Capital injections with negative surplus and delays:models and analysis

Zhuo JIN,George YIN

1.Centre for Actuarial Studies,Department of Economics,The University of Melbourne,VIC 3010,Australia;

2.Department of Mathematics,Wayne State University,Detroit,Michigan 48202,U.S.A.

Capital injections with negative surplus and delays:models and analysis

Zhuo JIN1?,George YIN2

1.Centre for Actuarial Studies,Department of Economics,The University of Melbourne,VIC 3010,Australia;

2.Department of Mathematics,Wayne State University,Detroit,Michigan 48202,U.S.A.

This work develops a new model to deal with the scenario that some companies can still run business even the surplus falls below zero temporarily.With such a scenario in mind,we allow the surplus process to continue in this negative-surplus period,during which capital injections will be ordered to assist in the stabilization of financial structure,until the financial status becomes severe enough to file bankruptcy.The capital injections will be modeled as impulse controls.By introducing the capital injections with time delays,optimal dividend payment and capital injection policies are considered.Using the dynamic programming approach,the value function obeys a quasi-variational inequality.With delays in capital injections,the company will be exposed to the risk of bankruptcy during the delay period.In addition,the optimal dividend payment and capital injection strategies should balance the expected cost of the possible capital injections and the time value of the delay periods.This gives rise to a stochastic control problem with mixed singular and delayed impulse controls.Under general assumptions,the lower capital injection barrier is determined,where bankruptcy occurs.The closed-form solution to the value function and corresponding optimal policies are obtained.

Stochastic control;Capital injection;Dividend policy;Delayed impulse control;Singular control

1 Introduction

Designing dividend payment policies has long been an important issue in finance and actuarial sciences.Because of the nature of their products,insurers tend to accumulate relatively large amounts of cash,cash equivalents,and investments in order to pay future claims and avoid insolvency.The payment of dividends to shareholders may reduce an insurer's ability to survive adverse investment and underwriting experience.A practitioner will manage the reserve and dividend payment against asset risks so that the company can satisfy itsminimum capital requirement.

Stochastic optimal control problems on dividend strategies for an insurance corporation have drawn increasing attention since the introduction of the optimal dividend payment model proposed in[1].There have been increasing efforts on using advanced methods of stochastic control to study the optimal dividend policy;see[2-4].As an extension of the previous work,dividend is assumed to be paid out with the constraint that a transaction cost must be paid.The studies related to optimal dividend problems with transaction costs and compound Poisson process can be referred to[5].To maximize the expected total discounted dividend payments,the company will bankrupt almost surely if the dividend payment is paid out as a barrier strategy.In practice,reference[6]suggested that capital injections can be taken into account to maintain the business when cashflow is insufficient.Furthermore,penalty will be paid when surplus falls below zero,which can be considered as the transaction cost of capital injection;see also[7-9].Whenever the company is on the verge of bankruptcy,the company has the opportunity to raise sufficient funds to survive.A natural payoff function is to maximize the difference between the expected total discounted dividend payment and the capital injections with costs until bankruptcy under the optimal controls.

In this work,we develop a model to deal with the scenario that some companies are not necessary to file bankruptcy with temporary negative surplus.In our model,negative surplus is not the end,and the shareholder will evaluate if it is profitable enough to rescue the company.If the financial deficit is severe enough,the shareholders will let the company go to bankruptcy to cut the losses.Otherwise,strategies will be taken to save the company such as capital injections,refinancing,or merger etc.With such a practice in mind,we distinguish between temporary deficit and bankruptcy,where bankruptcy occurs at a sufficiently low capital level.Recently,there have been surging work related to the optimal dividend and refinance policies after ruin.Reference[10]analyzed the absolute ruin probabilities in a jump diffusion risk,where companies will be not allowed to continue their business when surplus is below certain negative critical value.Reference[11]introduced the Omega model to distinguish between ruin and bankruptcy.Companies can still run the business normally until the financial status is severe enough to file bankruptcy.Reference[12]determined the expected discounted value of a penalty at bankruptcy and computed the probability of bankruptcy under the assumption of Brownian motion for the surplus.On the other hand,there have been resurgent efforts devoted to the study of time delay on stochastic models.Reference[13]considered the problem of a bank's optimal strategy of recapitalization with a fixed delay period.Reference[14]proposed a direct solution method for delayed impulse control problems of one-dimensional diffusions and solve an optimal labor force problem with firing delay.Reference[15]studied the optimal reinsurance strategy under fixed cost and delay.Considering such a delay in the capital injection makes our formulation more general and realistic.

With the classical capital injection policies,the company could run the business in the absence of risk of bankruptcy.However,empirical studies indicate that traditional surplus models with capital injections fail to capture the impact of regulatory processes of capital raising transactions.To better reflect reality,we have to consider the factor that the transactions of capital injections need certain amount of time to be carried out after the decision of injecting extra capitals is made.The time needed can be modeled by using delays.In the real world,the capital injections can never happen instantaneously.Time delays cannot be ignored and are unavoidable.

Time delays occur naturally in insurance decisionmaking problems such as improving the capital reserve to a nonnegative capital buffer level by capital injections.Many corporations face regulatory delays(e.g.,preparatory and administrative work),which need to be taken into account when the corporations make decisions under uncertainty of insolvency during the delays.The problem of finding the optimal strategy under the condition of delayed capital injections involves a stochastic delay system with impulse controls.In the presence of delay,the corporations will be exposed to a strictly positive probability of bankruptcy during the waiting period.The threshold of bankruptcy can be determined by the barrier where payoff function approaches zero.In addition,the dividend payment is not allowed during this waiting period.Unlike the models where capital injections can be implemented instantaneously to avoid the bankruptcy completely,the positive probability of liquidation risk in our model leads to the decision of capital injections with delays more realistic but more complicated.

This paper reveals clearly the difference of the strategies with and without delays for our model.In traditional models,negative surplus is not allowed,and capital injections without delays will only be implemented when surplus hits 0.The size of the capital injections is always a constant to increase the surplus to a positive capital buffer level,and payoff function will be positive even with zero initial surplus(see[9]).However,when temporary negative-surplus situation is allowed,capital injections without delays could always improve the surplus status instantaneously to avoid severe financial condition.Thus,payoff function or value function will always be positive and bankruptcy will never occur.

One of the novel contributions of the current paper is:Taking into account of the time delays,the impulse controls of capital injections depend on the surplus and can be very large.Together with the unrestricted dividend payment policy,using a quasi-variational inequality approach,we demonstrate that these state-dependent capital injections lead to the formulation of a free boundary problem.Under general assumptions,the analytic solution to the free boundary problem and the optimal statedependent 'threshold' strategies are obtained in this paper.Comparing with work of[13],our work further considers the case of negative reserve,which is naturally proposed when capital injections are included.The lower barrier of the capital injection region is a negative free parameter,which shows that the surplus status can only worsen up to atolerable status.With aflexible 'capital injection stop'-barrier,we fix the 'call for help'-state at 0 surplus.Choosing the newly constructed strategies will significantly effect the capital injection sizes and termination thresholds.

In addition,one of the new findings indicates that capital injections with delays are crucial in our model.Note that for the case of capital injection without delays,bankruptcy can be completely avoided.Thus,the company could run the business with negative capital reserve forever,which is not realistic.When delays are considered,the threshold of the bankruptcy can be determined and when surplus approaches the threshold of bankruptcy,the optimal capital injection goes to zero.It demonstrates that even capital injections are available,the insurance company will be unlikely to avoid the bankruptcy due to the delay when the surplus is sufficiently low.To the best of our knowledge,this has not been considered to date.

The model we constructed involves the consideration of an important factor-the delay in the capital injection process.It can be clearly seen that the delay factor leads to state-dependent optimal strategies,which provide insights for the insurance company in their decision making process and risk analysis.Not only are such results theoretically sound,but they are crucial in the insurance practice.Capital injections and dividend payment policies with transaction costs are introduced as impulse and singular stochastic controls.The imposed time delay on the capital injections makes the problem more complicated.By adopting a diffusion model,we obtain a quasi-variational inequality(QVI)in this paper.The closed-form solution to the QVI is obtained under certain general assumptions.The value function is verified to be a concave function and defined separately in three regions,which are capital injection region,continuation region,and dividend payment region.The capital injection barrier and dividend payment barrier are also given.The threshold of bankruptcy is also the lower barrier of capital injection region.Finally,the optimal capital injection and dividend payment strategies are obtained.

The rest of the paper is organized as follows.A formulation of optimal capital injection strategies and dividend payment policies is presented in Section 2.Section 3 deals with the construction of the value function and dividend payment strategy.Section 4 deals with the verification of the solution to the value function.Some limiting case of the solution is also considered.Finally,Section 5 gives some further remarks to conclude the paper.

2 Formulation

The surplus process is described by a Brownian motion.That is,

wherexis the initial surplus,μ is the expected return rate,σ represents the volatility,andW(t)is a standard Brownian motion.We are working with a filtered probability space(Ω,F,{Ft},P),where Ftis the σ-algebra generated byandis the associated filtration.

A dividend strategyZ(.)is an Ft-adapted process{Z(t):t≥0}corresponding to the accumulated amount of dividends paid up to timetsuch thatZ(t)is a nonnegative and nondecreasing stochastic process that is right continuous with left limits.Throughout the paper,we use the convention thatZ(0-)=0.The jump size ofZat timet≥0 is denoted by

denotes the continuous part ofZ(t).In this work,we assume the company could still work in the negativesurplus situation.That is,while the company is short of capital,the company is still able to run the business until bankruptcy occurs.When the surplus hits the threshold of disastrous financial status,bankruptcy is unavoidable.Note that the surplus levelX(t)=x0describes the severe financial condition leading to 'certain bankruptcy'.

On the other hand,capital injections will trigger the improvement of the capital structure.For the time cost of the money,extra capital will only be injected until surplus process falls below zero.LetX(t)=x0for allt> τx0,where τx0=inf{t≥ 0:X(t)

for allt<τx0.

We assume that the shareholders need to payK+ζ to meet the capital injection of ζ.K>0 is the fixed transaction costs.We omit the fixed transaction costs in the dividends payout process.Denote byr>0 the discounting factor.For an arbitrary admissible pairu=(Z,L),the performance function is

The pairu=(Z,L)is said to be admissible ifZandLsatisfy

i)Z(t)andL(t)are nonnegative for anyt≥0;

ii)Zis c`adl`ag(that is,it is right continuous and has left limits),nondecreasing and adapted to Ft;

iii)τnis a sequence of stopping time w.r.t.Ft,and 0≤ τ1<...< τn<...a.s.;

iv)ζnis measurable w.r.t.Ft;

vi)J(x,u)<∞for anyxand admissible pairu=(Z,L),whereJis the functional defined in(3).

In addition,we assume the admissible controlusatisfies

b)dZ(t)=0 for allt∈ [τn,τn+ Δ],n≥ 1.

Condition a)tells us a new capital injection should not be placed during the waiting period of the previous capital injection.Condition b)states the dividends may not be paid during the waiting periods of the capital injections.

Suppose that A is the collection of all admissible pairs.Define the value function as

When the surplus hits barrierthe financial condition is too severe to maintain the business and extra capital injections are useless.We imposex0=inf{x:V(x)≥0}.Intuitively,in the absence of time delay,on the boundary of the capital injection region,the value function obeys

then the optimal payoff or the value functionV(x)will not be 0 with the instantaneous capital injections,which could always guarantee the stability of the company's capital structure.However,with the capital injection delays,the company will violate the capital adequacy if the capital injection is hold while the surplus hits 0.Thus,taking into account the delay of the capital injection,the value of theV(x)on the boundary can be obtained as

For alldefine an operator L by

whereVxandVxx(x)denotes the first and second derivatives with respect tox.Define another capital injection operator M by

whereXΔis the value at time Δ ofXdefined by dX(t)= μdt+σdW(t),withX(0)=x.If the value functionVdefined in(4)is sufficiently smooth,by applying the dynamic programming principle([16]),we formally derive the following QVI:

Similar to[9],we divide the set of the surplus to three regions i)continuation region:C:{LV(x)-rV(x)=0,1

Boundary conditionsThe capital injection will be taken into account when there is not enough solvency capital to maintain the business.To make the company run continuously,the capital injections will definitely occur at the moments whenx<0.In addition,the capital injections also occur whenever the surplus is sufficiently low.The impulse control of the capital injections depends on the surplus states and leads to a free boundary of the capital injection region.We consider the dividend payout strategy with the delayed capital injection.Combing(9)and(6),the QVI with the boundary conditions follows

Remark 1The value functionV(x)is not necessarily smooth.In fact,the second derivative of the value function is not always continuous.In the absence of a classical solution to the QVI,one alternative definition for a solution to the quasi-variational inequalities(10)is the notion of a viscosity solution.However,we can interpret the differential generator in terms of left or right derivatives;see[13].

3 Value function and dividend strategy

To solve the quasi-variational inequality,we guess the form of a solution and verify the validation of the constructed solution in a general case.The solution can be formulated based on the strategies in each of the regions.Referring to[17],we consider the dividend payout strategy with the capital injection as a band strategy.The decision maker will take no action until the surplus hits 0,where an impulse control of capital injection will be taken.The dividend will be paid out immediately when the surplus reaches the upper barrier.Furthermore,the capital injections will only be considered withX(t)≥x0.That is,suppose there exist three thresholdsx0,0 andb0separating the four regions,where-∞

To proceed,we construct the solutions in the continuation region when the equality holds.Denoting the candidate solution in the continuation region byf(x).The equality in the continuation region then becomes

The solution to(11)is

Letg(x)be the solution to(13).It can easily be obtained that

whereacan be determined by the boundary condition of the dividend payment region.Based on the form of the solution,b0is the threshold to separate the continuation region and dividend payout region.Thus,the solution should satisfy both(11)and(13)atb0.On the other hand,the twice continuous differentiability ofg(x)atb0requires thatfx(b0)=1 andfxx(b0)=0.Imposing these boundary conditions on(12)yields

where

Note that we have usedf(x;b0)to denote the dependence on the parameterb0.Furthermore,substitutingfx(b0)=1 andfxx(b0)=0 into(11)yields

Moreover,subject to the boundary condition atb0,ais determined andg(x)becomes

Finally,we need only construct the solution in capital injection region.Assume a concave function denoted bythat satisfiesand(16).Because of the concavity of the functionh(x;b0),the supremum is achieved whens=b0-XΔ.Thus,(8)is simplified to

where τρ+x0represents the hitting time of ρ+x0.That is,

Define a Markov transition probability density functionp(Δ,x+ ρ,y)withy∈ [x0,∞),which is the density of the processthat starts atx+ ρ.Then,we have

Hence,(19)follows

Finally,we obtain the constructed function as follows:

where Φ(.)and φ(.)are the cumulative standard normal distribution and its density function,respectively.It is obvious that

On the other hand,the boundary condition ofh(x;x0,b0)on the boundary of the capital injection regionx0can be formulated as

The explicit expression of the region boundariesx0andb0are not easy to obtain because of the nonlinearity of the QVI.However,the existence ofx0andb0will be verified in the next section under conditions.Combining(15),(17)and(21),given the existence ofx0andb0,the value functionV(x)can be written as

4 Properties of the solution

4.1 Verification theorem

In this section,we will first verify the existence the boundaries of the capital injection and continuation regionsx0andb0.Under the general conditions,sufficient conditions of the existence ofx0andb0will be given.Moreover,the value functionV(x)defined in(24)will be verified as the solution to(10).Some limiting cases will also be discussed at last.

Lemma 1Letwe have

Proof

Hence,the equality is obtained.

Lemma 2If ρ+x0≥ 0,then ?x>0,we have that the constructed function in(21)follows

ProofIn view of(19),we have

For the first and second order derivatives ofh(x;x0,b0),we have

Sinceits derivatives satisfy

We obtain that

when ρ>-x0.This lemma shows the concavity of the value function in the capital injection region,which means the new capital issues can be optimized when ρ+x0>0.

Now,we will consider the barrier of continuation region.Define the functionas

Denoted a positive barrierb1which satisfies0.The we can deduce that

ProofDifferentiating(25)onb,we obtain

Moreover,it is shown that

Lemma 4If

ProofReferring to(18),by using Lemmas 1 and 2,we have

Thus,the inequality is verified.

Define a two component function?h(x,b)∈R+XR+as

Proof

Step 1In view of the expression(27),we haveAlso,from the definition.Suppose the conditionit implies that there exist some pointsuch thatOn the other hand,from Lemma 4,substitutinginwe haveThis shows thatmust crossfrom above at some pointx2in the interval(x0,b1).

Step 2For allsincewe haveIn addition,by Lemma 4.From(25)and(27),we getBy Lemma 2,ρ+x0≥ 0 implies thatis increasing and concave,so iswith respect tois also increasing inx.Thus,combining the previous inequality,we can always find a positiveb0in the interval(x0,b1)such that

Step 3In view of the results in Steps 1 and 2,following from the continuity ofandwith respect toxandb,there will exist ab0in the intervalsuch thatfor somewherefor allFor the(b2,b0)we have chosen,the continuous differentiability ofandwith respect toxyieldsThe equality is established because the two continuously differentiable lines have the same derivative if they coincide but not cross at some point.On the other hand,the capital will be injected with surplus hitting 0.Thus,b2=0.In view of the definition of the two functionsandwe find that(22)and(23)holds.

Theorem 1Assume a solution to(22)and(23)as defined in Lemma 5 exists andV(x)is defined in(24).ThenV(x)is a concave solution to(10).

ProofWe will prove the concavity ofV(x)in the three regions,respectively.In the dividend payout region,Vxx(x)= 0.In the capital injection region,h(x;x0,b0)is concave following from Lemma 2.In the continuation region,differentiatingf(x;b0)three times,it is shown thatonx∈[0,b0).Combining with the value of the second order derivative on the boundary thatwe havefxx(b0;b0)<0 for allx∈[0,b0).Hence,V(x)is concave in the continuation region[x0,b0).Thus,V(x)is concave.

To proveV(x)satisfying(10),we have four steps.

Step 1

Step 2Forx∈ [b0,∞),Vx(x)=1 by construction.Forx∈[0,b0),Vx(x)>1 following from the concavity ofV(x).

Step 3forx∈[x0,0)by construction.Forx∈[0,b0),we havefollowing from Lemma 5.Finally,forx∈ [b0,∞),we haveHence,V(x)≥M(x)globally.

Step 4ForDenote τ?= τε∧efor somee>0 such thatandFollowing from Dynkin's formula,we have

Thus,in the capital injection regionx∈[x0,0),we obtain

Taking the limit yields

whereForby construction.Forthen we have

4.2 Limiting case

In this section,we analyze the limiting case of immediate capital injections.That is,Δ=0.In the absence of any capital delays,the optimal dividend payout strategy is a barrier strategy,where the extra surplus will be paid out as dividend beyond some barrier level.To maximize the performance,the capital injection time will be postponed as much as possible because the capital injections can be effective immediately and always guarantee the continuity of the business even with the sufficient low surplus.Thus,the threshold of bankruptcy will be arbitrarily low if bankruptcy only occurs when payoff function hits zero.

Since the capital injection region is reduced to one arbitrarily low point,we only have one barrier in this limiting case.This barrier will separate the continuation region and dividend payout region.The value function is concave and monotone increasing.Furthermore,starting as a curve,the value function increases linearly after some barrier level,which means that the extra surplus will all be paid out as the dividend reaching certain barrier.

5 Conclusions

In this paper,we studied the optimal dividend and capital injection strategies with constant time delays.The time delay in this work is crucial not only because of its reality in modeling the capital injection process but also it provides practical criteria to determine the bankruptcy threshold.Under general assumptions,a closed-form solution to the value function and the optimal strategies are obtained.

The delayed capital injections are studied in this work.The time delays describe the regulatory process when extra capital is ordered to improve the company's financial status.Although the delay duration is generally uncertain and varies for different capital sources,the time delay is unlikely to vary in large range.We aim to find the closed-form solutions and ignore the impact of time-varying character of the delays in this paper.Treating time-varying delays is both theoretically interesting and practically useful.Considering a time-varying delay will make the model more realistic and versatile but more complicated,resulting in essential difficulty in finding the analytic solutions.The current paper focuses on obtaining closed-form solutions,whereas treating timevarying delay requires developing appropriate numerical algorithms,which is an important topic in the future study.

In addition to delayed capital injections,to better reflect the reality,regime-switching models for the asset can be considered.The regime-switching models are known to able to capture the extreme price movement such as market changes,which can be described by a continuous-time Markov chain.Using the dynamic programming approach,the value function obeys a coupled system of quasi-variational inequalities.Thus,the model becomes more versatile but more complicated.Together with the time delays of the impulse control,it is virtually impossible to obtain a closed-form solution.Nevertheless,numerical approximation can provide a viable alternative.

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22 April 2013;revised 26 February 2014;accepted 27 February 2014

DOI10.1007/s11768-014-0061-x

?Corresponding author.

E-mail:zjin@unimelb.edu.au.Tel.:+61 3 8344 4655;fax:+61 3 8344 6899.

The research of Z.Jin was supported by the Faculty Research Grant of University of Melbourne,and the research of G.Yin was partially supported by the National Science Foundation(No.DMS-1207667).

?2014 South China University of Technology,Academy of Mathematics and Systems Science,CAS,and Springer-Verlag Berlin Heidelberg

Zhuo JINreceived the B.S.degree in Mathematics from the Huazhong University of Science and Technology in 2005,and Ph.D.in Mathematics from Wayne State University in 2011.He joined the Centre for Actuarial Studies,Department of Economics,The University of Melbourne as a Lecturer in September2011.His research interests include numerical methods for stochastic systems,actuarial science and mathematical finance.Email:zjin@unimelb.edu.au.

George YINjoined Wayne State University in 1987 and became a professor in 1996.Working on stochastic systems,he is Chair of SIAM Activity Group in Control and Systems Theory and is one of the Board of Directors of American Automatic Control Council.He was Co-Chair of SIAM Conference on Control&Its Application,2011,Co-Chair of 1996 AMS-SIAM Summer Seminar and 2003 AMS-IMS-SIAM Summer Research Conference,Coorganizer of 2005 IMA Workshop on Wireless Communications.He chaired the SIAM W.T.and Idalia Reid Prize Committee,the SIAG/Control and Systems Theory Prize Committee,and the SIAM SICON Best Paper Prize Committee.He is an associate editor of Control Theory and Technology,SIAM Journal on Control and Optimization,and on the editorial board of many other journals and book series.He was an associate editor of Automatica and IEEE T-AC.He was President of Wayne State University's Academy of Scholars.He is a Fellow of IEEE.Email:gyin@math.wayne.edu.