a multivariate loss model based on markov processes...a multivariate loss model based on markov...

IntroductionRisk processes based on MMAP

Ruin probability analysis

A multivariate loss model based on MarkovProcesses

Jiandong Ren

University of Western Ontario, London, Ontario, Canada

Workshop on Insurance Mathematics

Université Laval 2014 A multivariate loss model based on Markov Processes



Univariate Risk models

I In collective risk model, the aggregate losses of a portfolioof insurance policies during a time interval (0, t ] ismodelled as

S(t) =

N(t)∑i=1

Xi ,

where {N(t), t ≥ 0} counts the number of claims in thetime interval (0, t ] and Xi , i = 1,2 · · · , are i.i.d. claim sizerandom variables and are independent of N(t).





I To measure the risk of S(t),I One could calculate the distribution of S(t) for some fixed

time t .I One could also study the probability of ruin for the surplus

process U(t) = u + ct − S(t)





If the time value of money is considered, one may also considerthe discounted loss process

Sδ(t) =

N(t)∑i=1

ν(Ti)Xi ,

where Ti is the time when the i th claim is paid and ν(Ti) is adiscounting factor.

Sδ(t) =

∫ t

0ν(s)XdN(s) =

∫ t

0X (s)dN(s)




Multivariate Risk models–literature review

I Hesselager (1996) considered aggregate losses for twolines of insurance businesses:

(X ,Y ) =

(N∑

i=0

Ui ,

M∑i=0

Vi

), (1)

where the claim frequencies (N,M) are dependent but theclaim sizes, Ui and Vi , are mutually independent and areindependent of the claim frequencies.

I The dependency of N and M were modelled by commonshock, common mixing variable and competing.

I The author derived a recursive formula for the jointdistribution of the aggregate losses for the two lines ofbusinesses.





I Sundt (1999) studied the joint distribution of the aggregatelosses:

(X1, · · · ,Xm) =N∑

i=0

(U1,i , · · · ,Um,i

), (2)

where the claim number is one–dimensional, but eachclaim generates an m–dimensional random vector, whichmay represent m > 1 types of dependent losses.

I For a general introduction to the aggregation of dependentrisk portfolios, one is referred to Wang (2008).





I Gerber and Shiu (1999), Li and Lu (2005), and Ji andZhang (2010) studied risk processes with two types oflosses:

U(t) = U(0) + ct −N1(t)∑i=1

Xi −N2(t)∑i=1

Yi , t ≥ 0,

I They studied the (competing) ruin probabilitiesE[I(T <∞,Z = z)|U(0) = u],u ≥ 0, where T is the time ofruin and Z is a random variable representing the type ofclaim that causes ruin.





I In Gerber and Shiu (1999), N1(t) and N2(t) areindependent Poisson process.

I In Li and Lu (2005), N1(t) is a Poisson process, and N2(t)is an independent renewal process with generalized Erlang(2) interclaim times.

I In Ji and Zhang (2010), N1(t) and N2(t) are independentrenewal processes with phase-type inter–claim times.





I Cossette and Marceau (2000) studied the ruin probabilityfor

U(t) = U(0) + ct −N1(t)∑i=1

Xi −N2(t)∑i=1

Yi , t ≥ 0,

when N1(t) and N2(t) are dependent through commonshocks.





I Gong, Badescu and Cheung (2012) considered themultivariate process

Ui(t) = Ui(0) + ci t −Ni (t)∑k=1

Xi,k , t ≥ 0, i = 1,2,

where the claim number process Ni(t), i = 1,2 aredependent through common shocks.

I They studied the multivariate ruin probabilitiesP(Tand <∞) and P(Tor <∞), where

Tor = inf{t ≥ 0 : min{Ui , i = 1,2} < 0}

andTand = inf{t ≥ 0 : max{Ui , i = 1,2} < 0}




This paper

I In this talk, I will:I introduce a multivariate risk model based on Markov

processes;I discuss how to compute the distribution of the (present

value of) losses within a fixed time interval;I discuss how to compute the (competing) ruin probabilities.




MMAP Risk process

I Assume that there are K categories of insurance claims ofinterest. Claims arrive in batches that may consist ofclaims from any of the K categories. Let the collection ofall possible combinations of claims be denoted as C0.

I For example, in auto insurance, let type 1 denote propertyclaims and type 2 denote bodily injury claims, then K = 2and C0 = {{1}, {2}, {12}}. In claim batch h = {1}, anaccident causes one property damage claim only; in claimbatch h = {12}, an accident causes one property damageclaim and one bodily injury claim.




MMAP Risk process

I Assume that the claim batches arrive according to amarked Markovian arrival process. (He and Neuts (1998)),which generalize the well–known multivariate risk models.

I Let {J(t)}t≥0 be a continuous time Markov process withfinite state space {1, · · · ,m} and infinitesimal generator D.A state transition in J may be accompanied by anoccurrence of a batch of claims. As such, D isdecomposed to {D0,Dh,h ∈ C0}, where each Dh forh ∈ C0 is a nonnegative matrix giving the transitionintensity of claim batch h.

I Let {Uh(t)}ij denote a random vector representing thesizes of the claims in claim batch h, arriving at time t , andaccompanied by an i → j transition in the process J. Let{fh,t (·)}ij denote the joint probability density function (p.d.f)of {Uh(t)}ij .




Some special cases of MMAP risk process

Special case 1:The simplest case of the model is when type 1 claims arriveaccording to a Poisson process with rates λ1 and type 2 claimsarrive according to an independent Poisson process with rateλ2. For this case, we have D0 = −λ1 − λ2, D1 = λ1 andD2 = λ2. This model was studied in Gerber and Shiu (1999) indetail.





Special case 1’:Type 1 claims arrive according to a Poisson process with ratesλ1 and type 2 claims arrive according to an independentPoisson process with rate λ2. Claim batches that have bothtype 1 and type 2 claim arrived at Poisson rate of λ3. For thiscase, we have D0 = −λ1 − λ2 − λ3, D1 = λ1, D2 = λ2 andD12 = λ3.





Special case 2:Type 1 claims arrive according to a Poisson process with rateλ1. type 2 claims arrive according to an ordinary renewalprocess with inter–arrival times following a generalized Erlangdistribution with rate λ2,1 and λ2,2. To represent the modelusing MMAP, we have

α = (1,0), D1 =

(λ1 00 λ1

), D2 =

(0 0λ2,2 0

),

and

D0 =

(−(λ1 + λ2,1) λ2,1

0 −(λ1 + λ2,2)

).





Special case 3:In this example, the interclaim times follow a two–stagegeneralized Erlang distribution with parameters λ1 andλ2,1 + λ2,2. Given that a claim has occurred, it is of type 1 withprobability λ2,1

λ2,1+λ2,2and of type 2 with probability λ2,2

λ2,1+λ2,2. In

MMAP representation,

α = (1,0), D1 =

(0 0λ2,1 0

), D2 =

(0 0λ2,2 0

),

and

D0 =

(−λ1 λ1

0 −(λ2,1 + λ2,2)

).





Special case 4:Assume that claim are influenced by the states of an externalenvironment, which evolves according to a continuous timeMarkov Chain E(t), with state space {N,R} and infinitesimalgenerator

Q =

(−q1 q1q2 −q2

).

The type 1 and 2 loss intensities in environment states N and Rare denoted by λN,1, λN,2, λR,1, and λR,2 respectively.

D0 =

(−q1 − λN,1 − λN,2 q1

q2 −q2 − λR,1 − λR,2

),

D1 =

(λN,1 0

0 λR,1

)and D2 =

(λN,2 0

0 λR,2

).




MMAP Risk process–the joint distribution ofaggregated losses

I Let X(t) = {X1(t), · · · ,XK (t)} be the vector of typek = {1, · · · ,K} aggregate losses in time interval (0, t ].

I For a fixed time t , let

Gij(x, t) = P{X(t) ≤ x , J(t) = j |J(0) = i},1 ≤ i , j ≤ m, (3)

be the distribution function of X(t) conditional on J(0) = iand J(t) = j .

I Let G(x, t) be a matrix with the ij th entry Gij(x, t).




MMAP Risk process–the joint distribution of theaggregated losses

I Then it is easy to verify that

∂

∂tG(x, t) = G(x, t)D0 +

∑h∈C0

∫G(x− yh, t)Dh(yh, t)dyh.

(4)where Dh(yh, t) is a matrix with the ij th element{Dh}ij{fh,t}ij .




MMAP Risk process–the Laplace transform of theaggregated losses

I LetG∗(ξ, t) =

∫e−ξ·xdG(x, t),

where ξ = {ξ1, · · · , ξK}, be the Laplace transform ofG(x, t).

I Then it satisfies∂

∂tG∗(ξ, t) = G∗(ξ, t)D0 + G∗(ξ, t)

∑h

D∗h(ξh, t), (5)

where D∗h(ξh, t) = {Dh}ij{f ∗h,t}ij with

f ∗h,t =

∫e−ξ·x{fh,t (x)}ijdx

being the Laplace transform of the joint p.d.f {fh,t}ij .Université Laval 2014 A multivariate loss model based on Markov Processes



MMAP Risk process–the moments of the aggregatedlosses

I For k = 1, · · · ,K , let

M(k)(t) = E [Xk (t)I(J(t) = j)|J(0) = i] ,

where I(·) denotes an indicator function, be the matrix ofthe first conditional moments of the type k aggregatelosses





I By differentiating (5) with respect to ξk , we have

ddt

M(k)(t) = M(k)(t)D + eDt D̄(k)(t), (6)

where D̄(k)(t) =∑

h(D̄(h,k)(t)),{D̄(h,k)(t)}ij = {Dh}ij{µ(h,k)(t)}ij , and {µ(h,k)(t)}ij is themean size of type k claim in claim batch h, arriving at timet , and accompanied by an i → j transition in process J.





I For 1 ≤ k1, · · · , kn ≤ K , let

M(k1···kn)(t) = E

∏k∈{k1,··· ,kn}

Xk (t)

× I(J(t) = j)|J(0) = i

be the matrix of the conditional joint moments of theaggregate type (k1 · · · kn) claims.





By differentiating (5) with respect to ξki and ξkj , we obtain

ddt

M(ki ,kj )(t) = M(ki ,kj )(t)D + M(ki )(t)D̄(kj )(t) + M(kj )(t)D̄(ki )(t)

+eD(t) ¯̄D(ki ,kj )(t), (7)

where ¯̄D(ki ,kj )(t) =∑

h¯̄D(h,ki ,kj )(t),

¯̄D(h,ki ,kj )(t) is a matrix havingmnth element {Dh}mn{µh,ki ,kj (t)}mn, and{µh,ki ,kj (t)}mn =

∫xy{fh,ki ,kj ,t (x , y)}mndxdy is the joint moment

of type ki and type kj claims in claim batch h, arriving at time t ,and accompanied by an m→ n transition in process J.




MMAP Risk process

I Calculations show that in example 3, the aggregate lossesfrom the two types of claims are negatively correlated, inExample 4, they are positively correlated.




MMAP Risk process– Example 1:

I let K = 2, C0 = {{1}, {2}, {12}}, and claim batches{1}, {2}, and {12} arrive at rates λ1, λ2 and λ3respectively. Therefore, we have in our notationsD{1} = λ1, D{2} = λ2, D{12} = λ3 and D0 = −λ1 − λ2 − λ3.The sizes of claims arriving in claim batches{1}, {2}, and {12} are denoted by U{1}(t), U{2}(t), andU{12}(t), which have univariate p.d.f. f{1},t (·), f{2},t (·) andbivariate p.d.f. f{12},t (·) respectively.

I Notice that this example generalizes the widely usedcommon–shock bivariate compound Poisson model (seefor example, Example 11.1 in Wang 1998)





I For this model, D = 0, and the differential equation (5) caneasily to solved to obtain the Laplace transform of the jointdistribution of the two types of aggregate losses, yielding:

G∗(ξ1, ξ2, t) = exp[λ1

∫ t

0(f ∗{1},s(ξ1)− 1)ds + λ2

∫ t

0(f ∗{2},s(ξ2)− 1)ds

+λ3

∫ t

0(f ∗{12},s(ξ1, ξ2)− 1)ds

]. (8)





I The mean of type 1 aggregate losses in the time interval(0, t ] is:

M(1)(t) =

∫ t

0D̄(1)(s)ds =

∫ t

0

[λ1µ{1},1(s) + λ3µ{12},1(s)

]ds.

(9)I When the nominal claim size distributions are

time–invariant, then µ{1},1(s) = µ{1},1e−δ1s,µ{12},1(s) = µ{12},1e−δ2s, and equation (9) becomes themean of the present value of type 1 aggregate losses:

M(1)(t) =1− e−δ1t

δ1(λ1µ{1},1 + λ3µ{12},1). (10)





I In this case, the variance of the present value of the type 1aggregate losses is obtained by

M(1,1)(t)−M2(1)(t) =

1− e−2δ1t

2δ1(λ1µ{1},1,1 + λ3µ{12},1,1).

(11)I The covariance between the present value of type 1 and 2

aggregate losses is

M(1,2)(t)−M(1)(t)M(2)(t) = λ3µ{12},1,21− e−(δ1+δ2)t

δ1 + δ2. (12)





I These results for the moments of present value ofaggregate losses generalize those for the univariatemodels in, for example, Willmot (1989).

I It is interesting to notice that if δ1 = δ2, then the correlationcoefficient of the discounted type 1 and 2 losses is given by

λ3µ{12},1,2√(λ1µ{1},1,1 + λ3µ{12},1,1) ∗ (λ2µ{2},2,2 + λ3µ{12},2,2)

,

and is independent of δ and t .




MMAP Risk process–Ruin Probability Analysis

I Define the surplus process

U(t) = U(0) + ct −N1(t)∑i=1

Xi −N2(t)∑i=1

Yi , t ≥ 0, (13)

where c is the premium rate, Xi (Yi) denote the ith type 1(2) claim size random variables. {N1(t) and N2(t) count thenumber of type 1 and type 2 claims, they follow a MMAPprocess with underlying Markov process J(t) andrepresentation (α,D0,D1,D2).

I Since we are interested in probability of ruin by types ofclaim, we assume that no two types of claims can occur ina claim batch.





I We note that as far as U(t) is concerned, it is a risk modelwith claims arrives according to a MAP process withrepresentation (α,D0,D1 + D2).

I So if we assume that

c > πM1(1)e>, (14)

where e> is a column vector of ones, then the ruinprobability is less than one (see for example, page 151 ofAsmussen, 2000).

I For results on ruin probabilities and the more generalGerber–Shiu functions for MAP risk models, see forexample, Albrecher and Boxma (2005) and Cheung andLandriault (2009).





I Let Z denote the cause–of–ruin random variable. Definefor i = 1, · · · ,m and z = 1,2,

φ(z,i,j)(u) = E[e−δT w(U(T−), |U(T )|)

I(T <∞,Z = z, J(T ) = j)|U(0) = u, J(0) = i] ,u ≥ 0,(15)

to be the expected discounted penalty function due toclaim type z.




Ruin Probability Analysis

The quantity φ(z,i,j)(u) satisfies:

cφ′(1,i,j)(u) = δφ(1,i,j)(u)−m∑

k=1

{D0}(i , k)φ(k ,j,1)(u)

−m∑

k=1

{D1}i,k[∫ u

0φ(k ,j,1)(u − x){f1(x)}ikdx + ω1,ik (u)

]

−m∑

k=1

{D2}i,k[∫ u

0φ(k ,j,1)(u − x){f2(x)}ikdx

], (16)

where ω1,ik (u) =∫∞

u w(u, x − u){f1(x)}ikdx .





I Taking Laplace transform on both sides of (16) andrearranging yields[

(cs − δ)I + (D0 + D̂f (s))]Φ̂1(s) = cΦ1(0)− D̂1ω(s), (17)





I LetLδ(s) = (cs − δ)I + D0 + D̂f (s).

It is well known in risk theory literature (see for example,Albreche and Boxma 2005, Cheung and Landriault 2009)that the generalized Lundberg equation

det Lδ(s) = 0 (18)

has m solutions with nonnegative real parts, say,ρ1, · · · , ρm when δ > 0 or when the positive loadingcondition (14) holds. For simplicity, we assume thatρ1, · · · , ρm are distinct.





I To determine Φ1(0), we follow the ideas in Section 4 ofAsmussen (1992). Particularly, because for i = 1, · · · ,m,det Lδ(ρi) = 0, zero is an eigenvalue of Lδ(ρi). Let vi be theleft eigenvector of the matrix Lδ(ρi) corresponding to theeigenvalue zero, then ~viLδ(ρi) = 0. Considering (17), wehave

cviΦ̂1(0) = viD̂1ω(ρi), i = 1, · · · ,m. (19)

This set of m equations can be used to obtain Φ1(0).





I Let Gz(x1, y1|u) denote a matrix with ij th elementgz,i,j(x1, y1|u).

I Then

G1(x1, y1|0) =1c

e−Qx1D1f (x1 + y1), (20)

where Q = V−1diag(ρi)V.I This result generalizes equation (4.7) of Gerber and Shiu

(1999).




Ruin Probability Analysis–A numerical example

I This continues the two–environment example discussedbefore. In this case, the equilibrium distribution of theunderlying Markov Chain J is π = (10/11,1/11). Thusassuming a security loading of 0.1, the premium rate is setat 420. Assuming that a discount rate of δ = 0.1.

I α = (1,0), q1 = 1, q2 = 10, λN,1 = 100, λN,2 = 10,λR,1 = 200, λR,2 = 20. In environment N, the sizes of type1 and 2 claims follow exponential distributions with mean 2and 4 respectively. In environment R, the size of type 1and 2 claims follow exponential distributions with mean 10and 20 respectively.





I Through inverting the Laplace transform formula in (17),we obtain the discounted probabilities of ruin due to type 1and type 2 claims as a function of the initial surplus u. It isshown in the figure next page.





Figure : Markov modulated Poisson model with two types of claims

0 5 10 15 20 25 30 35 40 45 500.2

0.25

0.3

0.35

0.4

0.45

0.5

Initial surplus

Rui

n pr

obab

ilitie

s

Type 1Type 2




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