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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
• 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)
Superadditivity:
 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/8thListederVortragende
  n.htm
Conclusions
• Difficult and necessary
• Risk measures for different purposes

				
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posted:7/7/2011
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