Pitfalls of inferring annual mortality from inspection of published survival curves

In many FU articles currently published, results are given primarily in the form of graphs of survival curves, rather than in the form of life table data. Sometimes the authors may comment on the slope of the survival curve as though it were equal to the annual mortality rate (after reversal of the minus sign to a plus sign). Even if no comment of this sort is made, medical directors and underwriters may be tempted to think along similar lines in trying to interpret the significance of the survival curve in terms of mortality. However it is a very serious error of life table methodology to conceive of mortality rate as equal to the negative slope of the survival curve. The nature of the error is demonstrated in this article. An annual mortality rate derived from the survival curve actually depends on two variables: a quotient with the negative slope (sign reversed), delta P/ delta as the numerator, and the survival rate, P, itself as the denominator. The implications of this relationship are discussed. If there are two "parallel" survival curves with the same slope at a given time duration, the lower curve will have a higher mortality rate than the upper curve. A constant slope with increasing duration means that the annual mortality rate also increases with duration. Some characteristics of high initial mortality are also discussed and their relation to different units of FU time.(ABSTRACT TRUNCATED AT 250 WORDS)     

Continuous-time multi-cohort mortality modelling with affine processes

Continuous-time mortality models, based on affine processes, provide many advantages over discrete-time models, especially for financial applications, where such processes are commonly used for interest rate and credit risks. This paper presents a multi-cohort mortality model for age-cohort mortality rates with common factors across cohorts as well as cohort-specific factors. The mortality model is based on well-developed and used techniques from interest rate theory and has many applications including the valuation of longevity-linked products. The model has many appealing features. It is a multi-cohort model that describes the whole mortality surface, it captures cohort effects, it allows for observed imperfect correlation between different cohorts, it is shown to fit historical data at pension-related ages very well, it has closed-form expressions for survival curves and we show that it outperforms a number of other commonly used discrete-time mortality models in forecasting future survival curves.     

Population-based mortality outcome of rheumatoid arthritis in females

Rheumatoid arthritis (RA) is a systemic disease whose morbidity exceeds its mortality. This abstract quantifies the mortality of RA in a general population over a 30-year period in females. The generic, across the spectrum of disease, mortality ratio is 136%; the excess death rate is 12. The mortality impact on males is minimal; the source publication noted a mortality ratio of 107%. Many selected cases of RA can be underwritten very favorably. PURPOSE: To quantify the excess mortality in females diagnosed with rheumatoid arthritis (RA) between 1955 to 1985. SUBJECTS AND METHODS: Medical records of all residents age 35 and over of Rochester, Minnesota, who met the American College of Rheumatology 1987 diagnostic criteria for RA were reviewed. Based on the comprehensive statistical base for residents in Rochester, virtually complete ascertainment of all clinically recognized cases of RA were identified. An incidence cohort identifying the same residents with new cases of RA occurring between January 1, 1955, and January 1, 1985, was created. Three, 10-year prevalence cohorts were assembled as of July 1, 1965, 1975 and 1985. Patients in each cohort were followed longitudinally until death or migration from Rochester. Data of disease characteristics, course, co-morbidity and death were collected. During the follow-up period, 6.9% moved out of the county, and 5.1% moved into the county after the diagnosis of RA had been made elsewhere. Expected survival was based on age and sex adjusted survival from the same community in the same time period. Mortality was described using the Kaplan-Meier product-limit method. Cox proportional hazards modeling was used to examine the effects of age, sex and rheumatoid factor on survival. DATA: In the 1965, 1975 and 1985 prevalence cohorts, there were 163, 235 and 272 cases of RA, respectively. Some individuals were present in more than one. Deaths in each cohort were 54, 93 and 111, respectively. Median follow-up was 12.7 years for the entire group with the earlier groups being longer. Mean follow-up was 15.1 years. Seventy-three percent of patients were females. The average age at diagnosis was 60.2 years.     

Reference mortality K2004 of personal life insurance policies in Finland

This article presents the reference mortality model K2004 approved by the Actuarial Society of Finland and the technique that was implemented in developing it. Initially, I will present the historical development of individual mortality rates in Finland. Then, the requirements posed for a modern mortality modelling will be presented. Reference mortality model K2004 is based on total population mortality rates, which were adjusted to correspond with that portion of the population that has a life insurance policy. First, the model presents a margin of the observed life insurance mortality rate in the total population with a Lee-Carter method together with a forecast, where the downward trend in mortality rates is expected to continue at the rate illustrated since the 1960s. Then, the mortality rate has been adjusted into life insurance mortality per age so that it corresponds to the differences observed between total population and the portion of population that has a life insurance during 1991–2001. Finally, a cohort and gender-specific functional margin will be presented to obtained data.     

Mycosis fungoides--an underwriting prospective with emphasis on staging and risk selection

Mycosis Fungoides is a T-cell lymphoma having a broad clinical spectrum ranging from localized cutaneous to rapidly fatal systemic disease. Early clinical presentation is non specific, delaying correct diagnosis. Compared to clinical, the insurable spectrum is narrow. Staging for skin, lymph node and other organ manifestations is presented. Factors which influence mortality within each stage are elucidated. The survival curves of stages and stage groupings are illustrated and discussed to facilitate risk classification. Cutaneous (T) and lymph node (LN) stages are the most important prognosticators. Substages T1/T2, LN1/LN2 without associated palpable adenopathy, eosinophilia, visceral and blood positive findings are insurable. It would be most appropriate to place them in a tumor class of mild/moderate risk after the initial excessive mortality period ends. Higher T and LN substages, adenopathy and visceral disease have highly adverse mortality. These ultimately reveal a flattening of survival curves at 8-10 years. Although numerous treatment modalities have been used, none appear to consistently prolong life expectancy except in the earliest stage skin disease.     

“A Direct Approach to the Discounted Penalty Function”, Hansjörg Albrecher,Hans U. Gerber,and Hailiang Yang,Volume 14, No. 4, 2010

Abstract

Mortality improvements, especially of the elderly, have been a common phenomenon since the end of World War II. The longevity risk becomes a major concern in many countries because of underestimating the scale and speed of prolonged life. In this study we explore the increasing life expectancy by examining the basic properties of survival curves. Specifically, we check if there are signs of mortality compression (i.e., rectangularization of the survival curve) and evaluate what it means to designing annuity products. Based on the raw mortality rates, we propose an approach to verify if there is mortality compression. We then apply the proposed method to the mortality rates of Japan, Sweden, and the United States, using the Human Mortality Database. Unlike previous results using the graduated mortality rates, we found no obvious signs that mortality improvements are slowing down. This indicates that human longevity is likely to increase, and longevity risk should be seriously considered in pricing annuity products.     

Effect of age on mortality experience in patients with hypertrophic cardiomyopathy

Medical authors typically combine all patient groups to increase the amount of data available for analysis. Use of this statistical methodology generally conceals higher mortality ratios at younger ages and masks survival differences related to disease severity and comorbid impairments. This paper discusses the effects of age and clinical characteristics on mortality experience in patients with hypertrophic cardiomyopathy. Limited data suggest the mortality pattern associated with this impairment is similar to that observed with most disorders: excess mortality (compared to the general population) that is high at younger ages, intermediate in middle-aged people, and minimal in the very elderly. Optimism regarding generally favorable mortality at older ages must be tempered with caution since studies report much poorer experience in certain subgroups of elderly patients with this impairment.     

Evaluation of survival in medically treated patients with homozygous beta thalassemia by the quick hit method

Articles published in medical journals often evaluate the survival and/or prognosis of a medical impairment. The conclusions of these articles can be very misleading if correct mortality methodology is not utilized. Authors evaluating survival in a cohort of individuals with Thalassemia Major concluded the prognosis was excellent. This was based solely on observed mortality. When the cohort is properly compared to the expected mortality of a similar nonimpaired population the result is quite different. Their survival is far from excellent. This article uses the quick hit method to evaluate the survival of this cohort having Thalassemia Major.     

Outcome of early surgery compared to surveillance in the management of small aortic aneurysms--a comparative mortality analysis

When underwriting individuals with aortic aneurysms, the following factors are used in risk selection: location, initial size, stability or velocity of change in size and co-morbid impairments. Treatment options based on clinical and ultrasonic findings are either surveillance or surgical resection. A recent study examined the 8-year outcome of early-surgery vs surveillance in small (4.6 cm) aortic aneurysms. Over the duration, the early-surgery group had a mortality ratio of 170% and excess death rate of 28; the surveillance group had a mortality ratio of 215% and excess death rate of 45. First-year mortality was expectedly high in the early-surgery group. Excluding first-year experience, the early-surgery group had an improved mortality ratio of 160% and excess death rate of 22. After 6 months, the death rate in the early-surgery group was about three fourths that of the surveillance group. However, due to high initial mortality cumulative survival curves did not cross in favor of early-surgery until about 3 years.     

An analytic method for longitudinal mortality studies

Our knowledge of mortality risks comes largely from longitudinal (cohort) studies. The most commonly used analytic tool is the Cox proportional hazards model for survival analysis. An alternative approach is a simple cross-sectional analysis of person-years. The key to the method is logistic regression, where the outcome variable is lived/died in the given year and the explanatory variables are age, sex, and other potential risk factors. This approach can be used to model any dichotomous outcome and has several important advantages over the more traditional survival analysis. As an example, we compare the two methods using a large data base of patients with spinal cord injury.     

债台高筑的美国经济

美国的财政赤字正以几何级数的速度在增长,但美国人自己并不为此而担心,他们甚至认为负债本身就是一种创造财富的过程。美国的决策者和经济学家们对于真正通过经济中的储蓄和投资所创造的财富和市场上依赖信贷政策鼓吹起来的资产泡沫所创造的"财富",是从来不加以区别的。居高不下的负债和不断增长的赤字消费,使美国成为一个债台高筑的国家。美国的赤字消费是影响全球经济的一个重大问题,对中国经济的影响也十分显著,从长远利益出发,我国的汇率、贸易及产业政策都应进行相应的调整,以确保中国经济长期稳定发展、企业竞争实力不断增强和人民生活水平不断提高。     

Note on a new method of construction of mortality tables when the number of lives exposed to risk is unknown

Abstract

It has been known for some time that the d^bx column in a mortality table can be considered as a compound frequency curve with a limited number of maxima and minima. From a theoretical point of view this is of course a self evident conclusion which follows directly from the so-called genetic theory of frequency originally introduced by Laplace. He showed that any frequency distribution can be considered or generated as the sum of a very large number of elementary errors, referrable to several sources of error, each group or error having its own peculiar law of error. While the pure theory of the generation of frequency curves from such secondary sources of elementary errors is simple enough, the inverse and essentially practical problem of decomposing a compound frequency curve into its component or constituent elements is by no means simple and often presents great difficulties, especially if certain restrictions are imposed upon the component curves. An example of such restrictions would be the requirement that all the component curves should be normal Laplacean probability curves.     

Iterative Adjustment of Survival Functions by Composed Probability Distortions

We introduce a parametric class of composite probability distortions that can be combined to converge to a target survival function. These distortions respect analytic invertibility and stability, which are shown to be relevant in many actuarial fields. We study the asymptotic impact of such distortions on hazard rates. The paper provides an estimation methodology, including hints for initialisation. Some applications to survival data bring results for catastrophic event impact modelling. We also obtain accurate parametric representations of the mortality trend over years. Finally, we suggest a prospective mortality simulation model that comes naturally from the above analysis.     

Liquidity,Reconstitution, and the Value of U.S. Treasury Strips

An apparent pricing anomaly exists in the market for U.S. Treasury strips: zero-coupon strips created from principal payments typically trade at significantly higher prices than otherwise identical zero-coupon strips created from coupon payments. In addition to documenting this phenomenon, this study demonstrates that differences in liquidity and differences in reconstitution characteristics explain much of this price variation.     

Redistribution and Insurance: Mandatory Annuitization With Mortality Heterogeneity

This article examines the distributional implications of mandatory longevity insurance when mortality heterogeneity exists in the population. Previous research has demonstrated the significant financial redistribution that occurs under alternative annuity programs in the presence of differential mortality across groups. This article embeds that analysis into a life‐cycle framework that allows for an examination of distributional effects on a utility‐adjusted basis. It finds that the degree of redistribution that occurs from the introduction of a mandatory annuity program is substantially lower on a utility‐adjusted basis than when evaluated on a purely financial basis. In a simple life‐cycle model with no bequests, complete annuitization is welfare enhancing even for those with higher‐than‐average expected mortality rates, so long as administrative costs are sufficiently low. These findings have implications for policy toward annuitization, particularly as part of a reformed Social Security system.     

Two-dimensional Hazard Estimation for Longevity Analysis

We investigate developments in Danish mortality based on data from 1974–1998 working in a two-dimensional model with chronological time and age as the two dimensions. The analyses are done with non-parametric kernel hazard estimation techniques. The only assumption is that the mortality surface is smooth. Cross-validation is applied for optimal bandwidth selection to ensure the proper amount of smoothing to help distinguishing between random and systematic variation in data. A bootstrap technique is used for construction of pointwise confidence bounds. We study the mortality profiles by slicing up the two-dimensional mortality surface. Furthermore we look at aggregated synthetic population metrics as ‘population life expectancy’ and ‘population survival probability’. For Danish women these metrics indicate decreasing mortality with respect to chronological time. The metrics can not directly be used for prediction purposes. However, we suggest that life insurance companies use the estimation technique and the cross-validation for bandwidth selection when analyzing their portfolio mortality. The non-parametric approach may give valuable information prior to developing more sophisticated prediction models for analysis of economic implications arising from mortality changes.     

COVID-19 and Ethnic Inequalities in England and Wales*

The economic and public health crisis created by the COVID-19 pandemic has exposed existing inequalities between ethnic groups in England and Wales, as well as creating new ones. We draw on current mortality and case data, alongside pre-crisis labour force data, to investigate the relative vulnerability of different ethnic groups to adverse health and economic impacts. After accounting for differences in population structure and regional concentration, we show that most minority groups suffered excess mortality compared with the white British majority group. Differences in underlying health conditions such as diabetes may play a role; so too may occupational exposure to the virus, given the very different labour market profiles of ethnic groups. Distinctive patterns of occupational concentration also highlight the vulnerability of some groups to the economic consequences of social distancing measures, with Bangladeshi and Pakistani men particularly likely to be employed in occupations directly affected by the UK's ‘lockdown’. We show that differences in household structures and inequalities in access to savings mean that a number of minority groups are also less able to weather short-term shocks to their income. Documenting these immediate consequences of the crisis reveals the potential for inequalities to become entrenched in the longer term.     

Temporal Evolution of Mortality Indicators

Abstract

In Spain, as in other developed countries, significant changes in mortality patterns have occurred during the 20th and 21st centuries. One reflection of these changes is life expectancy, which has improved in this period, although the robustness of this indicator prevents these changes from being of the same order as those for the probability of death. If, moreover, we bear in mind that life expectancy offers no information as to whether this improvement is the same for different age groups, it is important and necessary to turn to other mortality indicators whose past and future evolution in Spain we are going to study. These indicators are applied to Spanish mortality data for the period 1981–2008, for the age range 0–99. To study its future evolution, the mortality ratios have to be projected using an adequate methodology, namely, the Lee-Carter model. Confidence intervals for these predictions can be calculated using the methodology that Lee and Carter apply in their original article for expected lifetime confidence intervals, but they take into account only the error in the prediction of the mortality index obtained from the ARIMA model adjusted to its temporal series, excluding other sources of error such as that introduced by estimations of the other parameters in the model. That is why bootstrap procedures are preferred, permitting the combination of all sources of uncertainty.     

Interest Rate Risk and the Forward Premium Anomaly in Foreign Exchange Markets

This paper shows that even adjusted for the time-varying risk premiums implied by the yield curves across countries, uncovered interest parity is still strongly rejected by the data. Moreover, factors that predict the excess bond returns are found not significant at all in predicting the foreign exchange returns. These results reject the joint restrictions on the exchange rate and interest rates imposed by dynamic term-structure models, suggesting that foreign exchange markets and bond markets may not be fully integrated and we have to look beyond interest rate risk in order to understand the exchange rate anomaly.