Coherent Measures of Risk

1. Coherent Measures of Risk CAS Seminar on Dynamic Financial Analysis June 8, 2001 Glenn Meyers Insurance Services Office, Inc.

2. 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/CoherentMF.pdf • Coherent Measures of Risk - An Explanation for the Lay Actuary • Glenn Meyers • http://www.casact.org/pubs/forum/00sforum/meyers/Coherent%20Measures%20of%20Risk.pdf

3. A List of Loss Scenarios Define a measure of risk r(X) = Maximum{Xi}

4. Subadditivity r(X+Y)  r(X)+r(Y)

5. Monotonicity If X  Y for each scenario, then r(X)  r(Y)

6. Positive Homogeneity For all l  0 and random loss X, r(lX) = lr(Y)

7. Translation Invariance For all random losses X and constants a r(X+a) = r(X) + a

8. Axioms for Coherent Measures of RiskSatisfied by our example • Subadditivity – For all random losses X and Y, r(X+Y)  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(X+a) = r(X) + a

9. Value at Risk/Probability of Ruinis not coherent - violates subadditivity

10. Standard Deviation Principle is not coherent - violates monotonicity

11. 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

12. 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

13. 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

14. Proposed Measure of RiskTail Value at Risk Value at Risk Tail Conditional Expectation Tail Value at Risk

15. Tail Value at Risk is the average of all losses above the Value at Risk

16. VaR EPD Area TVaR

17. TVaR and Expected Policyholder Deficit The appeal of TVaR and EPD is that they both address the question -- How bad is bad?

18. 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 insurer’s entire portfolio of losses. • Capital determined by the risk measure. C = r(X) - E[X]

19. 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.

20. 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.

21. 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.

22. 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:

23. 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.

24. 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

25. Ways to Allocate Capital #3 • Line headed by CEO’s kid brother gets the marginal capital. Gross up all other lines. • Economic justification - ???

26. 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.

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