Transcript Let us consider two linear models, which described by the following

Modelling Multiple Lines of
Business: Detecting and using
correlations in reserve forecasting.
Presenter: Dr David Odell
Insureware, Australia
MULTIPLE TRIANGLE MODELLING ( or MPTF )
APPLICATIONS
• MULTIPLE LINES OF BUSINESS- DIVERSIFICATION?
• MULTIPLE SEGMENTS
– MEDICAL VERSUS INDEMNITY
– SAME LINE, DIFFERENT STATES
– GROSS VERSUS NET OF REINSURANCE
– LAYERS
• CREDIBILITY MODELLING
– ONLY A FEW YEARS OF DATA AVAILABLE
– HIGH PROCESS VARIABILITY MAKES IT DIFFICULT TO
ESTIMATE TRENDS
BENEFITS
• Level of Diversification- optimal capital
allocation by LOB
• Mergers/Acquisitions
• Writing new business- how is it related to
what is already on the books?
Model Displays for LOB1 and LOB3
LOB1 and LOB3 Weighted Residual
Plots for Calendar Year
Pictures shown above correspond to two linear models, which described
by the following equations
y 1  X 1β 1  ε 1 ,
y 2  X 2β 2  ε 2 ,
(1)
Eε i  0, i  1, 2; E ( ε 1 , ε T2 )  cov( ε 1 , ε 2 )  C; corr ( ε 1 , ε 2 )  R
Without loss of sense and generality two models in (1) could be
considered as one linear model:
 y1 
  
y2 
 X1 0   β 1   ε 1 

     
 0 X2  β2  ε 2 
(2)
Which could be rewritten as
y
Xβ
ε
For illustration of the most simple case we suppose that size of vectors y
in models (1) are the same and equal to n, also we suppose that
E ( ε i , ε Ti )  var( ε i )  I n i2 , i  1, 2;
C  I n 12
In this case
 I n 12
var( ε)  Σ  
 I n 12
I n 12 

2 
I n 2 
For example, when n = 3
  12 0 0  12 0 0 


2
 0  1 0 0  12 0 


2
0
0

0
0

1
12 
Σ
 12 0 0  22 0 0 


2
 0  12 0 0  2 0 
2 
 0 0 
0
0

12
2 

There is a difference between linear models in (1) and linear model (2).
In (1) we model separately and do not use additional information from
related trends, which we can do in model (2). To extract this additional
information we need to use proper methods to estimate the vector of
parameters . The general least squares (GLS) estimation equation
~
β  ( X T Σ 1 X) 1 X T Σ 1 y
enables us to achieve this.
However, it is necessary immediately to underline that we do not know
elements of the matrix  and we have to estimate them as well. So,
practically, we should build iterative process of estimations
~ (m) ~ (m)
β
,Σ
and this process will stop, when we reach estimations with satisfactory
statistical properties.
There are some cases, when model (2) provides the same results as
models in (1). They are:
1.
2.
Design matrices in (1) have the same structure ( they are the same
or proportional to each other ).
Models in (1) are non-correlated, another words
 12  0
However in situation when two models in (1) have common regressors
model (2) again will have advantages in spite of the same structure of design
matrices.
Correlation and Linearity
The idea of correlation arises naturally for two random
variables that have a joint distribution that is bivariate
normal. For each individual variable, two parameters a mean
and standard deviation are sufficient to fully describe its
probability distribution. For the joint distribution, a single
additional parameter is required – the correlation.
If X and Y have a bivariate normal distribution, the
relationship between them is linear: the mean of Y, given X,
is a linear function of X ie:
EY|X   α  βX
d
d
X  N (  X ,  ), Y  N ( Y ,  )
2
X
2
Y
d
X , Y  N 2 (  X , Y ,  X ,  Y ,  )
X  x, which is
x  X
X
s.d.'s away from mean.
How is the distribution of Y affected by this new information?
• Y|X = x has a normal distribution
Y | X  x  Y   Y
x  X
X
or
 Y
 Y
E (Y | X )  ( Y 
X )  (
)X
X
X
   X
and
Var (Y | X )   (1   )
2
Y
2
The slope  is determined by the correlation , and the standard
deviations and :
β   Y  X ,
where
  Cov( X , Y )  X Y .
The correlation between Y and X is zero if and only if the slope
 is zero.
Also note that, when Y and X have a bivariate normal
distribution, the conditional variance of Y, given X, is constant
ie not a function of X:
Var Y|X   
2
Y|X
This is why, in the usual linear regression model
Y =  + X + 
the variance of the "error" term  does not depend on X.
However, not all variables are linearly related. Suppose we have
two random variables related by the equation
S T
2
where T is normally distributed with mean zero and variance 1.
What is the correlation between S and T ?
Linear correlation is a measure of how close two random
variables are to being linearly related.
In fact, if we know that the linear correlation is +1 or -1,
then there must be a deterministic linear relationship
Y =  + X between Y and X (and vice versa).
If Y and X are linearly related, and f and g are functions,
the relationship between f( Y ) and g( X ) is not necessarily
linear, so we should not expect the linear correlation
between f( Y ) and g( X ) to be the same as between Y and
X.
A common misconception with correlated
lognormals
Actuaries frequently need to find covariances or correlations
between variables such as when finding the variance of a
sum of forecasts (for example in P&C reserving, when
combining territories or lines of business, or computing the
benefit from diversification).
Correlated normal random variables are well understood.
The usual multivariate distribution used for analysis of
related normals is the multivariate normal, where correlated
variables are linearly related. In this circumstance, the usual
linear correlation ( the Pearson correlation ) makes sense.
However, when dealing with lognormal random variables
(whose logs are normally distributed), if the underlying
normal variables are linearly correlated, then the correlation
of lognormals changes as the variance parameters change,
even though the correlation of the underlying normal does
not.
All three lognormals below are based on
normal variables with correlation 0.78,
as shown left, but with different standard
deviations.
We cannot measure the correlation on the log-scale and apply that
correlation directly to the dollar scale, because the correlation is
not the same on that scale.
Additionally, if the relationship is linear on the log scale (the
normal variables are multivariate normal) the relationship is no
longer linear on the original scale, so the correlation is no longer
linear correlation. The relationship between the variables in
general becomes a curve:
X1 , X 2 ~ N ( 1 ,  2 , 1 , 2 ,  )
Y1  exp( X 2 ), Y2  exp( X 2 )
corr( Y1 ,Y2 ) 
exp(  1 2 )  1
(exp(  12 )  1 )(exp(  22 )  1 )
Note that the correlation of Y1 and Y2 does not depend on the ’s
When the standard deviations are close to zero it is just below  but
decreases further as s.d.’s increase.
Weighted Residual Plots for LOB1 and LOB3 versus
Calendar Years
What does correlation mean?
Model Displays for LOB1 and LOB3 for Calendar Years
Model for individual iota parameters
ˆ1 ~ N 1 ,  12  ;
ˆ2 ~ N  2 ,  22  ;
ˆ1  0.1194; ˆ1  0.0331
ˆ 2  0.0814; ˆ 2  0.0321
 1 
 0.1194  ˆ  0.001097 0.000344 
  ~ N μ, Σ, μˆ  
, Σ  

 0.0814 
 0.000344 0.001027 
2 
  corr (1 ,2 ),
ˆ  0.359013
There are two types of correlations involved in
calculations of reserve distributions.
Weighted Residual Correlations between
datasets:
0.35 – is weighted residual correlation
between datasets LOB1 and LOB3;
Correlations in parameter estimates:
0.32 – is correlation between iota
parameters in LOB1 and LOB3.
These two types of correlations induce correlations
between triangle cells and within triangle cells.
Common iota parameter in both triangles
 ~ N  ,   ; ˆ  0.0996; ˆ  0.0267
2
Two effects:
Same parameter for each LOB increases
correlations and CV of aggregates
Single parameter for each line reduces CV of
aggregates
Forecasted reserves by accident year, calendar year and total
are correlated
Indicates dependency through residuals’ and parameters’
correlations
Indicates dependency through parameter estimate
correlations only
Dependency of aggregates in aggregate table
In each forecast cell and in aggregates
by accident year and calendar year
(and total)
Var(Aggregate) >> Var(LOB1) + Var(LOB3).
Correlation between reserve distributions is 0.82
Payment Stream Illustration
Year
0
1
…
t
Payment
Stream
100
100*exp()
…
100*exp(*t)
 ~ N (  ,  2 );   0.1,   0.027
Then
E100 * exp( * t )  100 * exp(  * t  0.5 * 2 * t 2 ),
StDev100 * exp( * t )  100 * exp(  * t   2 * t 2 ) 1  exp(  2 * t 2 ) .
Quantiles for aggregate of
both lines according to
model which accounts for
the correlation.
Quantiles for aggregate based
on summing two independent
models. (Note that these
values are lower.)
Simulations from lognormals correlated within LOB and
between LOBs
Density Comparisons (Acc. Year: Total)
Histogram
Kernel
Lognormal
Gamma
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
1.2 1.4 1.6 1.8
2
2.2 2.4 2.6 2.8
3
1 Unit = $1,000,000,000
Diagnostic for validity of using parametric distributions for forecast total.
GROSS VERSUS NET
Is your outward reinsurance program optimal?
(eg. Excess of Loss Reinsurance)
Model Display 1
6% stable trend in calendar year
Model Display 2
Zero trend in calendar year.
Therefore ceding part that is growing!
Weighted Residual Covariances Between Datasets
Weighted Residual Correlations Between Datasets
Weighted Residuals versus Accident Year
Gross
Weighted Residuals versus Accident Year
Net

Weighted Residual Normality Plot
Accident Year Summary
CV Net > CV Gross – Why? It is not good for cedant?
Example of Risk Capital Allocation
as a function of correlation
Simple example with 2 LOBs
• LOB1
Mean = 250
SD = 147
Loss
Probability
100
200
450
0.33333
0.33333
0.33333
100
450
800
0.33333
0.33333
0.33333
• LOB2
Mean = 450
SD = 285
Loss
Probability
Total of means = Mean Reserve = 700
Total losses for combined LOBs
LOB1
100
100
200
200
450
300
550
LOB2
450
550
650
900
(200)
800
900
(200)
1000
(300)
1250
(550)
The yellow cells are those in which the mean reserve has been exceeded.
The red numbers show the amount of this excess.
Probabilities in matrix of outcomes depend on
the correlation between the two LOBs
LOB1
100
LOB1
200
100
450
P=1/3
100
200
300
550
650
900
LOB2
LOB2
P=1/3
450
100
550
450
P=1/3
800
900
1000
“Correlation” =1.0
1250
Very poor diversification
800
200
450
P=1/9
P=1/9
P=1/9
200
300
550
P=1/9
P=1/9
P=1/9
550
650
900
P=1/9
P=1/9
P=1/9
900
1000
1250
“Correlation” =0.0
Good diversification
Probabilities in matrix of outcomes depend on
the correlation between the two LOBs
LOB1
100
LOB1
200
100
450
P=1/3
100
200
300
550
650
900
LOB2
LOB2
P=1/3
450
100
550
450
P=1/3
800
900
1000
“Correlation” =1.0
1250
Probability (Loss > Mean Reserve) = 1/3
Amount of excess loss = 550
LOB1 loss = 450 = 200 + mean(LOB1)
LOB2 loss = 800 = 350 + mean(LOB2)
Contribution to loss.
LOB1:LOB2 = 1:1.75
800
200
450
P=1/9
P=1/9
P=1/9
200
300
550
P=1/9
P=1/9
P=1/9
550
650
900
P=1/9
P=1/9
P=1/9
900
1000
1250
“Correlation” =0.0
Probabilities in matrix of outcomes depend on
the correlation between the two LOBs
LOB1
100
LOB1
200
100
450
P=1/3
100
200
300
550
650
900
LOB2
LOB2
P=1/3
450
100
550
450
P=1/3
800
900
1000
“Correlation” =1.0
1250
Probability (Loss > Mean Reserve) = 1/3
Mean Excess loss = 550 (E.S. at 67%)
LOB1 loss = 450 = 200 + mean(LOB1)
LOB2 loss = 800 = 350 + mean(LOB2)
Contributions to loss:
LOB1:LOB2 = 1:1.75
800
200
450
P=1/9
P=1/9
P=1/9
200
300
550
P=1/9
P=1/9
P=1/9
550
650
900
P=1/9
P=1/9
P=1/9
900
1000
1250
“Correlation” =0.0
Probability (Loss > Mean reserve)=4/9
Mean Excess loss = 312.5 (E.S. at 67%)
LOB1 Cond. Exp. Shortfall = 50
LOB2 Cond. Exp. Shortfall = 262.5
Contributions to mean loss:
LOB1:LOB2 = 1:5.25
Concept of Conditional Expected
Shortfall
• Loss for LOBi = li
i= 1,2,3..
• Expected Shortfall at nth percentile =
E(li -K| li > K), where Pr(li < K)=n/100.
• Assume LOBi is reserved at ri . Conditional
Expected Shortfall at nth percentile =
E(li - ri | l1 +l2+l3 ..> K),
where Pr(l1 +l2+l3 ..< K)=n/100.
Risk Capital allocation should depend on the expected
shortfalls (ES) of the LOBs under a given scenario.
Percentile of loss
67
78
89
ES(LOB2)/ES(LOB1) when corr =1
1.75
1.75
1.75
ES(LOB2)/ES(LOB1) when corr = 0
5.25
4.67
1.75
SD(LOB2)/SD(LOB1)
1.94
1.94
1.94
The contribution of the two lines depends on their correlation as
well as the percentile at which we are calculating the ES.
Example 2. Risk Capital Allocation ratio for two LOBs with
lognormal distributed losses.
LOB1 has 1=100, 1= 50
Ratio of Risk Capital allocation LOB2/LOB1
7
LOB2 has 2=150, 2=100
6
2/12.0
5
4
rho = 0.2
rho = 0.5
rho = 0.8
3
2
1
0.
6
0.
63
0.
66
0.
69
0.
72
0.
75
0.
78
0.
81
0.
84
0.
87
0.
9
0.
93
0.
96
0.
99
0.
51
0.
54
0.
57
0
Quantile loss for L1+L2
Results based on 200000 simulations.
Correlation of the
underlying normals = rho
Each LOB is reserved at the
mean.
MODELING LAYERS
• Similar Structure
• Highly Correlated
• Surprise Finding!
CV of reserves limited to $1M is the same as
CV of reserves limited to $2M !
Model Display for All 1M: PL(I)
Model Display for All 2M: PL(I)
Model Display for All 1Mxs1M: PL(I)
Note that All 1Mxs1M has zero inflation, and All 2M has lower inflation than
All 1M, and distributions of parameters going forward are correlated
Residual displays vs calendar
years show high correlations
between three triangles
.
Weighted Residual Covariances Between Datasets
Weighted Residual Correlations Between Datasets
Compare Accident Year Summary
Consistent forecasts based on composite model.
If we compare forecast by accident year for Limited 1M
and limited 2M it is easy to see that CV is the same.
Breaking up a triangle
0
data
=
0
+
data
1. Change of mix of business
2. Different development and/or inflation
3. Different process variability
Layer 1: Limit 100k
Composite (Layer 1 & Layer 2)
Layer 1: Limit 100k
Layer 2: Limit 200k
Layer 2: Limit 200k
Layer 1: Limit 100k
Layer 2: Limit 200k
Layer …
More information will be available
at the Warner Center
rd
3 floor
on Monday, September 10, 2006
7.30pm-12.00am
ICRFS-ELRF will be available for
FREE!