The Role of Risk Metrics in Insurer Financial Management

1. The Role of Risk Metricsin 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 NeedsLow Volatility

4. Volatility Determines Capital NeedsHigh 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: r(X) = A

7. Coherent Measures of Risk • 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 losses X r(lX) = lr(X) • Translation Invariance – For all random losses X and constants a r(X+a) =r(X) + a

8. 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

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: r(X) = A • If assets can vary, we have sufficient assets if: r(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, l, of the index.

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

13. Illustrate Implications with a Model • Calculate shares, l, of the asset index so that: TVaRa(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, b • Fix: • TVaR level a = 99%

15. Case 1Fixed 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 b = –2.00 • Stable Stock Index – Model with Std[I] = 0.02 • Stock prices increase with inflation • Model loss inflation with b = +2.00 • Volatile Stock Index – Model with Std[I] = 0.10 • Stock prices increase with inflation • Model loss inflation with b = +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.pdf • Meyers • Setting Capital Requirements with Coherent Measures of Risk – Part 1 and Part 2 • http://www.casact.org/pubs/actrev/aug02/latest.htm • http://www.casact.org/pubs/actrev/nov02/latest.htm