Report of the Correlation Working Party by armedman1

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									      Report of the
Correlation Working Party
         Glenn Meyers
 Insurance Services Office, Inc.
         April 27, 2004
  Charge of the Working Party
• ERM requires the quantification of the total
  risk of an enterprise. One must consider
  correlation to properly combine the
  individual risk components.
• Considerations
  – Theoretical
  – Empirical
  – Computational
   Theoretical Considerations
• Conclusion – No overriding “theory of
  correlation.”
• We will provide examples of multivariate
  models that exhibit correlation.
• Experts prefer the term “dependencies”
  rather than correlation.
  – I find myself reverting the common usage so
    nonexperts will know what I am talking about.
      Empirical Considerations
• Historical problem – lack of data
  – One observation per year
• If correlation matters, we should be able to
  find data that exhibits that correlation.
• One approach
  – Create a model that depends on a “driver” for
    correlation.
  – Use data from several insurers to parameterize
    the driver.
  – Example to follow
 Computational Considerations
• ERM demands the aggregation of
  segments.
• Simulation
  – Iman Conover and Copulas
• Fourier transforms
  – Faster than simulations, but less flexible and
    require more setup time.
            Chapters Written by
             Individual Authors
•   Common Shock Models – Glenn Meyers
•   The Iman-Conover Method – Stephen Mildenhall
•   Correlation over time – Hans Waszink
•   Aggregating Bivariate Distributions – David
    Homer
•   Dependency in Market Risk – Younju Lee
•   Modeling Time Series with Non-Constant
    Correlations – Dan Heyer
•   Correlations in a General Stochastic Setting –
    Lijia Guo
•   4 CAS Members and 3 non members
     From Meyers Chapter
The Negative Binomial Distribution

• Select a at random from a gamma
  distribution with mean 1 and variance c.
• Select the claim count K at random from
  a Poisson distribution with mean al
• K has a negative binomial distribution
  with:
     E K   l and Var K   l  c  l   2
 Multiple Line Parameter Uncertainty
 • Select b from a distribution with E[b] = 1
   and Var[b] = b.
 • For each line h, multiply each loss by b.
 • Can calculate r if desired.

Var  X   E b Var  X | b   Varb E  X | b 
                                                
Cov  X ,Y   E b Cov  X | b ,Y | b   Cov b E  X | b , E Y | b 
                                                                       
        Cov  X ,Y 
r
     Std  X   Std Y 
     Multiple Line Parameter Uncertainty

        A simple, but nontrivial example


         b1  1  3b , b 2  1, b3  1  3b

Pr b  b1  Pr b  b3   1/ 6 and Pr b  b2   2 / 3

               Eb = 1 and Var[b] = b
        Low Volatility
     b = 0.01 r = 0.50

                             Chart 3.3

             4,000
             3,500
             3,000
Y 2 = bX 2




             2,500
             2,000
             1,500
             1,000
              500
                0
                     0   1,000     2,000      3,000   4,000
                                 Y 1 = bX 1
        Low Volatility
     b = 0.03 r = 0.75

                             Chart 3.3

             4,000
             3,500
             3,000
Y 2 = bX 2




             2,500
             2,000
             1,500
             1,000
              500
                0
                     0   1,000     2,000      3,000   4,000
                                 Y 1 = bX 1
        High Volatility
     b = 0.01 r = 0.25

                             Chart 3.3

             4,000
             3,500
             3,000
Y 2 = bX 2




             2,500
             2,000
             1,500
             1,000
              500
                0
                     0   1,000     2,000      3,000   4,000
                                 Y 1 = bX 1
        High Volatility
     b = 0.03 r = 0.45

                             Chart 3.3

             4,000
             3,500
             3,000
Y 2 = bX 2




             2,500
             2,000
             1,500
             1,000
              500
                0
                     0   1,000     2,000      3,000   4,000
                                 Y 1 = bX 1
           About Correlation
• There is no direct connection between r
  and b.
• For the same value of b:
  – Small insurers have large process risk and
    hence smaller correlation
  – Large insurers have smaller process risk and
    hence larger correlations.
• Pay attention to the process that
  generates correlations.
           Estimating b From Data
Cov  X ,Y 
 Eb Cov  X =0Y | b   Cov b E  X | b , E Y | b 
             | b,                                     

 E  X   E Y   Cov  b , b 
 b  E  X   E Y  Thus:
                     x  E  X  y  E Y 
                  b            
                       EX        E Y 
    Reliable estimates of b are possible
    with lots of data.

   For example, 50 insurers with 10 years of
               50 
   data gives    10  12,250 observations.
              2

• Real estimates provided by Meyers, Klinker and
  Lalonde
   http://www.casact.org/pubs/forum/03sforum/03sf015.pdf
           Sample Calculations
  Common Shocks to Frequency and Severity
• Multiply expected claim count by a random
  shock.
  – Negative binomial count distributions
  – Var[Shock] called covariance generator
• Multiply scale of claim severity by a random
  shock.
  – Lognormal severity distributions
  – Var[Shock] called the mixing parameter
• Look at spreadsheet
Scroll Down
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            Parting Message
• Build models of underlying processes.
  – Common shock model illustrated here
  – Other chapters build other models
• Quantify parameters of models
  – Use data! (If data will never exist, why worry?)
  – Express parameters in a form that has intuitive
    meaning.
• Correlation is a consequence of the models.

								
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