We determine the optimal allocation of funds between the fixed and variable sub-accounts in a variable annuity with a GMDB (Guaranteed Minimum Death Benefit) clause featuring partial withdrawals by using a utility-based approach. In section two, the Merton method is applied by assuming that individuals allocate funds optimally in order to maximize the expected utility of lifetime consumption. It also reflects bequest motives by including the recipient's utility in terms of the policyholder's guaranteed death benefits. We derive the optimal transfer choice by the insured, and furthermore price the GMDB through maximizing the discounted expected utility of the policyholders and beneficiaries by investing dynamically in the fixed account and the variable fund and withdrawing optimally. In section three, we add fixed and stochastic income to the model and find that both human capital and the GMDB will influence the insured's allocation and withdrawal decisions. Section four explores the GMDB effects if there is also a term life policy available in the market. Our work suggests that if term life insurance is available and is continuously adjustable, fairly priced GMDBs may not be useful investments and the existence of GMDBs does not affect term life policy demand significantly.

The purpose of this study is to investigate the pricing of variable annuity embedded derivatives in a Lévy process setting. This is one of the practical issues that continues to face life insurers in the management of derivatives embedded within these products. It also addresses how such providers can protect themselves against adverse scenarios through a hedging framework built from the pricing framework. The aim is to comparatively consider the price differentials of a life insurer that prices its variable annuity guarantees under the more actuarially accepted regime-switching framework versus the use of a Lévy framework. The framework should address the inadequacies of conventional deterministic pricing approaches used by life insurers given the increasing complexity of the option-like products sold. The study applies finance models in the insurance context given the similarities in payoff structure of the products offered while taking into account the differences that may exist. The underlying Lévy process used in this study is the Variance-Gamma (VG) process. This process is useful in option pricing given its ability to model higher moments, skewness and kurtosis, and also incorporate stochastic volatility. The research results compare well with the regime-switching framework besides the added merit in the use of a more refined model for the underlying that captures most of the observed market dynamics. / Dissertation (MSc)--University of Pretoria, 2014. / tm2015 / Mathematics and Applied Mathematics / MSc / Unrestricted

Motivated by the Guaranteed Minimum Death Benefits (GMDB) in variable annuities, we are interested in valuing equity-linked options whose expiry date is the time of the death of the policyholder. Because the time-until-death distribution can be approximated by linear combinations of exponential distributions or mixtures of Erlang distributions, the analysis can be reduced to the case where the time-until-death distribution is exponential or Erlang. We present two probability methods to price American options with an exponential expiry date. Both methods give the same results. An American option with Erlang expiry date can be seen as an extension of the exponential expiry date case. We calculate its price as the sum of the price of the corresponding European option and the early exercise premium. Because the optimal exercise boundary takes the form of a staircase, the pricing formula is a triple sum. We determine the optimal exercise boundary recursively by imposing the “smooth pasting” condition. The examples of the put option, the exchange option, and the maximum option are provided to illustrate how the methods work. Another issue related to variable annuities is the surrender behavior of the policyholders. To model this behavior, we suggest using barrier options. We generalize the reflection principle and use it to derive explicit formulas for outside barrier options, double barrier options with constant barriers, and double barrier options with time varying exponential barriers. Finally, we provide a method to approximate the distribution of the time-until-death random variable by combinations of exponential distributions or mixtures of Erlang distributions. Compared to directly fitting the distributions, my method has two advantages: 1) It is more robust to the initial guess. 2) It is more likely to obtain the global minimizer.

A general methodology is described in which policyholder behaviour is decoupled from the pricing of a variable annuity based on the cost of hedging it, yielding two sequences of weakly coupled systems of partial differential equations (PDEs): the pricing and utility systems. The utility systems are used to generate policyholder withdrawal behaviour, which is in turn fed into the pricing systems as a means to determine the cost of hedging the contract. This approach allows us to incorporate the effects of utility-based pricing and factors such as taxation. As a case study, we consider the Guaranteed Lifelong Withdrawal and Death Benefits (GLWDB) contract. The pricing and utility systems for the GLWDB are derived under the assumption that the underlying asset follows a Markov regime-switching process. An implicit PDE method is used to solve both systems in tandem. We show that for a large class of utility functions, the two systems preserve homogeneity, allowing us to decrease the dimensionality of solutions. We also show that the associated control for the GLWDB is bang-bang, under which the work required to compute the optimal strategy is significantly reduced. We extend this result to provide the reader with sufficient conditions for a bang-bang control for a general variable annuity with a countable number of events (e.g. discontinuous withdrawals). Homogeneity and bang-bangness yield significant reductions in complexity and allow us to rapidly generate numerical solutions. Results are presented which demonstrate the sensitivity of the hedging expense to various parameters. The costly nature of the death benefit is documented. It is also shown that for a typical contract, the fee required to fund the cost of hedging calculated under the assumption that the policyholder withdraws at the contract rate is an appropriate approximation to the fee calculated assuming optimal consumption.

A general methodology is described in which policyholder behaviour is decoupled from the pricing of a variable annuity based on the cost of hedging it, yielding two sequences of weakly coupled systems of partial differential equations (PDEs): the pricing and utility systems. The utility systems are used to generate policyholder withdrawal behaviour, which is in turn fed into the pricing systems as a means to determine the cost of hedging the contract. This approach allows us to incorporate the effects of utility-based pricing and factors such as taxation. As a case study, we consider the Guaranteed Lifelong Withdrawal and Death Benefits (GLWDB) contract. The pricing and utility systems for the GLWDB are derived under the assumption that the underlying asset follows a Markov regime-switching process. An implicit PDE method is used to solve both systems in tandem. We show that for a large class of utility functions, the two systems preserve homogeneity, allowing us to decrease the dimensionality of solutions. We also show that the associated control for the GLWDB is bang-bang, under which the work required to compute the optimal strategy is significantly reduced. We extend this result to provide the reader with sufficient conditions for a bang-bang control for a general variable annuity with a countable number of events (e.g. discontinuous withdrawals). Homogeneity and bang-bangness yield significant reductions in complexity and allow us to rapidly generate numerical solutions. Results are presented which demonstrate the sensitivity of the hedging expense to various parameters. The costly nature of the death benefit is documented. It is also shown that for a typical contract, the fee required to fund the cost of hedging calculated under the assumption that the policyholder withdraws at the contract rate is an appropriate approximation to the fee calculated assuming optimal consumption.

A guaranteed minimum income benefit (GMIB) is a long-dated option that can be embedded in a deferred variable annuity. The GMIB is attractive because, for policyholders who plan to annuitize, it offers protection against poor market performance during the accumulation phase, and adverse interest rate experience at annuitization. The GMIB also provides an upside equity guarantee that resembles the benefit provided by a lookback option. We price the GMIB, and determine the fair fee rate that should be charged. Due to the long dated nature of the option, conventional hedging methods, such as delta hedging, will only be partially successful. Therefore, we are motivated to find alternative hedging methods which are practicable for long-dated options. First, we measure the effectiveness of static hedging strategies for the GMIB. Static hedging portfolios are constructed based on minimizing the Conditional Tail Expectation of the hedging loss distribution, or minimizing the mean squared hedging loss. Next, we measure the performance of semi-static hedging strategies for the GMIB. We present a practical method for testing semi-static strategies applied to long term options, which employs nested Monte Carlo simulations and standard optimization methods. The semi-static strategies involve periodically rebalancing the hedging portfolio at certain time intervals during the accumulation phase, such that, at the option maturity date, the hedging portfolio payoff is equal to or exceeds the option value, subject to an acceptable level of risk. While we focus on the GMIB as a case study, the methods we utilize are extendable to other types of long-dated options with similar features.

A guaranteed minimum income benefit (GMIB) is a long-dated option that can be embedded in a deferred variable annuity. The GMIB is attractive because, for policyholders who plan to annuitize, it offers protection against poor market performance during the accumulation phase, and adverse interest rate experience at annuitization. The GMIB also provides an upside equity guarantee that resembles the benefit provided by a lookback option. We price the GMIB, and determine the fair fee rate that should be charged. Due to the long dated nature of the option, conventional hedging methods, such as delta hedging, will only be partially successful. Therefore, we are motivated to find alternative hedging methods which are practicable for long-dated options. First, we measure the effectiveness of static hedging strategies for the GMIB. Static hedging portfolios are constructed based on minimizing the Conditional Tail Expectation of the hedging loss distribution, or minimizing the mean squared hedging loss. Next, we measure the performance of semi-static hedging strategies for the GMIB. We present a practical method for testing semi-static strategies applied to long term options, which employs nested Monte Carlo simulations and standard optimization methods. The semi-static strategies involve periodically rebalancing the hedging portfolio at certain time intervals during the accumulation phase, such that, at the option maturity date, the hedging portfolio payoff is equal to or exceeds the option value, subject to an acceptable level of risk. While we focus on the GMIB as a case study, the methods we utilize are extendable to other types of long-dated options with similar features.

台灣人口結構老化的趨勢，老人漸漸變成另一種社會不受重視的邊緣人。近年來，由於科技神速發展，職場人員替換週期愈來愈短，許多仍健壯的授薪階級，在智慧與經驗方面雖臻成熟，卻不得不被迫從職場上退下來。最近兩三年來，全球資本主義整體化的併購行為以及中國大陸整體市場崛起連帶引響台灣產業經濟蕭條，更是快速增加了淘汰職場的人數，對於大部分需要依賴勞保及公司退休金制度的勞工階層而言，退休金嚴重不足的情況，更是令人焦慮沮喪。 主管機關為因應社會變遷趨勢，於2000年陸續起開放利率變動型年金與變額年金保險商品於市場銷售，為老年化人口之財務規劃打開另一扇門，年金商品以及投資型保險之觀念架構，為國內消費者提供了多元化保險商品的選擇，化解保險公司利差損的營運壓力，提供創新營運的契機，同時也為壽險從業人員開闢了專業的「全方位金融理財顧問」生涯規劃。 由於投資型保險商品具備保戶可自行執行帳戶價值投資策略之特性，對消費者而言帳戶價值相對於傳統壽險有更大的想像空間，年金型商品由初期著重於銀行定存利率連結概念之利率變動型年金發展至變額年金，兩類商品不約而同的取代了傳統壽險及儲蓄險，對於壽險公司長期經營而言也具有消弭了利差損的風險之營運價值，使得短短的6年之間(2000年-2006年) 傳統壽險，意外險，醫療險之首年度保費佔有率逐年降至40%(中華民國人壽保險公會保費速報 2006.07)。 而不論是投資型保險或年金型保險，在市場之行銷活動均強調資產累積與退休規劃，而在台灣市場兩類主力銷售商品亦存在重疊特質，如變動不保證利率之帳戶價值，帳戶價值提領之彈性，長期運用的理財工具，可單筆大額資金購買亦可分期繳納等特質，使得商品設計多樣化，行政費用收取方式各有不同，行銷訴求則是推陳出新，對消費者而言更不易辨析商品之費用、價格、功能之間所存在價值差異，本研究希望對變額壽險與變額年金兩種商品從消費者需求、商品特性價格與費用等三方面分析此兩種商品在退休規劃之優劣比較。

為保險業跨業經營預先佈局、增加保單持有人之投資選擇權及因應市場利率之低糜，我國擬於近日推出投資型保險，然而投資型保險基於其特有之性質，其監理自應與一般傳統型保險有所不同。本文先就投資型保險作一定義，並論述其性質及特色，次敘述其歷史發展沿革，再予以介紹其主要之種類。而關於投資型保險監理部分，一般來說，保險監理可分為財務監理及業務監理二部分，本文即依循此二面向分別作一探討。就財務監理部分而言，分就分離帳戶之規範、資金運用規範、責任準備金提存之規範、解約金之規範及稅賦規範五個議題作分析探討，再行提出建議。另一方面，就業務監理部分而言，則分就經營資格規範、資訊揭露規範、商品規範及銷售規範四個議題作分析探討，再行提出建議；最後總結全文以提出結論並針對我國投資型保險之監理提出建議。 / For facilitating of cross-business operation of the insurance industry, increasing policyholders’ choices in financial services, and shifting attach of interest risk, investment linked insurance products will introduced into the Taiwan insurance market soon. Based on it’s special features, the regulation of investment linked insurance should be different from the regulation of traditional life insurance. This paper defines the scope of investment linked insurance and discourses it’s nature and special features in the first place, and then describes it’s historical evolution and main types. In terms of insurance regulation, it is generally divided into two broad categories: i.e. financial regulation and market conduct regulation. This paper studies on the issues of regulation of investment linked insurance in such an approach. With regard to financial regulation of investment linked insurance, five aspects of regulation of investment linked insurance are examined: separate accounts, investments, liability reserves, surrender cash value and tax. With regard to market conduct regulation, this paper includes four aspects: operational qualification, information discourse, products and marketing. Finally the author submits his conclusion and recommendations to the regulatory authority to enhance the framework of regulation of investment linked insurance in Taiwan.