Measuring
Correlation
by
Roger M. Hayne, FCAS, MAAA
Milliman
Casualty Loss Reserve Seminar
September 13-14, 2004
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Statement of Problem

How can we measure correlation
in ultimate loss estimates
– Among years from a single forecast
method
– Among forecast methods
– Across lines of business
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Basic Approach
Somewhat akin to bootstrap
 Given data and selections
determine hindsight “what if”
forecasts
 Measure the correlation of the
forecasts

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Simple Development
Example

Traditional methods used for
“selection”
 Consider the age-to-ultimate factors
implied by:
– Forecast by year
– Paid at a particular age

Apply these implied factors to current
paid to get alternate estimates
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Simple Development
Example
Months of Development
Year
12
24
36
48
2001
3,343
24,806
52,054
66,203
2002
3,847
34,171
59,232
2003
6,090
33,392
2004
5,451
Selected
66,203
76,901
82,611
84,110
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Simple Development
Example
Hindsight Factors
Year
12
2001
19.803
2.669
1.272
2002
19.990
2.250
1.298
2003
13.565
2.474
2004
15.430
To Date
24
5,451
33,392
36
59,232
48
1.000
66,203
Hindsight Estimates for Year
Based On
2004
2003
2002
2001
107,949
89,118
75,332
2002
108,965
75,148
76,901
2003
73,943
82,611
2004
84,110
2001
66,203
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Some Interpretation
Hindsight factors are age to
ultimate factors implied by paid to
date and ultimate selection
 Hindsight forecast for year x
based on year y is what y would
end up being if it developed as
did year x

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Simple Development
Example
Hindsight Estimates for Year
Based On
2004
2003
2002
2001
2001
107,949
89,118
75,332
66,203
2002
108,965
75,148
76,901
2003
73,943
82,611
2004
84,110
Matrix of Covariances
1.000
-0.065
1.000
-0.065
1.000
-1.000
1.000
-1.000
1.000
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Extensions
Not limited to development factor
(chain ladder)
 Will also work with any method
for which you can generate a
hindsight matrix
 Example – Berquist/Sherman
frequency and severity method

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Incremental Severities
Months of Development
Year
12
Selected
24
36
48
Average
2001
712
4,570
5,801
3,012
14,095
2002
749
5,905
4,880
3,421
14,976
2003
1,156
5,181
5,880
3,592
15,676
2004
1,236
5,874
6,174
3,772
19,073
Trend
8.3%
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Simple Severity Example
Hindsight Future Severities
Year
12
2001
13,383
8,813
3,012
2002
14,227
8,321
3,441
2003
14,520
9,339
2004
17,837
To Date
24
1,236
6,336
36
11,535
48
0
14,095
Hindsight Estimates for Year
Based On
2004
2003
2002
2001
85,675
78,325
69,504
2002
92,047
78,817
76,901
2003
89,394
82,611
2004
84,110
2001
66,203
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Some Interpretation

Hindsight severities are future average
cost per ultimate claim ultimate factors
implied by paid to date and ultimate
selection
 Hindsight forecast for year x based on
year y adjusts the hindsight severity
for year y to year x level using trend,
adds to severity to date and then
multiplied by claim count
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Simple Severity Example
Hindsight Estimates for Year
Based On
2004
2003
2002
2001
2001
85,675
78,325
69,504
66,203
2002
92,047
78,817
76,901
2003
89,394
82,611
2004
84,110
Matrix of Covariances
1.000
0.200
1.000
0.200
1.000
1.000
1.000
1.000
1.000
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Further Applications

The hindsight matrices can be used to
test the correlation of forecasts of
various methods
 If final selection is a weighted average
of methods by year, a weighted
average of the hindsight matrices
provides a view of the correlation of
selections over years
 Can also measure correlation among
lines of business
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Correlation of Selections
Averages of Development and Severity
Hindsight Estimates for Year
Based On
2004
2003
2002
2001
66,203
2001
96,812
83,721
72,418
2002
100,506
76,983
76,901
2003
81,668
82,611
2004
84,110
Matrix of Covariances
1.000
-0.527
1.000
-0.527
1.000
-1.000
1.000
-1.000
1.000
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Correlation of Methods
2004
2003
Chain
Based On
Ladder
Chain
Severity
Ladder
Severity
2001
107,949
85,675
89,118
78,325
2002
108,965
92,047
75,148
78,817
2003
73,943
89,394
Correlation
-0.071
-1.000
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An Application
Suppose you wish to model the
distribution of reserves using a
collective risk model
 Assume you have estimates of
distributions by accident year
including estimates of the
distributions of counts and
amounts

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Collective Risk Model

Basic collective risk model:
– Randomly select N, number of claims from
claim count distribution (often Poisson, but not
necessary)
– Randomly select N individual claims, X1, X2, …,
XN
 Xi
 Only necessary to estimate distributions for
number and size of claims
 Can get closed form expressions for
moments (under suitable assumptions)
– Calculate total loss as T =
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Adding Parameter
Uncertainty

Heckman & Meyers added parameter
uncertainty to both count and severity
distributions
 Modified algorithm for counts:
– Select  from a Gamma distribution with
mean 1 and variance c (“contagion”
parameter)
– Select claim counts N from a Poisson
distribution with mean  
– If c < 0, N is binomial, if c > 0, N is
negative binomial
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Adding Parameter
Uncertainty
Heckman & Meyers also incorporated
a “global” uncertainty parameter
 Modified traditional collective risk
model

– Select  from a distribution with mean 1
and variance b
– Select N and X1, X2, …, XN as before
– Calculate total as T =   Xi

Note  affects all claims uniformly
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Why Does This Matter?

Under suitable assumptions the
Heckman & Meyers algorithm gives the
following:
– E(T) = E(N)E(X)
– Var(T)= (1+b)E(X2)+2(b+c+bc)E2(X)

Notice if b=c=0 then
– Var(T)= E(X2)
– Average, T/N will have a decreasing
variance as E(N)= is large (law of large
numbers)
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Effects of Correlation

Generally for random variables X1 and
X2 having variances:
Var(X1+X2)=Var(X1)+Var(X2)
+2Cov(X1, X2)

If positively correlated, variance of the
sum exceeds the sum of the variances
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Incorporating Correlation

If Ri is the random variable for the
reserves for year i, and ρik is the
correlation coefficient between
years j and k, then the variance of
the total reserve is
Var  R  
n

i , j 1
ij
Var  Ri  Var  R j 
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Incorporating Correlation
We can use the prior equation to
calculate the expected variance
given the estimated correlations
between accident years
 This assumes we know Var(Ri)
for each year i

Milliman
Incorporating Correlation

If we use the Heckman & Meyers
algorithm with:
–
–
–
–

Expected severity for year i = i
Severity variance for year i = i
Expected counts for year i = i
Contagion parameter for year i = ci
Then we have:

Var  Ri   i   
2
i
2
i
c 
i
2
i
2
i
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Incorporating Correlation

With the mixing parameter b we
then have for total reserves we
combine the two
 n
 n
Var   Ri    ij Var  Ri  Var  R j 
 i 1  i , j 1
2
n
 n

 
  Var  Ri   b   Var  Ri     E  Ri   
 i 1

i 1
i 1




n
Milliman
Incorporating Correlation

We can then solve for b
n

b
i , j 1
i j
ij
Var  Ri  Var  R j 


Var  Ri     E  Ri  

i 1
 i 1

n
n
2
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Alternate Approach
Can use some of the methods
outlined in Wang (PCAS, 1998)
 Normal modeling may not directly
work
 Matrix of correlations may not be
positive definite

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