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David Schiess
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PublicationConsumption and protfolio optimisation at the end of the life-cycle( 2008-03-27)Consumption and portfolio optimisation during retirement has not received as much attention in financial research as optimisation prior to retirement. However, retirement planning is becoming more and more relevant for several reasons. The present paper is mainly based on Schiess (2008) and studies a pensioner deriving utility from a stream of consumption or an annuity and from bequeathing wealth to his heirs in a continuous-time framework. The task of finding the pensioner's optimal consumption, asset allocation and annuity decision rule leads to the interesting interplay of optimal control theory, optimal stopping theory and mortality issues or, technically speaking, to a combined optimal stopping and optimal control problem (COSOCP). Stabile (2006) solved this problem in an all-or-nothing framework assuming exponential mortality and power utility functions. In this paper we present the extensions of Schiess (2008) to the model of Stabile (2006):The essential inclusion of a bequest motive, the additional study of the economically interesting range of relative risk aversion levels greater than one and a new solution method for the COSOCP via duality arguments. For identical risk aversion levels Stabile (2006) finds that the pensioner either annuitises immediately or never which means that COSOCP reduces to a trivial or to a pure optimal control problem. In contrast to this, the annuitisation decision rule can become wealth-dependent in our more general model and consequently, a real COSOCP has to be dealt with. The main result is that longevity risk matters very much (quite attractive annuity market) even if we allow for a bequest motive.Type: conference paper
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PublicationConsumption and Portfolio Optimisation at the End of the Life-CycleThe thesis' focus is on the consumption/portfolio optimisation and the optimal annuitisation decision of a pensioner in a continuous time setting. Technically, this involves solving a combined optimal stopping and optimal control problem (COSOCP). The retiree faces the crucial question of how much to consume and how much to invest in the risky asset (financial market risk). This creates the optimal control aspect of the COSOCP. Any prior decisions on annuities and life insurance are taken as given. The second source of uncertainty is the pensioner's longevity risk, which is why we include an annuity market. The pensioner has to find the optimal time to annuitise his wealth. This constitutes the optimal stopping aspect of the COSOCP. Stabile (2006) provides an appropriate model to solve the mentioned COSOCP. Among other things we mainly contribute a new solution method for this COSOCP via duality arguments, the study of the economically interesting range of relative risk aversions greater than one and the essential inclusion of a bequest motive (annuitisation is in conflict with a potential bequest motive). The first part of the thesis lays down the necessary theoretical foundations for the COSOCP. We model the utility, which the pensioner derives from a stream of consumption or an annuity, and define his utility from bequests. Later, we will specify the pensioner's preferences to power utility and subsistence level utility functions. Afterwards, we discuss the three major ingredients for solving the pensioner's COSOCP: Optimal control theory, optimal stopping theory and mortality concepts. The second part of the thesis exploits the theoretical foundations of the first part. We only solve a pure optimal control problem under the Gompertz-Makeham mortality law. We are able to derive interesting comparisons; however, this problem is already quite involved and helps us to understand why we have to employ the less complicated exponential mortality law in a real COSOCP. The exponential mortality law has the great advantage of increased mathematical tractability. We use it to solve two different models. In the first model we impose that the relative risk aversion is the same for all utility functions: Utility from consumption, annuity and bequests. Most often the annuitisation decision is then of the now-or-never type: Depending on the model parameters, annuitisation either occurs immediately or never (reduction to a pure optimal control problem). We solve both cases and show how the annuitisation decision is influenced by the model parameters. Finally, the second model provides an extension to the first one by allowing for a higher relative risk aversion in the post-annuitisation phase. This last model leads to a real COSOCP in most cases. After exploiting some duality arguments, we arrive at a slightly nonlinear ordinary differential equation for the dual value function. While the no-bequest case allows a quasi-analytical solution, the bequest case has to be solved numerically. We give general characteristics of the optimal consumption and investment rule and numerically show how they depend on the parameters. Finally, we simulate the optimal annuitisation rule.Type: doctoral thesis
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PublicationAsset Allocation, Longevity Risk, Annuitisation and Bequests( 2009-04-14)Consumption and portfolio optimisation during retirement has not received as much attention in financial research as optimisation prior to retirement. However, retirement planning is becoming more and more relevant for several reasons. The present paper is mainly based on Schiess (2008) and studies a pensioner deriving utility from a stream of consumption or an annuity and from bequeathing wealth to his heirs in a continuous-time framework. The task of finding the pensioner's optimal consumption, asset allocation and annuity decision rule leads to the interesting interplay of optimal control theory, optimal stopping theory and mortality issues or, technically speaking, to a combined optimal stopping and optimal control problem (COSOCP). Stabile (2006) solved this problem in an all-or-nothing framework assuming exponential mortality and power utility functions. In this paper we present the extensions of Schiess (2008) to the model of Stabile (2006):The essential inclusion of a bequest motive, the additional study of the economically interesting range of relative risk aversion levels greater than one and a new solution method for the COSOCP via duality arguments. For identical risk aversion levels Stabile (2006) finds that the pensioner either annuitises immediately or never which means that COSOCP reduces to a trivial or to a pure optimal control problem. In contrast to this, the annuitisation decision rule can become wealth-dependent in our more general model and consequently, a real COSOCP has to be dealt with. The main result is that longevity risk matters very much (quite attractive annuity market) even if we allow for a bequest motive.Type: presentation
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PublicationAsset Allocation, Longevity Risk, Annuitisation and Bequests( 2008-03-27)Consumption and portfolio optimisation during retirement has not received as much attention in financial research as optimisation prior to retirement. However, retirement planning is becoming more and more relevant for several reasons. The present paper is mainly based on Schiess (2008) and studies a pensioner deriving utility from a stream of consumption or an annuity and from bequeathing wealth to his heirs in a continuous-time framework. The task of finding the pensioner's optimal consumption, asset allocation and annuity decision rule leads to the interesting interplay of optimal control theory, optimal stopping theory and mortality issues or, technically speaking, to a combined optimal stopping and optimal control problem (COSOCP). Stabile (2006) solved this problem in an all-or-nothing framework assuming exponential mortality and power utility functions. In this paper we present the extensions of Schiess (2008) to the model of Stabile (2006):The essential inclusion of a bequest motive, the additional study of the economically interesting range of relative risk aversion levels greater than one and a new solution method for the COSOCP via duality arguments. For identical risk aversion levels Stabile (2006) finds that the pensioner either annuitises immediately or never which means that COSOCP reduces to a trivial or to a pure optimal control problem. In contrast to this, the annuitisation decision rule can become wealth-dependent in our more general model and consequently, a real COSOCP has to be dealt with. The main result is that longevity risk matters very much (quite attractive annuity market) even if we allow for a bequest motive.Type: presentation
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PublicationOptimal Strategies During RetirementThe present paper studies a pensioner deriving utility from a stream of consumption or an annuity and from bequeathing wealth to his heirs in a continuous-time framework. The task of finding the pensioner's optimal consumption, asset allocation and annuity decision rule leads to the interesting interplay of optimal control theory, optimal stopping theory and mortality issues or, technically speaking, to a combined optimal stopping and optimal control problem (COSOCP). Stabile (2006) solved this problem in an all-or-nothing framework assuming exponential mortality and power utility functions. In this paper we extend his model in several dimensions: We contribute the essential inclusion of a bequest motive, we additionally study the economically interesting range of relative risk aversion levels greater than one and we provide a new solution method for the COSOCP via duality arguments. For identical risk aversion levels Stabile (2006) finds that the pensioner either annuitises immediately or never which means that COSOCP reduces to a trivial or to a pure optimal control problem. In contrast to this the annuitisation decision rule can become wealthdependent in our more general model and consequently, a real COSOCP has to be dealt with. The main result is that longevity risk matters very much (quite attractive annuity market) even if we allow for a bequest motive.Type: working paperIssue: Paper No. 70