Uncertainty on survival probabilities and solvency capital requirement: application to long-term care insurance

In this paper, we focus on uncertainty issues on disabled lives survival probabilities of LTC insurance policyholders and its consequences on solvency capital requirement. Among the risks affecting long-term care portfolios, special attention is addressed to the table risk, i.e. the risk of unanticipated aggregate mortality, arising from the uncertainty in modeling LTC claimants survival law. The table risk can be thought as the risk of systematic deviations referring not only to a parameter risk but, as well, to any other sources leading to a misinterpretation of the life table resulting for example from an evolution of medical techniques or a change in rules of acceptance. In fine, the idea is to introduce the risk of systematic deviations arising from the uncertainty on the disabled lives death probabilities directly. We analyze the consequences of an error of appreciation on the disabled lives survival probabilities in terms of level of reserves and describe a framework in an Own Risk and Solvency Assessment perspective to measure the gap between the risk profile from the standard formula to the risk analysis specific to the organism.     

Modeling exposure to losses on automobile leases

We present an integrated statistical model for assessing risk and projecting financial losses on automobile leases. The model employs nonstationary Markovian state transitions for active leases and hierarchical logistic and regression equations for different outcomes on termination. The model reveals that lower residual risks may partially offset higher credit risk for customers whose credit scores predict higher risk of default. It also reveals a risk profile that differs through time from other secured credits such as mortgages. A three-year follow-up of forecasts versus outcomes for 39,500 leasing contracts shows that the model predicted rates of repossession better than standard roll-rate models with stationary transition probabilities. It displayed similar accuracy in predicting unscheduled terminations and insurance settlements.      

Momentum and reversion in risk neutral martingale probabilities

A time homogeneous, purely discontinuous, parsimonous Markov martingale model is proposed for the risk neutral dynamics of equity forward prices. Transition probabilities are in the variance gamma class with spot dependent parameters. Markov chain approximations give access to option prices. The model is estimated on option prices across strike and maturity for five days at a time. Properties of the estimated processes are described via an analysis of return quantiles, momentum functions that measure the response of tail probabilities to such moves. Momentum and reversion are also addressed via the construction of reverse conditional expectations. Term structures for the moments of marginal distributions support a decay in skewness and excess kurtosis with maturity at rates slower than those implied by Lévy processes. Out of sample performance is additionally reported. It is observed that risk neutral dynamics by and large reflect the presence of momentum in numerous probabilities. However, there is some reversion in the upper quantiles of risk neutral return distributions.     

Longevity risk, retirement savings, and financial innovation

Over the last couple of decades unprecedented increases in life expectancy have raised important concerns for retirement savings. We solve a life-cycle model with longevity risk, which can be hedged through endogenous saving and retirement decisions. We investigate the benefits of financial assets designed to hedge the shocks to survival probabilities. When longevity risk is calibrated to match forward-looking projections, those benefits are substantial. This lends support to the idea that such hedging should be pursued by defined benefit pension plans on behalf of their beneficiaries. Finally, we draw implications for optimal security design.     

Strong stability in a two-dimensional classical risk model with independent claims

In this paper, we study the strong stability of ruin probabilities in risk models. The question of stability naturally arises in risk theory since the governing parameters in these models can only be estimated with uncertainty. Moreover, in most cases there are not explicit expressions known for the ruin probabilities. Our objective is to present the applicability of the strong stability method to the bivariate classical risk model with independent claims. After clarifying the conditions to approximate the two-dimensional risk model with disturbance parameters by the two-dimensional classical risk model, we obtain the stability inequalities with an exact computation of the constants.     

A common thread linking the design of guarantee and nonescalating payments of public annuities

Motivated by recent experiences in economies adopting the defined-contribution pension system, we study public annuities in the presence of survival probability heterogeneity. It is found that the difference of annuitization-weighted and unweighted averages of survival probabilities is a useful measure of the severity of adverse selection. We then examine public annuities with a guarantee feature which bundles annuity income and bequeathable wealth components. We show that when the heterogeneity in survival probability is limited, the magnitude of guarantee proportion is irrelevant. On the other hand, an increase in the guarantee proportion mitigates adverse selection when the extent of heterogeneity is sufficiently large, because the share of annuity purchase by retirees with lower (resp., higher) survival probabilities is increased (resp., decreased). We also obtain a similar set of results for public annuities with nonescalating payments. The results have useful implications regarding the design of public annuities.     

Constant Proportion Debt Obligations (CPDOs): modeling and risk analysis

Constant Proportion Debt Obligations (CPDOs) are structured credit derivatives that generate high coupon payments by dynamically leveraging a position in an underlying portfolio of investment-grade index default swaps. CPDO coupons and principal notes received high initial credit ratings from the major rating agencies, based on complex models for the joint transition of ratings and spreads for all names in the underlying portfolio. We propose a parsimonious model for analysing the performance of CPDO strategies using a top-down approach that captures the essential risk factors of the CPDO. Our approach allows us to compute default probabilities, loss distributions and other tail risk measures for the CPDO strategy and analyse the dependence of these risk measures on various parameters describing the risk factors. We find that the probability of the CPDO defaulting on its coupon payments can be made arbitrarily small—and thus the credit rating arbitrarily high—by increasing leverage, but the ratings obtained strongly depend on assumptions on the credit environment (high spread or low spread). More importantly, CPDO loss distributions are found to exhibit a wide range of tail risk measures inside a given rating category, suggesting that credit ratings are insufficient performance indicators for such complex leveraged strategies. A worst-case scenario analysis indicates that CPDO strategies have a high exposure to persistent spread-widening scenarios and that CPDO ratings are shown to be quite unstable during the lifetime of the strategy.     

Default Risk in Corporate Yield Spreads

An important research question examined in the credit risk literature focuses on the proportion of corporate yield spreads attributed to default risk. This topic is reexamined in light of the different issues associated with the computation of default probabilities obtained from historical default data. We find that the estimated default risk proportion in corporate yield spreads is sensitive to the ex ante estimated term structure of default probabilities used as inputs. This proportion can become a large fraction of the spread when sensitivity analyses are made with respect to the period over which the probabilities are estimated and the recovery rates.     

A Dynamic Analysis of Fixed- and Adjustable-Rate Mortgage Terminations

This paper provides a side-by-side comparison of loan-level statistical models for fixed- and adjustable-rate mortgages. Multinomial logit models for quarterly conditional probabilities of default and prepayment are estimated. We find that the estimated impacts of embedded option values for prepayment and default are generally quite similar across both FRM and ARM loans, providing additional empirical support for the basic predictions of the options theory. We also find that differences in estimates of conditional probabilities of prepayment and default associated with mortgage age, origination period, original LTV, and relative loan size, indicate the continued significance of these other economic and demographic factors for empirical models of mortgage terminations.     

Mortality Heterogeneity and the Distributional Consequences of Mandatory Annuitization

This article investigates the distributional consequences of mandatory annuitization. Using Health and Retirement Study (HRS) data and accounting for longevity risk pooling within marriage and preannuitized wealth, we find substantial redistribution away from disadvantaged groups in expected utility terms. Using HRS data on subjective survival probabilities, we construct a subjective life table for each individual in the HRS. We calculate the value each household would place on annuitization, based on the husband and wife's subjective life tables, and the household's degree of risk aversion and proportion of preannuitized wealth. A significant minority would perceive themselves as suffering a loss from mandatory annuitization.     

YOLO: Mortality Beliefs and Household Finance Puzzles

We study the effect of subjective mortality beliefs on life‐cycle behavior. With new survey evidence, we document that survival is underestimated (overestimated) by the young (old). We calibrate a canonical life‐cycle model to elicited beliefs. Relative to calibrations using actuarial probabilities, the young undersave by 26%, and retirees draw down their assets 27% slower, while the model's fit to consumption data improves by 88%. Cross‐sectional regressions support the model's predictions: Distorted mortality beliefs correlate with savings behavior while controlling for risk preferences, cognitive, and socioeconomic factors. Overweighting the likelihood of rare events contributes to mortality belief distortions.     

Contagion risk in the interbank market: a probabilistic approach to cope with incomplete structural information

One lesson of the financial crisis erupting in 2008 has been that domino effects constitute a serious threat to the stability of the financial sector, i.e. the failure of one node in the interbank network might entail the danger of contagion to large parts of the entire system. How important this effect is, depends on the exact topology of the network on which the supervisory authorities have typically very incomplete knowledge. In order to explore the extent of contagion effects and to analyse the effectiveness of macroprudential measures to contain such effects, a reconstruction of the quantitative features of the empirical network would be needed. We propose a probabilistic approach to such a reconstruction: we propose to combine some important known quantities (like the size of the banks) with a realistic stochastic representation of the remaining structural elements. Our approach allows us to evaluate relevant measures for the contagion risk after default of one unit (i.e. the number of expected subsequent defaults, or their probabilities). For some quantities we are able to derive closed form solutions, others can be obtained via computational mean-field approximations.     

The Cross‐Section of Credit Risk Premia and Equity Returns

We explore the link between a firm's stock returns and credit risk using a simple insight from structural models following Merton ( 1974 ): risk premia on equity and credit instruments are related because all claims on assets must earn the same compensation per unit of risk. Consistent with theory, we find that firms' stock returns increase with credit risk premia estimated from CDS spreads. Credit risk premia contain information not captured by physical or risk‐neutral default probabilities alone. This sheds new light on the “distress puzzle”—the lack of a positive relation between equity returns and default probabilities—reported in previous studies.     

Parameter estimation risk in asset pricing and risk management: A Bayesian approach

Parameter estimation risk is non-trivial in both asset pricing and risk management. We adopt a Bayesian estimation paradigm supported by the Markov Chain Monte Carlo inferential techniques to incorporate parameter estimation risk in financial modelling. In option pricing activities, we find that the Merton's Jump-Diffusion (MJD) model outperforms the Black-Scholes (BS) model both in-sample and out-of-sample. In addition, the construction of Bayesian posterior option price distributions under the two well-known models offers a robust view to the influence of parameter estimation risk on option prices as well as other quantities of interest in finance such as probabilities of default. We derive a VaR-type parameter estimation risk measure for option pricing and we show that parameter estimation risk can bring significant impact to Greeks' hedging activities. Regarding the computation of default probabilities, we find that the impact of parameter estimation risk increases with gearing level, and could alter important risk management decisions.     

Divergence of Opinion in Complete Markets: A Note

We consider an Arrow-Debreu model with agents who have different subjective probabilities. In general, asset prices will depend only on aggregate consumption and the distribution of subjective probabilities in each state of nature. If all agents have identical preferences then an asset with “more dispersed” subjective probabilities will have a lower price than an asset with less dispersed subjective probabilities if risk aversion does not decline too rapidly. It seems that this condition is likely to be met in practice, so that increased dispersion of beliefs will generally be associated with reduced asset prices in a given Arrow-Debreu equilibrium.     

A multiple state model for the working-age disabled population using cross-sectional data

A multiple state model describes the transitions of the disability risk among the states of active, inactive and dead. Ideally, estimations of transition probabilities and transition intensities rely on longitudinal data; however, most of the national surveys of disability are based on cross-sectional data measuring the disabled status of an individual at one point in time. This paper aims to propose a generic method of the estimation of the expected transition probabilities when the model allows recovery from disability using the UK cross-sectional data. The disability prevalence rates are modelled by taking into consideration the effect of age and time. Under some plausible assumptions concerning the death rates among inactive and active people, the estimated prevalence rates of disability are used to decompose survival probabilities in each state.     

Dynamic prediction of hedge fund survival in crisis-prone financial markets

This study focuses on dynamic changes in survival probabilities over the lifetimes of hedge funds. To model such probabilities, a mixed Cox proportional hazards (CPH) model-specifically, a survival/hazard model with time-varying covariates and fixed covariates- is employed. Resulting dynamic survival probabilities show that the mixed CPH model provides significantly higher accuracy in predicting hedge fund failure than other models in the literature, including fixed covariate CPH models and discrete logit models. Our results are useful to investors and regulators of hedge funds in crisis-prone financial markets.     

Life-cycle effects of health risk

Health status affects individuals׳ labor supply, asset accumulation and welfare through four main channels: productivity, medical expenditures, time endowments and survival probabilities. Using a life-cycle model calibrated to the U.S. for different education groups, I evaluate the relative importance of each channel and quantify their interactions. The productivity and time endowment channels dominate in importance and the risks implied by them contribute significantly to income inequality and precautionary savings. Health effects are larger for the non-college than college educated and account to a large extent for the lower labor supply and higher reliance on government transfers of the non-college group.     

Fast simulations in credit risk

We consider the problem of simulating tail loss probabilities and expected losses conditioned on exceeding a large threshold (expected shortfall) for credit portfolios. Our new idea, called the geometric shortcut, allows an efficient simulation for the case of independent obligors. It is even possible to show that, when the average default probability tends to zero, its asymptotic efficiency is higher than that of the naive algorithm. The geometric shortcut is also useful for models with dependent obligors and can be used for dependence structures modeled with arbitrary copulae. The paper contains the details for simulating the risk of the normal copula credit risk model by combining outer importance sampling with the geometric shortcut. Numerical results show that the new method is efficient in assessing tail loss probabilities and expected shortfall for credit risk portfolios. The new method outperforms all known methods, especially for credit portfolios consisting of weakly correlated obligors and for evaluating the tail loss probabilities at many thresholds in a single simulation run.