Minimizing the Probability of Lifetime Ruin under Random Consumption

Abstract

We determine the optimal investment strategy in a financial market for an individual whose random consumption is correlated with the price of a risky asset. Bayraktar and Young consider this problem and show that the minimum probability of lifetime ruin is the unique convex, smooth solution of its corresponding Hamilton-Jacobi-Bellman equation. In this paper we focus on determining the probability of lifetime ruin and the corresponding optimal investment strategy. We obtain approximations for the probability of lifetime ruin for small values of certain parameters and demonstrate numerically that they are reasonable ones. We also obtain numerical results in cases for which those parameters are not small.     

Minimizing the Probability of Ruin When Claims Follow Brownian Motion with Drift

Abstract

We extend the work of Browne (1995) and Schmidli (2001), in which they minimize the probability of ruin of an insurer facing a claim process modeled by a Brownian motion with drift. We consider two controls to minimize the probability of ruin: (1) investing in a risky asset and (2) purchasing quota-share reinsurance. We obtain an analytic expression for the minimum probability of ruin and the corresponding optimal controls, and we demonstrate our results with numerical examples.     

Minimizing the Probability of Lifetime Ruin When Shocks Might Occur: Perturbation Analysis

We determine the optimal investment strategy to minimize the probability of an individual’s lifetime ruin when the underlying model parameters are subject to a shock. Specifically, we consider two possibilities: (1) changes in the individual’s net consumption and mortality rate and (2) changes in the parameters of the financial market. We assume that these rates might change once at a random time. Changes in an individual’s net consumption and mortality rate occur when the individual experiences an accident or other unexpected life event, while changes in the financial market occur due to shifts in the economy or in the political climate. We apply perturbation analysis to approximate the probability of lifetime ruin and the corresponding optimal investment strategy for small changes in the model parameters and observe numerically that these approximations are reasonable ones, even when the changes are not small.     

On Discrete-Time Dynamic Programming in Insurance: Exponential Utility and Minimizing the Ruin Probability

This paper studies an insurance model where the risk process can be controlled by reinsurance and by investment in a financial market. The performance criterion is either the expected exponential utility of the terminal surplus or the ruin probability. It is shown that the problems can be imbedded in the framework of discrete-time stochastic dynamic programming but with some special features. A short introduction to control theory with infinite state space is provided which avoids the measure-theoretic apparatus by use of the so-called structure assumption. Moreover, in order to treat models without discount factor, a weak contraction property is derived. Explicit conditions are obtained for the optimality of employing no reinsurance.     

Optimal and Simple,Nearly Optimal Rules for Minimizing the Probability Of Financial Ruin in Retirement

Abstract

The increasing risk of poverty in retirement has been well documented; it is projected that current and future retirees’ living expenses will significantly exceed their savings and income. In this paper, we consider a retiree who does not have sufficient wealth and income to fund her future expenses, and we seek the asset allocation that minimizes the probability of financial ruin during her lifetime. Building on the work of Young (2004) and Milevsky, Moore, and Young (2006), under general mortality assumptions, we derive a variational inequality that governs the ruin probability and optimal asset allocation. We explore the qualitative properties of the ruin robability and optimal strategy, present a numerical method for their estimation, and examine their sensitivity to changes in model parameters for specific examples. We then present an easy-to-implement allocation rule and demonstrate via simulation that it yields nearly optimal ruin probability, even under discrete portfolio rebalancing.     

Two-Sided Bounds for the Finite Time Probability of Ruin

Explicit, two-sided bounds are derived for the probability of ruin of an insurance company, whose premium income is represented by an arbitrary, increasing real function, the claims are dependent, integer valued r.v.s and their inter-occurrence times are exponentially, non-identically distributed. It is shown, that the two bounds coincide when the moments of the claims form a Poisson point process. An expression for the survival probability is further derived in this special case, assuming that the claims are integer valued, i.i.d. r.v.s. This expression is compared with a different formula, obtained recently by Picard & Lefevre (1997) in terms of generalized Appell polynomials. The particular case of constant rate premium income and non-zero initial capital is considered. A connection of the survival probability to multivariate B -splines is also established.     

Optimal Investment Strategy to Minimize the Probability of Lifetime Ruin

Abstract

I study the problem of how individuals should invest their wealth in a risky financial market to minimize the probability that they outlive their wealth, also known as the probability of lifetime ruin. Specifically, I determine the optimal investment strategy of an individual who targets a given rate of consumption and seeks to minimize the probability of lifetime ruin. Two forms of the consumption function are considered: (1) The individual consumes at a constant (real) dollar rate, and (2) the individual consumes a constant proportion of his or her wealth. The first is arguably more realistic, but the second has a close connection with optimal consumption in Merton’s model of optimal consumption and investment under power utility.

For constant force of mortality, I determine (a) the probability that individuals outlive their wealth if they follow the optimal investment strategy; (b) the corresponding optimal investment rule that tells individuals how much money to invest in the risky asset for a given wealth level; (c) comparative statics for the functions in (a) and (b); (d) the distribution of the time of lifetime ruin, given that ruin occurs; and (e) the distribution of bequest, given that ruin does not occur. I also include numerical examples to illustrate how the formulas developed in this paper might be applied.     

Non-exponential Bounds for Ruin Probability with Interest Effect Included

In this paper, we consider a discrete time risk model. First we discuss the classical model, both exponential and non-exponential upper bounds for ruin probabilities are obtained by using martingale inequalities. Then similar results are obtained for the model with investment income.     

The Ruin Probability of a Discrete Time Risk Model under Constant Interest Rate with Heavy Tails

This paper investigates the ultimate ruin probability of a discrete time risk model with a positive constant interest rate. Under the assumption that the gross loss of the company within one year is subexponentially distributed, a simple asymptotic relation for the ruin probability is derived and compared to existing results.     

When winning is everything

In the heat of competition, executives can easily become obsessed with beating their rivals. This adrenaline-fueled emotional state, which the authors call competitive arousal, often leads to bad decisions. Managers can minimize the potential for competitive arousal and the harm it can inflict by avoiding certain types of interaction and targeting the causes of a win-at-all-costs approach to decision making. Through an examination of companies such as Boston Scientific and Paramount, and through research on auctions, the authors identified three principal drivers of competitive arousal: intense rivalry, especially in the form of one-on-one competitions; time pressure, found in auctions and other bidding situations, for example; and being in the spotlight--that is, working in the presence of an audience. Individually, these factors can seriously impair managerial decision making; together, their consequences can be dire, as evidenced by many high-profile business disasters. It's not possible to avoid destructive competitions and bidding wars completely. But managers can help prevent competitive arousal by anticipating potentially harmful competitive dynamics and then restructuring the deal-making process. They can also stop irrational competitive behavior from escalating by addressing the causes of competitive arousal. When rivalry is intense, for instance, managers can limit the roles of those who feel it most. They can reduce time pressure by extending or eliminating arbitrary deadlines. And they can deflect the spotlight by spreading the responsibility for critical competitive decisions among team members. Decision makers will be most successful when they focus on winning contests in which they have a real advantage--and take a step back from those in which winning exacts too high a cost.     

On The Decomposition Of The Ruin Probability For A Jump-Diffusion Surplus Process Compounded By A Geometric Brownian Motion

Abstract

If one assumes that the surplus of an insurer follows a jump-diffusion process and the insurer would invest its surplus in a risky asset, whose prices are modeled by a geometric Brownian motion, the resulting surplus for the insurer is called a jump-diffusion surplus process compounded by a geometric Brownian motion. In this resulting surplus process, ruin may be caused by a claim or oscillation. We decompose the ruin probability in the resulting surplus process into the sum of two ruin probabilities: the probability that ruin is caused by a claim, and the probability that ruin is caused by oscillation. Integro-differential equations for these ruin probabilities are derived. When claim sizes are exponentially distributed, asymptotical formulas of the ruin probabilities are derived from the integro-differential equations, and it is shown that all three ruin probabilities are asymptotical power functions with the same orders and that the orders of the power functions are determined by the drift and volatility parameters of the geometric Brownian motion. It is known that the ruin probability for a jump-diffusion surplus process is an asymptotical exponential function when claim sizes are exponentially distributed. The results of this paper further confirm that risky investments for an insurer are dangerous in the sense that either ruin is certain or the ruin probabilities are asymptotical power functions, not asymptotical exponential functions, when claim sizes are exponentially distributed.     

On the Joint Distributions of the Time to Ruin,the Surplus Prior to Ruin,and the Deficit at Ruin in the Classical Risk Model

Abstract

The seminal paper by Gerber and Shiu (1998) unified and extended the study of the event of ruin and related quantities, including the time at which the event of ruin occurs, the deficit at the time of ruin, and the surplus immediately prior to ruin. The first two of these quantities are fundamentally important for risk management techniques that utilize the ideas of Value-at-Risk and Tail Value-at-Risk. As is well known, calculation of these and related quantities requires knowledge of the associated probability distributions. In this paper we derive an explicit expression for the joint (defective) distribution of the time to ruin, the surplus immediately prior to ruin, and the deficit at ruin in the classical compound Poisson risk model. As a by-product, we obtain expressions for the three bivariate distributions generated by the time to ruin, the surplus prior to ruin, and the deficit at ruin. Finally, we consider mixed Erlang claim sizes and show how the joint (defective) distribution of the time to ruin, the surplus prior to ruin, and the deficit at ruin can be calculated.     

乐观喜人的消费形势

投资、消费和出口是拉动经济增长的"三架马车", 但是2004年上半年,经济运行中存在投资和出口快、消 费慢的问题。进入三季度以后,在国家宏观调控政策作 用下,部分行业投资过快增长的局面被有效抑制,但消 费需求增速不断加快,投资和消费不协调增长的问题有 所改善。2004年全年消费品零售额名义增长将达13.1％, 比2003年提高4个百分点,消费需求对经济增长的贡献 率将明显高于2003年。预计2005年消费品零售额增速 将略低于2004年,但最终消费对经济增长的贡献率将呈 上升趋势。     

When no news is good news

For as long as can be remembered, BestBaby Corporation, a manufacturer of baby equipment and furniture, has enjoyed a solid reputation with retailers, a good track record with consumers, and a supportive relationship with stockholders. But then the child of a celebrity is injured when her stroller tips over because its brakes failed. The media go wild, and CEO Greg James finds himself in uncharted territory. The morning after the accident, Greg calls an emergency meeting of his executive staff. As he searches his memory to prepare for it, he thinks about Arzep Enterprises, BestBaby's main provider of parts and materials. He remembers his COO, Keith Sigismund, telling him that Arzep had switched suppliers at some point in order to cut its own costs. Nevertheless, Keith had assured Greg that the new material, although not quite as sturdy, hadn't affected the quality of Arzep's components. By the time the meeting is set to begin, several employees have threatened to quit, and stories are surfacing in the press and on the Web about other consumers who have had problems with their strollers. Then in the meeting, Keith drops a bombshell: he reads from a year-old memo sent to him by an employee in manufacturing stating that the new brake fittings delivered by Arzep don't grab the front brakes as well as the ones previously supplied. The same employee, and others, had complained in the past that Keith hadn't adequately attended to concerns they brought up to him. In this fictional case study, four commentators offer advice to Greg on how BestBaby should respond to the victim's family, the media, the public, and the company's own employees during this PR crisis.     

When business is a confidence game.

When it comes to making business decisions, being overconfident about your choices can actually be more harmful than just guessing. Here's how managers can calibrate their confidence levels-and avoid being too sure in the wrong situations.     

运用收益法进行企业价值评估的相关问题探讨

企业价值评估的对象是企业整体价值、股东全部权益价值或股东部分权益价值.采用收益法进行的评估,思路是建立在资本的时间价值观念之上,将被评估企业预期收益予以资本化或折现.评估的关键一是量化企业的预期收益,按照不同的口径计算自由现金流量;二是合理确定资本化率或折现率;三是合理确定收益年期.     

Disruptive change. When trying harder is part of the problem

When a company faces a major disruption in its markets, managers' perceptions of the disruption influence how they respond to it. If, for instance, they view the disruption as a threat to their core business, managers tend to overreact, committing too many resources too quickly. But if they see it as an opportunity, they're likely to commit insufficient resources to its development. Clark Gilbert and Joseph Bower explain why thinking in such stark terms--threat or opportunity--is dangerous. It's possible, they argue, to arrive at an organizational framing that makes good use of the adrenaline a threat creates as well as of the creativity an opportunity affords. The authors claim that the most successful companies frame the challenge differently at different times: When resources are being allocated, managers see the disruptive innovation as a threat. But when the hard strategic work of discovering and responding to new markets begins, the disruptive innovation is treated as an opportunity. The ability to reframe the disruptive technology as circumstances evolve is not an easy skill to master, the authors admit. In fact, it might not be possible without adjusting the organizational structure and the processes governing new business funding. Successful companies, the authors have determined, tend to do certain things: They establish a new venture separate from the core business; they fund the venture in stages as markets emerge; they don't rely on employees from the core organization to staff the new business; and they appoint an active integrator to manage the tensions between the two organizations, to name a few. This article will help executives frame innovations in more balanced ways--allowing them to recognize threats but also to seize opportunities.     

当A国出现外汇支付危机时

经过几个月的艰苦谈判,2003年4月,一份由中国出口信用保险公司(以下简称"中国信保")和A国政府之间的债务重组协议终于签署.此后,A国开始照协议按期还款,中国信保及受益企业之一--中国某进出口集团(以下简称C公司),总算松了一口气.一次贸易伙伴国出现外汇支付危机带来的风波归于平静.     

在利差收窄趋势下追求以价值为导向的银行盈利模式——对现行利率和利差水平的实证分析

随着银行利差水平的不断收窄,现代商业银行的盈利模式应该是利息收入、非利息收入并重发展.本文对近年来商业银行的利差水平及变化趋势进行了分析,剖析了宏观经济环境、市场结构及流动性、汇率弹性、银行的管理和经营状况、银行自身发展战略及客户、产品结构等因素对利率及利差水平的影响,同时对银行在利差收窄趋势下如何改进盈利模式提出了对策建议.