# Jules by shuifanglj

VIEWS: 8 PAGES: 14

• pg 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
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
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

```
To top