Academic Articles & Supplements

Finite-Time Ruin Probabilities Using Bivariate Laguerre Series

Eric C.K. Cheung, Jae-Kyung Woo

Author

Eric C.K. Cheung, Jae-Kyung Woo

Title

Finite-Time Ruin Probabilities Using Bivariate Laguerre Series

Description

This notebook demonstrates how to calculate finite-time ruin probability in the compound Poisson risk model based on the publication with the same title (https://doi.org/10.1080/03461238.2022.2089051).

Category

Academic Articles & Supplements

Keywords

Actuarial science, Compound Poisson risk model, Finite-time ruin probability, Laguerre series.

URL

http://www.notebookarchive.org/2022-08-du0w05f/

DOI

https://notebookarchive.org/2022-08-du0w05f

Date Added

2022-08-30

Date Last Modified

2022-08-30

File Size

123.31 kilobytes

Supplements

Rights

CC BY-NC-SA 4.0

Finite-Time Ruin Probabilities Using Bivariate Laguerre Series

Eric C.K. Cheung* and Jae-Kyung Woo*

*School of Risk and Actuarial Studies, UNSW Business School, University of New South Wales, Sydney, Australia.

Abstract
This notebook was created by Eric Cheung and J.-K. Woo based on the paper “Cheung, E.C.K., Lau, H., Willmot, G.E. and Woo, J.-K. Finite-time ruin probabilities using bivariate Laguerre series. Scandinavian Actuarial Journal ( https://doi.org/10.1080/03461238.2022.2089051 )”. We demonstrate how to compute the finite-time ruin probabilities in the classical compound Poisson risk model when claims are distributed as a mixture of two exponentials (see Example 4.2 of the paper). Method 1(b), which is the best of the three methods in the paper, is utilized.

Setting up sparse matrices

and

We begin by setting up the sparse matrices

and

defined in Equations (34) and (39) for truncation points ranging from M = 5 to M = 100 (see Equation (45)).

In[]:=

matQ1element[i_,j_]:=matQ1element[i,j]=

G1-c	i1&&j1
-c	i1&&j≥2
G2	i≥2&&ji-1
G1	i≥2&&ji
λ*ΘpNN[i-j]	i≥3&&j≤i-2

In[]:=

matQ1[M_]:=matQ1[M]=SparseArray[{{i_,j_}matQ1element[i,j]},{M+1,M+1}]

In[]:=

Table[MatrixForm[matQ1[k]],{k,5,100}];

In[]:=

matQ2element[i_,j_]:=matQ2element[i,j]=

G3-c	i1&&j1
-c	i1&&j≥2
G4	i≥2&&ji-1
G3	i≥2&&ji
λ*ΘpNN[i-j]	i≥3&&j≤i-2

In[]:=

matQ2[M_]:=matQ2[M]=SparseArray[{{i_,j_}matQ2element[i,j]},{M+1,M+1}]

In[]:=

Table[MatrixForm[matQ2[k]],{k,5,100}];

Inputting the parameters

We input the relative security loading θ and the Poisson claim arrival rate λ. The density of the claim amounts is represented as p(y) =

∑

j=1



. Note that a scaling factor s/s* = 0.5 (called “claimscale” below) has been applied in the paper for a mixture of two exponentials. When this notebook is used for another claim distribution (that belongs to the class of combinations of exponentials), the scaling factor can be modified to aim for convergence with few terms (see Section 3.4 of the paper). For instance, when claims are distributed as a sum of two exponentials (also in Example 4.2 of the paper), one can set s/s* = 1.0, and then replace the 2nd and 4th lines below by

“claimscale=1.0;”and“z=2;

=1.5*claimrate;

=2.0;

=3.0*claimrate;

=-1.0;”

respectively.

In[]:=

θ=0.1;λ=1.0;

In[]:=

claimscale=0.5;

In[]:=

claimrate=1/claimscale;

In[]:=

z=2;

=(1.0/2)*claimrate;

=1.0/3;

=2.0*claimrate;

=2.0/3;

In[]:=

c=(1+θ)*λ*

∑

j=1

;

Laguerre coefficients for claim density and infinite-time ruin probability

The Laguerre coefficients

p,k

’s can be calculated explicitly since

p,k

∞

∫

p(y)

[y]y

, where

is the Laguerre function. Since the infinite-time ruin probability is well known to be a sum of exponential terms (with the calculations related to the Lundberg’s fundamental equation), its corresponding Laguerre coefficients can be calculated as well.

In[]:=

Θp[k_]:=Θp[k]=

z ∑ j=1 w j * β j β j + 1 2	k0
z ∑ j=1 w j * β j β j + 1 2 * k β j - 1 2 β j + 1 2	k≥1

In[]:=

ΘpN[k_]:=ΘpN[k]=

Θp[0]	k0
Θp[k]-Θp[k-1]	k≥1

In[]:=

ΘpNN[k_]:=ΘpNN[k]=

ΘpN[0]	k0
ΘpN[k]-ΘpN[k-1]	k≥1

In[]:=

NSolvec*s-λ+λ*

∑

j=1

0,s

Out[]=

{{s-3.0631},{s-0.118715},{s0.}}

In[]:=

R[i_]:=R[i]=-s/.NSolvec*s-λ+λ*

∑

j=1

0,s[[i]]

In[]:=

S[i_]:=S[i]=

R[i]

∑

j=1

-R[i]

∑

j=1



-R[i]

In[]:=

InfiniteRuinProb[u_]:=InfiniteRuinProb[u]=

∑

i=1

S[i]*

-R[i]*u



In[]:=

ΘPsi[m_]:=ΘPsi[m]=

z ∑ i=1 S[i]* 1 R[i]+ 1 2	m0
z ∑ i=1 S[i]* 1 R[i]+ 1 2 * m R[i]- 1 2 R[i]+ 1 2	m≥1

In[]:=

ΘPsiN[m_]:=ΘPsiN[m]=

ΘPsi[0]	m0
ΘPsi[m]-ΘPsi[m-1]	m≥1

Finite-time ruin probability

We first input the values of

and

according to Equations (32) and (37) so that

and

are now known. Then, the finite-time ruin probability can be calculated as ψ(u,t) = ψ(u) - h(u,t) in terms of the infinite-time ruin probability ψ(u) and the value of h(u,t) approximated via Equation (45) with a truncation point M.

In[]:=

G1=

c-1

-λ+λ*Θp[0];G2=

c+1

+λ+λ*(Θp[1]-2*Θp[0]);G3=

c+1

-λ+λ*Θp[0];G4=

c-1

+λ+λ*(Θp[1]-2*Θp[0]);

In[]:=

InversematQ1[M_]:=InversematQ1[M]=Inverse[matQ1[M]]

In[]:=

InversematQ1Q2[M_]:=InversematQ1Q2[M]=InversematQ1[M].matQ2[M]

In[]:=

matpsi[M_]:=matpsi[M]=Table[ΘPsiN[m-1],{m,M+1},{l,1}]

In[]:=

gmat[t_,M_]:=gmat[t,M]=-MatrixExp-t*

*IdentityMatrix[M+1]+Inverse[IdentityMatrix[M+1]-InversematQ1Q2[M]].InversematQ1Q2[M].Inverse[IdentityMatrix[M+1]-InversematQ1Q2[M]].InversematQ1[M].matpsi[M]

In[]:=

φ[j_,x_]:=φ[j,x]=LaguerreL[j,x]*



In[]:=

h[u_,t_,M_]:=h[u,t,M]=

∑

m=0

(gmat[t,M])[[m+1,1]]*φ[m,u]

In[]:=

FiniteRuinProb[u_,t_,M_]:=FiniteRuinProb[u,t,M]=InfiniteRuinProb[u]-h[u,t,M]

Table 10 of the paper

Here we compute the finite-time ruin probabilities ψ(u,t) for u, t = 2, 4, 6, 8, 10 using the truncation points M = 10, 20, 30, 40, 50, 60, 70, 80, 90, 100. In this example, the use of M = 50 is sufficient to lead to converging results at the 8th decimal place. Even the use of M = 20 is good enough as far as the 4th or the 5th decimal place is concerned.

In[]:=

TableForm[Table[NumberForm[FiniteRuinProb[claimscale*2*j,2*i,10],{8,8}],{i,1,5},{j,1,5}]]

Out[]//TableForm=

0.20089692	0.09225701	0.04018026	0.01862568	0.01009083
0.31145009	0.16429552	0.08487117	0.04462282	0.02374728
0.38107644	0.22072927	0.12523020	0.07068071	0.03928377
0.43026294	0.26557269	0.16025449	0.09529527	0.05546231
0.46740701	0.30194375	0.19055026	0.11808171	0.07156712

In[]:=

TableForm[Table[NumberForm[FiniteRuinProb[claimscale*2*j,2*i,20],{8,8}],{i,1,5},{j,1,5}]]

Out[]//TableForm=

0.20048555	0.09114582	0.04193977	0.01913525	0.00870810
0.31073268	0.16444438	0.08608829	0.04430243	0.02245497
0.38096467	0.22101718	0.12584609	0.07012709	0.03834473
0.43041487	0.26569392	0.16054146	0.09479592	0.05485195
0.46756007	0.30190050	0.19071521	0.11772347	0.07118755

In[]:=

TableForm[Table[NumberForm[FiniteRuinProb[claimscale*2*j,2*i,30],{8,8}],{i,1,5},{j,1,5}]]

Out[]//TableForm=

0.20051617	0.09111422	0.04195047	0.01915835	0.00867354
0.31072379	0.16443866	0.08610139	0.04429996	0.02243726
0.38095742	0.22102033	0.12584699	0.07012491	0.03833846
0.43041348	0.26569363	0.16053943	0.09479801	0.05484871
0.46755871	0.30189822	0.19071445	0.11772748	0.07118384

In[]:=

TableForm[Table[NumberForm[FiniteRuinProb[claimscale*2*j,2*i,40],{8,8}],{i,1,5},{j,1,5}]]

Out[]//TableForm=

0.20051464	0.09111569	0.04194911	0.01915939	0.00867332
0.31072444	0.16443824	0.08610151	0.04430014	0.02243695
0.38095752	0.22102030	0.12584704	0.07012482	0.03833851
0.43041359	0.26569367	0.16053932	0.09479802	0.05484883
0.46755887	0.30189820	0.19071436	0.11772755	0.07118390

In[]:=

TableForm[Table[NumberForm[FiniteRuinProb[claimscale*2*j,2*i,50],{8,8}],{i,1,5},{j,1,5}]]

Out[]//TableForm=

0.20051472	0.09111562	0.04194916	0.01915934	0.00867337
0.31072442	0.16443826	0.08610149	0.04430016	0.02243695
0.38095753	0.22102030	0.12584705	0.07012481	0.03833851
0.43041359	0.26569368	0.16053933	0.09479801	0.05484883
0.46755887	0.30189821	0.19071436	0.11772755	0.07118391

In[]:=

TableForm[Table[NumberForm[FiniteRuinProb[claimscale*2*j,2*i,60],{8,8}],{i,1,5},{j,1,5}]]

Out[]//TableForm=

0.20051472	0.09111562	0.04194916	0.01915934	0.00867337
0.31072442	0.16443826	0.08610149	0.04430015	0.02243695
0.38095753	0.22102030	0.12584705	0.07012481	0.03833851
0.43041359	0.26569368	0.16053933	0.09479801	0.05484883
0.46755887	0.30189821	0.19071436	0.11772755	0.07118391

In[]:=

TableForm[Table[NumberForm[FiniteRuinProb[claimscale*2*j,2*i,70],{8,8}],{i,1,5},{j,1,5}]]

Out[]//TableForm=

0.20051472	0.09111562	0.04194916	0.01915934	0.00867337
0.31072442	0.16443826	0.08610149	0.04430015	0.02243695
0.38095753	0.22102030	0.12584705	0.07012481	0.03833851
0.43041359	0.26569368	0.16053933	0.09479801	0.05484883
0.46755887	0.30189821	0.19071436	0.11772755	0.07118391

In[]:=

TableForm[Table[NumberForm[FiniteRuinProb[claimscale*2*j,2*i,80],{8,8}],{i,1,5},{j,1,5}]]

Out[]//TableForm=

0.20051472	0.09111562	0.04194916	0.01915934	0.00867337
0.31072442	0.16443826	0.08610149	0.04430015	0.02243695
0.38095753	0.22102030	0.12584705	0.07012481	0.03833851
0.43041359	0.26569368	0.16053933	0.09479801	0.05484883
0.46755887	0.30189821	0.19071436	0.11772755	0.07118391

In[]:=

TableForm[Table[NumberForm[FiniteRuinProb[claimscale*2*j,2*i,90],{8,8}],{i,1,5},{j,1,5}]]

Out[]//TableForm=

0.20051472	0.09111562	0.04194916	0.01915934	0.00867337
0.31072442	0.16443826	0.08610149	0.04430015	0.02243695
0.38095753	0.22102030	0.12584705	0.07012481	0.03833851
0.43041359	0.26569368	0.16053933	0.09479801	0.05484883
0.46755887	0.30189821	0.19071436	0.11772755	0.07118391

In[]:=

TableForm[Table[NumberForm[FiniteRuinProb[claimscale*2*j,2*i,100],{8,8}],{i,1,5},{j,1,5}]]

Out[]//TableForm=

0.20051472	0.09111562	0.04194916	0.01915934	0.00867337
0.31072442	0.16443826	0.08610149	0.04430015	0.02243695
0.38095753	0.22102030	0.12584705	0.07012481	0.03833851
0.43041359	0.26569368	0.16053933	0.09479801	0.05484883
0.46755887	0.30189821	0.19071436	0.11772755	0.07118391

Cite this as: Eric C.K. Cheung, Jae-Kyung Woo, "Finite-Time Ruin Probabilities Using Bivariate Laguerre Series" from the Notebook Archive (2022), https://notebookarchive.org/2022-08-du0w05f