本論文包含兩個研究主題。第一個主題旨在探討投資人在風險值限制下，如何在整個投資期間追求最適的動態資產配置。在投資期間中可以多次的交易情況下，本文所考慮的計算風險值期間是整個的投資期間。除外，不同於過去研究，本文並沒有作完全市場的假設。由於此問題不能利用標準的動態規劃或Martingale方法來求解；因此，本文提出一個數值演算法來解決這個困難的問題。
第二個主題探討在風險值限制下，設計出最適保險單型式，以滿足被保險人的預期期末財富最大化。本文發現，最適保險單可以藉由三種選擇權來複製—分別是買較小執行價的買權，賣較大執行價的買權，和賣現金或無買權（cash-or-nothing call option）。最後，本文畫出所有型式保單的效率前緣，圖形顯示最適保險單最有效率。

This dissertation includes two topics. The first topic focuses on the problem of investor optimization of dynamic asset allocation to maximize expected utility under the value at risk (VaR) constraint. Different to previous researches, this study considers a common realistic case where the VaR horizon is equal to the whole investment horizon without a complete market constraint. Since the problem cannot be solved using the standard dynamic programming method or the martingale method, this study particularly provides an algorithm to solve this difficult problem. Similar to the mean-variance frontier suggested by Markowitz (1952), this study draws the frontiers of dynamic and static asset allocations under the VaR constraint. The analytical results clearly show that the dynamic asset allocations are more efficient than the static asset allocations.
The second topic designs an optimal insurance policy form endogenously, assuming the objective of the insured is to maximize expected final wealth under the VaR constraint. The optimal insurance policy can be replicated using three options, including a long call option with a small strike price, a short call option with a large strike price, and a short cash-or-nothing call option. Moreover, expected wealth is increasing and concave in VaR and in significance level. Finally, Mean-VaR Frontiers are drawn, and reveal that the optimal insurance is more efficient than alternative insurance forms.

Chapter 1: Introduction 1
1.1. Motivation and contribution in the first topic 2
1.2. Motivation and contribution in the second topic 5
Chapter 2: Introduction to VaR and literature review 8
2.1. VaR Definition 8
2.2. Different approaches to measure VaR 10
2.3. Literature related to first topic 14
2.4. Literature related to second topic 17
Chapter 3: Optimal dynamic asset allocation under value
at risk 19
3.1. The model 19
3.2. Derivation of the optimal dynamic asset allocation 24
3.3. Simulation and numerical analysis 29
Chapter 4: Optimal insurance design under mean-VaR
framework 39
4.1. The model 39
4.2. Insurance policy under uniform distribution 50
4.3. Insurance policy under lognormal distribution 53
4.4. Numerical 56
Chapter 5: Conclusions 65
Appendix 68
Reference 73

Acerbi, Carlo and Dirk Tasche, 2002, On the Coherence of Expected Shortfall, Journal of Banking and Finance 26, 1487-1503.
Ahn, Dong-Hyun, Jacob Boudoukh, Matthew Richardson and Robert F. Whitelaw, 1999, Optimal Risk Management Using Options, Journal of Finance 54, 359-375.
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