ruin theory in risk models based on time series for count random … · 2017-09-28 · ruin theory...
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Ruin Theory in risk models based on time series forcount random variables
Etienne Marceau, Ph.D. A.S.A.
with H. Cossette, Florent Toureille
A celebration of Hans Gerber�s contributions (Waterloo, Canada)
Université Laval
June 18, 2013
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 1 / 54
1. Introduction
The objective of the talk is examine an extension of the classicaldiscrete-time risk model
Our extension is loosely inspired on Gerber�s own extension of thismodel
Gerber�s extension is based on gaussian linear time series
Our extension is based on time series for count data
We examine the computation of the following ruin measures :
Adjustment coe¢ cientGerber-Shiu fonction on �nite-time horizonJoint probabilities of time of ruin, de�cit at ruin and surplus before ruinDynamic VaR and TVaR
In this talk, we "celebrate" among numerous Gerber�s contributions
Gerber�s gaussian linear risk modelSo-called Gerber-Shiu functionComputation of various ruin measures
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 2 / 54
2. Classic discrete-time risk model
The classical discrete time risk model is due to De Finetti (1957) (seealso e.g. Bühlmann (1970), Gerber (1979) and Dickson (2005)).
We consider an insurance portfolio
Aggregate claim amount in period k 2 N+: r.v. Wk
0j0���| {z }W1
j1���| {z }W2
j2���| {z }W3
j3���| {z }W4
j4���| {z }W5
j5�
W = fWk , k 2 N+g = sequence of i.i.d. r.v.�sPremium income per period: π = (1+ η)E [W ]
Strictly positive security margin: η > 0
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 3 / 54
2. Classic discrete-time risk model
Surplus process: U = fUk , k 2 Ngfor k = 0, U0 = u = initial surplusfor k 2 N+, Uk = Uk�1 + π �Wk = u �∑kj=1
�Wj � π
�Time of ruin: r.v. τ with
τ =
(inf
k2f1,2,3,...gfk,Uk < 0g , if Uk goes below 0 at least once
∞, if Uk never goes below 0
Finite-time ruin probablity over n periods : ψ (u, n) = Pr (τ � n)In�nite-time ruin probability : ψ (u) = lim
n!∞ψ (u, n)
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 4 / 54
2. Classic discrete-time risk model
Net loss in period j 2 N+: Lj = (Wj � π)
L1, L2, ... i.i.d. with E [Lj ] = E [Lj ]� π < 0 (since η > 0)
Random walk with negative drift: Y = fYk , k 2 Ng: Y0 = 0 and
Yk =k
∑j=1Lj
Maximum net cumulative loss process: Z = fZk , k 2 Ng
Zk = maxj=0,1,2,..,k
fYjg
Finite-time ruin probablity over n periods:
ψ (u, n) = Pr (τ � n) = Pr (Zn > u) = 1� FZn (u)
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 5 / 54
2. Classic discrete-time risk model
The distribution of Zn has a mass at 0
Risk management: the knowledge of FZn can be used to determinethe initial capital
We consider 2 risk measures based on ZnDynamic VaRDynamic TVaR
Dynamic VaR :VaRκ (Zn) = F�1Zn (κ)
(see e.g. Albrecher (2009), Tru¢ n et al. (2010), Cossette et al.(2013)).
Interpretation: VaRκ (Zn) = capital required such that the probabilitythat the maximum Zn of the random walk Y over the �rst n periodsexceeds that capital is equal to 1� κ.
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 6 / 54
2. Classic discrete-time risk model
Dynamic TVaR :
TVaRκ (Zn) =1
1� κ
Z ∞
1�κVaR (d )v (Zn) dv
equals to
EhZn � 1fZn>VaRκ(Zn)g
i+ VaRu (Zn) (FZn (VaRκ (Zn))� κ)
1� κ
(Cossette et al. (2013))
Compared to the dynamic VaR, the dynamic TVaR has the advantageof being more sensitive to the stochastic behavior of Zn in the tail ofits distribution.
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 7 / 54
3. Gerber�s linear gaussian (discrete time) risk model
Some papers consider various models with temporal dependence.
Gerber (1982) proposes extension :
Title : Ruin Theory in the linear model, IME 1982The risk model is based on gaussian linear time series (e.g. AR,ARMA, etc.)Gerber examines the approximation of ruin probabilities
Variants and extension
Promislow (1991) : upper bounds for ψ (u) in a similar risk modelChrist and Steinebach (1995) : empirical mgf type estimator of theadjustment coe¢ cient in Gerber�s risk model.Yang and Zhang (2003) : both exponential and non-exponential upperbounds for ψ (u) in an extension to Gerber�s risk model.Tru�n et al (2010) : interesting application of the Gerber�s risk modelin the context of insurance cycles.
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 8 / 54
3. Gerber�s extension : Gerber�s linear gaussian risk model
As stated in almost all actuarial textbooks, compound distributionsare the corner stones of several risk models in risk theory.
The linear risk models such as the Gaussian AR(1) and ARMA(p, q)do not lead to compound distributions for the rvs W .
We adapt the idea of Gerber
We assume that Wk follows a compound distribution
Wk =
(∑Nkj=1 Bk ,j , Nk > 0
0, Nk = 0
r.v. Nk : number of claims in period k�Bk ,j , j 2 N+
= individual claim amounts in period k (positive r.v.�s)�
Bk ,j , j 2 N+= sequence of i.i.d. r.v�s and also independent of Nk
Premium income per period: π = (1+ η)E [W ]
Strictly positive security margin: η > 0
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 9 / 54
4. Ruin theory based on time series for count data
We consider risk models based on compound distributions assumingtime series models for count data for N = fNk , k 2 N+g.4 categories of time series models for count data for N.
Models based on thinning :
N = Poisson INAR, Poisson INMA, Poisson INARMA, etc.These models are based on appropriate thinning operations whichreplace the scalar multiplications by a fraction in the Gaussian ARMAtime series(see e.g. Mckenzie (1986, 1988, 2003), and Joe (1997))Quddus (2008) and Gourieroux and Jasiak (2004) : car accident countdataFreeland (1998) and Freeland and McCabe (2004) : Worker�sCompensation Board of British ColumbiaKremer (1995) adapts the theory of INAR processes to the context ofIBNR-predictions
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 10 / 54
4. Ruin theory based on count ...
Other types of models of times series for count data :
Models based on Markov chains. N = Markov chain of order 1 or more.Models based on a speci�c conditional distribution with stochasticparameters.
N has a dependence structure based on an underlying process ΘΘ = ARMA time series or hidden discrete time Markov chain de�nedon a �nite time spaceMalyshkina, Mannering and Tarko (2009) explore two-state Markovswitching count data models to study accident frequencies
Models based on copulas : marginals are �xed and the dependencestructure of N is based on a copula (see e.g. Frees and Wang (2006))
Reviews on time series models for count data :
McKenzie (2003)Cameron and Trivedi (1998)Kedem and Fokianos (2002)
In this talk, we focus on the case N = Poisson INAR1 process
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 11 / 54
Risk Model �Poisson INAR(1)
Dynamic of N = fNk , k 2 N+g
Nk = α �Nk�1 + εk , (1)
for k = 2, 3, ...ε = fεk , k 2 N+g : sequence of i.i.d. r.v.�s withεk � Pois ((1� α)λ) where α 2 [0, 1].α �Nk�1 :
The operator "�" is used in models based on thinningWe have
α �Nk�1 =Nk�1
∑i=1
δk�1,i ,
where fδk�1,i , i = 1, 2, ...g is a sequence of i.i.d Bernoulli r.v�s withmean α and independent of Nk�1α = dependence parameter
Given (1), N is stationary with Nk � Pois (λ).Étienne Marceau (Université Laval) Poisson frais June 18, 2013 12 / 54
Risk Model �Poisson INAR(1)
Interpretation in an insurance context :
nb of claimsin period k| {z }
Nk
=
nb of claimsfrom period k � 1leadind to claimsin period k| {z }α �Nk�1
+
nb of new claimsin period k| {z }
εk
Autocorrelation function for N
γN (h) = αh, h � 1,with γN (1) 2 [0, 1).Covariance between Wk and Wk+h :
Cov (Wk ,Wk+h) = λαhE [B ]2 ,
for h � 1.Étienne Marceau (Université Laval) Poisson frais June 18, 2013 13 / 54
Risk Model �Poisson INAR(1)
Paths of N (α = 0)
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 14 / 54
Risk Model �Poisson INAR(1)
Paths of N (α = 0.4)
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 15 / 54
Risk Model �Poisson INAR(1)
Paths of N (α = 0.99)
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 16 / 54
Risk Model �Poisson INAR(1)
Paths of V (α = 0)
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 17 / 54
Risk Model �Poisson INAR(1)
Paths of V (α = 0.4)
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 18 / 54
Risk Model �Poisson INAR(1)
Paths of V (α = 0.99)
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 19 / 54
Risk Model �Poisson INAR(1)
Paths of Z (α = 0)
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 20 / 54
Risk Model �Poisson INAR(1)
Paths of Z (α = 0.4)
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 21 / 54
Risk Model �Poisson INAR(1)
Paths of Z (α = 0.99)
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 22 / 54
Adjustment coe¢ cient and asymptotic expression
De�ne the convex function
cn (r) =1nln�EherVn
i�=1nln�Eher (Sn�nπ)
i�=, (2)
where Sn = ∑nk=1Wk
Lundberg adjusment coe¢ cient : ρ is the solution to
c (r) = limn!∞
cn (r) = 0. (3)
(see e.g. Nyrhinen (1998), Müller and P�ug (2001))Asymptotic Lundberg-type result :
limu!∞
� ln (ψ (u))u
= ρ,
Adjustment coe¢ cient ρ : measure of dangerousness of an insuranceportfolio.The expression of cn (r) depends on the temporal dependencestructure for W .
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 23 / 54
Adjustment coe¢ cient �Poisson INAR(1)
Poisson AR(1). Assuming that αMB (r) < 1, the expression forc (r) is given by
c (r) =(1� α)2 λMB (r)1� (αMB (r))
� (1� α) λ� rπ.
Poisson AR(1). Impact of dependence parameter α.If α < α0, then ρ > ρ0.Dangerousness increases with the dependence parameter α.
Poisson AR(1) + Exponential claims. Assume that B � Exp (β)with mean 1
β . Then, we have
ρ =(1� α) βη
1+ η, (4)
where α 2 [0, 1). Special case : if α = 0, ρ = βη(1+η)
(independence).
(Cossette et al. (2010)).
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 24 / 54
Adjustment coe¢ cient �Poisson INAR(1)
Example.
λ = 0.4, B � Exp�
β = 12
�and η = 0.25 ; c = 1
approximation based on 5000 crude MC simulations, 1000 periods
α E [Z1000 ] Var (Z1000) VaR0.95 (Z1000) VaR0.99 (Z1000)0.2 9.646 151.904 35.018 55.4930.8 34.777 1964.363 127.714 191.145
approximation with asymptotic expression
α ρ � 1ρ ln (0.05) � 1ρ ln (0.01)0.2 0.08 37.447 57.5650.8 0.02 149.787 230.258
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 25 / 54
Adjustment coe¢ cient �Poisson INAR(1)
Example (suite)
Approximation of ψ (u) using importance sampling
α ρ eψ (0) eψ (10) eψ (20) eψ (30)0 0.1 0.771769 0.280084 0.103552 0.0378060.2 0.08 0.816092 0.364378 0.164038 0.0723330.4 0.06 0.850753 0.463960 0.255592 0.140231
ψ (u) " as α "Adaptation from e.g. Asmussen (2003).
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 26 / 54
Finite-time ruin probability
We present recursive formulas for the computation of �nite-time ruinmeasuresAdditional assumptions : π 2 N+, u 2 N, rv B is de�ned on N (tosimplify the notation)All formulas can be easily adapted to the case when the support isf0, 1h, 2h, ...g with h > 1Notation :
Pr (N1 = j) = pj , j 2 N
Pr (N2 = k2 jN1 = k1) = pk1,k2 , k1, k2 2 N
We compute ψ (u, n) with
ψ (u, n) =∞
∑k1=0
pk1ψ (u, njN1 = k1) .
whereψ (u, njN1 = k1) = Pr (τ � njN1 = k1) .
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 27 / 54
Finite-time ruin probability
For n = 1, we have
ψ (u, 1jN1 = k1) =∞
∑l=u+c+1
f (l , k1),
where
f (j ; k1) =
8<:Pr (B1 + ...+ Bk1 = j) , if k1 > 01, if k1 = 0 and j = 00, if k1 = 0 and j 6= 0
.
For n � 2, we have
ψ (u, njN1 = k1) =∞
∑l=u+c+1
f (l , k1)
+∞
∑k2=0
pk1,k2u+c
∑j=0
f (j , k1)ψ (u + π � j , n� 1jN1 = k2) .
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 28 / 54
Finite-time ruin probability
Example :
Poisson parameter λ = 0.4; claim amount : B � Geom�13
�, with
E [B ] = 2premium income : 1
α E [Z20 ] Var (Z20) VaR0.95 (Z20) TVaR0.95 (Z20)0 4.2668398 33.264562 12 17.7237380.2 4.6316362 40.952906 13 19.6544150.5 5.4468555 63.290655 16 24.3475080.8 7.0412564 135.708264 22 35.378478
E [Z20 ], Var (Z20) and TVaR0.95 (Z20) increase as α " (to be shown)Interpretation :
if the dependence relation is stronger, than the risk process is riskierthen the required amount as initial capital has to be higher
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 29 / 54
Finite-time Gerber-Shiu Function
Additional assumptions : π 2 N+, u 2 N, rv B is de�ned on N (tosimplify the notation)
In�nite-time GS function(1998) :
mδ(u) = E�v τw (Uτ� , jU(τ)j) 1fτ<∞gjU(0) = u
�w(x , y), for x , y � 0, is the penalty function at the time of ruin for thesurplus prior to ruin and the de�cit at ruin,v = e�δ < 1 is the discounted factor,1A is the indicator function, such that 1A = 1, if the event A occurs,and equals 0 otherwise.
Special cases
when w(x , y) = 1, for all x , y � 0, mδ(u) = Laplace Transform (LT)of τif δ = 0 and w(x , y) = 1 for all x , y 2 R, mδ(u) corresponds to ψ(u)
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 30 / 54
Finite-time Gerber-Shiu Function
Extended version of the expected discounted penalty function over nperiods :
mδ(u, n) = E�v τw (Uτ� , jU(τ)j) 1fτ�ngjU(0) = u
�. (5)
(inspired by Morales (2010))
Special cases
when w(x , y) = 1 for all x , y � 0 and δ = 0, mδ(u, n) = ψ (u, n)
De�ne
mδ(u, n;N1 = k1) = E�v τw (Uτ� , jU(τ)j) 1fτ�ngjN1 = k1,U(0) = u
�.
We have
mδ(u, n) =∞
∑k1=0
pk1mδ(u, n;N1 = k1), (6)
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 31 / 54
Finite-time Gerber-Shiu Function
Proposition. We have
mδ(u, n;N1 = k1) = v∞
∑k2=0
pk1,k2u+π
∑j=0
f (j ; k1)mδ(u+π� j , n� 1;N1 = k2)
+v∞
∑j=u+π+1
f (j ; k1) w(u, j � u � π)
where
f (j ; k1) =
8<:Pr (B1 + ...+ Bk1 = j) , if k1 > 01, if k1 = 0 and j = 00, if k1 = 0 and j 6= 0
. (7)
for n 2 N+ and for k1 2 N
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 32 / 54
Bivariate and trivariate probabilities
De�nitions inspired from Alfa and Drekic (2007) (for n 2 N+)
Bivariate joint probabilities for (τ,Uτ�) :
ζn,i (u) = Pr (τ = n,Uτ� = i jU(0) = u)
Bivariate joint probabilities for (τ, jUτ j) :
ξn,j (u) = Pr (τ = n, jUτ j = j jU(0) = u)
Trivariate joint probabilities for (τ,Uτ�, jUτ j) :
$n,i ,j (u) = Pr (τ = n,Uτ� = i , jUτ j = j jU(0) = u)
for i = 0, 1, 2, ...; j = 1, 2, ...
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 33 / 54
Bivariate joint probabilities for time at ruin and de�cit atruin
We de�ne
ξ(k1)n,j (u) = Pr (τ = n, jUτj = j jU(0) = u,N1 = k1)
for n 2 N+, u 2 N, j 2 N
We have the following relationship
ξn,j (u) =∞
∑k1=0
pk1ξ(k1)n,j (u).
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 34 / 54
Bivariate joint probabilities for time at ruin and de�cit atruin
Proposition. Bivariate joint probabilities for (τ, jUτj).For n = 1, we have
ξ(k1)1,j (u) = f (u + c + j ; k1).
where
f (j ; k1) =
8<: Pr�B1 + ...+ Bk1 = j
�, if k1 > 0
1, if k1 = 0 and j = 00, if k1 = 0 and j 6= 0
.
For n � 2, we have
ξ(k1)n,j (u) =
∞
∑k2=0
Pr (N2 = k2 jN1 = k1)u+c
∑l=0
f (l , k1)ξ(k2)n�1,j (u + c � l).
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 35 / 54
Bivariate joint probabilities for time at ruin and de�cit atruin
Relations.
Ruin probabilities :
ψ(u, n) =n
∑k=1
∞
∑j=1
ξk ,j (u) .
Moments :
E [jUτ jm j τ � n,U(0) = u] =∑nk=1 ∑∞
j=1 jm ξk ,j (u)
ψ(u, n). (8)
More generally, for w (x , y) = ϕ (y) :
mδ(u, n) =n
∑k=1
∞
∑j=1
vk ϕ (j) ξk ,j (u) .
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 36 / 54
Trivariate joint probabilities for time at ruin, surplus beforeruin and de�cit at ruin
We de�ne
$(k1)n,i ,j (u) = Pr (τ = n,Uτ� = i , jUτj = j jU(0) = u,N1 = k1) ,
for n 2 N+, i = 0, 1, ..., j = 1, 2, ...
We have the following relationship
$n,i ,j (u) =∞
∑k1=0
pk1 � $(k1)n,i ,j (u).
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 37 / 54
Trivariate joint probabilities for time at ruin, surplus beforeruin and de�cit at ruin
Proposition. Trivariate joint probabilities for (τ,Uτ�, jUτj).For n = 1, we have
$(k1)1,i ,j (u) = δi ,u � f (u + π + j , k1),
where δi ,u = Kronecker delta of (i , u)
δi ,u =
�1, if i = u0, otherwise
.
For n � 2, we have
$(k1)n,i ,j (u) =
∞
∑k2=0
pk1,k2u+c
∑l=0
f (l , k1)$(k2)n�1,i ,j (u + π � l).
where
f (j ; k1) =
8<: Pr�B1 + ...+ Bk1 = j
�, if k1 > 0
1, if k1 = 0 and j = 00, if k1 = 0 and j 6= 0
.
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 38 / 54
Trivariate joint probabilities for time at ruin, surplus beforeruin and de�cit at ruin
Advantage :
We have all the probabilities to compute any quantities of interestGeneral relation :
mδ(u, n) =n
∑k=1
∞
∑i=0
∞
∑j=1
vkw(i , j) $k ,i ,j (u) , for u � 0 and n 2 N+
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 39 / 54
In�nite-time ruin measures
Expected discounted penalty function :
mδ(u) = E�v τw (Uτ� , jU(τ)j) 1fτ<∞gjU(0) = u
�For computation purposes, we assume that N is a discrete-timeMarkov Chain de�ned on Am = f0, 1, 2, ...,mg.We compute mδ(u) with
mδ(u) =m
∑k1=0
pk1 �mδ(u;N1 = k1), for u 2 N, (9)
where
mδ(u;N1 = k1) = E�v τw (Uτ� , jU(τ)j) 1fτ<∞gjU(0) = u,N1 = k1
�,
(10)for k1 2 Am
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 40 / 54
In�nite-time ruin measures
Recursive relation for mδ(u;N1 = k1) (u 2 N) :
mδ(u;N1 = k1) = vm
∑k2=0
pk1,k2u+c
∑j=0
f (j ; k1)mδ(u + π � j ;N1 = k2)
+ vγ(u + π;N1 = k1), for k1 2 Am ,
where
γ(u;N1 = k1) =∞
∑j=u+1
w(u � π, j � u)f (j ; k1),
and
f (j ; k1) =
8<:Pr (B1 + ...+ Bk1 = j) , if k1 > 01, if k1 = 0 and j = 00, if k1 = 0 and j 6= 0
.
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 41 / 54
In�nite-time ruin measures
Two approaches to compute the values mδ(u;N1 = k1)
Approach #1 : Laplace TransformApproach #2 : Recursive relation
In both cases, we need to �nd the values of mδ(u;N1 = k1) foru 2 f0, 1, ..., c � 1g and k1 2 AmIn the case when N is a Poisson INAR(1) process, we approximate Nby a Markov chain N (m) de�ned on Am .
The m�truncated joint probability are given by
Pr(N (m)1 = k1,N(m)2 = k2) =
Pr(N (m)1 = k1,N(m)2 = k2)
Pr(N (m)1 � m,N (m)2 � m), k1, k2 2 Am .
m is �xed such that Pr(N1 � m,N2 � m) is close to 1
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 42 / 54
Conclusion
Examine the impact of dependence of various (�nite- andin�nite-time) ruin measures.
When the claim amount Bc is a continuous rv,
approximate Bc by a discrete rv B using discretization technicsuse the recursive formulas
Computation time can be signi�cative.
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 43 / 54
Epilogue �Numerical evaluation of ruin probabilities basedon simulation
Here is a brief comment about another important Gerber�scontribution : application of theory of martingale in ruin theoryWe consider continuous-time risk models with dependence (see e.g.e.g. Albrecher et Teugels (2006), Boudreault et al. (2007), Cossetteet al. (2009))We consider an insurance portfolioNotations
continuous rv Xk : amount of the kth claim (k 2 N+)continuous rv Wk : time elapsed between claim #(k � 1) and claim#k (k 2 N+)
Illustration0j0���| {z }W1
X1jT1�����| {z }
W2
X2jT2��|{z}W3
X3jT3����| {z }
W4
X4jT4�|{z}W5
X5jT5�
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 44 / 54
Epilogue �Numerical evaluation of ruin probabilities basedon simulation
Assumption : f(Xk ,Wk ) , k 2 N+g forms a sequence of i.i.d. pairs ofrvs with (Xk ,Wk ) � (X ,W ) for k 2 N+.
Joint pdf of (X ,W ) : fX ,W (x ,w)
Joint cdf of (X ,W ) : FX ,W (x ,w)
Joint mgf of (X ,W ): MX ,W (x ,w)
Surplus process : U = fU (t) , t � 0g où
U (t) = u + ct �∑N (t)k=1 Xk , t > 0
U (0) = u = initial surplus
c = premium rate where c = (1+ η)E [X ]E [W ]
with η > 0
Claim number process N = fN (t) , t � 0g whereN (t) = ∑∞
k=1 1fW1+...+Wk�tg
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 45 / 54
Epilogue �Numerical evaluation of ruin probabilities basedon simulation
Random walk : Y = fYk , k 2 Ng where Y0 = 0 and
Yk = Yk�1 + (Xk � cWk ) , k = 1, 2, ...
In�nite-time ruin probability : ψ (u) = Pr (Z > u) whereZ = max
k2NfYkg.
Lundberg�s equation : Eher (X�cW )
i= MX ,W (r ,�cr)
Adjustment coe¢ cient : ρ is the unique strictly positive solution of
Eher (X�cW )
i= E
herLi= MX ,W (r ,�cr) = 1
where L = X � cW = net loss rv
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 46 / 54
Epilogue �Numerical evaluation of ruin probabilities basedon simulation
Time of ruin : rv τ
Using theory of martingale, Gerber obtained the the following result
Gerber�s exact expression for ψ (u) :
ψ (u) =e�ρu
E�e�ρU (τ)jτ < ∞
� ,for u � 0.
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 47 / 54
Epilogue �Numerical evaluation of ruin probabilities basedon simulation
3 approaches for numerical evaluation of ψ (u) based on simulation :
#1 : Crude Monte Carlo simulation method#2 : MC simulation method based on Gerber�s exact expression#3 : Importance sampling based change of measure
Nb of simulations : m
Approach #1 - Crude MC simulation : approximate ψ (u) by
eψ (u; ρ, n,m) = 1m
m
∑j=11fτ(j)<ng
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 48 / 54
Epilogue �Numerical evaluation of ruin probabilities basedon simulation
Approach #2 - MC simulation based on Gerber�s exact expressionψ (u) = e�ρu
E [e�ρU (τ)jτ<∞]:
Approximate Ehe�ρU (τ)jτ < ∞
iby
∑mj=1 e�ρU (j)(τ(j))1fτ(j)<ng∑mj=1 1fτ(j)<ng
We obtain
eψ (u; ρ, n,m) = e�ρu1m ∑mj=1 1fτ(j)<ng
1m ∑mj=1 e
�ρU (j)(τ(j))1fτ(j)<ng
It seems to correct the bias in the approximation under approach #1
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 49 / 54
Epilogue �Numerical evaluation of ruin probabilities basedon simulation
Approach #3 - Importance sampling
Based on another exact expression
ψ (u) = e�ρuE (ρ)heρU (τu )
ifor u � 0E (ρ) [...] : expectation de�ned on a new equivalent probability measure(see e.g. Asmussen & Albrecher (2009)).De�ne the rv L = X � cW with pdf fLThen, under new measure probability measure, we have
f (ρ)L (x) =eρx
E�eρL� fL (x)
for x 2 R.Di¢ culty : simulate under new probability measure
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 50 / 54
Epilogue �Numerical evaluation of ruin probabilities basedon simulation
Example : Distribution of (X ,W ) is based on fgm copula
Cα (u1, u2) = u1u2 + θu1u2 (1� u1) (1� u2) ,
and X � Exp (β), W � Exp (λ),Joint pdf of (X ,W ) :
fX ,W (x1, x2) = βe�βx1λe�λx2
+θβe�βx1λe�λx2 + θ2βe�2βx12λe�2λx2
�θ2βe�2βx1λe�λx2 � θβe�βx12λe�2λx2 .
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 51 / 54
Epilogue �Numerical evaluation of ruin probabilities basedon simulation
Example : (suite)
Joint pdf of (X ,W ) under change of measure : f (ρ)X ,W (x , y) = ...
(1+ θ)
E�eρL� � β
β� ρ
��λ
λ+ ρ
�(β� ρ) e�(β�ρ)x (λ+ ρ) e�(λ+ρ)y
� θ
E�eρL� � 2β
2β� ρ
��λ
λ+ ρ
�(2β� ρ) e�(2β�ρ)x (λ+ ρ) e�(λ+ρ)y
� θ
E�eρL� � β
β� ρ
��2λ
2λ+ ρ
�(β� ρ) e�(β�ρ)x (2λ+ ρ) e�(2λ+ρ)y
+θ
E�eρL� � 2β
2β� ρ
��2λ
2λ+ ρ
�(2β� ρ) e�(2β�ρ)x (2λ+ ρ) e�(2λ+ρ)y
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 52 / 54
Epilogue �Numerical evaluation of ruin probabilities basedon simulation
Example : (suite)
β = 1, λ = 1, premium rate c = 1.5Dependence parameter θ = 0.5ρ = 0.3788264025
u ψ (u) exact ψ (u) Gerber ψ (u) IS0 0.64512257776 0.644885890756 0.644888070605 0.09495341322 0.095648195486 0.0947083302910 0.01428545356 0.014594674121 0.0145949997620 0.00032334926 0.000380799477 0.00032934350
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 53 / 54
Conclusion #2
Approach #2 (with Gerber�s formula) is easy to apply.
It is possible to apply both approaches #2 and #3 for many jointdistributions for (X ,W )
Bivariate exponential distributions (Downton-Moran, Raftery, etc.)Bivariate gamma distributionsBivariate mixed exponentialBivariate mixed ErlangEtc.
It is possible to examine various dependence structure and the impactof dependence on ψ (u)
Using results from Schmidli (2010), we use approach #3 to evaluateG-S function.
Work in progress ..
Thank you for your attention !
Étienne Marceau (Université Laval) Poisson frais June 18, 2013 54 / 54