1 GIRO XXIX 2002 Convention 8-11 October 2002 Disneyland® Resort Paris.

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GIRO XXIX2002 Convention

8-11 October 2002

Disneyland® Resort Paris

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IASNEW ACTUARIAL TECHNIQUES REQUIRED

A. DESPEYROUX

C. LEVI

C. PARTRAT

10,11 October 2002

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Agenda

Introduction Run off Triangle Cash Flows Stochastic Methods Discount of Cash Flows Extreme Claims Risks Dependence References Conclusion

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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)

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Run Off Triangle Cash Flows

Date 31/12/n

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

Claims developing during (n+1) years

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0 j j' n Calendar year n0

i xij xi,n-i Calendar year (n+1)

KNOWN

UNKNOWN

i' Xi'j'

n Calendar year (2n)

n+1

n+k

2n

Underwriting

years

Following

years

Development

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• 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

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• 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

knjiijkn XCF

n

k knjiij

n

kkn XCFCF

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• 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..

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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

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Stochastic Methods

• 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() is the “variance function” of the family

jieXE ijij )(

)()( ijij VXV

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Stochastic Methods

• Parameters (factors)

mean

for year i

for delay j

(possibly=1) dispersion parameter

i

j

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Stochastic Methods

• 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 )

n

k knjiijXCF

1

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Stochastic Methods

D.f. FCF

m.g.f.

And inversion (Fast Fourier Transform)

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

n

k knjiX

sCFCF sMeEsM

ij1

)()()(

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Stochastic Methods

• Means Data : in the superior triangle

Maximum likehood method , we obtain

estimators of

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

njiijx )(

jieXE ij ˆˆˆ

)(ˆ

)ˆ(),ˆ(,ˆ ji )(),(, ji

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Stochastic Methods

for E(CF)

with uncertainty measure

or

more generally, we obtain

estimator of

n

k knjiijXECFE

1

)(ˆ)(ˆ

)](ˆ[ CFEV )](ˆ[ CFEMSE

])[(ˆ njiijX )( CFF

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Stochastic Methods

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

jieX ij ˆˆˆˆ

n

k knjiijXFC

1

ˆˆ

]ˆ[ FCV ]ˆ[ FCMSE

)](ˆ[ CFEV ]ˆ[ FCV

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Stochastic Methods

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

)( CFF

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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.

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• 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.

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• 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

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• 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

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• Example yield curve

Yield Curve

2,50%

3,00%

3,50%

4,00%

4,50%

5,00%

5,50%

6,00%

1 2 3 4 5 6 7 8 9 10

Increasing Yield Curve

Flat Yield Curve

Decreasing Yield Curve

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• Example : non discounted cash flows 1000Future Cash Flows

0

100

200

300

400

500

600

1 2 3 4 5 6 7 8 9 10

Série1

Série2

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• Impact of discounting ( long tail development)

0% 2%Decreasing yield curve 901 855Flat Yield curve 890 838Increasing yield curve 880 836

Risk Premium

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• Impact of payment pattern

short tail long tail

Decreasing yield curve 927 901Flat Yield curve 925 890Increasing yield curve 924 880

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• Profit and loss impact

Increase of rate => Profit recognition

Decrease of rate => Reduction of profit

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Extreme Claims

• MeasuresGiven 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

)()( xXPxF F

0)(lim

xFx

F

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Extreme Claims

Value-at-Risk (VaR)

(0.05;0.01;0.005;…)

VaR P(X>VaR)=

VaR x

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Extreme Claims

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 / Xu)

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

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Extreme Claims

These measures are used too for other problems:

Solvency Capital Allocation Coherent measures Etc..

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Extreme Claims

• Classification of theoritical distributions for modelling extreme claims.

Very light light medium heavy very heavy

Weibull(t) t>1

exponential=Weibull (t=1)

Gamma

Lognormal Weibull(t)

t<1

LogGamma() >1

Pareto() >1 Burr(,t) t>1

LogGamma() <1

Pareto() <1 Burr(,t) t<1

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Extreme Claims

• Uncertainty

F unknown

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

Interest measure (F)

Aims : estimation estimator

Estimation uncertainty :

standard error, confidence interval, analytic or bootstrap.

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Extreme Claims

• Return Period Claims frequency excluded

r.v. N(u)=min{i 1:Xiu}(rank of the smallest claim exceeding of u)

Return period of level u :

(in number of claims)

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

)(

1)]([)(

uFuNEuRP

100

1)( 100 uF

Time unit : yearX1 X2 Xn

0 1 2 n

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Extreme Claims

Including claims frequency

Assumption : Poisson process () for the claims frequency

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

(years) )(

1)]([)(

uFuYEuRP n

Time unit : yearX1 X2 Xn Xn+1

Occurrence time 0 T1 T2 Tn Tn+1

Interoccurence time Y0 Y1 Yn

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Extreme Claims

Extreme claim development ?

GLM : existence of the moments supposed

Heavy tail distribution : no assumption on the moments

0 1 …….. …… n-i n-i+1 ……. ….. n

i y0 y1 yn-i Yn-i+1 Yn

development

Observed r.v.

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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’)

0 j n0

KNOWN

Cash Flows CFn+k , CF

i Xij

n

0 j n0

KNOWN

Cash Flows CF'n+k , CF'

i X' ijn

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Risks dependence

• ModellingIf 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 (X ij ; 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 (X ij ; X’ij) Yij

Y'ijSij

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Risks dependence

Copula

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

FXij d.f. of Xij

FX'ij d.f. of X'ij

Bivariate d.f. of (Xij;X'ij)

CopulaMarginals

]1,0[]1,0[: 2 C

)]'(),([)',( '', xFxFCxxFijijijij XXXX

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Risks dependence

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)

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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

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References

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”

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References

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

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References

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.

1 GIRO XXIX 2002 Convention 8-11 October 2002 Disneyland® Resort Paris.

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Transcript of 1 GIRO XXIX 2002 Convention 8-11 October 2002 Disneyland® Resort Paris.