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When long-term savers plan for retirement they need to know their investment prospects in terms of real income (Merton, 2014). While inflation has traditionally been considered as a complication in financial analysis and financial practise, we obtain enhanced predictability and model fit if the real returns are targeted in conjunction with earnings-by-price minus inflation as predictor. For this latter case, we propose an investment strategy of updating the simple classical Merton proportion as we go along. This simple strategy is very close to the complicated theoretically optimal solution but has comparably much lower parameter uncertainty.

In this chapter the problems of the prevalence of esophageal cancer and the distribution of mortality rates from this disease are considered in the countries of Central and Eastern Europe. The rates of mortality from esophageal cancer are analyzed by statistical methods, econometrics, spatial and time series analysis. A study is carried out of the features of the tendencies of mortality rates from esophageal cancer in six EU countries and in Ukraine. The problems of the prevalence of esophageal cancer and the spatial distribution of mortality rates from this disease are considered using as examples the NUTS 2 regions in six countries of Central and Eastern Europe (Austria, Germany, the Czech Republic, Poland, Slovakia and Hungary). The spatial distribution of the rates of mortality from esophageal cancer is analyzed. It allows us to determine more and less epidemiologically affected regions and to carry out more detailed studies on the link between the mortality rates from esophageal cancer and various factors. It is emphasized that various factors, such as the environmental situation, socio-demographic characteristics of the population, culture and nature of nutrition, the general health status of the population, the availability of resources and the level of healthcare in the region should be took into account for development of complex of treatment and care and for the programs for the prevention of prevalence of esophageal cancer.

The paper shows how to reform the platform of pension products so that pension savers, professional financial advisors, actuaries and investment experts intuitively understand the underlying financial risk of the optimal investment profile. It is also pointed out that an excellent optimal investment strategy can destroy the future expected utility of a pension saver if the financial communication is wrong. It is shown that a simple system with an upper and a lower bound, originally inspired by Merton (2014), which can be executed easily using fintech, can replace complicated power utility optimization for the pension saver so that everyone can exactly understand the amount of financial risk taken. The paper focuses on investing money as a lump sum because being able to communicate the associated financial risk can serve as the first step towards communicating more complex pension saving structures.

This abstract relates to the following paper: Gerrard, R., Hiabu, M., Kyriakou, I. and Nielsen, J. P. (2018) Self-selection and risk sharing in a modern world of life-long annuities. British Actuarial Journal. Cambridge University Press, 23. doi: 10.1017/S135732171800020X.

Communicating a pension product well is as important as optimising the financial value. In a recent study, we showed that up to 80% of the value of a pension lump sum could be lost if customer communication failed. In this paper, we extend the simple customer interaction of the earlier contribution to the more challenging lifetime annuity case. Using a simple mobile phone device, the pension customer can select the life-long optimal investment strategy within minutes. The financial risk trade-off is presented as a trade-off between the pension paid and the number of years the life-long annuity is guaranteed. The pension payment decreases when investment security increases. The necessary underlying mathematical financial hedging theory is included in the study.

Many methods of measuring niche overlap have been proposed, such as those of Renkonen, Morisita, and Horn. In this paper, conditions are put forward which overlap indices should reasonably be expected to satisfy. A general description of a good index is given, which covers as special cases those of the well-known indices which satisfy all the conditions. A mixture of the Renkonen and van Belle & Ahmad indices is suggested, as it has many desirable properties. The problem of estimating overlap from data is discussed briefly

Background: There are different approaches to the assessment of the severity of trauma in a victim and to the provision of specialized health care. Some of them are based on the development of scales and logistic models, using expert systems or statistical methods, to assess the severity of injury and the probability of a particular outcome. This article presents the results of a study on the feasibility of developing and applying various statistical models in order to predict the outcome in the case of different types of trauma, based on data on the status of victims with severe trauma. Patients and methods: We present selected information about 373 victims, admitted and treated at the Department of Traumatic Shock of the GI «V.T. Zaycev Kharkiv Research Institute of General and Emergency Surgery» of NAMS of Ukraine; the records, which relate to patients with severe and combined trauma, were made between 1985 and 2015. The initial database contained 263 victims who had positive outcomes (survived), while 110 had fatal outcomes. Most of the patients presented with an open trauma (285 cases), then there were 80 cases with a closed injury and only 8 cases with a combined injury. Results: To estimate the probability of the outcome for various types of trauma we have developed a predictive model, based on a logistic relationship. Categorical variables, indicating the presence or absence of various types of trauma, were used in the model. Information about the eventual outcome for a given victim with the indicated type of trauma was used as the dependent variable. The logit model which we obtained has a high predictive accuracy in predicting positive outcomes. Thus, based on the a posteriori analysis, 92% of cases in which victims survived were correctly recognized by the model. In view of the fact that abdominal trauma is the commonest of all trauma mechanisms, we constructed a predictive model to estimate the probability of various outcomes in the case of abdominal trauma or injury to certain organs of the abdominal cavity. Linear discriminant functions were developed by us and used for the classification of possible outcomes depending on the condition of the victim and the resuscitation measures carried out. The model presented has a high predictive accuracy: on the basis of a posteriori analysis using data of discriminant functions, correct conclusions were drawn in 90% of cases when there was a positive outcome, and in 75% of cases when the outcome was fatal. Conclusion: We conclude that it is reasonable to use the statistical model developed, along with other qualitative and quantitative methods of prognostic determination of outcomes for victims with severe injuries. As different models have different predictive accuracy and require the provision of different information, it is necessary to use a sufficiently large number of techniques to derive accurate predictions and to choose the right tactics for diagnosis and treatment.

In this paper we consider the problem of hedging an arithmetic Asian option with discrete monitoring in an exponential Levy model by deriving backward recursive integrals for the price sensitivities of the option. The procedure is applied to the analysis of the performance of the delta and delta–gamma hedges in an incomplete market; particular attention is paid to the hedging error and the impact of model error on the quality of the chosen hedging strategy. The numerical analysis shows the impact of jump risk on the hedging error of the option position, and the importance of including traded options in the hedging portfolio for the reduction of this risk.

Modeling and forecasting extreme co-movements in financial market is important for conducting stress test in risk management. Asymptotic independence and asymptotic dependence behave drastically different in modeling such co-movements. For example, the impact of extreme events is usually overestimated whenever asymptotic dependence is wrongly assumed. On the other hand, the impact is seriously underestimated whenever the data is misspecified as asymptotic independent. Therefore, distinguishing between asymptotic independence/dependence scenarios is very informative for any decision-making and especially in risk management. We investigate the properties of the limiting conditional Kendall’s tau which can be used to detect the presence of asymptotic independence/dependence. We also propose nonparametric estimation for this new measure and derive its asymptotic limit. A simulation study shows good performances of the new measure and its combination with the coefficient of tail dependence proposed by Ledford and Tawn (1996, 1997). Finally, applications to financial and insurance data are provided.

Multivariate extremes behave very differently under asymptotic dependence as compared to asymptotic independence. In the bivariate setting, we are able to characterise the extreme behaviour of the asymptotic dependent case by using the concept of the copula. As a result, we are able to identify the properties of the boundary cases, that are asymptotic independent but still have some asymptotic dependent features. These situations are the most problematic in statistical extreme, and, for this reason, distinguishing between asymptotic dependence and asymptotic independence represents a difficult problem. We propose a simple test to resolve this issue which is an alternative to the procedure based on the classical coefficient of tail dependence. In addition, we are able to identify the worst/least asymptotic dependence (in the presence of asymptotic dependence) that maximises/minimises the probability of a given extreme region if tail dependence parameter is fixed. It is found that the perfect extreme association is not the worst asymptotic dependence, which is consistent with the existing literature. We are able to find lower and upper bounds for some risk measures of functions of random variables. A particular example is the sum of random variables, for which a vivid academic effort has been noticed in the last decade, where bounds for a sum of random variables are sought. It is numerically shown that our approach provides a great improvement of the existing methods, which reiterates the sensible conclusion that any additional piece of information on dependence would help to reduce the spread of these bounds.

In this article the problems of the prevalence of esophageal cancer and the spatial distribution of mortality rates from this disease are considered using as examples the NUTS 2 regions in six countries of Central and Eastern Europe (Austria, Germany, the Czech Republic, Poland, Slovakia and Hungary). The rates of mortality from esophageal cancer are analyzed by statistical methods and by spatial econometrics. A study is carried out of the features of the spatial distribution of the rates of mortality from esophageal cancer. It allows us to determine more and less epidemiologically affected regions and to carry out more detailed studies on the link between the mortality rates from esophageal cancer and various factors, such as the environmental situation, socio-demographic characteristics of the population, culture and nature of nutrition, the general health status of the population, the availability of resources and the level of healthcare in the region. By means of the multifactor regression model we forecast the rates of mortality from esophageal cancer, taking into account characteristics of the countries, the dynamics of the number of patients with diseases of the esophagus and the general time trend.

In the context of decision making for retirees of a defined contribution pension scheme in the de-cumulation phase, we formulate and solve a problem of finding the optimal time of annuitization for a retiree having the possibility of choosing her own investment and consumption strategy. We formulate the problem as a combined stochastic control and optimal stopping problem. As criterion for the optimization we select a loss function that penalizes both the deviance of the running consumption rate from a desired consumption rate and the deviance of the final wealth at the time of annuitization from a desired target. We find closed-form solutions for the problem and show the existence of three possible types of solutions depending on the free parameters of the problem. In numerical applications we find the optimal wealth that triggers annuitization, compare it with the desired target and investigate its dependence on both parameters of the financial market and parameters linked to the risk attitude of the re...

In many problems of risk analysis, failure is equivalent to the event of a random risk factor exceeding a given threshold. Failure probabilities can be controlled if a decisionmaker is able to set the threshold at an appropriate level. This abstract situation applies, for example, to environmental risks with infrastructure controls; to supply chain risks with inventory controls; and to insurance solvency risks with capital controls. However, uncertainty around the distribution of the risk factor implies that parameter error will be present and the measures taken to control failure probabilities may not be effective. We show that parameter uncertainty increases the probability (understood as expected frequency) of failures. For a large class of loss distributions, arising from increasing transformations of location-scale families (including the log-normal, Weibull, and Pareto distributions), the article shows that failure probabilities can be exactly calculated, as they are independent of the true (but unknown) parameters. Hence it is possible to obtain an explicit measure of the effect of parameter uncertainty on failure probability. Failure probability can be controlled in two different ways: (1) by reducing the nominal required failure probability, depending on the size of the available data set, and (2) by modifying of the distribution itself that is used to calculate the risk control. Approach (1) corresponds to a frequentist/regulatory view of probability, while approach (2) is consistent with a Bayesian/personalistic view. We furthermore show that the two approaches are consistent in achieving the required failure probability. Finally, we briefly discuss the effects of data pooling and its systemic risk implications.

We consider a collective insurance risk model with a compound Cox claim process, in which the evolution of a claim intensity is described by a stochastic differential equation driven by a Brownian motion. The insurer operates in a financial market consisting of a risk-free asset with a constant force of interest and a risky asset which price is driven by a Levy noise. We investigate two optimization problems. The first one is the classical mean-variance portfolio selection. In this case the efficient frontier is derived. The second optimization problem, except the mean-variance terminal objective, includes also a running cost penalizing deviations of the insurer’s wealth from a specified profit-solvency target which is a random process. In order to find optimal strategies we apply techniques from the stochastic control theory.

We consider the problem of optimal funding and asset allocation for a defined benefit pension scheme by assuming that the pension fund can be invested in a risk-free asset and a risky asset whose return follows a jump diffusion process. We extend existing literature which mainly assumes that the risky asset’s return follows a pure diffusion process. In a stochastic analysis of the optimal policies we show that the optimal contribution and asset allocation policies have similar forms as in the pure diffusion approaches, but with a modification for the effect of jumps. These results hold under both constant pension scheme benefit outgo and stochastic pension scheme benefit outgo. Using a sensitivity analysis of the effect of the mean jump magnitude on the asset allocation policy, we show that increasing (in absolute terms) the mean jump magnitude reduces the allocation in the risky asset and increases the allocation in the risk-free asset.

In this article the problems of the prevalence of esophageal cancer and the spatial distribution of mortality rates from this disease are considered using as examples the NUTS 2 regions in six countries of Central and Eastern Europe (Austria, Germany, the Czech Republic, Poland, Slovakia and Hungary). The rates of mortality from esophageal cancer are analyzed by statistical methods and by spatial econometrics. A study is carried out of the features of the spatial distribution of the rates of mortality from esophageal cancer. It allows us to determine more and less epidemiologically affected regions and to carry out more detailed studies on the link between the mortality rates from esophageal cancer and various factors, such as the environmental situation, socio-demographic characteristics of the population, culture and nature of nutrition, the general health status of the population, the availability of resources and the level of healthcare in the region. By means of the multifactor regression model we forecast the rates of mortality from esophageal cancer, taking into account characteristics of the countries, the dynamics of the number of patients with diseases of the esophagus and the general time trend.

We solve a portfolio selection problem of an investor with a deterministic savings plan who aims to have a target wealth value at retirement. The investor is an expected power utility-maximizer. The target wealth value is the maximum wealth that the investor can have at retirement. By constraining the investor to have no more than the target wealth at retirement, we find that the lower quantiles of the terminal wealth distribution increase, so the risk of poor financial outcomes is reduced. The drawback of the optimal strategy is that the possibility of gains above the target wealth is eliminated.

In this work, we consider the premium setting and the associated risk with regard to a life insurance policy when the interest rate process is a diffusion. Explicit formulas and numerical examples are given for the premiums determined by the equivalence principle under some well-known interest rate models such as the Vasicek model and the CIR model. Two types of premium design, single deposit premium and continuous premium, are considered. A martingale related to the discounted reserve process is introduced. With this martingale, an analysis of the risk of ruin is also carried out on a collective basis when the premium is the natural premium plus an initial positive loading. A bound of exponential type is derived for the ruin probability.

Modeling and forecasting extreme co-movements in financial market is important for conducting stress test in risk management. Asymptotic independence and asymptotic dependence behave drastically different in modeling such co-movements. For example, the impact of extreme events is usually overestimated whenever asymptotic dependence is wrongly assumed. On the other hand, the impact is seriously underestimated whenever the data is misspecified as asymptotic independent. Therefore, distinguishing between asymptotic independence/dependence scenarios is very informative for any decision-making and especially in risk management. We investigate the properties of the limiting conditional Kendall's tau which can be used to detect the presence of asymptotic independence/dependence. We also propose nonparametric estimation for this new measure and derive its asymptotic limit. A simulation study shows good performances of the new measure and its combination with the coefficient of tail dependence proposed by Ledford and Tawn (1996, 1997). Finally, applications to financial and insurance data are provided.

We focus on automatic strategies to optimize life cycle savings and investment. Classical optimal savings theory establishes that, given the level of risk aversion, a saver would keep the same relative amount invested in risky assets at any given time. We show that, when optimizing lifecycle investment, performance and risk assessment have to take into account the investor’s risk aversion and the maximum amount the investor could lose, simultaneously. When risk aversion and maximum possible loss are considered jointly, an optimal savings strategy is obtained, which follows from constant rather than relative absolute risk aversion. This result is fundamental to prove that if risk aversion and the maximum possible loss are both high, then holding a constant amount invested in the risky asset is optimal for a standard lifetime saving/pension process and outperforms some other simple strategies. Performance comparisons are based on downside risk-adjusted equivalence that is used in our illustration.

Methods of funding pension funds which amortize inter-valuation gains or losses over a fixed number of years have been considered by Dufresne (1988, 1989) and Haberman (1990, 1991, 1994). We consider the effects of such a method in the case where the rate of return on investments behaves as a first-order autoregressive process. This means that the actuarial loss process has the structure of a non-linear Time Series, where an autoregressive component is multiplied by the autoregressive rate of return. We obtain a recursive formula for the expected actuarial loss in a given year and for the expectation of the square of this quantity and prove that, for suitable values of the parameters, these expectations converge in time to limits which, though not explicitly obtainable, can nevertheless be found by a simple numerical procedure in any given case.

In the context of decision making for retirees of a defined contribution pension scheme in the de-cumulation phase, we formulate and solve a problem of finding the optimal time of annuitization for a retiree having the possibility of choosing her own investment and consumption strategy. We formulate the problem as a combined stochastic control and optimal stopping problem. As criterion for the optimization we select a loss function that penalizes both the deviance of the running consumption rate from a desired consumption rate and the deviance of the final wealth at the time of annuitization from a desired target. We find closed form solutions for the problem and show the existence of three possible types of solutions depending on the free parameters of the problem. In numerical applications we find the optimal wealth that triggers annuitization, compare it with the desired target and investigate its dependence on both parameters of the financial market and parameters linked to the risk attitude of the retiree. Simulations of the behaviour of the risky asset seem to show that under typical situations optimal annuitization should occur a few years after retirement.

In this paper we consider the problem of hedging an arithmetic Asian option with discrete monitoring in an exponential Levy model by deriving backward recursive integrals for the price sensitivities of the option. The procedure is applied to the analysis of the performance of the delta and delta-gamma hedges in an incomplete market; particular attention is paid to the hedging error and the impact of model error on the quality of the chosen hedging strategy. The numerical analysis shows the impact of jump risk on the hedging error of the option position, and the importance of including traded options in the hedging portfolio for the reduction of this risk.

In this paper we deal with contribution rate and asset allocation strategies in a pre-retirement accumulation phase. We consider a single cohort of workers and investigate a retirement plan of a defined benefit type in which an accumulated fund is converted into a life annuity. Due to the random evolution of a mortality intensity, the future price of an annuity, and as a result, the liability of the fund, is uncertain. A manager has control over a contribution rate and an investment strategy and is concerned with covering the random claim. We consider two mean–variance optimization problems, which are quadratic control problems with an additional constraint on the expected value of the terminal surplus of the fund. This functional objectives can be related to the well-established financial theory of claim hedging. The financial market consists of a risk-free asset with a constant force of interest and a risky asset whose price is driven by a Levy noise, whereas the evolution of a mortality intensity is described by a stochastic differential equation driven by a Brownian motion. Techniques from the stochastic control theory are applied in order to find optimal strategies.

The aim of the paper is to lay the theoretical foundations for the construction of a flexible tool that can be used by pensioners to find optimal investment and consumption choices in the distribution phase of a defined contribution pension plan. The investment/consumption plan is adopted until the time of compulsory annuitization, taking into account the possibility of earlier death. The effect of the bequest motive and the desire to buy a higher annuity than the one purchasable at retirement are included in the objective function. The mathematical tools provided by dynamic programming techniques are applied to find closed-form solutions: numerical examples are also presented. In the model, the tradeoff between the different desires of the individual regarding consumption and final annuity can be dealt with by choosing appropriate weights for these factors in the setting of the problem. Conclusions are twofold. First, we find that there is a natural time-varying target for the size of the fund, which acts as a sort of safety level for the needs of the pensioner. Second, the personal preferences of the pensioner can be translated into optimal choices, which in turn affect the distribution of the consumption path and of the final annuity.

In defined contribution pension schemes, the financial risk is borne by the member. Financial risk occurs both during the accumulation phase (investment risk) and at retirement, when the annuity is bought (annuity risk). The annuity risk faced by the member can be reduced through the “income drawdown option”: the retiree is allowed to choose when to convert the final capital into pension within a certain period of time after retirement. In some countries, there is a limiting age when annuitization becomes compulsory (in UK this age is 75). In the interim, the member can withdraw periodic amounts of money to provide for daily life, within certain limits imposed by the scheme’s rules (or by law). In this paper, we investigate the income drawdown option and define a stochastic optimal control problem, looking for optimal investment strategies to be adopted after retirement, when allowing for periodic fixed withdrawals from the fund. The risk attitude of the member is also considered, by changing a parameter in the disutility function chosen. We find that there is a natural target level of the fund, interpretable as a safety level, which can never be exceeded when optimal control is used. Numerical examples are presented in order to analyse various indices — relevant to the pensioner — when the optimal investment allocation is adopted. These indices include, for example, the risk of outliving the assets before annuitization occurs (risk of ruin), the average time of ruin, the probability of reaching a certain pension target (that is greater than or equal to the pension that the member could buy immediately on retirement), the final outcome that can be reached (distribution of annuity that can be bought at limit age), and how the risk attitude of the member affects the key performance measures mentioned above.

Animal movements are frequently circumscribed, especially at the time of breeding, hence it is often true that close relatives are forced into direct competition. Whereas some inhibition of aggression is expected in conflict among relatives, a higher escalation of conflict seems inevitable if the same individuals repeatedly contest resources, and if it pays a loser in one fight to behave as a subordinate in subsequent ones. This higher escalation may be incorporated into game theory by assuming that strangers not only compete for a particular resource, but also for dominant status in future contests with the same opponent. When these two opposing influences of population structure are compared in the Hawk-Dove game and the War of Attrition, it turns out that the ESS is strongly influenced by both. For populations which are highly viscous, as regards both genes and individuals, the evolutionarily stable investment in fighting is particularly sensitive to small changes in either type of viscosity. In competition, close relatives are therefore expected to switch between the behavioural extremes of altruism and selfish aggression after only minor changes of population structure.

The classical condition for regularity of a Markov chain is extended to include semi-Markov chains. In addition, for any given semi-Markov chain, we find Markov chains which exhibit identical regularity properties. This is done either (i) by transforming the state space or, alternatively, (ii) by imposing conditions on the holding-time distributions. Brief consideration is given to the problem of extending the results to processes other than semi-Markov chains.

The ways in which different animal and resource distributions affect contest behaviour and emigration thresholds are discussed, using a model which allows successful individuals to claim exclusive control of resources. Three parameters are distinguished which influence an individual's ability to exploit resources: the expected search time for a resource, its extraction rate, and the frequency of challenges for ownership. Two types of contest are proposed, those in which fights continue until the expected gain from fighting equals that obtained from backing down and searching, and those in which fight time is considered negligible. Four main conclusions are drawn. The emigration threshold should be delayed when challenges are frequent, particularly in conditions favouring lengthy contests, because the expected long-term gain rate is reduced. Resources which can be extracted quickly are valued by how rapidly they can be exploited, but resources requiring a long extraction are valued by how readily they can be utilized under the threat of take-overs. For resources which have an exponential decay in extraction rate, the resource value for a contest will be peaked when plotted against search time or extraction rate; it will decline when challenges are frequent. The length of a fight is related in a simple way to resource value and the expected long-term gain rate.