Ruin analysis of a discrete-time dependent Sparre Andersen model with external financial activities and randomized dividends

Kim, Sung Soo; Drekic, Steve Article Ruin analysis of a discrete-time dependent Sparre Andersen model with external financial activities and randomized dividends Risks Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Kim, Sung Soo; Drekic, Steve (2016) : Ruin analysis of a discrete-time dependent Sparre Andersen model with external financial activities and randomized dividends, Risks, ISSN 2227-9091, MDPI, Basel, Vol. 4, Iss. 1, pp. 1-15, https://doi.org/10.3390/risks4010002 This Version is available at: https://hdl.handle.net/10419/167876 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ Article Ruin Analysis of a Discrete-Time Dependent Sparre Andersen Model with External Financial Activities and Randomized Dividends Sung Soo Kim and Steve Drekic * Department of Statistics & Actuarial Science, University of Waterloo, 200 University Avenue West, Waterloo, ON N2L 3G1, Canada *Correspondence: sdr[email protected]; Tel.: +1-519-888-4567 (ext. 35550); Fax: +1-519-746-1875 Academic Editor: Mogens Steffensen Received: 18 September 2015; Accepted: 20 January 2016; Published: 3 February 2016 Abstract: We consider a discrete-time dependent Sparre Andersen risk model which incorporates multiple threshold levels characterizing an insurer’s minimal capital requirement, dividend paying situations, and external financial activities. We focus on the development of a recursive computational procedure to calculate the finite-time ruin probabilities and expected total discounted dividends paid prior to ruin associated with this model. We investigate several numerical examples and make some observations concerning the impact our threshold levels have on the finite-time ruin probabilities and expected total discounted dividends paid prior to ruin. Keywords: Sparre Andersen model; randomized dividends; ruin probability; threshold level 1. Introduction The classical Cramér–Lundberg model is a foundational mathematical representation of an insurer’s surplus process in risk theory. Despite its tractability, however, the model has limitations in terms of applications. Certainly, more complex models are desirable in modern industrial settings. This paper strives to contribute to the ever-growing literature on insurance risk models that reflect more general and realistic modelling approaches to ruin theory. Bruno de Finetti [ 1 ] first introduced the notion of a dividend strategy and the idea of finding an optimal dividend payment strategy for the insurance risk model. This was followed by numerous other researchers who further explored the problem in a variety of contexts (for reviews of the area, the interested reader is directed to Albrecher and Thonhauser [ 2 ] and Avanzi [ 3 ]). In particular, Drekic and Mera [ 4 ] considered the ruin analysis of a threshold-based dividend payment strategy in a discrete-time Sparre Andersen model. Their analysis was an extension of the work by Alfa and Drekic [ 5 ], in which a Sparre Andersen insurance risk model in discrete time was analyzed as a doubly-infinite Markov chain to establish a computational procedure for calculating the joint probability distribution of the time of ruin, the surplus immediately prior to ruin, and the deficit at ruin. In this paper, we focus on the development of a recursive computational procedure to calculate the finite-time ruin probabilities and expected total discounted dividends paid prior to ruin associated with a model which generalizes the single threshold-based risk model introduced by Drekic and Mera [ 4 ]. In actual fact, three additional threshold levels are introduced to depict a minimum surplus level control strategy and external financial activities related to both investment and loan undertakings. Readers are referred to, for example, Cai and Dickson [ 6 ] and Li [ 7 ] for other general investment strategies found in insurance risk models, where the latter paper examined an insurance risk model with risky investments under the assumption that the risky assets follow a Wiener process, and the former paper considered a Markov chain based interest rate model. Korn and Wiese [ 8 ] studied Risks 2016,4, 2; doi:10.3390/risks4010002 www.mdpi.com/journal/risks Risks 2016,4, 2 2 of 15 optimal investment strategies in an insurance risk model where they also assumed that the risky assets follow a Wiener process. The remainder of the paper is organized as follows. In Section 2, we introduce notation and specify the fundamental components underlying our threshold-based risk model. Section 3 details the derivation of a recursive formula (namely, Equation (9)) for the finite-time ruin probability associated with our proposed risk model and demonstrates the simplification of the result to that of Drekic and Mera [ 4 ]. Section 4 presents the derivation of a similar recursive formula (namely, Equation (13)) to compute the expected total discounted dividends paid prior to ruin and likewise demonstrates the simplification of the result to that of Drekic and Mera [ 4 ]. Finally, Section 5 discusses some numerical examples and related findings. 2. Model Description and Assumptions For t∈N (with N being the set of non-negative integers), we define Ut as the insurer’s amount of surplus at time t (measured in discrete monetary units) and Ft as the amount of funds present at time t in the external fund of the insurer, a separate monetary account the insurer holds to better manage its reserve through both investment activities and loan undertakings. In actual fact, Ut represents the surplus level at the end of the time interval (t− 1, t] , t∈Z+ (with Z+ being the set of positive integers), at which point any premiums, deposits, claims, or withdrawals corresponding to this time interval have been received/paid out. We adopt the convention that premiums are received at (t− 1 )+ and any claims are applied at t−. In what follows, we assume that it is the insurer’s policy to pay out all the outstanding debt before resuming investment activities, and that the insurer first utilizes its investment assets to make any adjustments to its surplus level before engaging in loan activities. To differentiate between investment activities and loan activities with respect to the external fund, we split the support set of {Ft , t∈N} into two disjoint sets, namely Ft∈( 0, ∞) and Ft∈[β , 0 ] , where β is a non-positive integer representing the minimum support value of {Ft , t∈N} . In fact, β is one of four threshold levels we feature in our model with the understanding that −β represents the borrowing limit of the insurer. When Ft> 0, Ft represents the insurer’s investment activities in which interest is assumed to be earned at a constant rate of κ1> 0 per period. Conversely, when β≤Ft≤ 0, Ft represents the insurer’s loan activities and interest expense accumulates at a constant rate of κ2>0 per period. We next introduce the remaining three threshold levels (to be denoted by `1 , `2 , and `3 ) and explain how they, along with β , define our risk model. To aid in the understanding, let Ut− and Ft− represent the surplus and external fund levels, respectively, immediately after a claim instance but before a withdrawal instance. Herein, withdrawal refers to any cash inflow from the external fund to the surplus process (whereas deposit refers to any cash outflow from the surplus process to the external fund). Firstly, the threshold `1 represents the insurer’s minimum acceptable surplus level, and if Ut− (corresponding to the time interval (t− 1, t] ) is below `1 due to a claim, we withdraw or borrow from Ft− to bring Ut− up to level `1 . However, if Ft−=β , then we can neither withdraw nor borrow more from Ft− even if Ut− is below `1 . In addition, if Ft− (corresponding to the time interval (t− 1, t] ) drifts below β due to interest expense accumulation, we use Ut− to pay back the difference at t−as a form of deposit so that Ftis at least kept at its minimum support value of β. On the other hand, `2 is a trigger point for investment activities. If Ut≥`2 , a constant deposit of size d is paid to the external fund at t+ . Note that the deposit and withdrawal amounts are also stochastic in the sense that they are dependent on the surplus process, and that a deposit can be made at both the left and right limits of a time interval. We denote the left limit deposit amount corresponding to the time interval (t , t+ 1 ] to be D1,t , the right limit deposit amount corresponding to the time interval (t− 1, t] to be D2,t , and the withdrawal amount corresponding to the time interval (t−1, t]to be Wt. Lastly, as in Drekic and Mera [ 4 ], if Ut≥`3 , a random dividend is paid out to shareholders at t+ . We denote the random dividend paid at time t by Dt and assume that {Dt , t∈N} is a conditionally Risks 2016,4, 2 3 of 15 independent and identically distributed (iid) sequence of random variables given {Ut , t∈N} . As a final requirement, we assume that `1≤`2≤`3. To sum up, premiums and left limit deposits corresponding to the time interval (t , t+ 1 ] , t∈N , are collected and paid out at t+according to the following respective (random) rates: Pt=(cif Ut< `3, Xtif Ut≥`3, and D1,t=(0 if Ut< `2, dif Ut≥`2, where Xt=c− Dt and bi=Pr{Xt=i} , i=c1 , c1+ 1, . . . , c2 , denotes the probability mass function (pmf) of Xt . We refer to c∈Z+ as the pure (constant) premium and assume that c1 , c2∈ {d , d+ 1, . . . , c} where d≤c , c1≤c2 , and ∑c2 i=c1bi= 1. Clearly, c1 and c2 are the respective lower and upper support values of the distribution of the random premium amount Xt . Note that, by assumption, the probability distribution of Xtis identical for all values of t∈N. Let µ=E{X0}denote the common mean. Withdrawals and right limit deposits corresponding to the time interval (t− 1, t] , t∈Z+ , are made at t−according to the following respective (random) rates: Wt=(0 if Ut−≥`1, min{`1−Ut−, max{0, Ft−−β}} if Ut−< `1, and D2,t=max{0, β−Ft−}. For illustrative purposes, Figures 1and 2depict an example of the simultaneous evolution of both the surplus process and that of the external fund. t 012345 Ut 0+1+2+3−3+4−4+5− `1 v `2 `3 Premium Premium - Left deposit Claim Claim Withdrawal Withdrawal Claim + Right deposit, causing Ruin Premium Initial surplus Figure 1. Sample evolution of the surplus process {Ut,t∈N}. Risks 2016,4, 2 4 of 15 t 012 3 45 1+2−2+3−4−5− Ft β g Initial fund amount Left deposit Interest income Interest income Withdrawal Withdrawal Interest expense Right deposit Figure 2. Sample evolution of the external fund process {Ft,t∈N}. Beginning at time 0 with an initial surplus level of v∈ {`1 , `1+ 1, . . .} and an initial external fund amount of g∈N, the insurer’s amount of surplus at time tis expressible as Ut=v+ t−1 ∑ i=0 Pi− t−1 ∑ i=0 D1,i+ t ∑ i=1 Wi− t ∑ i=1 D2,i− Nt ∑ i=1 Yi,t∈N, (1) where Nt is the number of claims occurring by time t and individual claim sizes {Yi , i∈Z+} are assumed to form an iid sequence of positive, integer-valued random variables. We assume that the number of claims process is a discrete-time renewal process with independent, positive, integer-valued interclaim times {Wi , i∈Z+} , where Wi is the time between the (i− 1 ) -th and i -th claims (with the understanding that the 0-th claim occurs at time 0). In particular, {Wi , i∈Z+} forms an iid sequence of positive random variables with common pmf ak=Pr{Wi=k} , k= 1, 2, . . . , na where na∈Z+ , and corresponding survival function Ak=Pr{Wi>k}= 1 −∑k j=1aj . Furthermore, we assume that the pairs {(Wi,Yi),i∈Z+}are iid, so that the joint pmf of (Wi,Yi)is given by Pr{Wi=k,Yi=j}=akαj(k), where αj(k) = Pr{Yi=j|Wi=k} denotes the conditional pmf of Yi given Wi . Such a structurization allows for possible dependence between interclaim times and claim sizes in Sparre Andersen models (e.g., see Cheung et al. [9]). 3. Calculation of Finite-Time Ruin Probabilities We begin by examining the finite-time ruin probabilities associated with the risk model defined by Equation (1). To start with, ruin occurs if and only if Ut< 0 for some t∈Z+ and we denote T to be the time of ruin. In other words, T=min{t∈Z+|Ut<0}with T=∞if Ut≥0∀t∈Z+. Risks 2016,4, 2 5 of 15 In what follows, we are interested in computing the quantity Ψ(v,g,n) = Pr{T≤n|U0=v,F0=g}=1−Pr{T>n|U0=v,F0=g},n∈N, which we refer to as the finite-time ruin probability. To aid in the computation of this quantity, we introduce the following related function: σ(u,f,n,m) = Pr{T>n|U0=u,F0=f,M0=m},n∈N,u∈Z,f∈ {β,β+1, . . . }, where Z denotes the set of all integers and Mt , referred to as the elapsed waiting time counter, represents the elapsed time at time t since the most recent claim occurrence and its values lie in the set {0, 1, . . . , na−1}. With the introduction of this function, we remark that Ψ(v,g,n) = 1−σ(v,g,n, 0). First of all, assuming the occurrence of no claims and no right limit deposits, we need to identify when Ut≥`2 and Ut≥`3 for the first time. We introduce two functions to denote these time points, namely: zt,u=(0 if u≥`2, min{b`2−u−1 cc+1, t}if u< `2, and z0 t,u=(0 if u≥`3, min{b`3−u−czt,u−1 c−dc+zt,u+1, t}if u< `3, where bxc, referred to as the floor function of x, yields the largest integer less than or equal to x. To aid us in obtaining a mathematical expression for σ(u , f , n , m) , we have to examine how the process {Ft , t∈N} evolves over time. Let us first assume that there are no claims or withdrawals to consider. Clearly, Ft is a non-decreasing function of t if f∈N . On the other hand, if f/∈N , Ft could either be a strictly decreasing function of t or perhaps a convex function depending on the values of f , κ2 and d . Consequently, if Ft drifts below β , a deposit is forced to be made and this may cause ruin. Thus, in this model, ruin can occur due to either a claim or a deposit. This certainly adds more complexity in deriving a formula for σ(u , f , n , m) , and as a result, we have to introduce a few more functions. One such function is denoted by ot,u,f , representing the time point s∈ { 1, 2, . . . , t} at which Fs is set to become greater than or equal to 0 for the first time. Obtaining this value is not difficult since, assuming the occurrence of no claims and that Fs≥β∀s≤ot,u,f , Fs becomes non-stochastic, the form of which we denote by: ˜ Fs,u,f=(f(1+κ2)s+dκ2 zs,u,sif f<0, f(1+κ1)s+dκ1 zs,u,sif f≥0, where dκi k,l , k , l∈N , represents the future value of deposits made at times k+ , (k+ 1 )+ , . . . , (l− 1 )+ with respect to interest rate κiper period, i=1, 2. Clearly, we have dκi k,l=0 for l≤kand dκi k,l=d(1+κi)l−k+d(1+κi)l−k−1+· · · +d(1+κi) = d(1+κi)[(1+κi)l−k−1] κi ,l>k. It subsequently follows that ot,u,f=(tif ˜ Fi+1,u,f<0∀i∈ {0, 1, . . . , t}, min{i∈ {0, 1, . . . , t}| ˜ Fi+1,u,f≥0}otherwise. Risks 2016,4, 2 6 of 15 In defining ot,u,f above, we utilize the value of ˜ Fi+1,u,f instead of ˜ Fi,u,f . This is because for f< 0, the function ˜ Ft,u,f , t∈N , up-crosses level 0 only if a positive amount of deposit is made to the external fund. We stated earlier that left deposits are made at the left limit point of a discrete-time interval. Thus, ˜ Fi+1,u,f≥ 0 for the first time implies that at time i+ , there was a left deposit made to the external fund. With the introduction of ot,u,f, we henceforth express the non-stochastic form of Ftas ˆ Ft,u,f=b(f(1+κ2)ot,u,f+dκ2 zot,u,f,u,ot,u,f)(1+κ1)t−ot,u,f+dκ1 max{zt,u,ot,u,f},tc,t∈N. (2) Note that Equation (2) involves the use of the floor function to calculate the (non-stochastic) value of the external fund at time t . Such an assumption can be viewed as conservative in nature, since any non-integer value of the external fund (which can arise due to interest accumulation) is essentially rounded down. There is another important function we introduce next, as it represents the earliest time point when Ft falls below β (again assuming the occurrence of no claims) due to interest expense accumulation. We denote this time point by ct,m,u,fand refer to it as a calling point. It is given by ct,m,u,f=(min{na−m,t}if ˆ Fi,u,f≥β∀i∈ {1, 2, . . . , min{na−m,t}}, min{i∈ {1, 2, . . . , min{na−m,t}}| ˆ Fi,u,f<β}otherwise. Note that the above function depends on both t and m . As introduced earlier in this section, m represents the elapsed time at time 0 since the most recent claim occurrence. With these preliminaries in place, we adopt the principle of conditioning on the first claim time as in Cossette et al. [ 10 ] or Drekic and Mera [ 4 ]. Measured from our initial time point (which we label as time 0), the lower limit of the time until the first claim occurs is 1, but its pmf is now conditional on the value of m . Morever, in evaluating σ(u , f , n , m) , we condition on first claim times ranging from 1 up to cn,m,u,f , and on the event that the time until the first claim occurs is greater than cn,m,u,f . For first claim time instances which take place at or before the calling point, the recursive process used is very similar to that of Drekic and Mera [ 4 ]. However, in the event that the time until the first claim occurs is greater than cn,m,u,f , the recursive process is performed differently. By doing so, we are essentially denoting cn,m,u,f to be the “new” initial time point, updating the parameters of the function σ , and proceeding with the recursive process. We further explain this situation after introducing some necessary boundary conditions for σ(u,f,n,m), namely: σ(u,f,n,m) = (0 if u∈Z−or m=na, 1 if u∈N,n=0, and m=0, 1, . . . , na−1, (3) where Z− in Equation (3) denotes the set of negative integers. By conditioning on the events outlined above, we get σ(u,f,n,m) = cn,m,u,f ∑ k=1 ak+m Am Pr{T>n|U0=u,F0=f,M0=m,W1(m) = k} +Acn,m,u,f+m Am Pr{T>n|U0=u,F0=f,M0=m,W1(m)>cn,m,u,f}, where W1(m) is the duration from our initial time point until the first claim occurs given that the elapsed waiting time at time 0 since the most recent claim is m. Risks 2016,4, 2 7 of 15 At time k∈ { 1, 2, . . . , cn,m,u,f} , the elapsed waiting time counter is reset to 0 for the next recursion, n is reduced by k , and the “new” initial surplus and external fund amounts are determined by the size of the incurred claim and the premiums received up to time k. Specifically, we obtain Pr{T>n|U0=u,F0=f,M0=m,W1(m) = k} = (k−z0 k,u)c2 ∑ l=(k−z0 k,u)c1 bl,k−z0 k,u u+cz0 k,u−d(k−zk,u)+l+ˆ Fk,u,f−β ∑ j=1 αj(k+m)σ(u?,f?,n−k, 0), where u?=u+cz0 k,u−d(k−zk,u) + l−j +min{ˆ Fk,u,f−β, max{0, j−[u+cz0 k,u−d(k−zk,u) + l−`1]}}, (4) f?=max{β, min{ˆ Fk,u,f,ˆ Fk,u,f−j+ [u+cz0 k,u−d(k−zk,u) + l−`1]}}, (5) and l denotes the value of the sum of the random premiums received up to time k , with corresponding pmf bl,k−z0 k,u representing the (k−z0 k,u) -fold convolution of bl with itself. To evaluate the pmf bl,r , r∈N, we define bl,0 =δl,0 (where δi,j, in general, denotes the Kronecker delta function of iand j), bl,1 =(blif l=c1,c1+1, . . . , c2, 0 otherwise, and for r=2, 3, . . . , bl,r=   ∑c2 j=c1bj,1bl−j,r−1if l=rc1,rc1+1, . . . , rc2, 0 otherwise. The reasoning behind the definitions of the above parameters is that we first consider whether the claim size is substantial enough for the surplus process to fall below its minimum support level `1 . If so, then j−[u+cz0 k,u−d(k−zk,u) + l−`1] is a positive quantity and we consider whether the external fund is able to support the surplus process. We do this by comparing ˆ Fk,u,f−β and j−[u+cz0 k,u−d(k−zk,u) + l−`1] , and choosing the minimum of the two quantities to ensure that the external fund does not fall below the maximum level of external funding allowed, β . If j−[u+ cz0 k,u−d(k−zk,u) + l−`1] is a non-positive quantity, then the surplus process is greater than or equal to `1 after the claim, in which case, we only need to consider whether ˆ Fk,u,f is below β . If so, ˆ Fk,u,f−β is less than 0, and we would subtract |ˆ Fk,u,f−β| from the surplus process and add it to the external fund to bring it up to β. In situations when W1(m)>cn,m,u,f, we perform a recursion at cn,m,u,fto similarly acquire Pr{T>n|U0=u,F0=f,M0=m,W1(m)>cn,m,u,f} = (cn,m,u,f−z0 cn,m,u,f,u,f)c2 ∑ l=(cn,m,u,f−z0 cn,m,u,f,u)c1 bl,cn,m,u,f−z0 cn,m,u,f,u,fσ(u0,f0,n−cn,m,u,f,cn,m,u,f+m), (6) where u0=u+cz0 cn,m,u,f,u−d(cn,m,u,f−zcn,m,u,f,u) + l+min{0, ˆ Fcn,m,u,f,u,f−β}(7) Risks 2016,4, 2 8 of 15 and f0=max{ˆ Fcn,m,u,f,u,f,β}. (8) We remark that when W1(m)>cn,m,u,f , there is no claim size to consider at time cn,m,u,f . Thus, all we need to account for is whether ˆ Fcn,m,u,f,u,f falls below β . In this case, just enough funds would be withdrawn from the surplus process and added to the external fund to bring it up to β . However, note that ˆ Fcn,m,u,f,u,f may not necessarily be below β . If ˆ Fcn,m,u,f,u,f≥β , then cn,m,u,f=min{na−m , n} and this implies that either n−cn,m,u,f= 0 or cn,m,u,f+m=na in Equation (6). This yields an interesting outcome. Given that W1(m)≥cn,m,u,f and ˆ Fcn,m,u,f,u,f≥β , it must be that u0≥ 0 at time cn,m,u,f . However, if cn,m,u,f+m=na , then σ(u0 , f0 , n−cn,m,u,f , na) is set equal to 0 via Equation (3). On the other hand, if n<na−m so that cn,m,u,f=n , then σ(u0 , f0 , 0, cn,m,u,f+m) = 1 from Equation (3). Putting it altogether, we establish the following final formula for σ(u,f,n,m): σ(u,f,n,m) = cn,m,u,f ∑ k=1 ak+m Am (k−z0 k,u)c2 ∑ l=(k−z0 k,u)c1 bl,k−z0 k,u u+cz0 k,u−d(k−zk,u)+l+ˆ Fk,u,f−β ∑ j=1 αj(k+m)σ(u?,f?,n−k, 0) +Acn,m,u,f+m Am (cn,m,u,f−z0 cn,m,u,f,u)c2 ∑ l=(cn,m,u,f−z0 cn,m,u,f,u)c1 bl,cn,m,u,f−z0 cn,m,u,f,uσ(u0,f0,n−cn,m,u,f,cn,m,u,f+m). (9) Note that the determination of σ(u , f , n , m) via Equation (9) requires a double recursion in both n and m , with boundary conditions given by Equation (3) serving as the starting point. Moreover, if we assume that f= 0, d= 0, `1= 0, β= 0, m= 0, and αj(k) = Pr{Yi=j|Wi=k}=Pr{Yi=j}=αj (i.e., Yi is independent of Wi∀i∈Z+ ), then the model under consideration is equivalent to the independent Sparre Andersen model studied by Drekic and Mera [ 4 ]. To verify this, we first observe that cn,0,u,0 =min{n,na} ∀ n∈Z+. If n<na, then σ(u0,f0, 0, n) = 1 and An A0 Pr(T>n|U0=u,F0=0, M0=0, W1(0)>n) = An (n−z0 n,u)c2 ∑ l=(n−z0 n,u)c1 bl,n−z0 n,u=An. Conversely, if n≥na, then σ(u0,f0,n−na,na) = 0 and Ana A0 Pr(T>n|U0=u,F0=0, M0=0, W1(0)>na) = Ana×0=0. Thus, Equation (9) simplifies to become σ(u, 0, n, 0) = An+ min{n,na} ∑ k=1 ak (k−z0 k,u)c2 ∑ l=(k−z0 k,u)c1 bl,k−z0 k,u u+cz0 k,u+l ∑ j=1 αjσ(u+cz0 k,u+l−j, 0, n−k, 0), which is consistent with the result in Drekic and Mera [4], p. 744. 4. Calculation of Expected Total Discounted Dividend Payments Our next objective is to derive a corresponding recursive formula to compute the expected total discounted dividend payments made prior to ruin. The approach we employ essentially borrows from that of Dickson and Waters [ 11 ], Section 5. Let E{Dv,g} denote the expected total discounted (i.e., to time 0 according to discount factor ν∈( 0, 1 ) per unit of time) dividends paid prior to ruin, where the random variable Dv,g represents the total discounted dividends paid before ruin starting from an initial surplus of v and an initial level of g in the external fund. Moreover, we also introduce the Risks 2016,4, 2 15 of 15 6. Conclusions In this paper, we considered a discrete-time dependent Sparre Andersen risk model featuring multiple threshold levels in an effort to characterize an insurer’s minimal capital requirement, dividend paying scenarios, and external financial activities. In analyzing this model, we developed recursive computational procedures to calculate two particular performance measures of interest, namely finite-time ruin probabilities and expected total discounted dividends paid prior to ruin. Through a variety of numerical experiments performed, we were able to make some observations concerning the impact our threshold levels have on both of these performance measures. Acknowledgments: Steve Drekic acknowledges the financial support from the Natural Sciences and Engineering Research Council of Canada through its Discovery Grants program (#238675-2010-RGPIN). 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