本文連結當今世界中貧富差距和高齡化兩大社會問題。本研究的主要目的也試圖同時解決這兩個問題。本文包含兩個主題:
首先，在第二章中討論在課徵富人稅下的最優資產配置。本研究探討在牽涉課徵富人稅下的選擇最優投資組合，隨機財富演化過程中以隨機微分方程來捕捉財富的累積，並允許億萬富翁投資策略(a)在行為決策上係風險趨避，如逃稅或避稅 (b)資產暴露在股市的波動中及(c)資產配置和多樣化避險，實證結果證實針對富人課稅會產生逃稅的效果。此外，全球億萬富翁的資產對該國政府稅收沒有顯著影響。模型中以課富人稅制度下之基本假設參數來描述並凸顯最優的資產配置策略，但本文認為在試圖降低貧富不均的情況，目前對富人課稅在實務上的做法是適當的程序，合宜的富人稅制度可以減少課稅對投資行為所造成的扭曲。
其次，在第三章、第四章中探索房屋淨值轉換商品(以下簡稱ERPs)議題。特別對已經退休的財產的所有權人而言，該產品已被推行做為連結住宅鎖定淨值計劃，但是ERPs(亦稱反向抵押貸款)囊括一些風險因子：臨界風險(Crossover Risk)，房價貶值風險和利率風險。在歐式選擇權價格和抵押貸款保險的精算公平價格的基礎上，本文中推導出一個新的近似封閉解，同時涉及布朗運動過程中的假設和反向抵押貸款(reverse mortgages)保險人的再保險政策。模擬方法中考慮住房價格的隨機過程，以歐式選擇權保險定價模型的架構下分析了政府保險的反向抵押貸款，數值結果證實，反向抵押貸款以比例再保險合約對房價格跳躍擴散的過程非常敏感。為面對長者的長壽風險，本文以Black-Scholes模型選擇權權來評價預測未來的死亡率預期損失的價值。據我所知，本文是首篇房屋淨值轉換產品以再保險策略來分散風險的研究。

This dissertation links the two leading social issues in the world today, i.e. wealth inequality and aging societies. The major aim of this study attempts to simultaneously solve these two problems. This dissertation contains two topics.
First, in Chapter 2, I discuss the optimal asset allocation under a system that taxes the rich. A random wealth evolution process capturing wealth accumulation by a stochastic differential equation allows the billionaires to investment strategies (a) behavioral decision-making for risk aversion, such as tax evasion or avoidance, (b) exposing assets to stock market fluctuations, and (c) using asset allocation and diversification to hedge. The empirical results demonstrate that taxation on the rich will generate the tax evasion effect. Furthermore, the net worth of global billionaires is an insignificant influence on the government revenue. The model sheds light on the origin of the optimal asset allocation strategy described by base-case parameters under a taxing-the-rich system. Thus, I consider whether the current practice of taxing the rich is an appropriate procedure when trying to reduce wealth inequality and whether a suitable billionaire tax system could be adapted to reduce the taxation system’s distortion of the investor’s portfolio behavior.
Second, in Chapter 3 and Chapter 4, I explore the Home Equity Release Products (hereafter ERPs) issue. The products have been promoted as a scheme of accessing equity locked up in a residence, particularly after the property owner has retired. However, ERPs constitute some risk factors: crossover risk, housing price depreciation risk and interest rate risk. In this article, I derive a new closed form approximation for European option prices and an actuarially fair price of mortgage insurance based on the Brownian motion process assumption and a reverse mortgage insurer’s risk with a reinsurance policy. The simulation method considers the stochastic processes of housing prices, and I analyze the insurer's risk of government insured reverse mortgages after development of the insurance pricing model that applies the framework of European put options. The numerical results confirm that the reverse mortgages with proportional reinsurance contracts were highly sensitive to the housing price jump- diffusion process. To deal with longevity risk for the elderly, I evaluate the Black-Scholes model option to project future mortality to determine the values of expected losses. To my best knowledge, this paper is the first study to look at the risk diversification of ERPs via the reinsurance strategy.

論文審定書 i
謝辭 ii
摘要 iii
Abstract iv
List of Tables viii
List of Figures ix
Chapter 1 Introduction 1
1.1 Motivation 1
1.2 Overview of the Thesis 3
Chapter 2 Research on the Optimal Asset Allocation as Taxing Billionaires 6
2.1 Introduction 6
2.1.1 Literature Review 8
2.2 Economic Model of Optimal Asset Allocation for Billionaires 11
2.2.1 Optimal Asset Allocation for Billionaires under Tax System 11
2.2.2 Portfolio Strategy Solution 11
2.2.3 Theoretical Model of Wealth Accumulation 11
2.2.4 Solving the Optimal Portfolio Problem 12
2.3 Empirical Strategy and Analytical Framework 15
2.3.1 Methodology 15
2.3.2 Data 17
2.4 Analysis of Results 18
2.4.1 Descriptive Statistics and Data Analysis 18
2.4.2 Correlating Wealth with the Economy 29
2.4.3 Robust Test 32
2.4.4 Simulation of Optimal Strategy: A case of Exponential Utility 33
2.5 Conclusion and Future Research 39
Chapter 3 The Reverse Mortgage Model 43
3.1 An Overview of Housing Equity 43
3.2 Mortgage with Value-Maximizing Equity Policy 44
Chapter 4 Research on the Equity Release Products with Financial Innovation 54
4.1 Introduction 54
4.1.1 Equity Release Products’ Trend in the World and Their Feature 54
4.1.2 Highlight Innovation on Home Equity Release Products 55
4.1.3 Literature Review 58
4.2 The Model for the Risk of RMs 60
4.2.1 Insurer’s Risks 60
4.2.2 Pricing Contracts with Jump Diffusion 60
4.2.3 Pricing ERPs or RMs with Proportional Reinsurance Contracts 63
4.3 The Evaluation of Actuarially Fair Price (AFP) 66
4.4 The Numerical Results 67
4.4.1 The Impact of Option Valuation with Different Jump 68
4.4.2 Implications for Pricing Option Contract with Reinsurance Mechanism 70
4.4.3 The Implications of the Research Results 71
4.5 Conclusion 72
Chapter 5 Discussion and Future Research 74
5.1 Further Measure of Capital Gains Tax Approaches for the Wealth 74
5.2 Innovation on Longevity Risk for Survivor Annuities 74
Bibliography 76
Appendix 82
Appendix A.I Portfolio Strategy Solution 82
Appendix A.II Interpretation Relation to the Exponential Distribution 83
Appendix A.III Exponential Utility Function Case 84
Appendix B Derivation Process 86
Appendix C.I An Overview of Equity Release in United Stated, South Africa, and U.K. 89
Appendix C.II Proof of Proposition 4.1. 90
Appendix C. III Proof of Poisson distribution. 93

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