An Introduction to Stochastic Reserve Analysis

1
An Introduction to Stochastic Reserve Analysis

• Gerald Kirschner, FCAS, MAAA
• Deloitte Consulting
• Casualty Loss Reserve Seminar
• September 2004

2
Presentation Structure

• Background
• Chain-ladder simulation methodology
• Bootstrapping simulation methodology

3
Arguments against simulation

• Stochastic models do not work very well when data
  is sparse or highly erratic.
• Stochastic models overlook trends and patterns in
  the data that an actuary using traditional
  methods would be able to pick up and incorporate
  into the analysis.

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Why use simulation in reserve analysis?

• Provide more information than traditional
  point-estimate methods
• More rigorous way to develop ranges around a best
  estimate
• Allows the use of simulation-only methods such as
  bootstrapping

5
Simulating reserves stochastically using a
chain-ladder method

• Begin with a traditional loss triangle
• Calculate link ratios
• Calculate mean and standard deviation of the link
  ratios

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Simulating reserves stochastically using a
chain-ladder method

• Think of the observed link ratios for each
  development period as coming from an underlying
  distribution with mean and standard deviation as
  calculated on the previous slide
• Make an assumption about the shape of the
  underlying distribution easiest assumptions are
  Lognormal or Normal

7
Simulating reserves stochastically using a
chain-ladder method

• For each link ratio that is needed to square the
  original triangle, pull a value at random from
  the distribution described by
• Shape assumption (i.e. Lognormal or Normal)
• Mean
• Standard deviation

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Simulating reserves stochastically using a
chain-ladder method
Simulated values are shown in red
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Simulating reserves stochastically using a
chain-ladder method

• Square the triangle using the simulated link
  ratios to project one possible set of ultimate
  accident year values. Sum the accident year
  results to get a total reserve indication.
• Repeat 1,000 or 5,000 or 10,000 times.
• Result is a range of outcomes.

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Enhancements to this methodology

• Options for enhancing this basic approach
• Logarithmic transformation of link ratios before
  fitting, as described in Feldblum et al 1999
  paper
• Inclusion of a parameter risk adjustment as
  described in Feldblum, based on Rodney Kreps 1997
  paper Parameter Uncertainty in (Log)Normal
  distributions

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Simulating reserves stochastically via
bootstrapping

• Bootstrapping is a different way of arriving at
  the same place
• Bootstrapping does not care about the underlying
  distribution instead bootstrapping assumes that
  the historical observations contain sufficient
  variability in their own right to help us predict
  the future

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Simulating reserves stochastically via
bootstrapping

1. Keep current diagonal intact
2. Apply average link ratios to back-cast a series
   of fitted historical payments

Ex 1,988 2,3001.157
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Simulating reserves stochastically via
bootstrapping

1. Convert both actual and fitted triangles to
   incrementals
2. Look at difference between fitted and actual
   payments to develop a set of Residuals

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Simulating reserves stochastically via
bootstrapping

• Adjust the residuals to include the effect of the
  number of degrees of freedom.
• DF adjustment
• where n data points and p parameters to
  be estimated

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Simulating reserves stochastically via
bootstrapping

1. Create a false history by making random draws,
   with replacement, from the triangle of adjusted
   residuals. Combine the random draws with the
   recast historical data to come up with the false
   history.

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Simulating reserves stochastically via
bootstrapping

1. Calculate link ratios from the data in the
   cumulated false history triangle
2. Use the link ratios to square the false history
   data triangle

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Simulating reserves stochastically via
bootstrapping

• Could stop here this would give N different
  possible reserve indications.
• Could then calculate the standard deviation of
  these observations to see how variable they are
  BUT this would only reflect estimation variance,
  not process variance.
• Need a few more steps to finish incorporating
  process variance into the analysis.

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Simulating reserves stochastically via
bootstrapping

1. Calculate the scale parameter F.

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Simulating reserves stochastically via
bootstrapping

1. Draw a random observation from the underlying
   process distribution, conditional on the
   bootstrapped values that were just calculated.
2. Reserve sum of the random draws

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Pros / Cons of each method

• Chain-ladder Pros
• More flexible - not limited by observed data
• Chain-ladder Cons
• More assumptions
• Potential problems with negative values

• Bootstrap Pros
• Do not need to make assumptions about underlying
  distribution
• Bootstrap Cons
• Variability limited to that which is in the
  historical data

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Selected References for Additional Reading

• England, P.D. Verrall, R.J. (1999). Analytic
  and bootstrap estimates of prediction errors in
  claims reserving. Insurance Mathematics and
  Economics, 25, pp. 281-293.
• England, P.D. (2001). Addendum to Analytic and
  bootstrap estimates of prediction errors in
  claims reserving. Actuarial Research Paper
  138, Department of Actuarial Science and
  Statistics, City University, London EC1V 0HB.
• Feldblum, S., Hodes, D.M., Blumsohn, G. (1999).
  Workers compensation reserve uncertainty.
  Proceedings of the Casualty Actuarial Society,
  Volume LXXXVI, pp. 263-392.
• Renshaw, A.E. Verrall, R.J. (1998). A
  stochastic model underlying the chain-ladder
  technique. B.A.J., 4, pp. 903-923.