A Top-Down Approach to Quantifying Parameter Risk

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A Top-Down Approach to Quantifying Parameter Risk
Alice UnderwoodExecutive Vice President, Willis
Re
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Definition of Terms

• Process Risk
• inherent variability in random process
• fluctuation about the mean
• Parameter Risk
• possibility that parameters are misestimated
• e.g., incorrect mean
• Model Risk
• possibility that the mathematical model of the
  process is inappropriate

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Types of Risk Considerations

• Process Risk
• diversifiable
• foundation of the insurance business
• Parameter Risk
• systemic
• affects all estimates using this parameter
• may be correlated across years / companies
• Model Risk
• a type of operational risk

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Other Liability Occurrence Change from OLR to
ULR
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Other Liability Occurrence Change from OLR to
ULR
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Other Liability Occurrence Industry OLR vs. ULR

• Clearly some error is not diversifying away
• Magnitude and direction of industry error change
  over time

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Sources of Parameter Risk in Actuarial Analysis

• Data issues
• finite sample
• flawed data
• Projection (as if) issues
• loss trend development
• premium on-level
• Judgment factors
• development method selected
• inclusion of soft factors
• External influences
• law changes, coverage changes

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LR Parameter Risk Bottom-Up Approach

• Identify potential sources of risk
• Data / projection issues, judgment factors,
  external influences, etc
• Quantify potential error arising from each
• Address correlations
• Roll up all these estimates of parameter error
  arising from various sources to quantify
  parameter error for loss ratio

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Top-Down Approach Business Interpretation
Plan Loss Ratio
True Mean of True Loss Ratio Distribution
Ultimate Loss Ratio
process risk
parameter risk
Parameter error ratio True Mean / PLR ? ULR
/ OLR
diversifiable over long time period / large
companies
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Quantifying Parameter Risk Top-Down Approach

• Parameter error ratio R ULR / OLR
• For a single company average over a long time
  frame yields company bias
• (should be 1.0 but may not be, depending on
  planning strategy)
• For a single accident year average over a large
  number of companies yields industry delusion
  for that accident year
• Difference between industrys initial view of
  loss potential for that AY and true loss potential

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Quantifying Parameter Risk Top-Down Approach
Key findings Other Liability Occurrence For
each accident year the R values are lognormally
distributed The mean and standard deviation of
these lognormal distributions are
correlated The lognormal s parameter can be
approximated as a function of the µ
parameter The µ parameter can be analyzed using
time series methods
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Time Series Analysis of µ
possible future trajectories for µ
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Time Series Analysis of µ
percentiles of simulation based on time series
analysis
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Back-Testing

• Compared µ values fitted to data to the
  percentiles of the forecast distributions

• Observations fit theoretical quartiles reasonably
  well
• Forecast may be somewhat too conservative but
  hard to tell given small data set

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Summary Top-Down Approach

• Time series analysis of µ
• Simulation of ULR/OLR

• Distribution of forecast average 2007 parameter
  error is skew to the right
• Median approximately 1.0
• But significant chance of large upward deviation

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Summary Top-Down Approach

• Do not necessarily expect future µ values to fall
  at the center of forecast distribution
• Not a precise point estimate of future µ
• Not a crystal ball to predict shifts in market
• However, useful in predicting the range of future
  µ values
• How likely is the industry to get it wrong, and
  by how much?

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Caveats

• Imperfect data
• Law of large numbers assumption residual process
  error
• ULR approximation
• OLR approximation
• Company-specific behavior

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Legal Disclaimers

• In preparing this Presentation, Willis Re has
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