1. The Role of Risk Metrics in Insurer Financial Management Glenn Meyers Insurance Services Office, Inc. Joint CAS/SOS Symposium on Enterprise Risk Management July 29, 2003

2. 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.

3. Volatility Determines Capital Needs Low Volatility

4. Volatility Determines Capital Needs High Volatility

5. 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 premium reserves, the rest must come from insurer capital.

6. Coherent Measures of Risk Axiomatic Approach Use to determine needed insurer assets, A X is random variable for insurer loss An insurer has sufficient assets if:  (X) = A

7. Coherent Measures of Risk Subadditivity – For all random losses X and Y,  (X+Y)   (X)+  (Y) Monotonicity – If X  Y for each scenario, then  (X)   (Y) Positive Homogeneity – For all 0 and random losses X  ( X) = (X) Translation Invariance – For all random losses X and constants   (X+  ) =  (X) + 

8. Examples of Coherent Measures of Risk Simplest – Maximum loss  (X) = Max(X) Next simplest - Tail Value at Risk  (X) = Average of top (1-  )% of losses

9. Examples of Risk that are Not Coherent Standard Deviation –Violates monotonicity –Possible for E[X] + T×Std[X] > Max(X) Value at Risk/Probability of Ruin –Not subadditive –Large X above threshold –Large Y above threshold –X+Y not above threshold

10. But – Assets Can Vary! If assets are fixed, we have sufficient assets if:  (X) = A If assets can vary, we have sufficient assets if:  (X – A) = 0 If assets are fixed, the new criteria reduce to the old because of translation invariance.

11. Illustrate Implications with a Model Losses, L, have lognormal distribution –Mean 10,000 –Standard deviation will depend on example Asset Index, I, has lognormal distribution –Mean 10,000 –Standard deviation will depend on example Assets are a multiple,, of the index.

12. Illustrate Implications with a Model Random effect, E, of economic conditions Assets A = I×(1+E) Losses X = L×(1+  E) Loss volatility multiplier –  E drives the correlation between assets and liabilities

13. Illustrate Implications with a Model Calculate shares,, of the asset index so that: TVaR  (X–A) = 0 Also look at standard deviation risk metric with T satisfying: E[X–A] + T×Std[X–A] = 0 Normally T is fixed. Here I calculate the implied T as a way to compare risk metrics.

14. Illustrate Implications with a Model Select sample of 1000 L’s, I’s and E’s Six cases varying: –Standard deviation of L –Standard deviation of I –Standard deviation of E –Loss volatility multiplier,  Fix: –TVaR level  = 99%

15. Case 1 Fixed Assets and Volatile Losses Required assets are larger than expected loss

16. Case 2 Fixed Assets and Less Volatile Losses Value of assets smaller than Case 1. Implied T smaller than that of Case 1. –TVaR is more sensitive the large loss potential

17. Case 3 Variable Assets Introducing asset variability increases expected value of assets – a bit.

18. Asset Risk and Economic Variability Model with Std[E] = 2% When economic inflation is high Bond Index – Model with Std[I] = 0.02 –Interest rates are high and bond prices drop –Model loss inflation with  = –2.00 Stable Stock Index – Model with Std[I] = 0.02 –Stock prices increase with inflation –Model loss inflation with  = +2.00 Volatile Stock Index – Model with Std[I] = 0.10 –Stock prices increase with inflation –Model loss inflation with  = +2.00

19. Case 4 Variable Assets – Bond Index When assets move in the opposite direction of losses, you need assets with higher expected value.

20. Case 5 Variable Assets – Stable Stock Index You need assets with lower expected value than with Case 4 because stocks move in the same direction as losses.

21. Case 6 Variable Assets – Volatile Stock Index Higher expected value with volatile stocks Perhaps this explains why PC insurers stay out of stocks despite the wrong correlation.

22. Summary – Risk Metrics Introduced the latest and greatest (??) risk metric – TVaR Compared it to the current champion (??) TVaR –Has a strong axiomatic foundation –Does more to discourage risky business

23. Summary – Using Risk Metrics Use to determine the amount of assets needed to support insurance liabilities Takes into account –Insurance risk –Asset risk –Correlation between the two

24. References Artzner, Delbaen, Eber and Heath –Coherent Measures of Risk –Original paper –http://www.math.ethz.ch/~delbaen/ftp/preprints/CoherentMF.pdfhttp://www.math.ethz.ch/~delbaen/ftp/preprints/CoherentMF.pdf Meyers –Setting Capital Requirements with Coherent Measures of Risk – Part 1 and Part 2 –http://www.casact.org/pubs/actrev/aug02/latest.htmhttp://www.casact.org/pubs/actrev/aug02/latest.htm –http://www.casact.org/pubs/actrev/nov02/latest.htmhttp://www.casact.org/pubs/actrev/nov02/latest.htm