How the co-integration analysis can help in mortality forecasting.

series of age-specific log-mortality rates as a sum of an independent of time age-specific component and a bilinear term in which one of the component is a time-varying parameter reflecting general change in mortality and the second one is an age-specific factor. Such a rigid model structure implies that on average the mortality improvements for different age groups should be proportional, regardless the calendar period. In this paper we investigate whether the mortality data for England and Wales follow this property or not. We perform the analysis by applying the concept of the Engle and Granger co-integration to the time series of log-mortality rates. We investigate the goodness of fit of the predictions to the historical data. We find that a lack of co-integration indeed can cause some problems in performance of the model. In the last section we propose several opportunities to omit the pitfalls.Data; Forecasting; Goodness of fit; Model; Opportunities; Performance; Predictions; Problems; Structure; The Lee-Carter model; Time; Time series; Time series analysis;

On the distribution of life annuities with stochastic interest rates.

In the traditional approach to life contingencies only decrements are assumed to be stochastic. In this contribution we consider the distribution of a life annuity (and a portfolio of life annuities) when also the stochastic nature of interest rates is taken into account. Although the literature concerning this topic is already quite rich, the authors usually restrict themselves to the computation of the first two or three moments. However, if one wants to determine e.g. capital requirements using more sofisticated risk measures like Value-at-Risk or Tail Value-at-Risk, more detailed knowledge about underlying distributions is required. For this purpose, we propose to use the theory of comonotonic risks developed in Dhaene et al. (2002a and 2002b), which has to be slightly adjusted to the case of scalar products. This methodology allows to obtain reliable approximations of the underlying distribution functions, in particular very accurate estimates of upper quantiles and stop-loss premiums. Several numerical illustrations confirm the very high accuracy of the methodology.Comonotonicity; Life annuity; Stochastic interest rates; Stop-loss premium;

Comparing approximations for risk measures of sums of non-independent lognormal random variables.

In this paper, we consider different approximations for computing the distribution function or risk measures related to a sum of non-independent lognormal random variables. Approximations for such sums, based on the concept of comonotonicity, have been proposed in Dhaene et al. (2002a,b). These approximations will be compared with two well-known moment matching approximations: the lognormal and the reciprocal Gamma approximation. We find that for a wide range of parameter values the comonotonic lower bound approximation outperforms the two classical approximations.Approximation; Choice; Comonotonicity; Criteria; Decision; Distribution; Dual theory; Lognormal; Market; Moment matching; Optimal; Order; Portfolio; Problems; Random variables; Reciprocal gamma; Research; Risk; Risk measure; Selection; Simulation; Theory; Time; Value; Variables;

Closed-form approximations for constant continuous annuities.

Abstract: For a series of cash flows, its stochastically discounted or compounded value is often a key quantity of interest in finance and actuarial science. Unfortunately, even for most realistic rate of return models, it may be too difficult to obtain analytic expressions for the risk measures involving this discounted sum. Some recent research has demonstrated that in the case where the return process follows a Brownian motion, the so-called comonotonic approximations usually provide excellent and robust estimates of risk measures associated with discounted sums of cash flows involving log-normal returns. In this paper, we derive analytic approximations for risk measures in case one considers the continuous counterpart of a discounted sum of log-normal returns. Although one may consider the discrete sums as providing a more realistic situation than its continuous counterpart, considering in this case, the continuous setting leads to more tractable explicit formulas and may therefore provide further insight necessary to expand the theory and to exploit new ideas for later developments. Moreover, the closed-form approximations we derive in this continuous set-up can then be compared more effectively with some exact results, thereby facilitating a discussion about the accuracy of the approximations. Indeed, in the discrete setting, one always must compare approximations with results from simulation procedures which always give rise to room of debate. Our numerical comparisons reveal that the comonotonic 'maximal variance' lower bound approximation provides an excellent fit for several risk measures associated with integrals involving log-normal returns. Similar results as we derive here for continuous annuities can also be obtained in case of continuously compounding which therefore opens a roadmap for deriving closed-form approximations for the prices of Asian options. Future research will also focus on optimal portfolio slection problems.Approximation; Choice; Comonotonicity; Criteria; Decision; Distribution;

Bounds for stop-loss premiums of stochastic sums (with applications to life contingencies).

In this paper we present in a general setting lower and upper bounds for the stop-loss premium of a (stochastic) sum of dependent random variables. Therefore, use is made of the methodology of comonotonic variables and the convex ordering of risks, introduced by Kaas et al. (2000) and Dhaene et al. (2002a, 2002b), combined with actuarial conditioning. The lower bound approximates very accurate the real value of the stop-loss premium. However, the comonotonic upper bounds perform rather badly for some retentions. Therefore, we construct sharper upper bounds based upon the traditional comonotonic bounds. Making use of the ideas of Rogers and Shi (1995), the first upper bound is obtained as the comonotonic lower bound plus an error term. Next this bound is refined by making the error term dependent on the retention in the stop-loss premium. Further, we study the case that the stop-loss premium can be decomposed into two parts. One part which can be evaluated exactly and another part to which comonotonic bounds are applied. As an application we study the bounds for the stop-loss premium of a random variable representing the stochastically discounted value of a series of cash flows with a fixed and stochastic time horizon. The paper ends with numerical examples illustrating the presented approximations. We apply the proposed bounds for life annuities, using Makeham's law, when also the stochastic nature of interest rates is taken into account by means of a Brownian motion.Comonotonicity; Life annuity; Stochastic time horizon; Stop-loss premium;

Visual monitoring of water deficit stress using infra-red thermography in wheat

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