Variance and Covariance Due to Inflation
David R. Clark
Casualty Loss Reserve Seminar – September 2006
Agenda
Statement of the Correlation Problem
3
Graphical Description of Inflation Applied to Loss Payout
5
The Mean-Reverting Random Walk Model
10
What Can Be Accomplished with this Model
15
2
Variance and Covariance Due to Inflation
The Problem:
 We wish to estimate the covariance between two or
more reserve segments.
But…
 The development triangles themselves do not
provide a sufficient basis for reliably estimating this
covariance.
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Variance and Covariance Due to Inflation
A Solution:
 From first principles, we begin by asking “why do we
think that there should be covariance and not simple
independence?”
 One reason for thinking this is that there are external
forces, such as inflation, that will affect all of the
reserve segments to greater or lesser degrees.
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Variance and Covariance Due to Inflation
Inflation Variability for Sample Loss Payout
25.0%
200
180
160
140
15.0%
120
100
20.4%
10.0%
80
Inflation Index
Percent of Reserve Paid by Year
20.0%
18.4%
15.3%
60
11.5%
5.0%
40
8.6%
6.5%
4.8%
20
3.6%
2.7%
2.0%
0.0%
0
12
24
36
48
60
72
Payment Year
84
96
108
120
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Variance and Covariance Due to Inflation
Two observations as to how inflation should behave:
 There is more variability in the inflation index as the
time horizon increases.
 There is an implicit correlation between payments.
For example, if the ninth year is higher than average,
then the tenth year should also be higher than
average.
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Variance and Covariance Due to Inflation
The actual inflation index could be higher or lower than
expected, and each annual payment would move
with the corresponding point on the index.
Similarly, if there are two payout patterns, then the
annual payment for each of the two patterns would
move with the corresponding point on the index.
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Variance and Covariance Due to Inflation
Inflation Variability for Sample of Two Loss Payouts
25.0%
200
180
160
140
15.0%
120
100
23.5%
10.0%
80
Inflation Index
Percent of Reserve Paid by Year
20.0%
60
12.8%
10.6%
5.0%
40
8.9%
7.4%
6.1%
5.1%
4.3%
20
3.6%
3.0%
0.0%
0
12
24
36
48
60
72
Payment Year
84
96
108
120
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Variance and Covariance Due to Inflation
We can start by thinking about this in a simulation
context. We generate a set of simulated inflation
projections and then adjust the payout patterns by
each of those projections.
So we must define a model from which to pull our
simulated values.
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Variance and Covariance Due to Inflation
For the inflation model, we will use a mean-reverting
random walk.
X t 1 
Projected
Inflation Rate
for Time t+1
X t  r    (1  r )  et 1
Projected
Inflation Rate
for Time t
Long-Run
Average
Inflation Rate
Normal Error
Term N 0, 


Note: Xt will actually be log(1+inflation rate at time t) rather than the inflation rate
directly.
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Variance and Covariance Due to Inflation
The idea of the mean-reverting random walk is that the
inflation rate for one year is dependent on the
inflation rate in the immediately prior year, but with a
tendancy back towards some long-run average rate.
This is sometimes illustrated by a drunk whose every
step could be in a different direction, but who has at
least a vague idea of how to stagger back home.
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Variance and Covariance Due to Inflation
The mean-reverting random walk model is also known
as a first order autoregressive model, or AR(1).
More simply, it can be recognized as simple linear
regression. All of the parameters, including the
standard error, are easily calculated via least
squares.
X t 1 
X t  r    (1  r )  et 1
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Variance and Covariance Due to Inflation
For a simulation model, the steps are therefore:
 Perform linear regression on an historical inflation index to
estimate the model parameters.
 Generate a sequence of normal random variables, one for
each year in the payment stream.
 Plug the random numbers into the regression model to
calculate an inflation index.
 Apply this inflation index to the corresponding future
payments.
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Variance and Covariance Due to Inflation
Even though we have explained this model by talking
about simulated values, that type of approximation is
not necessary. The variance and covariance terms
can be calculated directly; which is given in my call
paper.
Don‘t simulate if you can calculate.
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Variance and Covariance Due to Inflation
Given the parameters of the inflation model, and
expected payout patterns by LOB, we can:
 Estimate variability solely due to inflation
 Mix in other sources of variability
 Estimate sensitivity of reserves to length of payout
 Create a correlation matrix for all reserve segments
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Thank you very much for your attention.
David R. Clark
© Copyright 2006 Munich Reinsurance America, Inc. All rights reserved.
The material in this presentation is provided for your information only, and is not permitted to be
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not intended to be legal, underwriting, financial, or any other type of professional advice.