Multiperiod risk measures and capital management

1. Multiperiod risk measures and capital management Philippe Artzner artzner@math.u-strasbg.fr CAS/SOA/GSU professional education symposium ERM 2004, Chicago, April 26-27

2. Outline • Multiperiod - risk - measurement • Risk supervision, out of risk measurement • Simple examples of risk-adjustment • Measure of risk and capital • Capital management • The case of CTE / TailVaR

3. Multiperiod - risk • A three coins example: excess of “heads” over “tails” matters, or last throw matters • Multiperiod information: take early care of future possible bad news: 20 states of nature at date 2, P/L seen at date 1 as (-10, 9, 12, ….…) or as (-20, 19, 22, ….…) • Multiperiod actions, strategies

4. Multiperiod - measurement • Measurement at the initial date: adjustment0(X) (or 0(XN) ? )of the (random) surplus process (or surplus XN of some future date N) • Measurements at a future date n: adjustment n(X) (or n(XN) ? ) contingent on future events, viewed as of initial date

5. Risk supervision • Concern: future solvency (ies) “ = ” current acceptability: positivity of (current) 0(X) ( of many n(X) ? ) • Future acceptability: 0 tells about 1: points to recursive character of the  measurement; definition/computation

6. Some required properties Monotony: X ≤ Y, 0(X) ≤ 0(Y) Translation: 0(X+ a) = 0(X) + a Homogeneity: 0(X) = 0(X) Superadditivity: 0(X+ Y ) ≥ 0(X) + 0(Y) and more … for numeraire invariance (?) and multiperiod -measurement

7. Some examples • 0(X) = p0X0 + p1uX1u + p1dX1d + p2uuX2uu + p2udX2ud + p2ddX2dd and a minimum of such combinations as the test probability (p0 , … , p2dd ) varies • 0(X) = E [ X  ] ,  a stopping time • 0(X) = E [X0+X1+ … +XN ] / (N+1) • 0(f) = 0( n(f) ), f surplus at N

8. Measure of risk and capital • Initial and final dates • Intermediate dates!

9. Capital management • Should the supervisor access the strategies ? • DFA compares for various « strategies », actually various assets portfolios, (estimates of) distributions of future surplus: surplus’ adjustment can only depend on distribution but remember the three coins example!

10. Capital management, ctd. • Is decentralization of portfolio composition, under supervisory constraint(s) possible? • Treasurer’s job in a (re-)insurance company • Internal trading of risk limits over states of nature and dates

11. The case of TailVaR • A favorite in actuarial circles: IAA RBC WP, OSFI, FOPI; NAIC 1994! • The case of CTE / TailVaR TailVaR0(X) is in fact TailVaR0(XN) • TailVaR1(X2) may be a positive random variable, with TailVaR0(X2) negative: no “time consistency” • Example: 20 states of nature at date 2, P/L seen at date 1 as (-10, 11, 12, ….…) or as (-20, 21, 22, ….…) and = 20%

12. The case of TailVaR ctd. • Notice that with (-10, 9, 12, 13, ...…) and (-20, 19, 22, 21, .…) and = 20% there should be trouble from date 0 on, but the one-period TailVaR at date 0 would probably NOT notice it! • Case of given liabilities and date 1 and assets decided upon by throw of a coin

13. References • The world according to Steve Ross, hedging vs. reserving : http://www.DerivativesStrategy.com/magazine/archive/1998/0998qa.asp • Internal market for capital: http://symposium.wiwi.uni-karlsruhe.de/8thListederVortragenden.htm

14. Conclusions • Difficult and necessary • Risk measures for different purposes