where 𝑞 = 1 − CDF
𝑋
𝑇 .
𝑁
𝑆
= 𝑁 − 𝑁
𝐿
.
Draw 𝑋
1,𝐿
, … , 𝑋
𝑁,𝐿 large losses. 𝑍
𝐿
= 𝑋
1,𝐿
+ ⋯ +𝑋
𝑁,𝐿
Draw 𝑍
𝑆 from Conditional Aggregate Distribution (eg, lognormal) matching moments of 𝑍
𝑆
|𝑁
𝑆
.
𝑘 ≥ 2
𝑍 = + 𝑍
𝐿
• H-R Paper:
𝑍 𝐸 𝑍 , 𝐸 𝑍
𝑆
( 𝑍
𝐿
𝐸 𝑍
𝐿
D
) → 𝐺 . This generalizes the “severity is irrelevant” result. Also, the method generates the correct dependence between large and small losses
Parameterizing Collective Risk Models 11

• Limited Info CAD (Don’t have 𝑁, 𝑋 )
Draw from 𝐺 only.
Draw 𝑁
𝐿 from 𝑃𝑜𝑖𝑠𝑠𝑜𝑛 𝐸 𝑁
𝐿
𝐺
-
Draw large losses as previously.
Draw 𝑍
𝑆 from CAD matching first two moments of 𝑍
𝑆
|𝐺
•
Minimum Parameterization : 𝐺 𝑐 , 𝐸 𝑁
𝐿
, 𝑋
𝐿
, 𝐸 𝑍 , 𝝂 𝑍
• Can then eliminate severity, 𝑁
𝑆 from equations for first two moments of 𝑍
𝑆
|𝐺.
•
To wit, 𝐸 𝑍
𝑆
|𝐺 = 𝐺𝐸 𝑍
𝑆
, 𝝂 𝑍
𝑆
|𝐺 = 𝝂 2 𝑍
𝑆
− 𝑐 𝐺
• But, it is not automatic that this minimal parameterization is consistent with CRM
Parameterizing Collective Risk Models 12

• To address, suppose we have all the minimal parameters except 𝝂 𝑍 . We can then evaluate the lhs and rhs of inequality (*) 𝑐 +
𝐸 2
𝐸 2
𝑍
𝐿
𝑍 𝝂 2 𝑍
𝐿
−𝑐
≤ 𝝂 𝑍 ≤ 𝑐 +
𝐸 2
𝐸 2
𝑍
𝐿
𝑍 𝝂 2 𝑍
𝐿
−𝑐
+
𝑇
𝐸 𝑍
1−
𝐸 𝑍
𝐸 𝑍
𝐿
• Any choice for 𝝂 𝑍 within this interval is a) possible and b) consistent with MP CRM.
Parameterizing Collective Risk Models 13

2
Loss Parameters
Non-Cat
LoB
GL
WC
CAL
Umb
PropNon-Cat
Total Non-Cat
Cat
SmallCat
MajorCat (Net)
Total Inc Cat
Premium
110,000,000
90,000,000
40,000,000
9,000,000
300,000,000
549,000,000
65,000,000
45,000,000
22,000,000
6,500,000
175,000,000
313,500,000
E(Z) Loss Ratio n
(Z)
59.10%
50.00%
55.00%
72.20%
58.30%
57.10%
0.2000
0.2200
0.2750
0.5200
0.1600
0.1139
T
1,000,000
1,000,000
1,000,000
1,000,000
1,000,000
549,000,000 40,000,000
549,000,000
549,000,000
25,000,000
378,500,000
7.30% 0.4300 2,000,000
4.60%
68.94%
1.9000
0.1685
Inf c
0.03
0.02
0.04
0.02
0.02
E(N
L
)
3.500
3.000
0.250
3.000
14.000
23.750
E(Z
L
)
5,457,138
6,568,231
512,500
4,248,825
30,534,169
47,320,864
X
L
Empirical
Empirical
Empirical
Empirical
Empirical n
(Z
L
) E(Z
S
)
0.7349
0.7604
3.2929
59,542,862
38,431,769
21,487,500
0.7444
2,251,175
0.3734
144,465,831
0.2877
266,179,136 n
(Z
S
)
0.194
0.2065
0.2668
0.4525
0.1513
0.1096
0.16
1
10.000
-
33.750
4,000,000
-
51,320,864
Lognormal 0.43
-
N\A 25,000,000
0.2847
291,179,136
-
1.9
0.195
Parameterizing Collective Risk Models 14

2
Parameterizing Collective Risk Models 15

• Correlate LoBs modeled with MP CRM/CAD method.
• LoBs are organized into covariance groups. Only Lobs within the same covariance group co-vary with one another.
• Frequency , “severity”, and serial common shock.
Parameterizing Collective Risk Models 16

• General Idea: Common draw from mixing distribution.
• Need to allow that LoBs might have different mixing distributions.
• Solution is draw common uniforms and use these to invert the mixing distributions
( 𝑔 = 𝐹 −1
𝐺 𝑢 ).
• Remaining problem is that this will tend to generate very high correlation.
• Usual solution is to assume that 𝐺 is an independent product, ie
𝐺 𝑐 = 𝐺
1 𝑐
1
𝐺
2 𝑐
2
Then apply common shock only to 𝐺
1
.
Note that 𝑐 = 𝑐
1
+ 𝑐
2
+ 𝑐
1 𝑐
2
Parameterizing Collective Risk Models 17

• Variant is the “twisted product” 𝐺 𝑐 = 𝐺
1 𝑐
1
⋉ 𝐺
2 𝑐
2 defined by 𝐺 = 𝐺
1
𝐺
2
𝑐
2
/
Parameterizing Collective Risk Models 18

• Bring in uniforms necessary to invert 𝐺
1 covariance group and year.
’s for frequency c.s. These vary by
• Also bring in uniforms for 𝐺
2
’s – varying by LoB and year.
• Reason for 𝐺
2
’s is generate sufficient correlation between years but within LoB.
• Flip a weighted coin.
• For year 𝑗, 𝑗 ≥ 2 , if coin flip comes up “heads” use the uniforms from year 𝑗 − 1 .
Otherwise use year 𝑗 .
• Parameter – FrSerialCoVarWt – the weight for the coin flip. Can vary by covariance group or LoB. Usually by covariance group.
Parameterizing Collective Risk Models 19

• Summary
𝐺
1 correlates non-identical LoBs, both within-year and serially.
𝐺
2
- serial correlation for identical LoBs.
Serial correlation decays by FrSerialCoVarWt.
Parameterizing Collective Risk Models 20

• Really it’s c.s. applied to the conditional aggregate distribution generating 𝑍
𝑆
.
• By HR, the particular distribution family used doesn’t matter.
• Assume lognormal, with 𝝁, 𝝈 the conditional parameters.
• Parameters : ZSCoVarWt, ZSSerialCoVarWt.
• Express CAD as a product of Lognormals
• 𝐶𝐴𝐷 = logn .5𝝁, 𝝈 𝑍𝑆𝐶𝑜𝑉𝑎𝑟𝑊𝑡 logn .5𝝁, 𝝈 1 − 𝑍𝑆𝐶𝑜𝑉𝑎𝑟𝑊𝑡
• Play same game as previously.
Parameterizing Collective Risk Models 21

• Example: Identical LoBs LoB1, LoB2
•
• 𝐹𝑟𝐶𝑜𝑉𝑎𝑟𝑊𝑡 = .85
, 𝑍𝑆𝐶𝑜𝑉𝑎𝑟𝑊𝑡 = 0 , 𝐺
1
= 1 ± 𝑐 , with probabiltiy .5
.
𝑐 = 𝑂 𝝂 2
- High Correlation 𝑐 = 0 𝝂 2 ≫ 𝑐 - No Correlation
Parameterizing Collective Risk Models 22

• 𝐹𝑟𝐶𝑜𝑉𝑎𝑟𝑊𝑡 = 0 , 𝑍𝑆𝐶𝑜𝑉𝑎𝑟𝑊𝑡 = .85
, 𝑐 = 0
Parameterizing Collective Risk Models 23

• For Identical LoBs:
2
2
Parameterizing Collective Risk Models 24

• Can increase skewness by adding shift to mixing distributions.
Shifted lognormal: 𝐺 = 𝑠 + 𝐿𝑜𝑔𝑛 𝑙𝑛 1−𝑠 2 𝑐+ 1−𝑠 2
, 𝑙𝑛 1 + 𝑐
1−𝑠 2
Skewness = 𝑐
1−𝑠
3 + 𝑐
1−𝑠 2
• Can use discrete mixing distribution to create a mass at 0, for example.
Parameterizing Collective Risk Models 25

2
Parameterizing Collective Risk Models 26

2
• Casualty lines co-vary. (Covariance group 1)
• Non-Cat Property, Cats are independent (CoVar groups 2-4).
• Mixing Distributions all of form G = 𝑙𝑜𝑔𝑛 ⋉ 𝑙𝑜𝑔𝑛 except Major Cat
• Major Cat (Net of Cat XoL)
From Cat modelling know 𝑃𝑟𝑜𝑏 0 , 𝐸 𝑀𝑎𝑗𝑜𝑟 𝐶𝑎𝑡 , 𝝂, 𝛄 = 𝑠𝑘𝑒𝑤𝑛𝑒𝑠𝑠 .
Modeling Solution: G = 𝐷𝑖𝑠𝑐𝑟𝑒𝑡𝑒 𝑈𝑛𝑖𝑓𝑜𝑟𝑚 ∗ 𝑠ℎ𝑖𝑓𝑡 𝑙𝑜𝑔𝑛 , 𝑐 = 1
Parameters of Discrete Uniform (including 𝑐
1 mass at 0 .
= 𝐹𝑟𝐶𝑜𝑉𝑎𝑟𝑊𝑡) set up to match probability
Shifted lognormal set up to match skewness.
• Umbrella: parameters set up to give higher correlation with GL than other casualty lines.
Parameterizing Collective Risk Models 27

Parameterizing Collective Risk Models 28

2
• Casualty lines co-vary. (Covariance group 1)
• Non-Cat Property, Cats are independent (CoVar groups 2-4).
• Mixing Distributions all of form G = 𝑙𝑜𝑔𝑛 ⋉ 𝑙𝑜𝑔𝑛 except Major Cat
• Major Cat (Net of Cat XoL)
From Cat modelling know 𝑃𝑟𝑜𝑏 0 , 𝐸 𝑀𝑎𝑗𝑜𝑟 𝐶𝑎𝑡 , 𝝂, 𝛄 = 𝑠𝑘𝑒𝑤𝑛𝑒𝑠𝑠 .
Modeling Solution: G = 𝐷𝑖𝑠𝑐𝑟𝑒𝑡𝑒 𝑈𝑛𝑖𝑓𝑜𝑟𝑚 ∗ 𝑠ℎ𝑖𝑓𝑡 𝑙𝑜𝑔𝑛 , 𝑐 = 1
Parameters of Discrete Uniform (including 𝑐
1 mass at 0 .
= 𝐹𝑟𝐶𝑜𝑉𝑎𝑟𝑊𝑡) set up to match probability
Shifted lognormal set up to match skewness.
• Umbrella: parameters set up to give higher correlation with GL than other casualty lines.
Parameterizing Collective Risk Models 29

• NPV basis (Have also developed payout patterns by LoB).
• Low parameter model allows for efficient sensitivity testing.
• Key Stats (Reinsurer PoV):
Key Stats w\Sensitivity Testing
NPV(Profit/Loss)
Prob(Econ. Loss)
TVaR(95)
TVaR(97.5)
RoRaC (95)
RoRaC (97.5)
ERD
Base
12,851,332
11.59%
(35,406,559)
(47,754,569)
9.46%
7.53%
-4.40% c's, CVs Up
9,773,105
17.51%
(50,849,025)
(66,138,266)
6.00%
5.08%
-8.00%
CoVar Wts Up
12,281,670
12.32%
(41,391,711)
(56,187,593)
8.08%
6.48%
-5.23%
Skewness Up
12,776,752
11.73%
(35,944,163)
(48,446,883)
9.34%
7.45%
-4.52%
Parameterizing Collective Risk Models 30

R 2
Richard Rosengarten
E: Richard.Rosengarten@jltre.com
T: 215 246 1711
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Presentation Title 31