Risk Dependency Research:
A Progress Report
Enterprise Risk Management Symposium
Washington DC July 30, 2003
B. John Manistre FSA, FCIA, MAAA
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
2
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
3
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”
4
Phase 1: Risk Measures

Project focusing on risk measures defined by an
increasing distortion function g : [0,1]  [0,1].

For a random variable X risk measure is given by
 g ( X )   xdg[ F ( x)],
where

F ( x)  Pr( X  x).
Capital is usually taken to be the excess of the risk
measure over the mean
C g ( X )   g ( X )  E ( X )   xd ( g[ F ( x)]  F [ x]).
5
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
0 t  1  
g (t )  
1 t  1  
g (t )  [ 1 (t )   1 ( )]
g (t )  t 1 /(1 )
 0 t  1

g (t )   t  
t  1
1  
6
Phase 1: Risk Measures
X  X  X Z

For any Normal Risk X,

Risk measure is mean plus a multiple of the std deviation

 g ( X )   xdg[(
x  X
X

 X  X
)],

 zdg[( z )],

 X  X Kg

Can use Kg as a tool to understand the risk measure
7
Phase 1: Risk Measures
`

Kg 
 xdg[ ( x)]

Significance Level 
Risk Measure
25%
50%
75%
90%
95%
99%
VaR
(0.67)
0.00
0.67
1.28
1.64
2.33
Wang Transform
(0.67)
0.00
0.67
1.28
1.64
2.33
Block Maximum
0.25
0.57
0.92
1.54
1.87
2.52
CTE
0.42
0.80
1.27
1.75
2.06
2.67
8
Phase 1: Risk Measures - Aggregating Normal
Risks


X   Xi
Suppose all risks normal and
Then
i
 g ( X )  E ( X )  K g X
 E( X )  K g
  
ij
i
j
i, j
 E( X ) 

ij
( K g  i )( K g  j )
i, j

For any g conclude
Cg ( X ) 
 C
ij
g
( X i )C g ( X j )
i, j

This is “An exact solution to an approximate problem”.
9
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
10
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
11
Phase 1:Copulas
Independence Copula
1.2
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
12
Phase 1:Copulas
Gaussian Copula =1/3, =1/2
1.2
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
13
Phase 1:Copulas
Sample from t-Copula with 2 deg. of freedon
1.2
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
14
Phase 1:Copulas
Clayton Copula
1.2
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2 15
Phase 1: Formula Approximations

“Simple” Investment Problem. Let X   eiU i
i

Fix the joint distribution of the Ui and consider
C (e1 ,..., en )   xdg[ F ( x)]  E ( X )

Capital function is homogeneous of degree 1 in the
exposure variables C (e1, e2 ,..., en )  C (e1, e2 ,..., en )

Choose a target mix of risks

Put C0  C(e10 ,..., en0 ), Ci  C / ei0 , Cij   2C / ei0 e 0j
e10 , e20 ,..., en0
16
Phase 1: Formula Approximations


Theoretical Result: The first two derivatives are given by
C
  E[U i | X  x]dg[ F ( x)],
ei 

 2C
  Cov[(U i ,U j ) | X  x] f X ( x)dg [ F ( x)].
ei e j 

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.
17
Phase 1: Formula Approximations

0
Let ri be a vector such that K  C0   ri ei  0
homogeneous formula approximationi
Cˆ (e1 , e2 ,..., en )   ri ei 
i
[ KC
ij
then the
 (Ci  ri )(C j  r j )]ei e j
i, j
agrees with the capital function and its first two
derivatives at the target risk mix e10 , e20 ,..., en0.

If ri is a vector such that K  C0   ri ei0  0 then a
i
homogeneous formula approximation is
Cˆ (e1 , e2 ,..., en , )   ri ei 
i
[ KC
ij
 (Ci  ri )(C j  r j )]ei e j
i, j
18
Phase 1: Formula Approximation #1

When ri =0
Cg ( X ) 
 (C C
0
ij
 Ci C j )ei e j
i, j


(C 0 Cij  Ci C j )
ci c j
i, j

 ˆ C
ij
g
(ci ei )(c j e j )
( X i )C g ( X j )
i, j

Suggests definition of “tail correlation”.
ˆ ij 
(C0 Cij  Ci C j )
ci c j
19
Phase 1: Formula Approximation #2

Some simple choices
– ri =0
– ri = Ci
– ri = ci=Cg (Ui)

When ri =0
Cg ( X ) 
 (C C
0
ij
 Ci C j )ei e j
i, j


(C 0 Cij  Ci C j )
ci c j
i, j

 ˆ C
ij
g
(ci ei )(c j e j )
( X i )C g ( X j )
i, j

Exact for Normal Risks
20
Phase 1: Formula Approximation #2

When ri = Ci formula is essentially first order
C g ( X )   Ci ei
i


“Factors “ Ci < ci already reflect diversification.
Suggests many existing capital formulas are as good (or
bad) as first order Taylor Expansions.
21
Phase 1: Formula Approximation #3

When ri = ci we get
C g ( X )   ci ei 
i
0
((
c
e
  k k  C0 )Cij  (Ci  ci )(C j  c j ))ei e j
i, j
  ci ei 
i
k

[(  ck e 0 k  C0 )Cij  (Ci  ci )(C j  c j )]
ci c j
i, j
 Cg ( X i ) 
 C
ij
i

k
g
(ci ei )(c j e j )
( X i )C g ( X j )
i, j
Undiversified capital less an adjustment determined by
“inverse correlation”
ij 
[( ck e 0 k  C0 )Cij  (Ci  ci )(C j  c j )]
k
ci c j
22
Phase 1: Formula Approximations

Practical work so far suggests
Cg ( X ) 
 ˆ
ij
C g ( X i )C g ( X j )
i, j

is a more robust approximation. In particular, when the
risks are normal
 ˆ
i, j

ij
C g ( X i )C g ( X j )   C g ( X i ) 
i

ij
C g ( X i )C g ( X j )
i, j
Other homogeneous approximations are possible.
23
Phase 1: Numerical Example: Inputs

Three Pareto Variates U1 ,U 2 ,U 3 combined with t-copula
Input Parameters
U1
U2
Marginal Distribution Parameters
Mean
1.00
1.00
Std Dev'n
0.10
0.15
Shape -0.20
0.00
Risk Measure: CTE @
Exposures ei
1.00
95%
1.00
U3
U1
t-Copula Parameters
Kendal's 
1.00
0.30
0.10
1.00
0.20
0.30
Kg=
1.00
Deg. Of Fr'dm
5.00
Sin(ij*/2)
1.00
0.45
0.16
U2
U3
0.30
1.00
0.20
0.10
0.20
1.00
0.45
1.00
0.31
0.16
0.31
1.00
2.06
24
Phase 1: Numerical Example: Results

Simulation
Results
Sample Size
10,000
Standard Measures
U 3 X=  e i U i
U1
U2
1.00
Std Dev'n 0.10
Emp. Shape ^ - 0.22
1.00
0.15
0.01
1.00
0.19
0.24
0.44
1.00
0.31
0.14
0.31
1.00
Mean u
Corr(Ui ,U j )
1.00
0.44
0.14
Tail Measures
3.00
0.33
0.23
U1
U2
U3
X=  e i U i
c i =E(U i - u i |U i >u)
C i=E(U i - u i |X>x )
Risk Mult
0.27
0.14
1.30
0.45
0.34
1.47
0.62
0.49
1.54
1.34
0.967
1.44
TailCorr(U i ,U j )
0.85
0.48
0.10
0.48
0.95
0.26
0.10
0.26
1.03
0.969
0.45
0.16 - 0.02
0.16
0.22 - 0.08
- 0.02 - 0.08
0.20
0.965
Inverse Corr(U i ,U j )
25
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.
26
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.
27
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.
28
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.
29
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.
30
Literature Survey:
Formula Approximation
 Tasche, D.,”Risk Contributions and Performance
Measurement”, Preprint 2000.
31