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B. John Manistre FSA, FCIA, MAAA Risk Dependency Research: A Progress Report Enterprise Risk...
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Transcript of B. John Manistre FSA, FCIA, MAAA Risk Dependency Research: A Progress Report Enterprise Risk...
B. John Manistre FSA, FCIA, MAAA
Risk Dependency Research:A Progress Report
Enterprise Risk Management Symposium
Washington DC July 30, 2003
2
Agenda
Nature of the project
Tool Development:
– Risk Measures
– Special Results for Normal Risks
– Extreme Value Theory
– Copulas
Formula Approximations
Toward Real Application
Literature Survey
3
Nature of the Project
Response to SoA’s Request for Proposal on “RBC Covariance”
Broad Mandate: “determine the covariance and correlation among various insurance and non-insurance risks generally, particularly in the tail”.
Phase 1: Theoretical Framework/Literature Search
Phase 2: Data Collection/Analysis - the practical element
Project organized at University of Waterloo
– J Manistre (Aegon USA), H Panjer(U of W) & graduate students J Rodriguez, V Vecchione
4
Phase 1: Theoretical Framework
Tools:
– Risk Measures
– Extreme Value Theory
– Copulas
Formula Approximations to Risk Measures
– New results
– Formula Approximations suggest measures of “tail covariance and correlation”
5
Phase 1: Risk Measures
Project focusing on risk measures defined by an increasing distortion function
For a random variable X risk measure is given by
where
Capital is usually taken to be the excess of the risk measure over the mean
].1,0[]1,0[: g
,)]([)( xFxdgXg
).Pr()( xXxF
]).[)]([()()()( xFxFgxdXEXXC gg
6
Phase 1: Risk Measures- Examples
Project does not take a position on which risk measure is best
Planning to work with the following:
– Value at Risk
– Wang Transform
– Block Maximum
– Conditional Tail Expectation
11
10)(
t
ttg
)]()([)( 11 ttg
11
10)(
tt
ttg
)1/(1)( ttg
7
Phase 1: Risk Measures
For any Normal Risk X,
Risk measure is mean plus a multiple of the std deviation
Can use Kg as a tool to understand the risk measure
ZX XX
gXX
XX
X
Xg
K
zzdg
xxdgX
,)]([
)],([)(
8
Phase 1: Risk Measures
`
S ig n if ic a n c e L e v e l R is k M e a s u re
2 5 % 5 0 % 7 5 % 9 0 % 9 5 % 9 9 %
V a R (0 .6 7 ) 0 .0 0 0 .6 7 1 .2 8 1 .6 4 2 .3 3
W a n g T ra n s fo rm (0 .6 7 ) 0 .0 0 0 .6 7 1 .2 8 1 .6 4 2 .3 3
B lo c k M a x im u m 0 .2 5 0 .5 7 0 .9 2 1 .5 4 1 .8 7 2 .5 2
C T E 0 .4 2 0 .8 0 1 .2 7 1 .7 5 2 .0 6 2 .6 7
)]([ xxdgK g
9
Phase 1: Risk Measures - Aggregating Normal Risks
Suppose all risks normal and
Then
For any g conclude
This is “An exact solution to an approximate problem”.
i
iXX
ji
jgigijg XCXCXC,
)()()(
))(()(
)(
)()(
,
,
jgigji
ij
jiji
ijg
Xgg
KKXE
KXE
KXEX
10
Phase 1:Extreme Value Theory
EVT applies when distribution of scaled maxima converge to a member of the three parameter EVT family
Works for most ‘standard’ distributions e.g. normal, lognormal, gamma, pareto etc.
Key Result is the “Peaks Over Thresholds” approximation
– When EVT applies excess losses over a suitably high threshold have an approximate generalized pareto distribution
– Suggests that a generalized pareto distribution should be a reasonable model for the tail of a wide range of risks
11
Phase 1:Copulas
A tool for modeling the dependency structure for a set of risks with known marginal distributions
Technically a probability distribution on the unit n-cube
Large academic literature
Some sophisticated applications in P&C reinsurance
Project is concentrating on
– t- copulas
– Gumbel copulas
– Clayton copulas
12
Phase 1:Copulas
Independence Copula
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2
13
Phase 1:Copulas
Gaussian Copula =1/3, =1/2
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2
14
Phase 1:Copulas
Sample from t-Copula with 2 deg. of freedon
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2
15
Phase 1:Copulas
Clayton Copula
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2
16
Phase 1: Formula Approximations
“Simple” Investment Problem. Let
Fix the joint distribution of the Ui and consider
Capital function is homogeneous of degree 1 in the exposure variables
Choose a target mix of risks
Put
i
iiUeX
)()]([),...,( 1 XExFxdgeeC n
002
01 ,...,, neee
00200010 /,/),,...,( jiijiin eeCCeCCeeCC
),...,(),...,( 2,12,1 nn eeeCeeeC
17
Phase 1: Formula Approximations
Theoretical Result: The first two derivatives are given by
Some challenges in using these results to estimate derivatives. Second derivatives harder to estimate.
Some risk measures easier to work with than others.
Project team is working with a number of approaches.
.)]([)(]|),[()],([]|[2
xFgdxfxXUUCovee
CxFdgxXUE
e
CXji
jii
i
18
Phase 1: Formula Approximations
Let ri be a vector such that then the homogeneous formula approximation
agrees with the capital function and its first two derivatives at the target risk mix .
If ri is a vector such that then a homogeneous formula
approximation is
000
iiierCK
ji
jijjiiiji
iin eerCrCKCereeeC,
21 )])(([),...,,(ˆ
002
01 ,...,, neee
000
iiierCK
ji
jijjiiiji
iin eerCrCKCereeeC,
21 )])(([),,...,,(ˆ
19
Phase 1: Formula Approximation #1
When ri =0
Suggests definition of “tail correlation”.
)()(ˆ
))(()(
)()(
,
,
0
,0
jXCXC
ececcc
CCCC
eeCCCCXC
gigji
ij
jjiiji ji
jiij
jiji
jiijg
ji
jiijij cc
CCCC )(ˆ 0
20
Phase 1: Formula Approximation #2
Some simple choices
– ri =0
– ri = Ci
– ri = ci=Cg (Ui)
When ri =0
Exact for Normal Risks
)()(ˆ
))(()(
)()(
,
,
0
,0
jXCXC
ececcc
CCCC
eeCCCCXC
gigji
ij
jjiiji ji
jiij
jiji
jiijg
21
Phase 1: Formula Approximation #2
When ri = Ci formula is essentially first order
“Factors “ Ci < ci already reflect diversification.
Suggests many existing capital formulas are as good (or bad) as first order Taylor Expansions.
i
iig eCXC )(
22
Phase 1: Formula Approximation #3
When ri = ci we get
Undiversified capital less an adjustment determined by “inverse correlation”
)()()(
))(()])(()[(
)))(()(()(
,
,
00
,0
0
jXCXCXC
ececcc
cCcCCCecec
eecCcCCCececXC
gigji
iji
ig
jjiiji ji
jjiiijk
kk
iii
jiji
jjiiijk
kki
iig
ji
jjiiijk
kk
ij cc
cCcCCCec )])(()[( 00
23
Phase 1: Formula Approximations
Practical work so far suggests
is a more robust approximation. In particular, when the risks are normal
Other homogeneous approximations are possible.
)()(ˆ)(,
jXCXCXC gig
jiijg
)()()()()(ˆ,,
jjXCXCXCXCXC gig
jiij
iiggig
jiij
24
Phase 1: Numerical Example: Inputs
Three Pareto Variates combined with t-copula321 ,, UUU
Input ParametersU 1 U 2 U 3 U 1 U 2 U 3
Marginal Distribution Parameters t-Copula ParametersMean 1.00 1.00 1.00 Kendal's 1.00 0.30 0.10
Std Dev'n 0.10 0.15 0.20 0.30 1.00 0.20Shape -0.20 0.00 0.30 0.10 0.20 1.00
Deg. Of Fr'dm 5.00
Risk Measure: CTE @ 95% K g = 2.06 Sin(ij*/2) 1.00 0.45 0.16
Exposures ei 1.00 1.00 1.00 0.45 1.00 0.31 0.16 0.31 1.00
25
Phase 1: Numerical Example: Results
Simulation Results Sample Size 10,000
Standard Measures Tail Measures
U 1 U 2 U 3 X= e i U i U 1 U 2 U 3 X= e i U i
Mean u 1.00 1.00 1.00 3.00 c i =E(U i - u i |U i >u) 0.27 0.45 0.62 1.34
Std Dev'n 0.10 0.15 0.19 0.33 C i=E(U i - u i |X>x ) 0.14 0.34 0.49 0.967 Emp. Shape ^ 0.22- 0.01 0.24 0.23 Risk Mult 1.30 1.47 1.54 1.44
Corr(Ui ,U j ) 1.00 0.44 0.14 TailCorr(U i ,U j ) 0.85 0.48 0.10 0.969 0.44 1.00 0.31 0.48 0.95 0.26 0.14 0.31 1.00 0.10 0.26 1.03
Inverse Corr(U i ,U j ) 0.45 0.16 0.02- 0.965 0.16 0.22 0.08- 0.02- 0.08- 0.20
26
Phase 2: Real Application
Phase 2 not yet begun
Will not be totally objective
Process:
– Develop high level models for individual risks e.g. model C-1 losses with a pareto dist’n.
– Assume a copula consistent with “expert” opinion
– Adopt a measure of “tail correlation” and calculate
– Make subjective adjustments to final results as nec.
27
Literature Survey: Risk Measures
Artzner, P., Delbaen, F., “Thinking Coherently”, Eber, J-M., Heath, D., “Thinking Coherently”, RISK (10), November: 68-71.
Artzner, P, “Application of Coherent Risk Measures to Capital Requirements in Insurance”, North American Actuarial Journal (3), April 1999.
Wang,S.S., Young, V.R. , Panjer, H.H., “Axiomatic Characterization of Insurance Prices”, Insurance Mathematics and Economics (21) 171-183.
Acerbi, C., Tasche, D., “On the Coherence of Expected Shortfall”, Preprint, 2001.
28
Literature Survey:Measures and Models of Dependence (1)
Frees, E.W., Valdez,E.A., “Understanding Relationships Using Copulas”, North American Actuarial Journal (2) 1998, pp 1-25.
Embrechts, P., NcNeil, A., Straumann, D., “Correlation and Dependence in Risk Mangement: Properties and Pitfalls”, Preprint 1999
Embrechts, P., Lindskog, F., McNeil, A., “Modelling Dependence with Copulas and Applications to Risk Management”, Preprint 2001.
McNeil, A., Rudiger, F., “Modelling Dependent Defaults”, Preprint 2001.
29
Literature Survey:Measures and Models of Dependence (2)
Lindskog, F., McNeil, A., “Common Poisson Shock Models: Applications to Insurance and Credit Risk Modelling”, Preprint 2001.
Joe, H, 1997 “Multivariate Models and Dependence”, Chapman-Hall, London
Coles, S., Heffernan, J., Tawn, J. “Dependence Measures for Extreme Value Analysis”, Extremes 2:4, 339-365, 1999.
Ebnoether, S., McNeil, A., Vanini, P., Antolinex-Fehr, P., “Modelling Operational Risk”, Preprint 2001.
30
Literature Survey:Extreme Value Theory
King, J.L., 2001 “Operational Risk”, John Wiley & Sons UK.
McNeil,A., “Extreme Value Theory for Risk Managers”, Preprint 1999.
Embrechts, P. Kluppelberg, C., Mikosch, T. “Modelling Extreme Events”, Springer – Verlag, Berlin, 1997.
McNeil, A., Saladin, S., “The Peaks over Thresholds Method for Estimating High Quantiles of Loss Distributions”, XXVII’th International ASTIN Colloquim, pp 22-43.
McNeil, A., “On Extremes and Crashes”, RISK, January 1998, London: Risk Publications.
31
Literature Survey:Formula Approximation
Tasche, D.,”Risk Contributions and Performance Measurement”, Preprint 2000.