Competing Risks Model for Corporate Exit Analysis : Discrete Hazard Model and Extension with Stochastic Frailties

Publicly traded companies can leave a public system by bankruptcy or an exit due to merger. Discrete sub-hazard functions are modeled with a multinomial logit model. For 12,571 U.S. industrial firms spanning 1980 to 2004, quarterly firm specific financial variables and macroeconomic variables are available. Time-varying baseline hazard functions may capture unobservable or missing macroeconomic information. We examine the relationship between the effect of smoothing baseline hazard estimators and inclusion of macroeconomic variables. Smoothed nonparametric estimates of previous quarters baseline hazard functions are used for one step ahead predictions. A prediction power association methods are used currently in the literature. We propose a more direct method that yields an estimate of the probability of bankruptcy/merger for each company and hence a prediction of bankruptcy/merger in the next quarter. Optimal roughness penalties are chosen to minimize the sum of Shannon's entropies for all prediction periods. _________________________________________________ On the regulator-insurer interaction in a structural model. Carole Bernard, University of Waterloo An Chen, University of Amsterdam Abstract: In this paper, we provide a new insight of previous works of Briys and de In this paper, we provide a new insight of previous works of Briys and de Varenne [1997] and Grosen and Jà ̧rgensen [2002]. Firstly, we investigate the impact of regulatory authoritiesâ€TM rules on the fair value of companyâ€TMs liabilities and assets. In particular, we study how to choose regulation intervention levels in order to control for instance the shortfall probability of the issuing company. But market values of (long term) liabilities and regulators' rules are determined by assuming a constant volatility and no management of the insurer's portfolio. However, insurance companies follow dynamic investment strategies. Therefore, secondly, we study the interaction between the regulator who determines the regulation rule and the insurance companyâ€TMs risk management. Following the recent work of Ballotta, Haberman and Wang [2005] and the guidelines of the IASB, we develop an analysis of the model error when the insurance company is informed of regulators' rules and trades according to a certain discrete risk management hedging strategy instead of staying passive until the contractâ€TMs maturity. _________________________________________________ A nonparametric test for comparing the riskiness of portfolios Vytaras Brazauskas University of Wisconsin-Milwaukee Abstract: Inspired by the problem of testing hypotheses about the equality of several risk measure values, we find that the ""nested L-statistic"" -a notion introduced herein -is natural and particularly convenient. Indeed, the test statistic that we explore in this paper is a nested L-statistic. We discuss large-sample properties of the statistic, investigate its performance using a simulation study, and consider an example involving the comparison of risk measure values where the risks of interest are those associated with tornado damage in different time periods and different regions. Inspired by the problem of testing hypotheses about the equality of several risk measure values, we find that the ""nested L-statistic"" -a notion introduced herein -is natural and particularly convenient. Indeed, the test statistic that we explore in this paper is a nested L-statistic. We discuss large-sample properties of the statistic, investigate its performance using a simulation study, and consider an example involving the comparison of risk measure values where the risks of interest are those associated with tornado damage in different time periods and different regions. _________________________________________________ Moments of a Regime-Switching Stochastic Interest Rate Model with Randomized Regimes James G. Bridgeman University of Connecticut Abstract: A regime-switching stochastic model with randomized regime parameters creates a more plausible set of extreme paths than do the usual stochastic interest rate models. Generalizing the Black-Karasinski model by randomizing the mean reversion target provides an example. Better to understand such models, as well as to calibrate their parameters without trial and error runs of stochastic model, we use an asymptotic expansion to estimate the moments of the integrated stochastic process over time. _________________________________________________ A regime-switching stochastic model with randomized regime parameters creates a more plausible set of extreme paths than do the usual stochastic interest rate models. Generalizing the Black-Karasinski model by randomizing the mean reversion target provides an example. Better to understand such models, as well as to calibrate their parameters without trial and error runs of stochastic model, we use an asymptotic expansion to estimate the moments of the integrated stochastic process over time. _________________________________________________ Modeling mortality with jumps: transitory effects and pricing implication to mortality securitization Hua Chen, Georgia State University Samuel Cox, University of Manitoba Abstract: In this paper, we incorporate a jump-diffusion process into the original LeeCarter model, and use it to forecast mortality rates and analyze mortality securitization. Mortality jumps must be modeled explicitly in mortality securitization, because the rationale behind the design of mortality securities is to hedge mortality risks. We examine the outlier-adjusted Lee-Carter model to provide further evidence of mortality jumps. We also explore alternative models with transitory jump effects versus permanent jump effects, and find that modeling the mortality via permanent jump effects induces errors in parameter estimation and distortions in security valuation consequently. We use the Swiss Re mortality bond as an example to show how to apply our model and the distortion measure approach to value mortality-linked securities. Pricing the Swiss Re mortality bond is difficult because the mortality index is correlated across countries and over time. Cox, Lin and Wang (2006) employ the normalized multivariate exponential tilting to take into account correlations across countries. We show in this paper how to account for correlations of the mortality index over time by simulating the mortality index and changing the measure on paths. _________________________________________________ In this paper, we incorporate a jump-diffusion process into the original LeeCarter model, and use it to forecast mortality rates and analyze mortality securitization. Mortality jumps must be modeled explicitly in mortality securitization, because the rationale behind the design of mortality securities is to hedge mortality risks. We examine the outlier-adjusted Lee-Carter model to provide further evidence of mortality jumps. We also explore alternative models with transitory jump effects versus permanent jump effects, and find that modeling the mortality via permanent jump effects induces errors in parameter estimation and distortions in security valuation consequently. We use the Swiss Re mortality bond as an example to show how to apply our model and the distortion measure approach to value mortality-linked securities. Pricing the Swiss Re mortality bond is difficult because the mortality index is correlated across countries and over time. Cox, Lin and Wang (2006) employ the normalized multivariate exponential tilting to take into account correlations across countries. We show in this paper how to account for correlations of the mortality index over time by simulating the mortality index and changing the measure on paths. _________________________________________________ An option-based operational risk management on pandemics Hua Chen, Georgia State University Samuel Cox, University of Manitoba Abstract: An enormous literature on the dynamics of epidemics has been developed in epidemiology and mathematical biology, but little has been studied about epidemic risks for private enterprises. To plan for the possible outbreak of an influenza pandemic evolving from the current avian flu, the CDC has provided guidelines for large businesses and instructed firms to set triggers for activating and terminating optimal response policies, including shutting down parts of their businesses in affected areas. In this paper, we develop a two-stage model to help firms determine these triggers in the event of an influenza pandemic. In the first stage, we present a stochastic model to simulate the spread of the virus, depending on the regime that the firm is currently in. In the second stage, we view the reactivation decision as a call option and the suspension decision as a put option, and employ the theory of real option valuation and the regime switching model to determine the optimal switching thresholds. Our numerical experiments suggest that given the parameter values in our paper, it is optimal for the firm to suspend the business (or parts of its business) when the fraction of infected employees is higher than 18%, and to reactivate the operation anytime the fraction drops to 3%. When considering the uncertainty in the future, firms are more conservative about the decisions of suspension and reactivation. If the firm incurs switching costs, the suspension threshold increases with costs, while the reactivation threshold decreases with costs. By implementing policies to control the disease, firms can meet their social obligations and in the meantime, increase their values in both regimes. _________________________________________________ An enormous literature on the dynamics of epidemics has been developed in epidemiology and mathematical biology, but little has been studied about epidemic risks for private enterprises. To plan for the possible outbreak of an influenza pandemic evolving from the current avian flu, the CDC has provided guidelines for large businesses and instructed firms to set triggers for activating and terminating optimal response policies, including shutting down parts of their businesses in affected areas. In this paper, we develop a two-stage model to help firms determine these triggers in the event of an influenza pandemic. In the first stage, we present a stochastic model to simulate the spread of the virus, depending on the regime that the firm is currently in. In the second stage, we view the reactivation decision as a call option and the suspension decision as a put option, and employ the theory of real option valuation and the regime switching model to determine the optimal switching thresholds. Our numerical experiments suggest that given the parameter values in our paper, it is optimal for the firm to suspend the business (or parts of its business) when the fraction of infected employees is higher than 18%, and to reactivate the operation anytime the fraction drops to 3%. When considering the uncertainty in the future, firms are more conservative about the decisions of suspension and reactivation. If the firm incurs switching costs, the suspension threshold increases with costs, while the reactivation threshold decreases with costs. By implementing policies to control the disease, firms can meet their social obligations and in the meantime, increase their values in both regimes. _________________________________________________ The Cost Control on a DB underpin hybrid pension plan Kai Chen & Mary Hardy University of Waterloo Abstract: The Defined Benefit (DB) underpin Defined Contribution hybrid pension plan offers an attractive combination of benefit security and upside potential for pension plan members, and offers cost containment for employers, whilst still ensuring the provision The Defined Benefit (DB) underpin Defined Contribution hybrid pension plan offers an attractive combination of benefit security and upside potential for pension plan members, and offers cost containment for employers, whilst still ensuring the provision of adequate equitable benefits. The DB underpin guarantee is valued and hedged as a financial option, within the traditional funding paradigms of actuarial science. Assuming fixed interest rates, and assuming that salaries can be treated as a tradable asset, contribution rates are developed for Projected Unit Credit and Traditional Unit Credit funding methods. In addition, for the accruals methods, we demonstrate the implied hedging strategy. The traditional unit credit approach shows promise as a funding method in terms of average costs and in the incidence of costs, on average. However, the average monthly hedging costs are time dependent and can be unstable when employees are close to retirement. Some cost control approaches will be proposed to reduce the volatility of hedging cost _________________________________________________ Modelling the Cumulative Cases from SARS Renbao Chen & Ping Wang National University of Singapore Abstract: In this paper, we have derived a model of for the cumulative cases of SARS (severe acute respiratory syndrome). To test the model, we applied it to the actual data from the 2003 SARS outbreak. The model proved a good fit for the four areas we examined, China (mainland), Canada, Hong Kong and Singapore, as evident from the high R2 values and the randomness of the residuals. In this paper, we have derived a model of for the cumulative cases of SARS (severe acute respiratory syndrome). To test the model, we applied it to the actual data from the 2003 SARS outbreak. The model proved a good fit for the four areas we examined, China (mainland), Canada, Hong Kong and Singapore, as evident from the high R2 values and the randomness of the residuals. In addition, we obtained second order derivatives of our curve fits and proposed the use of T*, the time in days taken to slow the rate of the increase in the number of cumulative SARS cases, as a measure of the effectiveness of control measures in each country. We then incorporated explanatory variables to link the model parameters to demographic indicators of areas hit heavily by SARS. The work makes it possible to predict the future behavior of SARS or other epidemics should they ever strike again. _________________________________________________ Negative Effects of the GIS Clawback and Possible Solutions Diana K. Chisholm University of Waterloo Abstract: In Canada, there are three main sources of government-provided retirement income: the Canada Pension Plan (CPP), which is based on pre-retirement income and contributed to over the course of employment; the Old Age Security (OAS), which is a fixed amount for most but drops off for extremely high-income individuals; and the Guaranteed Income Supplement (GIS), which is designed to supplement those with extremely low income. The annual GIS amount is reduced (or clawed back) by 50 cents for every dollar of annual income in retirement, including CPP and Registered Retirement Savings Plans (RRSP). The result of this is that low-income individuals who In Canada, there are three main sources of government-provided retirement income: the Canada Pension Plan (CPP), which is based on pre-retirement income and contributed to over the course of employment; the Old Age Security (OAS), which is a fixed amount for most but drops off for extremely high-income individuals; and the Guaranteed Income Supplement (GIS), which is designed to supplement those with extremely low income. The annual GIS amount is reduced (or clawed back) by 50 cents for every dollar of annual income in retirement, including CPP and Registered Retirement Savings Plans (RRSP). The result of this is that low-income individuals who aim for a 70% replacement rate actually see a decrease in government-provided funding the more they save for retirement. In fact, savings in an RRSP can effectively be taxed at more than 100% through corresponding reductions in the GIS, social housing, home care, GAINS (Guaranteed Annual Income Supplement), and other benefits based on retirement income. It would be better for these individuals to put their money in a proverbial mattress than to save it in an RRSP! This paper explores a basic GIS exemption, a tax rate of lower than 50%, and various combinations of the two as alternative policies for the GIS. We find that these new methods can reduce the disincentive to save money without dramatically increasing the overall cost of the GIS. _________________________________________________ Bounds for Ruin Probabilities and Value at Risk Samuel Cox, University of Manitoba Yijia Lin, University of Nebraska Ruilin Tian, Georgia State University Luis F. Zuluaga, University of New Brunswick Abstract: In many situations, complete information about a rare event is not available, meaning the underlying probability distribution is not completely specified. This paper finds the best one can do when the incomplete information consists of estimates of the first two moments of the distribution. These are called semi-parametric lower and upper bounds on the probability of the rare event. We apply it to the a rare event hitting an insurer for which losses are extraordinary high and investment income is low. We refer to this as “ruin†although the company may survive; it is just a convenient way to describe a rare event that would threaten a companyâ€TMs solvency. We also consider value-at-risk (VaR) in the sense that we find bounds a portfolio return, given only the first two moments of the portfolio components. In many situations, complete information about a rare event is not available, meaning the underlying probability distribution is not completely specified. This paper finds the best one can do when the incomplete information consists of estimates of the first two moments of the distribution. These are called semi-parametric lower and upper bounds on the probability of the rare event. We apply it to the a rare event hitting an insurer for which losses are extraordinary high and investment income is low. We refer to this as “ruin†although the company may survive; it is just a convenient way to describe a rare event that would threaten a companyâ€TMs solvency. We also consider value-at-risk (VaR) in the sense that we find bounds a portfolio return, given only the first two moments of the portfolio components. In order to numerically solve the semiparametric bounds considered here, we reformulate the corresponding semiparametric bound problem as a sum of squares (SOS) program. However, a SOS program is an optimization problem where the variables are coefficients of polynomials, the objective is a linear combination of the variable coefficients, and the constraints are given by the polynomials being SOS. This form of the problem allows us to use one of several readily available SOS programming solvers to solve numerically the SOS programming formulation of the moment problem. _________________________________________________ Mortality Modelling using Projection Pursuit Regression Steven Craighead, TP – Tillinghast Tom Edwalds, Munich American Reassurance Company Abstract: In this paper, we develop sixteen separate mortality models based on the current SOA mortality data being used for the preferred life study. Our models are In this paper, we develop sixteen separate mortality models based on the current SOA mortality data being used for the preferred life study. Our models are constructed using projection pursuit regression. The sixteen models arise from all combinations of sex, smoker status, select and ultimate status, and per life vs. per dollar mortality rates. In general our models confirm the following facts: 1. Mortality rates increase as attained age increases 2. Mortality rates increase as duration increases. 3. Mortality rates decrease as Face Amount Bands increase. 4. Mortality rates decrease if preferred underwriting is present. _________________________________________________ Stochastic Life Annuities Daniel Dufresne University of Melbourne, Australia Abstract: This paper gives analytic approximations for the distribution of a stochastic life annuity. It is assumed that returns follow a geometric Brownian motion. The distribution of the stochastic annuity may be used to answer questions such as ``What is the probability that an amount F is sufficient to fund a pension with annual amount y to a pensioner aged x?"" The main idea is to approximate the future lifetime distribution with a combination of exponentials, and then apply a known formula (due to Yor) related to the integral of geometric Brownian motion. The approximations are very accurate in the cases studied _________________________________________________ This paper gives analytic approximations for the distribution of a stochastic life annuity. It is assumed that returns follow a geometric Brownian motion. The distribution of the stochastic annuity may be used to answer questions such as ``What is the probability that an amount F is sufficient to fund a pension with annual amount y to a pensioner aged x?"" The main idea is to approximate the future lifetime distribution with a combination of exponentials, and then apply a known formula (due to Yor) related to the integral of geometric Brownian motion. The approximations are very accurate in the cases studied _________________________________________________ Fourier inversion formulas in option pricing and insurance Daniel Dufresne, University of Melbourne, Australia Jose Garrido, Concordia Unvi., Montreal Canada Manuel Morales, University of Montreal, Canada Abstract: Several authors have used Fourier inversion to compute prices of puts and calls, some using Parseval’s theorem. The expected value of max(S K,0) also arises in excess-of –loss of stop-loss insurance and we show that Fourier methods may be used to compute them. In this paper, we take the idea of using Parseval’s theorem further: (1) formulas requiring weaker assumptions; (2) relationship with classical inversion theorems for probability distributions; (3) formulas for pay-offs which occur in insurance. Numerical examples are provided. _________________________________________________ Several authors have used Fourier inversion to compute prices of puts and calls, some using Parseval’s theorem. The expected value of max(S K,0) also arises in excess-of –loss of stop-loss insurance and we show that Fourier methods may be used to compute them. In this paper, we take the idea of using Parseval’s theorem further: (1) formulas requiring weaker assumptions; (2) relationship with classical inversion theorems for probability distributions; (3) formulas for pay-offs which occur in insurance. Numerical examples are provided. _________________________________________________ Quantitative Risk Measures of Individual Life Settlement Purchases Shamita Dutta Gupta Pace University Abstract: Life Settlement market provides life insurance policyholders an opportunity to realize the values embedded in their policies. It offers liquidity to policyholder and at the same time it creates a new asset class for investors seeking uncorrelated risks. The market has been growing rapidly in recent years. In this paper (talk), we will provide a few summary quantitative risk measures in investing in life settlement policies. It is obvious that the risk of investor reduces as the portfolio size increases. But in this work, we will focus on measures that relate only to individual policy purchases. As examples, some of the measures are, the cost and benefit ratio, the breakeven mortality multiple, the breakeven mortality year, and the breakeven probability. The conclusion is that picking a policy with less cost of carry is as important as picking a policy with high mortality multiple. Lower the cost of carry, Smaller the risk. _________________________________________________ Life Settlement market provides life insurance policyholders an opportunity to realize the values embedded in their policies. It offers liquidity to policyholder and at the same time it creates a new asset class for investors seeking uncorrelated risks. The market has been growing rapidly in recent years. In this paper (talk), we will provide a few summary quantitative risk measures in investing in life settlement policies. It is obvious that the risk of investor reduces as the portfolio size increases. But in this work, we will focus on measures that relate only to individual policy purchases. As examples, some of the measures are, the cost and benefit ratio, the breakeven mortality multiple, the breakeven mortality year, and the breakeven probability. The conclusion is that picking a policy with less cost of carry is as important as picking a policy with high mortality multiple. Lower the cost of carry, Smaller the risk. _________________________________________________ Modeling treatment costs associated with a multi-stage pandemic Michael Ekhaus Gibraltar Analytics Abstract: Introduced is an Interacting Particle System (IPS) approach to modeling multiple stage pandemics. The model population receives differing treatments with possibly differing costs depending on the stage of exposure/infection. The total cost of treatment is a time varying utility function of the ensemble population for which this approach uses particle system methods to simulate the cost of maintaining survival of the population. Interacting Particle Systems are a class of spatial-temporal stochastic processes suitable for studying the spread of infectious diseases and other interaction phenomena. Introduced is an Interacting Particle System (IPS) approach to modeling multiple stage pandemics. The model population receives differing treatments with possibly differing costs depending on the stage of exposure/infection. The total cost of treatment is a time varying utility function of the ensemble population for which this approach uses particle system methods to simulate the cost of maintaining survival of the population. Interacting Particle Systems are a class of spatial-temporal stochastic processes suitable for studying the spread of infectious diseases and other interaction phenomena. The paper gives a brief background on particle systems with specific focus on ``the contact process'', followed by expanding the contact process to represent more realistic state transitions. _________________________________________________ Generalized Gerber-Shiu Function in Piecewise-deterministic Markov