Statistical Inference for Lee-Carter Mortality Model and Corresponding Forecasts

Although the Lee-Carter model has become a benchmark in modeling mortality rates, forecasting mortality risk, and hedging longevity risk, some serious issues exist on its inference and interpretation in the actuarial science literature. After pointing out these pitfalls, this article proposes a modified Lee-Carter model, provides a rigorous statistical inference, and derives the asymptotic distributions of the proposed estimators and unit root test when the mortality index is nearly integrated and errors in the model satisfy some mixing conditions. After a unit root hypothesis is not rejected, future mortality forecasts can be obtained via the proposed inference. An application of the proposed unit root test to U.S. mortality rates rejects the unit root hypothesis for the female and combined mortality rates but does not reject the unit root hypothesis for the male mortality rates.     

Forecasting Mortality Trends Allowing for Cause-of-Death Mortality Dependence

Longevity risk is among the most important factors to consider for pricing and risk management of longevity products. Past improvements in mortality over many years, and the uncertainty of these improvements, have attracted the attention of experts, both practitioners and academics. Since aggregate mortality rates reflect underlying trends in causes of death, insurers and demographers are increasingly considering cause-of-death data to better understand risks in their mortality assumptions. The relative importance of causes of death has changed over many years. As one cause reduces, others increase or decrease. The dependence between mortality for different causes of death is important when projecting future mortality. However, for scenario analysis based on causes of death, the assumption usually made is that causes of death are independent. Recent models, in the form of Vector Error Correction Models (VECMs), have been developed for multivariate dynamic systems and capture time dependency with common stochastic trends. These models include long-run stationary relations between the variables and thus allow a better understanding of the nature of this dependence. This article applies VECMs to cause-of-death mortality rates to assess the dependence between these competing risks. We analyze the five main causes of death in Switzerland. Our analysis confirms the existence of a long-run stationary relationship between these five causes. This estimated relationship is then used to forecast mortality rates, which are shown to be an improvement over forecasts from more traditional ARIMA processes, which do not allow for cause-of-death dependencies.     

Mortality Change and Forecasting

Abstract

Prospects of longer life are viewed as a positive change for individuals and as a substantial social achievement but have led to concern over their implications for public spending on old-age support. This paper makes a critical assessment of knowledge about mortality change. It is oriented toward the problem of forecasting the course of mortality change and the potential of existing work to contribute to the development of useful forecasts in Canada, Mexico, and the U.S.

We first examine broad patterns in the historical decline in death rates in the three countries, the effect of these on trends in life expectancy, and the epidemiological transition. Next we review theories of the age pattern and evolution of mortality, including graduations, evolutionary theory, reliability models, dynamic models, and relational models.

The analysis and forecasting of mortality change have been shaped largely by some key historical lessons, which we summarize next. We emphasize issues that have been or are likely to be significant in mortality analysis, especially the questions of the age pattern and time trend in mortality at old ages; we distinguish patterns and facts that are established from those that remain uncertain. Next, we consider mortality differentials in characteristics such as sex, marital status, education, and socioeconomic variables; we summarize their key features and also point to the substantial gaps in our understanding of their determinants.

Finally, we review methods of forecasting, including the scenario method used by the U.S. Social Security Administration and the time series method of Lee and Carter. We set out some important recommendations for forecasters: forecasting assumptions should be made more formal and explicit; there should be retrospective evaluations of forecast performance; and greater attention should be paid to the assessment and consequences of forecast uncertainty.     

The CBD Mortality Indexes: Modeling and Applications

Most extrapolative stochastic mortality models are constructed in a similar manner. Specifically, when they are fitted to historical data, one or more series of time-varying parameters are identified. By extrapolating these parameters to the future, we can obtain a forecast of death probabilities and consequently cash flows arising from life contingent liabilities. In this article, we first argue that, among various time-varying model parameters, those encompassed in the Cairns-Blake-Dowd (CBD) model (also known as Model M5) are most suitably used as indexes to indicate levels of longevity risk at different time points. We then investigate how these indexes can be jointly modeled with a more general class of multivariate time-series models, instead of a simple random walk that takes no account of cross-correlations. Finally, we study the joint prediction region for the mortality indexes. Such a region, as we demonstrate, can serve as a graphical longevity risk metric, allowing practitioners to compare the longevity risk exposures of different portfolios readily.     

On the Modeling and Forecasting of Socioeconomic Mortality Differentials: An Application to Deprivation and Mortality in England

In any country, mortality rates and indices such as life expectancy usually differ across subpopulations, for example, defined by gender, geographic area, or socioeconomic variables (e.g., occupation, level of education, or income). These differentials, and in particular those related to socioeconomic circumstances, pose important challenges for the design of public policies for tackling social inequalities, as well as for the design of pension systems and the management of longevity risk in pension funds and annuity portfolios. We discuss the suitability for the modeling and forecasting of socioeconomic differences in mortality of several multiple population extensions of the Lee-Carter model, including a newly introduced relative model based on the modeling of the mortality in socioeconomic subpopulations alongside the mortality of a reference population. Using England mortality data for socioeconomic subpopulations defined using a deprivation index, we show that this new relative model exhibits the best results in terms of goodness of fit and ex post forecasting performance. We then use this model to derive projections of deprivation specific mortality rates and life expectancies at pensioner ages and analyze the impact of socioeconomic differences in mortality on the valuation of annuities.     

A Data-Analytic Method for Forecasting Next Record Catastrophe Loss

We develop in this article a data‐analytic method to forecast the severity of next record insured loss to property caused by natural catastrophic events. The method requires and employs the knowledge of an expert and accounts for uncertainty in parameter estimation. Both considerations are essential for the task at hand because the available data are typically scarce in extreme value analysis. In addition, we consider three‐parameter Gamma priors for the parameter in the model and thus provide simple analytical solutions to several key elements of interest, such as the predictive moments of record value. As a result, the model enables practitioners to gain insights into the behavior of such predictive moments without concerning themselves with the computational issues that are often associated with a complex Bayesian analysis. A data set consisting of catastrophe losses occurring in the United States between 1990 and 1999 is analyzed, and the forecasts of next record loss are made under various prior assumptions. We demonstrate that the proposed method provides more reliable and theoretically sound forecasts, whereas the conditional mean approach, which does not account for either prior information or uncertainty in parameter estimation, may provide inadmissible forecasts.     

Case-based Reasoning and Multiple-agent Systems for Accounting Regulation Systems with Extensions

One of the areas of judgment research in accounting and financial applications is that of accounting regulation. Previously, artificial intelligence efforts at modeling human judgment in accounting regulation systems have concentrated on rule-based expert systems. In those systems, general heuristic knowledge was captured using ‘if … then …’ rules in order to model particular decision processes. Recent developments in artificial intelligence have focused on case-based reasoning (CBR) and multiple-agent intelligent systems (MAIS). The ideas behind CBR are that ‘if it worked once then remember to use it again’ and ‘if it did not work before, then remember to not use it again‘. MAIS assumes that many organizational systems can be treated as computational models of multiple-interacting intelligent agents. Typically, solutions may be derived using some form of negotiations between agents to accomplish single global or separate individual interacting goals. This paper argues that many accounting regulation judgment processes can be modeled using CBR and MAIS. As a result, it summarizes some examples of both CBR and MAIS useful in accounting regulation and extends those to other accounting applications. In addition, it describes the results of some previously developed systems that employ CBR or MAIS.     

抽样调查方法在多种经济成分统计中的应用

全面报表不能取得准确数据的各种经济成分（如个体经济,私营经济,外资经济等）,或某些统计项目,如何采用抽样调查方法收集数据。针对长沙市的特殊情况,介绍了分层抽样和复合抽样两种抽样方法,特别是复合抽样方法,既较好地解决了确定抽样框的困难,又避免了采用放回（重复）抽样方法。     

Historical and Projected Mortality for Mexico,Canada, and the United States

Abstract

This paper presents historical death rates for Canada, Mexico, and the U.S. by sex and broad age group. The time period for this historical analysis begins with 1900 (1930 for Mexico). These data provide a quite consistent basis from which experts can develop and contrast their expectations for future mortality trends. Official mortality projections developed by government agencies of each of the three countries provide a starting point for this discussion.

During this century, death rates declined fairly rapidly in all three countries. However, the rate of mortality improvement has varied considerably across time periods: distinct periods of rapid and slow improvement are evident in the data, but are not consistent across the countries and have not yet been explained.

The historical rates of improvement in mortality have also varied greatly by age and sex: younger age groups have shown the most rapid proportional improvement in mortality in all three countries, and mortality improvement during this century has generally been greater for females than for males. However, the data provide evidence that this difference in the rates of mortality improvement between men and women has recently slowed, and even reversed, in the U.S. and Canada. Historical experience and projections are provided in graphs, in which death rates are plotted on a logarithmic scale. This approach allows easy detection of the extent to which rates of improvement have been changing (death rates with constant rates of improvement would be plotted as straight lines).

The official projections supplied for comparison provide strikingly similar outlooks for future potential mortality improvement. In each case, the relatively average rapid rate of mortality improvement experienced so far this century is assumed to slow in the future. In addition, rates of improvement are projected to be much more similar for all three countries across age groups and between the sexes.     

Coherent Modeling and Forecasting of Mortality Patterns for Subpopulations Using Multiway Analysis of Compositions: An Application to Canadian Provinces and Territories

Mortality levels for subpopulations, such as countries in a region or provinces within a country, generally change in a similar fashion over time, as a result of common historical experiences in terms of health, culture, and economics. Forecasting mortality for such populations should consider the correlation between their mortality levels. In this perspective, we suggest using multilinear component techniques to identify a common time trend and then use it to forecast coherently the mortality of subpopulations. Moreover, this multiway approach is performed on life table deaths by referring to Compositional Data Analysis (CoDa) methodology. Compositional data are strictly positive values summing to a constant and represent part of a whole. Life table deaths are compositional by definition because they provide the age composition of deaths per year and sum to the life table radix. In bilinear models the use of life table deaths treated as compositions generally leads to less biased forecasts than other commonly used models by not assuming a constant rate of mortality improvement. As a consequence, an extension of this approach to multiway data is here presented. Specifically, a CoDa adaptation of the Tucker3 model is implemented for life table deaths arranged in three-dimensional arrays indexed by time, age, and population. The proposed procedure is used to forecast the mortality of Canadian provinces in a comparative study. The results show that the proposed model leads to coherent forecasts.     

Obesity,Mortality, and the Obesity Paradox

The percentage of the population who are obese has grown dramatically on a worldwide basis over the last several decades, although the growth in the prevalence of obesity has slowed recently at a high level in the United States. Although there have been numerous studies of the effect of this trend on mortality, the findings have been inconsistent and controversial, in part because of methodological differences and the complexity of the relationships between obesity and mortality. The objective of this article is to discuss the issues surrounding these relationships and to shed light on the likely effects of the obesity epidemic on mortality. Of particular interest is the so-called obesity-mortality paradox, where mortality experience is lower for overweight and in some cases obese individuals than for those of normal weight. Although more recent studies of the relationship between mortality and obesity seem to indicate those who are obese have experienced a reduced percentage of additional mortality, this may in part be due to the shorter average time those currently obese have been exposed to their condition, the heterogeneity of the normal and obese populations, measurement issues including treatment of smokers and those who are ill, and study design limitations. An increased number of premature deaths may arise as more individuals who are obese are exposed for a longer period to excess adiposity. Although public policy issues surrounding obesity are being addressed with a great deal of activity and publicity, they have and will continue to prove quite challenging for both individuals and society to manage and overcome. The prevalence of obesity has had and will continue to have a significant effect on the mortality experience in most areas of actuarial practice. As a result, it is important for actuaries to enhance their understanding of these effects.     

Nonlinear Models in Corporate Finance Research: Review,Critique, and Extensions

Since the work of Morck, Shleifer and Vishny (1988), nonlinear model specification has gained more attention in corporate finance research. In this paper, we provide a detailed review of the previous studies that have examined nonlinear relations in corporate finance. We review the theory and evidence in these studies and discuss the advantages and disadvantages of the various methodologies used to detect nonlinearity. We also suggest two possible methodological extensions, which we apply in the empirical analysis of R&D investment and firm value.     

Financial Forecasting,Risk and Valuation: Accounting for the Future

Valuation involves forecasting payoffs and discounting expected payoffs for risk. Forecasting is often seen as the province of the statistician, risk determination the province of asset pricing. This paper elaborates on the idea that financial forecasting, risk determination and valuation are a matter of accounting. Accounting not only provides information to forecast payoffs but also specifies the payoffs to be forecasted. Further, accounting determines the transition from the present to the future and thus implicitly the evolutionary parameters that a statistician might estimate for forecasting. Accounting also bears on risk determination in the way it handles uncertainty. Accordingly, accounting is involved in both the numerator and the denominator of a valuation model. Indeed, a valuation model is a model of accounting for the future, and the effectiveness of a valuation model rides on the accounting principles employed.     

The Discounted Economic Stock of Money with VAR Forecasting

We measure the economic capital stock of money implied by the Divisia monetary aggregate service flow, in a manner consistent with asset pricing theory. Based on Barnett’s (Monetary policy on the 75th anniversary of the Federal Reserve System, pp. 232–244, Kluwer, Boston, 1991) definition of the economic stock of money, we estimate the expected discounted flow of expenditure on the services of monetary assets, where expenditure on monetary services is evaluated at the user costs of the monetary components. We use forecasts based on martingale expectations, asymmetric vector autoregressive expectations, and the Bayesian vector autoregressive expectations. We find the resulting capital-stock index to be surprisingly robust to the modeling of expectations.     

Forecasting risk,informed speculation,and financial innovation

Speculators who prey on hedgers can stifle financial innovation in the sense that new markets can fail. In this paper I analyze whether a profit maximizing exchange nonetheless chooses to open markets for speculative securities and if so, how to circumvent the problem of market failure. I find that the optimal financial innovation takes two forms. The first is a market structure consisting of hedge instruments, traded in low volume at stable asset prices. The second is a market structure consisting of speculative instruments, traded in greater volume at volatile asset prices. These strategies are derived within the same framework where the cost and the quality of the speculators' information set and the hedgers risk aversion ultimately determine which is the optimal one.     

Criteria for the Stochastic Ordering of Random Sums,with Actuarial Applications

It is shown that vectors ( S M 1 , … , S Mn ) and ( S' M'1 , …, S' M'n ) of random sums of positive random variables are stochastically ordered by upper orthant dependence, lower orthant dependence, concordance or by the supermodular ordering whenever their corresponding random numbers of terms ( M 1 , … , M n ) and ( M' 1 , … , M' n ) are themselves ordered in this fashion. Actuarial applications of these results are given to different dependence structures for the collective risk model with several classes of business.     

Revenue Forecasting Practices: Differences across Countries and Consequences for Forecasting Performance*

This paper reviews the practice and performance of revenue forecasting in selected OECD countries. It turns out that the cross‐country differences in the performance of revenue forecasting are first of all associated with uncertainty about the macroeconomic fundamentals. To some extent, they are also driven by country characteristics such as the importance of corporate and (personal) income taxes. Also, differences in the timing of the forecasts prove important. However, controlling for these differences, we find that the independence of revenue forecasting from possible government manipulation exerts a robust, significantly positive effect on the accuracy of revenue forecasts.