1 giro xxix 2002 convention 8-11 october 2002 disneyland® resort paris
TRANSCRIPT
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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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Run Off Triangle Cash Flows
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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Run Off Triangle Cash Flows
• 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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Run Off Triangle Cash Flows
• 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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Run Off Triangle Cash Flows
• 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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Discounting Cash Flows
• 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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Discounting Cash Flows
• 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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Discounting Cash Flows
• 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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Discounting Cash Flows
• 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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Discounting Cash Flows
• 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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Discounting Cash Flows
• 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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Discounting Cash Flows
• 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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Discounting Cash Flows
• 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.
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