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SEMINAIRE SCIENTIFIQUE 29/01/2003 METHOLOGIES NON VIE DANS LA PERSPECTIVE DES NORMES IFRS/IAS DESPEYROUX, GIE AXA PowerPoint Presentation
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SEMINAIRE SCIENTIFIQUE 29/01/2003 METHOLOGIES NON VIE DANS LA PERSPECTIVE DES NORMES IFRS/IAS DESPEYROUX, GIE AXA C. LEVI, GuyCarpenter C. PARTRAT, ISFA Université Lyon 1 Reprennent leur présentation à la

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SEMINAIRE SCIENTIFIQUE 29/01/2003

METHOLOGIES NON VIE DANS LA PERSPECTIVE DES NORMES IFRS/IAS

DESPEYROUX, GIE AXA

C. LEVI, GuyCarpenter

C. PARTRAT, ISFA Université Lyon 1

Reprennent leur présentation à la

GIRO Convention des Actuaires non vie britanniques, octobre 2002

agenda
Introduction

Run off Triangle Cash Flows

Stochastic Methods

Discount of Cash Flows

Extreme Claims

Risks Dependence

References

Conclusion

Agenda
introduction
DSOP : Entity Specific Value

Assessment date : 31/12/n

Assets

Cash

Securities (Bonds, equities) in Fair Value

Real estate

Liabilities

Yearly cash flows

run off (no new business)

gross paid claims (for contracts in force before 31/12/n)

Calculations on gross paid claims (no reinsurance taken into account)

Introduction
run off triangle cash flows
Date 31/12/n

1 set of contracts (no new business, no renewals)

Claims developing during (n+1) years

Run Off Triangle Cash Flows
run off triangle cash flows2
Data in the rectangle are incremental values

xij = claims amounts paid for underwriting year i during development year j

Data : xij i+j n

Unknown : Xij i+j >n

Run Off Triangle Cash Flows
run off triangle cash flows3
Future cash flows (without discounting)

For k=1,…,n and year (n+k)

Total

To be compared to available assets A at 31/12/n

Run Off Triangle Cash Flows
run off triangle cash flows4
For evaluation of CFn+k or CF, we can use the same approaches (deterministic or stochastic) and the same methods as for reserving.

Deterministic methods :

Chain Ladder

Separation (arithmetic) because diagonals effects

etc..

Run Off Triangle Cash Flows
stochastic methods
Modelling

more possibilities including uncertainty measures on results

but specification error risk

Thanks to the City University group (England, Haberman, Renshaw, Verrall) and T. Mack for their work on stochastic reserving

Stochastic Methods
stochastic methods1
For each model

Assumption 1 :

For i,j = 1,..,n, Xij are independent random variables(r.v.)

Standard models now : Generalized Linear Models

(with the support of Genmod procedure in SAS)

Assumption 2 :

For i,j = 1,..,n, distribution of Xij belongs to the same exponential family with

where V(m) is the “variance function” of the family

Stochastic Methods
stochastic methods2
Parameters (factors)

mean

for year i

for delay j

(possibly=1) dispersion parameter

Stochastic Methods
stochastic methods3
Aims

Let FCF distribution function (d.f.) of the r.v.

(FCF) selected parameter to be estimated (risk?)

Central values : average E(CF), median, fractiles…

Dispersion : V(CF)

Insufficiency probability : P(CF>A)

Tail : VaR with P(CF>VaR)=

Expected shortfall E(CF/ CF> VaR )

Stochastic Methods
stochastic methods4
D.f. FCF

m.g.f.

And inversion (Fast Fourier Transform)

Determining a predictor of Xij(i+j>n), CFn+k then CF

Stochastic Methods
stochastic methods5
Means

Data : in the superior triangle

Maximum likehood method , we obtain

estimators of

For i+j>n estimator of E(Xij)

Stochastic Methods
stochastic methods6
for E(CF)

with uncertainty measure

or

more generally, we obtain

estimator of

Stochastic Methods
stochastic methods7
is a predictor of Xij and

for CF,

with uncertainty measure or

Difficult to obtain analytic expression (even with some approximation) of

and

Easier by bootstrapping

Stochastic Methods
stochastic methods8
Bootstrapping Pearson’s residuals after modelling the superior triangle gives

Confidence interval for the parameter

Prediction interval

Estimation of probability distribution of CF

finding again insufficiency probability, VaR..

Cf England, Verrall, 1999

Pinheiro et al., 2001

Stochastic Methods
discounting cash flows
Which risk / discount rate

Risk free

Risk premium for liabilities risk

Risk premium for assets risk

others?

IASB current proposal

Risk free

Plus eventually premium independent of assets

dependent of liabilities if not reflected in the market value margin.

Discounting Cash Flows
discounting cash flows1
Market Value Margin

There is always some risk or uncertainty about future cash flows, because of

occurrence risk

severity risk

development risk

Adjustment for risks and uncertainty must be reflected preferably in the cash flows.

Discounting Cash Flows
discounting cash flows2
How evaluate the discount rate

risk free

Market value of discount rate (yield curve)

models(like Vasicek/ Cox Ingersoll Ross/ Wilkie…)

risk adjusted discount rate

CAPM

State price deflators

Discounting Cash Flows
discounting cash flows3
State Price Deflators

State price deflators can be thought of as stochastic discount factors

allow for

investment risk

time value of money

a cash flow at date t has a value E[DtCt]/D0

Dt are random variable, vary with scenarios

Discounting Cash Flows
discounting cash flows8
Profit and loss impact

Increase of rate => Profit recognition

Decrease of rate => Reduction of profit

Discounting Cash Flows
extreme claims
Measures

Given a line (natural events, casualty,…)

X r.v. claim amount

D.f. F

Tail Distribution

Speed of convergence of (x) to 0

closely linked with the existence of moments of X

Extreme Claims
extreme claims1
Value-at-Risk (VaR)

e (0.05;0.01;0.005;…)

VaRe P(X>VaR)=

e

VaRe

x

Extreme Claims
extreme claims2
Tail VaR - Mean excess

Tail VaR 0.01=E( X / X  VaR 0.01)

More generally

Mean excess : e such that e(u)= E( X-u / Xu)

Remark : the d.f. of X can be derived from e.

Extreme Claims
extreme claims3
These measures are used too for other problems:

Solvency

Capital Allocation

Coherent measures

Etc..

Extreme Claims
extreme claims5
Uncertainty

F unknown

Historical data : x1,…,xn realization of X1,…,Xn (n-sample)

Interest measure p(F)

Aims : estimation estimator

Estimation uncertainty :

standard error, confidence interval, analytic or bootstrap.

Extreme Claims
extreme claims6
Return Period

Claims frequency excluded

r.v. N(u)=min{i  1:Xiu}

(rank of the smallest claim exceeding of u)

Return period of level u :

(in number of claims)

u100 such that E[N(u100)]=100 =>

Extreme Claims
extreme claims7
Including claims frequency

Assumption : Poisson process (l) for the claims frequency

Yn(u)=r.v. interoccurence time between two claims u

(years)

Extreme Claims
extreme claims8
Extreme claim development ?

GLM : existence of the moments supposed

Heavy tail distribution : no assumption on the moments

Extreme Claims
risks dependence
2 sub-lines of business “Claims correlated” give 2 run-off triangles of increments.

Aims : modelling stochastic dependence to obtain the bivariate distribution of (CF;CF’)

Risks dependence
risks dependence1
Modelling

If we need to go over correlation

Assumption :

dependence is just between Xij and X’ij (i,j=0,…,n)

we need the bivariate distribution of (Xij ; X’ij)

Common shock models

Xij = Yij + Sij Yij , Y’ij , Sij independent r.v.

X’ij = Y’ij + Sij Sij : common shock.

Dist. of dist of (Xij ; X’ij)

Risks dependence
risks dependence2
Copula

Nelsen R. B. (1999) : “An introduction to Copulas” Springer

Risks dependence
risks dependence3
Methods developped in an actuarial dissertation :

Gillet A., Serra B. (2002) : “Effets de la dépendance entre différentes branches sur le calcul des provisions “ ENSAE

Presented to the Institut des Actuaires for AA (next November)

Paper submitted to Astin Colloquium (Berlin, August 2003)

Risks dependence
references
Blondeau J., Partrat C. (2002) : “La réassurance : approche technique “ Economica (to be published)

Embrechts P., Kluppelbegr C., Mikosh T. (1997) “Modelling extremal events for insurance and finance” Springer

Daykin C.D. , Hey G.B. (1991) : “ A management model of General Insurance Company using Simulation Techniques in Managing the Insolvency Risk of Insurance Companies” eds : Cummins J.D et al., Kluwer Academic Publ.

Daykin C.D., Pentikäinen T., Pesonen M. (1994) : “Practical Risk Theory for Actuaries” Chapman & Hall.

Duffie D. (1994) “Modèles dynamiques d’évaluation” PUF

References
references1
Efron B., Tibshirani P.J. (1993) : “An introduction to the Bootstrap” Chapman & Hall.

England P.D., Verrall R.J. (1999) : “Analytic Bootstrap estimates of prediction error in claims reserving” Insurance : Math. and Econ. Vol. 25, 281-293.

England P.D., Verrall R.J. (2002) : “Stochastic claims reserving in General Insurance” Institute of Actuaries.

IASB (2001) : “Draft Statements of Principles”

Jarvis S., Southall F., Varnell E. (2001) “Modern Valuation Techniques”

References
references2
Kaufman R., Gardmer A., Klett R.(2001) : “Introduction to Dynamic Financial Analysis” Astin Bull. Vol.31,217-253.

KPMG (2002) : “Study into the methodologies to assess the overall financial position of an insurance undertaking from the perspective of prudential supervisor” Report for European Commission.

Kaas R., Goovaerts M., Dhaene J., Denuit M.(2001) : “Modern Actuarial Risk Theory” Kluwer Academic Publ.

Mack T. (1993) : “Distribution free calculation of the standard error of Chain Ladder reserve estimates” Astin Bull. Vol.23, 213-225.

McCullagh P.,Nelder J.A. (1985) : “Generalized Linear Models” 2e ed. Chapman & Hall.

Quittard-Pinon F. (1993) “Marchés des capitaux et théorie financière” Economica

References
references3
Shao J., Tu D. (1995) : “The Jackknife and Bootstrap “ Springer.

Pinheiro P., Andrade e Silvo J., Centeno M. (2002) “Bootstrap methodology in claims reserving” Astin Colloquium Washington.

Taylor G. (2002) : “Loss reserving - An actuarial Perpective” Kluwer Academic Publ.

References