Risk and Return - Part 1 Introduction to VaR and RAROC

1
Risk and Return - Part 1Introduction to VaR and
RAROC

• Glenn Meyers - Insurance Services Office
• Tim Freestone/Wei-Keung Tang
• Seabury Insurance Capital LLC
• Peter Nakada - eRisk, Inc.

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Risk and Return - Part 1Introduction to VaR and
RAROC

• The purpose of Part 1 is to provide an overview
  of the issues involved in determining the cost of
  capital for an insurer.
• We dont all agree on how to deal with these
  issues.
• Go to Part 2 to see some different points of view
  on this issue.

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Determine Capital Needs for an Insurance Company

• The insurer's risk, as measured by its
  statistical distribution of outcomes, provides a
  meaningful yardstick that can be used to set
  capital needs.
• A statistical measure of capital needs can be
  used to evaluate insurer operating strategies.

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Volatility Determines Capital NeedsLow Volatility
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Volatility Determines Capital NeedsHigh
Volatility
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Define Risk

• A better question - How much money do you need to
  support an insurance operation?
• Look at total assets.
• Some of the assets can come from unearned
  premium reserves and loss reserves, the rest must
  come from insurer capital.

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Coherent Measures of Risk

• Axiomatic Approach
• Use to determine insurer assets
• X is random variable for insurer loss
• r(X) Total Assets
• Capital r(X) Reserves(X)

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Coherent Measures of Risk

• Subadditivity For all random losses X and Y,
• r(XY) ? r(X)r(Y)
• Monotonicity If X ? Y for each scenario, then
• r(X) ? r(Y)
• Positive Homogeneity For all l ? 0 and random
  losses X
• r(lX) lr(X)
• Translation Invariance For all random losses X
  and constants a
• r(Xa) r(X) a

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Examples of Coherent Measures of Risk

• Simplest Maximum loss
• r(X) Max(X)
• Next simplest - Tail Value at Risk
• r(X) Average of top (1-a) of losses

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Examples of Risk that are Not Coherent

• Standard Deviation
• Violates monotonicity
• Possible for EX TStdX gt Max(X)
• Value at Risk/Probability of Ruin
• Not subadditive
• Large X above threshold
• Large Y above threshold
• XY not above threshold

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Representation Theorems

• Artzner, Delbaen, Eber and Heath

Maximum of a bunch of generalized scenarios

• Wang, Young and Panjer

Expected value of X with probabilities distorted
by g, where g(0)0, g(1)1 and g is concave down.
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CorrelationMultiple Line Parameter Uncertainty

• Select b from a distribution with Eb 1 and
  Varb b.
• For each line h, multiply each loss by b.
• Generates correlation between lines.

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Multiple Line Parameter UncertaintyA simple,
but nontrivial example
Eb 1 and Varb b
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Correlation and Capital b 0.00
Chart 3.4
Correlated Losses
7,000
6,000
5,000
4,000
Sum of Random Losses
3,000
2,000
1,000
0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
Random Multiplier
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Correlation and Capital b 0.03
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Positive Correlation Means More Capital

• A good insurer strategy will try to reduce
  correlation between its insureds.
• Unless the price is right
• Example Avoid geographic concentration in
  catastrophe-prone areas.

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Long-Tailed Lines of Insurance

• Uncertainty in loss reserve must be supported by
  capital.
• Release capital over time as uncertainty is
  reduced.

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Reinsurance

• Reduces capital needs
• Reduces the cost of capital
• Adds reinsurance transaction costs
• Insurer strategy - Minimize the combined capital
  and reinsurance transaction costs.

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Allocating Capital

• Actually Allocate the cost of capital
• In total, the cost of capital must come from the
  profit provisions of individual insurance
  policies.
• Allocate capital implicitly, or explicitly.
• See session C-3.

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Measure Risk/Determine Capital

• Build insurers aggregate loss distribution.
• Claim count distribution
• Claim severity distribution
• Dependencies/Correlation
• Catastrophes
• Reinsurance
• Hard part is to get the information.
• Should be fast as to evaluate various
  line/reinsurance strategies.

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Measure Risk/Determine Capital

• For various line/reinsurance strategies
• Calculate your favorite measure of risk/needed
  assets/capital.
• Allocate cost of capital to business segments.
• Compare resulting costs with market driven
  premiums.
• Select the most desirable strategy

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Measure Risk/Determine Capital

• Links to a comprehensive example
• The Cost of Financing Insurance
• CAS Ratemaking Seminar
• http//www.casact.org/coneduc/ratesem/2002/handout
  s/meyers1.ppt
• Papers
• http//www.casact.org/pubs/forum/01spforum/meyers/
  index.htm