Optimal risk probability for first passage models in semi-Markov decision processes

This paper studies the risk minimization problem in semi-Markov decision processes with denumerable states. The criterion to be optimized is the risk probability (or risk function) that a first passage time to some target set doesn't exceed a threshold value. We first characterize such risk functions and the corresponding optimal value function, and prove that the optimal value function satisfies the optimality equation by using a successive approximation technique. Then, we present some properties of optimal policies, and further give conditions for the existence of optimal policies. In addition, a value iteration algorithm and a policy improvement method for obtaining respectively the optimal value function and optimal policies are developed. Finally, two examples are given to illustrate the value iteration procedure and essential characterization of the risk function.     

Dynamic pricing and warranty policies for products with fixed lifetime

We consider a repairable product with known market entry and departure times. A warranty policy is offered with product purchase, under which a customer can have a failed item repaired free of charge in the warranty period. It is assumed that customers are heterogeneous in their risk attitudes toward uncertain repair costs incurred after the warranty expires. The objective is to determine a joint dynamic pricing and warranty policy for the lifetime of the product, which maximizes the manufacturer’s expected profit. In the first part of the analysis, we consider a linearly decreasing price function and a constant warranty length. We first study customers’ purchase patterns under several different pricing strategies by the manufacturer and then discuss the optimal pricing and warranty strategy. In the second part, we assume that the warranty length can be altered once during the product lifetime in developing a joint pricing and warranty policy. Numerical studies show that a dynamic warranty policy can significantly outperform a fixed-length warranty policy.     

Importance sampling for integrated market and credit portfolio models

A sophisticated approach for computing the total economic capital needed for various stochastically dependent risk types is the bottom-up approach. In this approach, usually, market and credit risks of financial instruments are modeled simultaneously. As integrating market risk factors into standard credit portfolio models increases the computational burden of calculating risk measures, it is analyzed to which extent importance sampling techniques previously developed either for pure market portfolio models or for pure credit portfolio models can be successfully applied to integrated market and credit portfolio models. Specific problems which arise in this context are discussed. The effectiveness of these techniques is tested by numerical experiments for linear and non-linear portfolios.     

Affordable and adequate annuities with stable payouts: Fantasy or reality?

This paper introduces a class of unit-linked annuities that extends existing annuities by allowing portfolio shocks to be gradually absorbed into the annuity payouts. Consequently, our new class enables insurers to offer an affordable and adequate annuity with a stable payout stream. We show how to price and adequately hedge the annuity payouts in a general financial environment. In particular, our model accounts for various stylized facts of stock returns such as asymmetry and heavy-tailedness. Furthermore, the generality of our framework makes it possible to explore the impact of a parameter misspecification on the annuity price and the hedging performance.     

A class of risk processes with reserve-dependent premium rate: sample path large deviations and importance sampling

Let (X(t)) be a risk process with reserve-dependent premium rate, delayed claims and initial capital u. Consider a class of risk processes {(X ^ε (t)): ε > 0} derived from (X(t)) via scaling in a slow Markov walk sense, and let Ψ_ε(u) be the corresponding ruin probability. In this paper we prove sample path large deviations for (X ε (t)) as ε → 0. As a consequence, we give exact asymptotics for log Ψ_ε(u) and we determine a most likely path leading to ruin. Finally, using importance sampling, we find an asymptotically efficient law for the simulation of Ψ_ε(u). AMS Subject Classifications 60F10, 91B30 This work has been partially supported by Murst Project “Metodi Stocastici in Finanza Matematica”     

Decision making under uncertainty with fuzzy targets

This paper discusses the issue of how to use fuzzy targets in the target-based model for decision making under uncertainty. After introducing a target-based interpretation of the expected value on which it is shown that this model implicitly assumes a neutral behavior on attitude about the target, we examine the issue of using fuzzy targets considering different attitudes about the target selection of the decision maker. We also discuss the problem for situations on which the decision maker’s attitude about target may change according to different states of nature. Especially, it is shown that the target-based approach can provide an unified way for solving the problem of fuzzy decision making with uncertainty about the state of nature and imprecision about payoffs. Several numerical examples are given for illustration of the discussed issues.     

Theory of optimal consumption and portfolio selection under a Capital-at-Risk (CaR) and a Value-at-Risk (VaR) constraint

The solution to the optimal portfolio selection and consumptionrule subject to Capital-at-Risk and Value-at-Risk constraintsis derived via the use of stochastic dynamic programming.     

Multivariate Gaussian criteria in SMAA

We consider stochastic multicriteria decision-making problems with multiple decision makers. In such problems, the uncertainty or inaccuracy of the criteria measurements and the partial or missing preference information can be represented through probability distributions. In many real-life problems the uncertainties of criteria measurements may be dependent. However, it is often difficult to quantify these dependencies. Also, most of the existing methods are unable to handle such dependency information.In this paper, we develop a method for handling dependent uncertainties in stochastic multicriteria group decision-making problems. We measure the criteria, their uncertainties and dependencies using a stochastic simulation model. The model is based on decision variables and stochastic parameters with given distributions. Based on the simulation results, we determine for the criteria measurements a joint probability distribution that quantifies the uncertainties and their dependencies. We then use the SMAA-2 stochastic multicriteria acceptability analysis method for comparing the alternatives based on the criteria distributions. We demonstrate the use of the method in the context of a strategic decision support model for a retailer operating in the liberated European electricity market.     

Asset and liability modelling for participating policies with guarantees

We study the problem of asset and liability management of participating insurance policies with guarantees. We develop a scenario optimization model for integrative asset and liability management, analyze the tradeoffs in structuring such policies, and study alternative choices in funding them. The nonlinearly constrained optimization model can be linearized through closed form solutions of the dynamic equations. Thus large-scale problems are solved with standard methods. We report on an empirical analysis of policies offered by Italian insurers. The optimized model results are in general agreement with current industry practices. However, some inefficiencies are identified and potential improvements are highlighted.     

Applications of a multi-state risk factor/mortality model in life insurance

Mortality rates are known to depend on socio-economic and behavioral risk factors, and actuarial calculations for life insurance policies usually reflect this. It is typically assumed, however, that these risk factors are observed only at policy issue, and the impact of changes that occur later is not considered. In this paper, we present a discrete-time, multi-state model for risk factor changes and mortality. It allows one to more accurately describe mortality dynamics and quantify variability in mortality. This model is extended to reflect health status and then used to analyze the impact of selective lapsation of life insurance policies and to predict mortality under reentry term insurance.