This article presents a method for estimating the variance of the firm or enterprise value distribution by incorporating temporal dependencies in cash flows using ARMA models. The analysis highlights the importance of considering these dependencies, as neglecting them can lead to a significant increase in variance and subsequent erroneous decision-making. By utilizing ARMA models, decision-makers can obtain a more accurate assessment of the underlying risks and make informed investment decisions based on a comprehensive understanding of the firm's value distribution. The proposed method provides valuable insights for evaluating the uncertainty associated with future cash flows and enhances the accuracy of investment decision processes.

The old-age dependency ratios are indicators of the number of elderly people who are generally economically inactive compared to the number of people of working age. They significantly affect the financial burden of social public pension schemes, making it essential to analyze the influence of mortality on this ratio. In this paper, the Gompertz model is used to investigate the effect of mortality and fertility on the old-age dependency ratio, with a focus on the impact of changes in life expectancy. Elasticity formulas are derived to analyze this effect, and the results indicate that an increase in life expectancy leads to a considerable rise in the old-age dependency ratio.

In 1761, the German demographer Johann Peter Süßmilch published a simple population growth model that starts with a couple, in the eighth chapter of his book “Die göttliche Ordnung” (The Divine Order). With the help of the Swiss mathematician Leonhard Euler, the population has been projected for 300 years. Euler and Süßmilch demonstrate that after that time the population will be growing approximately geometrically. In this paper, the population projection of Euler and Süßmilch is reanalyzed using matrix algebra. Graphs and tables show the time series of the population and its growth rates. Age structures of selected years are presented. The solution of the projection equation is derived. It is shown that the projection model can be described by a geometric trend model which is superimposed by six cyclical components. In the long run, the population time series can be explained quite well by the sum of only two components, the trend component and one component with explosive cycles of a period of about 24 years. In the very long run, the influence of the cyclical component diminishes, and the series can be solely explained by its geometric trend component, as has been also recognized by Euler and Süßmilch.

This article analyzes the stochastic aspects of a tontine using a Gompertz distribution. In particular, the probabilistic and demographic risks of a tontine investment are examined. The expected value and variance of tontine payouts are calculated. Both parameters increase with age. The stochastic present value of a tontine payout is compared with the present value of a fixed annuity. It is shown that only at very high ages the tontine is more profitable than an annuity. Finally, the demographic risks associated with a tontine are discussed. Elasticities are used to calculate the impact of changes in modal age on the tontine payout. It is shown that the tontine payout is very sensitive to changes in modal age.

INTRODUCTION:(Excess) mortality and years of life lost are important measures of health risks from the Corona pandemic. The aim of this paper was to identify methodological factors that affect the calculation of mortality and further to point out possible misinterpretations of years of life lost. METHODOLOGY:Standardized mortality ratios (SMRs) can be used to compare mortalities (e. g., an SMR of 1.015 means excess mortality of 1.5%, an SMR of 0.990 means that mortality is reduced by 1.0%). In this study, SMRs as a measure of association for mortality in Germany were calculated for 2020 using different methods. In particular, the influence of different data sources and reference periods was examined. Furthermore, its influence on the calculated mortality was also examined to take into account increasing life expectancy. In addition, published results on years of life lost were critically analyzed. RESULTS:Using January 2022 data from the Federal Statistical Office on mortality for 5-year age groups resulted in higher SMR values than using preliminary data from February 2021 with 20-year age groups (SMR=0.997, 95% confidence interval (CI): 0.995-0.999 versus SMR=0.976 (95% CI: 0.974-0.978)). The choice of the reference period had a large impact on calculated mortality (for men, SMR=1.024 (95% CI: 1.022-1.027) with 2019 as the reference year versus SMR=0.998 (95% CI: 0.996-1.001) with 2016 to 2019 as the reference period). Analyses in which declining mortality in 2016 to 2019 was carried forward into 2020 when calculating expected deaths resulted in significantly higher SMR values (for men SMR=1.024 (95% CI: 1.021-1.026) with, and SMR=0.998 (95% CI: 0.996-1.001) without carrying forward declining mortality). Figures for pandemic-related years of life lost per person who died from COVID-19 should be interpreted with caution: Calculation from remaining life reported in mortality tables can lead to misleading results. CONCLUSION:When calculating mortality and years of life lost during the pandemic, a number of methodological assumptions must be made that have a significant impact on the results and must be considered when interpreting the results.

Zusammenfassung Einleitung (Über)sterblichkeit und verlorene Lebensjahre sind wichtige Maße für gesundheitliche Risiken durch die Corona-Pandemie. Das Ziel dieses Beitrags ist es, methodische Faktoren zu benennen, die die Berechnung der Sterblichkeit beeinflussen, und auf mögliche Fehlinterpretationen von verlorenen Lebensjahren hinzuweisen. Methodik Standardisierte Mortalitätsratios (SMRs) können für den Vergleich von Sterblichkeiten verwendet werden (z. B. bedeutet ein SMR von 1,015 eine Übersterblichkeit von 1,5%, ein SMR von 0,990 eine Untersterblichkeit von 1,0%). In dieser Studie werden SMRs als Assoziationsmaße für die Sterblichkeit in Deutschland mit unterschiedlicher Methodik für das Jahr 2020 berechnet. Insbesondere wird der Einfluss unterschiedlicher Datenquellen und Referenzperioden untersucht. Ferner wird geprüft, welchen Einfluss es auf die berechnete Sterblichkeit hat, die steigende Lebenserwartung zu berücksichtigen. Darüber hinaus werden publizierte Ergebnisse zu verlorenen Lebensjahren kritisch diskutiert. Ergebnisse Die Nutzung aktueller Daten des Statistischen Bundesamts vom Januar 2022, in denen die Sterblichkeit für 5-Jahres-Altersgruppen berichtet wird, führt zu höheren SMR-Werten als die Nutzung vorläufiger Daten vom Februar 2021 mit 20-Jahres-Altersklassen (SMR=0,997, 95% Konfidenzintervall (KI): 0,995–0,999 versus SMR=0,976 (95% KI: 0,974–0,978)). Die Wahl des Referenzzeitraums hat großen Einfluss auf die berechnete Sterblichkeit (für Männer: SMR=1,024 (95% KI: 1,022–1,027) mit 2019 als Referenzjahr versus SMR=0,998 (95% KI: 0,996–1,001) mit 2016 bis 2019 als Referenzzeitraum). Analysen, in denen bei der Berechnung erwarteter Sterbefälle die sinkende Mortalität in den Jahren 2016 bis 2019 in das Jahr 2020 fortgeschrieben wird, führen zu deutlich höheren SMR-Werten (für Männer SMR=1,024 (95% KI: 1,021–1,026) mit, und SMR=0,998 (95% KI: 0,996–1,001) ohne Fortschreibung der sinkenden Mortalität). Zahlen zu pandemiebedingten verlorenen Lebensjahren pro an COVID-19 Verstorbenem sind mit Vorsicht zu interpretieren: Eine Berechnung aus der in Sterbetafeln angegebenen verbleibenden Lebenszeit führt zu irreführenden Ergebnissen. Schlussfolgerung Bei Berechnung zur Sterblichkeit und zu verlorenen Lebensjahren während der Pandemie sind eine Reihe methodischer Annahmen zu treffen, die erheblichen Einfluss auf die Ergebnisse haben und bei der Interpretation der Ergebnisse beachtet werden müssen.

Introduction Excess mortality is a suitable indicator of health consequences of COVID-19 because death from any cause is clearly defined contrary to death from Covid-19. We compared the overall mortality in 2020 with the overall mortality in 2016 to 2019 in Germany, Sweden and Spain. Contrary to other studies, we also took the demographic development between 2016 and 2020 and increasing life expectancy into account. Methods Using death and population figures from the EUROSTAT database, we estimated weekly and cumulative Standardized Mortality Ratios (SMR) with 95% confidence intervals (CI) for the year 2020. We applied two approaches to calculate weekly numbers of death expected in 2020: first, we used mean weekly mortality rates from 2016 to 2019 as expected mortality rates for 2020, and, second, to consider increasing life expectancy, we calculated expected mortality rates for 2020 by extrapolation from mortality rates from 2016 to 2019. Results In the first approach, the cumulative SMRs show that in Germany and Sweden there was no or little excess mortality in 2020 (SMR = 0.976 (95% CI: 0.974–0.978), and 1.030 (1.023–1.036), respectively), while in Spain the excess mortality was 14.8% (1.148 (1.144–1.151)). In the second approach, the corresponding SMRs for Germany and Sweden increased to 1.009 (1.007–1.011) and 1.083 (1.076–1.090), respectively, whereas results for Spain were virtually unchanged. Conclusion In 2020, there was barely any excess mortality in Germany for both approaches. In Sweden, excess mortality was 3% without, and 8% with consideration of increasing life expectancy.

August Zillmer (1831-1893) was a German life insurance actuary in Berlin. He is credited for one of the first German textbooks on actuarial mathematics. His name is associated with the Zillmer method of calculating life insurance reserves. In this paper, August Zillmer's early contribution to demographic analysis, which is virtually unknown, is described and appreciated. In 1863 he published a discrete population model, and produced several age distributions using a given life table and different population growth rates. He showed that the resulting age distributions will eventually become stable. Although the stable model in demography can be traced back to Euler in 1760, Zillmer's model is the first dynamic analysis of the influence of changes in population growth rates on the age distribution and population parameters such as mean age. His results and conclusions are discussed and compared with modern demographic methods. Finally, Zillmer's model is considered as a tool for special population forecasts, where new inputs (births) do not depend on the population size of other age-groups.

Es wird zunächst ein Überblick über die gängigen Methoden der Sterblichkeitsmodellierung gegeben. Von besonderem Interesse sind Vorhersagen. Ein Schwerpunkt der Arbeit liegt in diesem Zusammenhang auf dem Brass-Verhältnismodell. In einer ex-post Studie an deutschen Daten führt dieser Ansatz zu ebenso guten Prognosen wie die normalerweise verwendeten Vorhersagemodelle. Bei Prognosen für das Jahr 2050 liegen die Brass-Vorhersagen für die Lebenserwartung deutlich über denen des Statistischen Bundesamtes. Es gibt jedoch auch Stimmen, welche die amtlichen Prognosen als zu niedrig einstufen. Über diesen Aspekt kann daher diskutiert werden.

Forecasts for the number of students in Germany are conducted by the Kultusministerkonferenz. They use a transition model which does not allow for prediction intervals and therefore lack a measure of uncertainty of the forecast. Since the uncertainty is high for such forecasts, this lack is of importance.

Die stochastische Investitionsrechnung, ob sie nun simulativ oder analytisch durchgeführt wird, ist ein geeignetes Instrument zur Identifizierung und Bewertung von Risiken bei Investitionsentscheidungen. Dieser Beitrag soll die Grundzüge der stochastischen Investitionsrechnung darstellen und an einem einfachen Beispiel demonstrieren.

The objective of this paper is to examine the effect of migration on the size and the age structure of the population in Germany by applying population projection models.

"The use of the Box-Jenkins approach for forecasting the population of the United States up to the year 2080 is discussed. It is shown that the Box-Jenkins approach is equivalent to a simple trend model when making long-range predictions for the United States. An investigation of forecasting accuracy indicates that the Box-Jenkins method produces population forecasts that are at least as reliable as those done with more traditional demographic methods."