Optimal asset allocation for participating contracts with mortality risk under minimum guarantee

Abstract

We investigate an optimal investment problem of participating insurance contracts with mortality risk under minimum guarantee. The insurer aims to maximize the expected utility of the terminal payoff. Due to its piecewise payoff structure, this optimization problem is a non-concave utility maximization problem. We adopt a concavification technique and a Lagrange dual method to solve the problem and derive the representations of the optimal wealth process and trading strategies. We also carry out some numerical analysis to show how the portfolio insurance constraint impacts the optimal terminal wealth.     

Optimal investment-consumption and life insurance with capital constraints

Abstract

The aim of this paper is to solve an optimal investment, consumption and life insurance problem when the investor is restricted to capital guarantee. We consider an incomplete market described by a jump-diffusion model with stochastic volatility. Using the martingale approach, we prove the existence of the optimal strategy and the optimal martingale measure and we obtain the explicit solutions for the power utility functions.     

Best quadratic predictions in finite populations

For a finite population and the resulting linear model Y=Xβ+e, the problem of the optimal invariant quadratic predictors including optimal invariant quadratic unbiased predictor and optimal invariant quadratic (potentially) biased predictor for the population quadratic quantities, f(H)=Y′HY , is of interest and has been previously considered in the literature for the case of HX=0. However, the special case does not contain all of situations at all. So, predicting f(H) in general situations may be of particular interest. In this paper, we make an effort to investigate how to offer a good predictor for f(H), not restricted yet to the mentioned case. Permutation matrix techniques play an important role in handling the process. The expected predictors are finally derived. In addition, we mention that the resulting predictors can be viewed as acceptable in all situations.     

Asymptotically Optimal Design Points for Rejection Algorithms

ABSTRACT

Very fast automatic rejection algorithms were developed recently which allow us to generate random variates from large classes of unimodal distributions. They require the choice of several design points which decompose the domain of the distribution into small sub-intervals. The optimal choice of these points is an important but unsolved problem. Therefore, we present an approach that allows us to characterize optimal design points in the asymptotic case (when their number tends to infinity) under mild regularity conditions. We describe a short algorithm to calculate these asymptotically optimal points in practice. Numerical experiments indicate that they are very close to optimal even when only six or seven design points are calculated.     

Generalized value at risk forecasting

Abstract

In this paper, using estimating function approach, a new optimal volatility estimator is introduced and based on the recursive form of the estimator a data-driven generalized EWMA model for value at risk (VaR) forecast is proposed. An appropriate data-driven model for volatility is identified by the relationship between absolute deviation and standard deviation for symmetric distributions with finite variance. It is shown that the asymptotic variance of the proposed volatility estimator is smaller than that of conventional estimators and is more appropriate for financial data with larger kurtosis. For IBM, Microsoft, Apple stocks and SP 500 index the proposed method is used to identify the model, estimate the volatility, and obtain minimum mean square error(MMSE) forecasts of VaR.     

Optimal control strategies for dividend payments and capital injections in compound Markov binomial risk model with penalties for deficits

We consider the compound Markov binomial risk model. The company controls the amount of dividends paid to the shareholders as well as the capital injections in order to maximize the cumulative expected discounted dividends minus the discounted capital injections and the discounted penalties for deficits prior to ruin. We show that the optimal value function is the unique solution of an HJB equation, and the optimal control strategy is a two-barriers strategy given the current state of the Markov chain. We obtain some properties of the optimal strategy and the optimal condition for ruining the company. We offer a high-efficiency algorithm for obtaining the optimal strategy and the optimal value function. In addition, we also discuss the optimal control problem under a restriction of bounded dividend rates. Numerical results are provided to illustrate the algorithm and the impact of the penalties.     

EUROPEAN OPTION PRICING WITH GENERAL TRANSACTION COSTS AND SHORT-SELLING CONSTRAINTS

In this paper, we study the problem of European Option Pricing in a market with short-selling constraints and transaction costs having a very general form. We consider two types of proportional costs and a strictly positive fixed cost. We study the problem within the framework of the theory of stochastic impulse control. We show that determining the price of a European option involves calculating the value functions of two stochastic impulse control problems. We obtain explicit expressions for the quasi-variational inequalities satisfied by the value functions and derive the solution in the case where the parameters of the price processes are constants and the investor's utility function is linear. We use this result to obtain a price for a call option on the stock and prove that this price is a nontrivial lower bound on the hedging price of the call option in the presence of general transaction costs and short-selling constraints. We then consider the situation where the investor's utility function has a general form and characterize the value function as the pointwise limit of an increasing sequence of solutions to associated optimal stopping problems. We thereby devise a numerical procedure to calculate the option price in this general setting and implement the procedure to calculate the option price for the class of exponential utility functions. Finally, we carry out a qualitative investigation of the option prices for exponential and linear-power utility functions.     

Planning step-stress test under Type-I censoring for the exponential case

We consider in this work a k-level step-stress accelerated life-test (ALT) experiment with unequal duration steps τ=(τ₁, …, τ_k). Censoring is allowed only at the change-stress point in the final stage. An exponential failure time distribution with mean life that is a log-linear function of stress, along with a cumulative exposure model, is considered as the working model. The problem of choosing the optimal τ is addressed using the variance-optimality criterion. Under this setting, we then show that the optimal k-level step-stress ALT model with unequal duration steps reduces just to a 2-level step-stress ALT model.     

On pre-test estimation of parametric functions in the general Gauss-Markov model with quadratic risk

The approach to preliminary test estimation based on comparing the weighted quadratic risk function of two competing estimators of β under the linear regression model {y,Xβ, σ² I} is extended to the case when a given vector of parametric functions κ=Kβ is to be estimated under the general Gauss-Markov model.     

Implementation of ridge analysis on quadratic response surfaces

A computer program that performs ridge analysis on quadratic response surfaces is presented in this paper, the primary goal of which is to seek the estimated optimum operating conditions inside a spherical region of experimentation during the stage of process optimization. The computational algorithm is developed based upon the trust-region methods in nonlinear optimization and guarantees the resulting operating conditions to be globally optimal without any priori assumption on the structure of response functions. Under a particular condition termed the "hard case" arising from the trust region literature, the conventional ridge analysis procedure fails to provide a set of acceptable optimum operating settings, yet the proposed algorithm has the capability of locating a pair of non-unique global solutions achieved on an identical estimated response value. Two illustrative examples taken from the response surface methodology (RSM) literature are given to demonstrate the effectiveness and efficiency of the method addressed in the paper.     

PITMAN-CLOSENESS AS A MEASURE TO EVALUATE THE QUALITY OF FORECASTS

ABSTRACT

We consider Pitman-closeness to evaluate the performance of univariate and multivariate forecasting methods. Optimal weights for the combination of forecasts are calculated with respect to this criterion. These weights depend on the assumption of the distribution of the individual forecasts errors. In the normal case they are identical with the optimal weights with respect to the MSE-criterion (univariate case) and with the optimal weights with respect to the MMSE-criterion (multivariate case). Further, we present a simple example to show how the different combination techniques perform. There we can see how much the optimal multivariate combination can outperform different other combinations. In practice, we can find multivariate forecasts e.g., in econometrics. There is often the situation that forecast institutes estimate several economic variables.     

THE SIMPLE LINEAR CALIBRATION PROBLEM AS AN OPTIMAL EXPERIMENTAL DESIGN

ABSTRACT

The aim of this paper is to consider the linear calibration problem through an optimal design approach, to evaluate the approximate variance of the calibrating value and provide approximate confidence intervals. The sequential approach through Stochastic Approximation is also applied to obtain a D-optimal design is considered.     

Designs for weighted least squares regression, with estimated weights

We study designs, optimal up to and including terms that are O(n ^?1), for weighted least squares regression, when the weights are intended to be inversely proportional to the variances but are estimated with random error. We take a finite, but arbitrarily large, design space from which the support points are to be chosen, and obtain the optimal proportions of observations to be assigned to each point. Specific examples of D- and I-optimal design for polynomial responses are studied. In some cases the same designs that are optimal under homoscedasticity remain so for a range of variance functions; in others there tend to be more support points than are required in the homoscedastic case. We also exhibit minimax designs, that minimize the maximum, over finite classes of variance functions, value of the loss. These also tend to have more support points, often resulting from the breaking down of replicates into clusters.     

Risk minimization for an insurer with investment and reinsurance via g-expectation

Abstract

This paper is devoted to the study of a risk-based optimal investment and proportional reinsurance problem. The surplus process of the insurer and the risky asset process in the financial market are assumed to be general jump-diffusion processes. We use a convex risk measure generated by g-expectation to describe the risk of the terminal wealth with investment and reinsurance. Under the aim of minimizing the risk, the problem is solved by using techniques of stochastic maximum principles. Two interesting special cases are studied and the explicit expressions for optimal strategies and corresponding minimal risks are derived.     

Admissible and optimal sampling strategy for estimating finite population mean in randomized response surveys with multiple responses

ABSTRACT

We consider the problem of estimation of a finite population mean (or proportion) related to a sensitive character under a randomized response model when independent responses are obtained from each sampled individual as many times as he/she is selected in the sample and prove the admissibility of a sampling strategy in a class of comparable linear unbiased strategies. We prove that the admissible strategy is also optimal in this class under a super-population model.     

Outcome-adaptive allocation with natural lead-in for three-group trials with binary outcomes

ABSTRACT

Just as Bayes extensions of the frequentist optimal allocation design have been developed for the two-group case, we provide a Bayes extension of optimal allocation in the three-group case. We use the optimal allocations derived by Jeon and Hu [Optimal adaptive designs for binary response trials with three treatments. Statist Biopharm Res. 2010;2(3):310–318] and estimate success probabilities for each treatment arm using a Bayes estimator. We also introduce a natural lead-in design that allows adaptation to begin as early in the trial as possible. Simulation studies show that the Bayesian adaptive designs simultaneously increase the power and expected number of successfully treated patients compared to the balanced design. And compared to the standard adaptive design, the natural lead-in design introduced in this study produces a higher expected number of successes whilst preserving power.     

Asymptotic results for model robust regression

Since the mid 1980's many statisticians have studied methods for combining parametric and nonparametric models to improve the quality of fits in a regression problem. Notably Einsporn (1987) proposed the Model Robust Regression 1 estimate (MRRl) in which the parametric function, f, and the nonparametric functiong were combined in a straightforward fashion via the use of a mixing parameter, λ This technique was studied extensively atsmall samples and was shown to be quite effective at modeling various unusual functions. In this paper we have asymptotic results for the MRRl estimate in the case where λ is theoretically optimal, is asymptotically optimal and data driven, and is chosen with the PRESS statistic (Allen, 1971) We demonstrate that the MRRl estimate with λchosen by the PRESS statistic is slightly inferior asymptotically to the other two estimates, but, nevertheless possesses positive asymptotic qualities.     

Local asymptotic normality for long-memory process with strong mixing noises

Abstract

We will establish the local asymptotic normality (LAN) for fractional autoregressive long memory model in the case of strong mixing noises. This opens the way in future work to construct an adaptive estimator and construct optimal tests for the parameters. To check the feasibility and validity of our theoretical results a simulations study is considered.     

Optimal investment strategy for a DC pension plan with mispricing under the Heston model

Abstract

In this article, we consider the optimal investment problem for a defined contribution (DC) pension plan with mispricing. We assume that the pension funds are allowed to invest in a risk-free asset, a market index, and a risky asset with mispricing, i.e. the prices are inconsistent in different financial markets. Assuming that the price process of the risky asset follows the Heston model, the manager of the pension fund aims to maximize the expected utility for the power utility function of terminal wealth. By applying stochastic control theory, we establish the corresponding Hamilton-Jacobi-Bellman (HJB) equation. And the optimal investment strategy is obtained for the power utility function explicitly. Finally, numerical examples are provided to analyze effects of parameters on the optimal strategy.     

Sub-optimal investment for insurers

Abstract

We consider the investment problem for a non-life insurance company seeking to minimize the ruin probability. Its reserve is described by a perturbed risk process possibly correlated with the financial market. Assuming exponential claim size, the Hamilton-Jacobi-Bellman equation reduces to a first order nonlinear ordinary differential equation, which seems hard to solve explicitly. We study the qualitative behavior of its solution and determine the Cramér-Lundberg approximation. Moreover, our approach enables to find very naturally that the optimal investment strategy is not constant. Then, we analyze how much the company looses by adopting sub-optimal constant (amount) investment strategies.