B. John Manistre FSA, FCIA, MAAA Risk Dependency Research: A Progress Report Enterprise Risk Management Symposium Washington DC July 30, 2003

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


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


`

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


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


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


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


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


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


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.

Documents

B. John Manistre FSA, FCIA, MAAA Risk Dependency Research: A Progress Report Enterprise Risk Management Symposium Washington DC July 30, 2003