Multiperiod risk measures and capital management

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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. Outline. Multiperiod - risk - measurement Risk supervision, out of risk measurement Simple examples of risk-adjustment

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## Multiperiod risk measures and capital management

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### 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

Outline
• Multiperiod - risk - measurement
• Risk supervision, out of risk measurement
• Measure of risk and capital
• Capital management
• The case of CTE / TailVaR
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
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
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
Some required properties

Monotony: X ≤ Y, 0(X) ≤ 0(Y)

Translation: 0(X+ a) = 0(X) + a

Homogeneity: 0(X) = 0(X)

0(X+ Y ) ≥ 0(X) + 0(Y)

and more … for numeraire invariance (?) and multiperiod -measurement

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
Measure of risk and capital
• Initial and final dates
• Intermediate dates!
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!
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
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%
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
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
Conclusions
• Difficult and necessary
• Risk measures for different purposes