Transcript Document

Correlation and Aggregation in
Stochastic Loss Reserve Models
Glenn Meyers
ISO Innovative Analytics
CLRS – September 2007
Drivers of Correlation
A Simple Example
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Select X1 and X2 independently at random
Select B1, B2 and B at random
B1X1 and B2X2 are not correlated
BX1 and BX2 are correlated
– Bad things (high B) happen at the same time
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B * X2
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B2 * X2
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Drivers of Correlation
B Causes Correlation
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B1 * X1
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B * X1
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Drivers of Correlation
B Increases Volatility of Sum
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0 50
Frequency
Histogram of B1 * X1 + B2 * X2
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B1 * X1 + B2 * X2
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0
Frequency
Histogram of B * X1 + B * X2
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B * X1 + B * X2
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A Stochastic Loss Reserve Model
• XAY,Lag is the loss paid in (AY,Lag)
• E[XAY,Lag] = f(PremiumAY,ELR,a,b)
– PremiumAY is given
– ELR = Expected Loss Ratio
– a and b are parameters of the b distribution
• XAY,Lag ~ Collective Risk Model
– Count distribution is
• Negative binomial or
• Negative multinomial
– Claim severity increases with settlement lag
Expected Loss Model
E  X AY ,Lag   PremiumAY  ELR  Dev Lag
Dev Lag
 Lag

 Lag  1

 β
| a, b   β 
| a, b 
 10

 10

b is the cdf of the beta distribution
Negative Binomial Model
for Claim Count
• Given expected claim count lAY,Lag
• Select CAY,Lag from a gamma distribution
with mean 1 and variance c
• NAY,Lag~ Poisson with mean CAY,Lag lAY,Lag
• NAY,Lag’s are:
– independent for all (AY,Lag) pairs
Negative Multinomial Model
for Claim Count
• Given expected claim count lAY,Lag
• Select CAY from a gamma distribution
with mean 1 and variance c
• NAY,Lag~ Poisson with mean CAY lAY,Lag
• NAY,Lag’s are:
– Correlated within a given AY
– Independent between different AY’s
Using Model to Predict the
Distribution of Outcomes
• Model predicts the distribution of each XAY,Lag
• We need to find sum over future payments:
Outcome 
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10
 
AY  2 Lag 12 AY
X AY ,Lag
• To evaluate the distribution of the sum we
need to consider correlations
• More precisely we need to consider the
structure of the drivers of correlation
Using FFT’s to Calculate
Distribution of Sums
Independent
FFT  X  Y   FFT  X   FFT Y 
Dependence Driven by q
FFT  X  Y   Eθ FFT  X | θ   FFT Y | θ 
Parameters Used in Example
• Realistic - Derived from fitting real data
– My call paper from last year’ CLRS
“Estimating Predictive Distributions for Loss
Reserve Models”
• Parameters
– Premium = $50 million
– Claim severity distributions given
– c = 0.01
– ELR = 0.807
– a = 1.654
– b = 4.780
Negative Binomial/Multinomial
Independent by Lag
Correlated by Lag
Probability Density
Thicker tails for the
negative multinomial
model
Correlation makes a
difference!
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Reserve Outcomes (000,000)
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Correlation Driven by
Parameter/Model Risk
• The choice of ELR, a and b parameters
affects all XAY,Lag’s
• A good estimation procedure provides a
distribution of possible estimates
– Used Gibbs sampler (a Bayesian method)
– To be described in future paper
• Random parameters drives correlation
Parameters Generated by
Gibbs Sampler
Histogram of a
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10 15 20 25
Frequency
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5 10
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Frequency
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Histogram of ELR
0.78
0.82
0.86
1.50
1.60
1.70
1.80
ELR
a
Histogram of b
Sample Development Factors
2.0
0.0
1.0
Incremental Paid %
25
15
5
0
Frequency
3.0
0.74
4.0
4.5
5.0
b
5.5
2
4
6
Settlement Lag
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10
Negative Multinomial Model
Probability Density
Without Parameter Risk
With Parameter Risk
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Reserve Outcomes (000,000)
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Parameter Risk Across Two Lines
• For demonstration purposes, treat the above as
two separate lines.
• Parameters sets can be selected:
– Simultaneously
Eθ FFT   X AY ,Lag | θ   FFT   X AY ,Lag | θ  
– Independently
Eθ FFT   X AY ,Lag | θ    Eθ FFT   X AY ,Lag | θ 
Combining Two Identical Lines
Probability Density
Independent Sum
Para. Paired Sum
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Reserve Outcomes (000,000)
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Summary Statistics Relevant
to Capital Management
• Standard Deviation Capital @ 99.5%
= 2.576 × Standard Deviation
• Tail Value at Risk @ 99%
= Average Loss over the 99th percentile
• TVaR Capital = TVaR @ 99% – Expected Value
Summary Statistics for Each Model
Model
Neg. Bin & Fixed Parm
Neg Mult & Fixed Parm
Neg Mult & Random Parm
Mean
83.76
83.76
83.92
Random Sum
Paired Sum
167.84
167.84
Std Capital TVaR Capital
17.07
18.51
19.66
21.50
23.25
25.75
32.89
37.25
35.78
40.37
• Model dependencies (i.e. correlation) can have
a significant effect in capital management