scholarly journals On the time value of absolute ruin with debit interest

2007 ◽  
Vol 39 (02) ◽  
pp. 343-359 ◽  
Author(s):  
Jun Cai
Keyword(s):  
Differential Equations ◽  
Ruin Probability ◽  
Expected Discounted Penalty Function ◽  
Surplus Process ◽  
Absolute Ruin ◽  
The Laplace Transform ◽  
The Absolute ◽  
Heavy Tailed ◽  
Premium Income ◽  
Debit Interest

Assume that the surplus of an insurer follows a compound Poisson surplus process. When the surplus is below zero or the insurer is on deficit, the insurer could borrow money at a debit interest rate to pay claims. Meanwhile, the insurer will repay the debts from her premium income. The negative surplus may return to a positive level. However, when the negative surplus is below a certain critical level, the surplus is no longer able to be positive. Absolute ruin occurs at this moment. In this paper, we study absolute ruin questions by defining an expected discounted penalty function at absolute ruin. The function includes the absolute ruin probability, the Laplace transform of the time to absolute ruin, the deficit at absolute ruin, the surplus just before absolute ruin, and many other quantities related to absolute ruin. First, we derive a system of integro-differential equations satisfied by the function and obtain a defective renewal equation that links the integro-differential equations in the system. Second, we show that when the initial surplus goes to infinity, the absolute ruin probability and the classical ruin probability are asymptotically equal for heavy-tailed claims while the ratio of the absolute ruin probability to the classical ruin probability goes to a positive constant that is less than one for light-tailed claims. Finally, we give explicit expressions for the function for exponential claims.

scholarly journals On the time value of absolute ruin with debit interest

2007 ◽  
Vol 39 (2) ◽  
pp. 343-359 ◽  
Author(s):  
Jun Cai
Keyword(s):  
Differential Equations ◽  
Ruin Probability ◽  
Expected Discounted Penalty Function ◽  
Surplus Process ◽  
Absolute Ruin ◽  
The Laplace Transform ◽  
The Absolute ◽  
Heavy Tailed ◽  
Premium Income ◽  
Debit Interest

Assume that the surplus of an insurer follows a compound Poisson surplus process. When the surplus is below zero or the insurer is on deficit, the insurer could borrow money at a debit interest rate to pay claims. Meanwhile, the insurer will repay the debts from her premium income. The negative surplus may return to a positive level. However, when the negative surplus is below a certain critical level, the surplus is no longer able to be positive. Absolute ruin occurs at this moment. In this paper, we study absolute ruin questions by defining an expected discounted penalty function at absolute ruin. The function includes the absolute ruin probability, the Laplace transform of the time to absolute ruin, the deficit at absolute ruin, the surplus just before absolute ruin, and many other quantities related to absolute ruin. First, we derive a system of integro-differential equations satisfied by the function and obtain a defective renewal equation that links the integro-differential equations in the system. Second, we show that when the initial surplus goes to infinity, the absolute ruin probability and the classical ruin probability are asymptotically equal for heavy-tailed claims while the ratio of the absolute ruin probability to the classical ruin probability goes to a positive constant that is less than one for light-tailed claims. Finally, we give explicit expressions for the function for exponential claims.

scholarly journals Estimates for the Absolute Ruin Probability in the Compound Poisson Risk Model with Credit and Debit Interest

2008 ◽  
Vol 45 (03) ◽  
pp. 818-830 ◽  
Author(s):  
Jinxia Zhu ◽  
Hailiang Yang
Keyword(s):  
Ruin Probability ◽  
Risk Model ◽  
Compound Poisson ◽  
Compound Poisson Risk Model ◽  
Absolute Ruin ◽  
Constant Rate ◽  
Negative Constant ◽  
The Absolute ◽  
Debit Interest ◽  
Poisson Risk Model

In this paper we consider a compound Poisson risk model where the insurer earns credit interest at a constant rate if the surplus is positive and pays out debit interest at another constant rate if the surplus is negative. Absolute ruin occurs at the moment when the surplus first drops below a critical value (a negative constant). We study the asymptotic properties of the absolute ruin probability of this model. First we investigate the asymptotic behavior of the absolute ruin probability when the claim size distribution is light tailed. Then we study the case where the common distribution of claim sizes are heavy tailed.

scholarly journals Insurance with borrowing: first- and second-order approximations

2009 ◽  
Vol 41 (04) ◽  
pp. 1141-1160 ◽  
Author(s):  
A. A. Borovkov
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Ruin Probability ◽  
Joint Distribution ◽  
Critical Level ◽  
Order Approximation ◽  
Second Order ◽  
Surplus Process ◽  
Absolute Ruin ◽  
The Absolute

We consider the operation of an insurer with a large initial surplus x>0. The insurer's surplus process S(t) (with S(0)=x) evolves in the range S(t)≥ 0 as a generalized renewal process with positive mean drift and with jumps at time epochs T 1,T 2,…. At the time T η(x) when the process S(t) first becomes negative, the insurer's ruin (in the ‘classical’ sense) occurs, but the insurer can borrow money via a line of credit. After this moment the process S(t) behaves as a solution to a certain stochastic differential equation which, in general, depends on the indebtedness, -S(t). This behavior of S(t) lasts until the time θ(x,y) at which the indebtedness reaches some ‘critical’ level y>0. At this moment the line of credit will be closed and the insurer's absolute ruin occurs with deficit -S(θ(x,y)). We find the asymptotics of the absolute ruin probability and the limiting distributions of η(x), θ(x,y), and -S(θ(x,y)) as x → ∞, assuming that the claim distribution is regularly varying. The second-order approximation to the absolute ruin probability is also obtained. The abovementioned results are obtained by using limiting theorems for the joint distribution of η(x) and -S(T η(x)).

scholarly journals Insurance with borrowing: first- and second-order approximations

2009 ◽  
Vol 41 (4) ◽  
pp. 1141-1160 ◽  
Author(s):  
A. A. Borovkov
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Ruin Probability ◽  
Joint Distribution ◽  
Critical Level ◽  
Order Approximation ◽  
Second Order ◽  
Surplus Process ◽  
Absolute Ruin ◽  
The Absolute

We consider the operation of an insurer with a large initial surplus x>0. The insurer's surplus process S(t) (with S(0)=x) evolves in the range S(t)≥ 0 as a generalized renewal process with positive mean drift and with jumps at time epochs T1,T2,…. At the time Tη(x) when the process S(t) first becomes negative, the insurer's ruin (in the ‘classical’ sense) occurs, but the insurer can borrow money via a line of credit. After this moment the process S(t) behaves as a solution to a certain stochastic differential equation which, in general, depends on the indebtedness, -S(t). This behavior of S(t) lasts until the time θ(x,y) at which the indebtedness reaches some ‘critical’ level y>0. At this moment the line of credit will be closed and the insurer's absolute ruin occurs with deficit -S(θ(x,y)). We find the asymptotics of the absolute ruin probability and the limiting distributions of η(x), θ(x,y), and -S(θ(x,y)) as x → ∞, assuming that the claim distribution is regularly varying. The second-order approximation to the absolute ruin probability is also obtained. The abovementioned results are obtained by using limiting theorems for the joint distribution of η(x) and -S(Tη(x)).

scholarly journals Estimates for the Absolute Ruin Probability in the Compound Poisson Risk Model with Credit and Debit Interest

2008 ◽  
Vol 45 (3) ◽  
pp. 818-830 ◽  
Author(s):  
Jinxia Zhu ◽  
Hailiang Yang
Keyword(s):  
Ruin Probability ◽  
Risk Model ◽  
Compound Poisson ◽  
Compound Poisson Risk Model ◽  
Absolute Ruin ◽  
Constant Rate ◽  
Negative Constant ◽  
The Absolute ◽  
Debit Interest ◽  
Poisson Risk Model

In this paper we consider a compound Poisson risk model where the insurer earns credit interest at a constant rate if the surplus is positive and pays out debit interest at another constant rate if the surplus is negative. Absolute ruin occurs at the moment when the surplus first drops below a critical value (a negative constant). We study the asymptotic properties of the absolute ruin probability of this model. First we investigate the asymptotic behavior of the absolute ruin probability when the claim size distribution is light tailed. Then we study the case where the common distribution of claim sizes are heavy tailed.

scholarly journals Randomized Dividends in a Discrete Insurance Risk Model with Stochastic Premium Income

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Wenguang Yu
Keyword(s):  
Ruin Probability ◽  
Risk Model ◽  
Insurance Risk ◽  
Ruin Time ◽  
Expected Discounted Penalty Function ◽  
Recursion Formulas ◽  
Compound Binomial ◽  
Discounted Penalty Function ◽  
Premium Income ◽  
Insurance Risk Model

The compound binomial insurance risk model is extended to the case where the premium income process, based on a binomial process, is no longer a constant premium rate of 1 per period and insurer pays a dividend of 1 with a probabilityq0when the surplus is greater than or equal to a nonnegative integerb. The recursion formulas for expected discounted penalty function are derived. As applications, we present the recursion formulas for the ruin probability, the probability function of the surplus prior to the ruin time, and the severity of ruin. Finally, numerical example is also given to illustrate the effect of the related parameters on the ruin probability.

scholarly journals The Absolute Ruin Insurance Risk Model with a Threshold Dividend Strategy

Symmetry ◽  
2018 ◽  
Vol 10 (9) ◽  
pp. 377 ◽  
Author(s):  
Wenguang Yu ◽  
Yujuan Huang ◽  
Chaoran Cui
Keyword(s):  
Risk Model ◽  
Insurance Companies ◽  
Insurance Company ◽  
Insurance Risk ◽  
Threshold Dividend Strategy ◽  
Dividend Payments ◽  
Absolute Ruin ◽  
The Absolute ◽  
Debit Interest ◽  
Insurance Risk Model

The absolute ruin insurance risk model is modified by including some valuable market economic information factors, such as credit interest, debit interest and dividend payments. Such information is especially important for insurance companies to control risks. We further assume that the insurance company is able to finance and continue to operate when its reserve is negative. We investigate the integro-differential equations for some interest actuarial diagnostics. We also provide numerical examples to explain the effects of relevant parameters on actuarial diagnostics.

scholarly journals On the absolute ruin in a MAP risk model with debit interest

2011 ◽  
Vol 43 (01) ◽  
pp. 77-96 ◽  
Author(s):  
Zhimin Zhang ◽  
Hailiang Yang ◽  
Hu Yang
Keyword(s):  
Risk Model ◽  
Penalty Functions ◽  
Renewal Equation ◽  
Arrival Process ◽  
Size Distributions ◽  
Absolute Ruin ◽  
Asymptotic Formulae ◽  
Discounted Penalty Function ◽  
Heavy Tailed ◽  
Debit Interest

In this paper we consider a risk model where claims arrive according to a Markovian arrival process (MAP). When the surplus becomes negative or the insurer is in deficit, the insurer could borrow money at a constant debit interest rate to repay the claims. We derive the integro-differential equations satisfied by the discounted penalty functions and discuss the solutions. A matrix renewal equation is obtained for the discounted penalty function provided that the initial surplus is nonnegative. Based on this matrix renewal equation, we present some asymptotic formulae for the discounted penalty functions when the claim size distributions are heavy tailed.

scholarly journals On the absolute ruin in a MAP risk model with debit interest

2011 ◽  
Vol 43 (1) ◽  
pp. 77-96 ◽  
Author(s):  
Zhimin Zhang ◽  
Hailiang Yang ◽  
Hu Yang
Keyword(s):  
Risk Model ◽  
Penalty Functions ◽  
Renewal Equation ◽  
Arrival Process ◽  
Size Distributions ◽  
Absolute Ruin ◽  
Asymptotic Formulae ◽  
Discounted Penalty Function ◽  
Heavy Tailed ◽  
Debit Interest

In this paper we consider a risk model where claims arrive according to a Markovian arrival process (MAP). When the surplus becomes negative or the insurer is in deficit, the insurer could borrow money at a constant debit interest rate to repay the claims. We derive the integro-differential equations satisfied by the discounted penalty functions and discuss the solutions. A matrix renewal equation is obtained for the discounted penalty function provided that the initial surplus is nonnegative. Based on this matrix renewal equation, we present some asymptotic formulae for the discounted penalty functions when the claim size distributions are heavy tailed.

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