Coherent Measures of Risk

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Coherent Measures of Risk

• CAS Seminar on
• Dynamic Financial Analysis
• June 8, 2001
• Glenn Meyers
• Insurance Services Office, Inc.

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New Papers

• Coherent Measures of Risk
• Philippe Artzner, Freddy Delbaen, Jean-Marc Eber
  and David Heath, Math. Finance 9 (1999), no. 3,
  203-228
• http//www.math.ethz.ch/delbaen/ftp/preprints/Coh
  erentMF.pdf
• Coherent Measures of Risk - An Explanation for
  the Lay Actuary
• Glenn Meyers
• http//www.casact.org/pubs/forum/00sforum/meyers/C
  oherent20Measures20of20Risk.pdf

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A List of Loss Scenarios
Define a measure of risk r(X) MaximumXi
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Subadditivity
r(XY) ? r(X)r(Y)
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Monotonicity
If X ? Y for each scenario, then r(X) ? r(Y)
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Positive Homogeneity
For all l ? 0 and random loss X, r(lX) lr(Y)
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Translation Invariance
For all random losses X and constants a r(Xa)
r(X) a
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Axioms for Coherent Measures of RiskSatisfied by
our example

• 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
  loss X
• r(lX) lr(Y)
• Translation Invariance For all random losses X
  and constants a
• r(Xa) r(X) a

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Value at Risk/Probability of Ruinis not coherent
- violates subadditivity
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Standard Deviation Principle is not coherent -
violates monotonicity
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The Representation Theorem

• Let ? denote a finite set of scenarios.
• Let X be a loss associated with each scenario.
• A risk measure, ?, is coherent if and only if
  there exists a family, ?, of probability measures
  defined on ? such that

i.e. the maximum of a bunch of generalized
scenarios
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Probability Measures?The Easiest Example

• Let A Ai be the set of one element subsets of
  W. Let Xi be the loss for ai.

• Then

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Probability Measures?The Next Easiest Example

• Let A Ai be the set of n element subsets of W.
  Let Xw be the loss for w?W

• Then

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Proposed Measure of RiskTail Value at Risk
Value at Risk
Tail Conditional Expectation Tail Value at Risk
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Tail Value at Risk is the average of all losses
above the Value at Risk
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(No Transcript)
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TVaR and Expected Policyholder Deficit
The appeal of TVaR and EPD is that they both
address the question -- How bad is bad?
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Determine the Amount of Capital

• Decide on a measure of risk
• Tail Value at Risk
• Average of the top 1 of aggregate losses
• Note that the measure of risk is applied to the
  insurers entire portfolio of losses.
• Capital determined by the risk measure.
• C r(X) - EX

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Step 2Allocate Capital

• How are you going to use allocated capital?
• Use it to set profitability targets.

• How do you allocate capital?
• Any way that leads to correct economic decisions,
  i.e. the insurer is better off if you get your
  expected profit.

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Better Off?

• Let P Profit and C Capital. Then the insurer
  is better off by adding a line/policy if

? Marginal return on new business ? return
on existing business.
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OK - Set targets so that marginal return on
capital equal to insurer return on Capital?

• If risk measure is subadditive then
• Sum of Marginal Capitals is ? Capital
• Will be strictly subadditive without perfect
  correlation.
• If insurer is doing a good job, strict
  subadditivity should be the rule.

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OK - Set targets so that marginal return on
capital equal to insurer return on Capital?
If the insurer expects to make a return, e P/C
then at least some of its operating divisions
must have a return on its marginal capital that
is greater than e. Proof by contradiction If
then
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Ways to Allocate Capital 1

• Gross up marginal capital by a factor to force
  allocations to add up.
• Economic justification - Long run result of
  insurers favoring lines with greatest return on
  marginal capital in their underwriting.
• Appropriate for stock insurers.
• I use it because it is easy.

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Ways to Allocate Capital 2

• Average marginal capital, where average is taken
  over all entry orders.
• Shapley Value
• Economic justification - Game theory
• Appropriate for mutual insurers

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Ways to Allocate Capital 3

• Line headed by CEOs kid brother gets the
  marginal capital. Gross up all other lines.
• Economic justification -

???
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Conclusion on Allocating Capital

• Axioms for coherent measures of risk do not
  prescribe a unique allocation method.
• Additional economic and/or fairness assumptions
  are needed.

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Papers on Coherent Allocation

• Coherent Allocation of Risk Capital
• Michael Denault
• http//www.risklab.ch/ftp/papers/CoherentAllocatio
  n.pdf
• The Cost of Financing Insurance - V 2.0
• Glenn Meyers
• http//www.casact.org/pubs/forum/01spforum/meyers/
  index.htm