近幾年，風險管理受到大家廣為重視，Value-at-Risk (VaR)則是最常用來衡量風險的工具。Basak and Shapiro (2001)是首位將涉險值（VaR）的限制式納入效用函數內，再極大化投資人之效用函數而求出最適資產配置。依據他們的方法，本文的第一部分（見第二章）探討當資產報酬分配呈左偏和肥尾時，對風險管理者資產配置之影響。許多實證研究顯示資產報酬分配呈左偏和肥尾。本文採用Gram-Charlier expansion近似資產報酬分配，探討當資產報酬分配在非常態分配下，其資產配置的變化。對風險管理者而言，最重要的工作就是準確預測損失與發生損失的機率。瞭解資產報酬的型態將有助於準確的預測損失，我們無法降低損失，但可以降低發生損失的機率，本文建議可以降低 值（期末財富損失大於VaR之機率）來達成，而降低 值會使期末財富在好的狀態與壞的狀態的財富稍減。利率是影響使用金融工具的主要因素，本文的第二部分（見第三章）探討VaR風險管理者當考慮利率風險時如何配置其資產，本文採用Vasicek-type模型描述隨機利率，探討在隨機利率的情況下，財富配置於現金、股票與債券之比例。本文將這些參數以數值代入，分析VaR風險管理者期末財富的分配情況以及期中現金、股票與債券之配置情形。本文的第三部分（見第四章）探討VaR風險管理者當考慮利率與通膨風險時如何配置其資產。本文採用correlated Ornstein-Uhlenbeck過程描述隨機實質利率與通膨率，探討當考慮利率與通膨風險的情況下，VaR風險管理者財富配置於現金、股票與債券之比例。對風險管理者而言，最重要的工作就是準確預測期末財富與損失。研究發現忽略通膨風險將使風險管理者嚴重低估期末財富與損失。 / Risk management has received much attention in the last few years. Value-at-Risk (VaR) is widely used by corporate treasurers, fund managers and financial institution (Hull, 2000). A vast amount of literature considered a simple one-period asset allocation problem under VaR constraint. Furthermore, the aggregation of single-period optimal decisions across periods might not be optimal for multi-period as a whole. Basak and Shapiro (2001) were the first to address VaR-related issue in a dynamic general equilibrium setting. This dissertation builds upon the work of Basak and Shapiro (2001) to discuss three issues about dynamic asset allocation. The first topic focuses on how deviations from normality affect asset choices made by risk managers. This study utilizes the Gram-Charlier expansion to approximate asset returns with negatively skewed and excess kurtosis. This work examines how negatively skewed and excess kurtosis affects asset allocations when investors manage market-risk exposure using Value-at-Risk-based risk management (VaR-RM). It is important for risk managers to precisely forecast the loss. The analytical results imply that the impact of leptokurtic asset returns is based on the shape of asset returns, and a correct measurement of leptokurtic asset returns is helpful to risk managers seeking to precisely forecast the loss. A risk manager cannot reduce the loss in bad states, but can reduce the value of , the probability that a loss exceeds VaR, and the agent will suffer from reduced terminal wealth in both the good and bad states. The second topic solves an optimal investment problem involving a VaR risk manager who must allocate his wealth among cash, stocks and bonds. This study incorporates a stochastic interest rate process into the optimization problem. A Vasicek（1977）one-factor model governed the dynamics of the term structure of interest rates and risk premia are constant. Closed form formulate for the optimal investment strategy are obtained by assuming complete financial markets. Moreover, this study provides numerical examples to analyze the optimal terminal wealth and portfolio weights in stocks and bonds of the VaR risk manager. This work demonstrated the bond-stock allocation puzzle of Canner et al. (1997) that the bond-to-stock weighting ratio increases with risk aversion in popular investment advice in contradiction with standard two fund separation. Finally, this work derives the optimal portfolio selection of the VaR manager by assuming complete financial markets and that the inflation and real interest rates follow correlated Ornstein-Uhlenbeck processes. This study provides numerical examples to analyze the optimal terminal real wealth and optimal portfolio in stocks and two nominal bonds with different maturities. Furthermore, this work studies the influence of the parameters of inflation on the solution. This work illustrated that the younger VaR agent who has a long investment horizon invests the fraction of wealth in stock varies with the state price. It is not consistent with the Samuelson puzzle.

本研究以隨機利率模型應用至淨現值法、邊際年利率法、比較利率法及內部報酬率法等保單成本評價方法中，藉此觀察多期保單年度的價值變化，進而找出保單的報酬率、成本值或指數所呈現的趨勢，並考量在相同的情況下，比較各險種的成本或報酬優劣，最後希望消費者能在合理的隨機利率下更清楚了解保單成本的概念，並基於消費者對於合理的保單成本分析需求，能提供主管機關對於揭露保單成本的規範有更多的參考。 / 本研究發現各種保單在隨機利率變化的情況下，分紅終身壽險於各種成本分析方法中皆有良好的表現，不論是在考量淨成本結構的淨現值法，或是考量儲蓄性質、投資報酬為主的比較利率法以及各年度的邊際報酬利率等方法，整體而言分紅終身壽險對於消費者及保險公司應該是最優質的選擇。 / 再者以內部報酬率法應用隨機利率模型分析年金保險，可得知傳統遞延年金的報酬優於利率變動型年金。另外由於各種成本評價方法所著重的要素不同，想要了解保單完整全面性的評價，透過數個不同性質的保單成本分析方法計算較能呈現客觀且適切評價結果。 / This research is applied in the Stochastic Interest Model to the appraised method of insurance policy cost, such as Net Present Value Method, Marginal Yield Method, Comparative Interest Rate Method and Internal Rate of Return…etc., so as to observe the annual variation of value for different term of insurance policies, and then find out the rate of returns, cost value or trend appeared of index of the insurance policy, and consider it in the same cases to compare the good and bad from the cost or remuneration of every insurance. Hope consumers can finally clearer understand the concept of the insurance policy cost under the rational Stochastic Interest Rate, and on the basis of consumers’ demand for the rational insurance policy cost analysis can offer the competent authority more reference in revealing norms of the insurance policy cost. / In this research discovered that various insurance policies in changing of Stochastic Interest Rate, its Participating Whole-Life Insurance in varied cost analytical methods has good representation, no matter in considering the Net Present Value Method of the net cost structure, or considering Comparative Interest Rate Method of the main nature of deposits or main invest remuneration, and the annual marginal return interest rate…etc., the Participating Whole-Life Insurance should be generally the most high-quality choice to consumer and insurance company. / Moreover, according to the Internal Rate of Return, using the Stochastic Interest Model to analyze the annuity insurance can learn the remuneration of the Traditional Deferred Annuity is superior to the Interest Sensitive Annuity. In addition, as various cost appraised methods focused on different elements, if want to comprehension overall appraisal of insurance policy, then it can represent more objective and appropriate calculation through the analytical method of several different nature insurance policy cost.

隨著金融市場的開放，越來越多的金融商品被開發出來以迎合市場參予者的需求，利率衍生性金融商品是一種以利率為標的的一種新金融商品，而這種新金融商品的交易量也是相當的可觀。我們在設計金融商品的第一步就是要去定價，在現實社會中利率是隨機波動的而不是像在B-S的選擇權公式中是固定的。隨機利率模型的用途就是在描述利率隨機波動的行為，進而對利率衍生性金融商品定價。本文嘗試以隨機利率模型估計台灣公債市場的殖利率曲線，而殖利率曲線的建立對於固定收益證券及其衍生性金融商品的定價是很重要的。在台灣大部分的利率模型的研究都是利用模擬的方式做比較，這也許是因為資料取得上的問題，本文利用CKLS(1992)所提出的方式以GMM(Generalized Method of Moment)的估計方法，利用隨機利率模型估計出台灣公債市場的殖利率曲線。本文中將三種隨機利率模型做比較他們分別為: Vasicek model (Vasicek 1977),、隨機均數的Vasicek 模型 (BDFS 1998) ，以及隨機均數與隨機波動度的Vasicek 模型 (Chen,Lin 1996). 後面兩個模型是首次出現在台灣的研究文獻中。在本文的附錄中將提出如何利用偏微分方程式(PDE)的方法求解出這三個模型的零息債券價格的封閉解(Closed-Form Solution)。文中利用台灣商業本票的價格當作零息債券價格的近似值，再以RMSE (Root mean squared Price Prediction Error)作為利率模型配適公債市場價格能力的指標。本文的主要貢獻在於嘗試以隨機利率模型估計出台灣公債市場的殖利率曲線，以及介紹了兩種首次在台灣研究文獻出現的利率模型，並且詳細推導其債券價格的封閉解，這對於想要建構一個新的隨機利率模型的研究人員而言，這是一個相當好的一個練習。 / With the growth in the area of financial engineering, more and more financial products are designed to meet demands of the market participants. Interest rate derivatives are those instruments whose values depend on interest rate changes. These derivatives form a huge market worth several trillions of dollars. The first step to design or develop a new financial product is pricing. In the real world interest rate is not a constant as in the B-S option instead it changes over time. Stochastic interest rate models are used for capturing the volatile behavior of interest rate and valuing interest rate derivatives. Appropriate models are necessary to value these instruments. Here we want to use stochastic interest rate models to construct the yield curve of Taiwan Government Bond (TGB) market. It is important to construct yield curve for pricing some financial instruments such as interest rate derivatives and fixed income securities. 　In Taiwan Although most of the research surrounding interest rate models is intended towards studying their usefulness in valuing and hedging complex interest rate derivatives by simulation. But just a few papers focus on empirical study. Maybe this is due to the problems for data collection. In this paper we want to use stochastic interest models to construct the yield curve of Taiwan’s Government Bond market. The estimation method that we use in this paper is GMM (Generalized Method of Moment) followed CKLS (1992). I introduce three different interest rate model, Vasicek model (Vasicek 1977), Vasicek with stochastic mean model (BDFS 1998) and Vasicek with stochastic mean and stochastic volatility model (Chen,Lin 1996). The last two models first appear in Taiwan’s research. In the Chapter 3, I will introduce these models in detail and in the appendix of my thesis I will show how to use PDE approach to derive each model’s zero coupon bond price close-form solution. In this paper we regard Taiwan CP (cmmercial Paper) rates as a proxy of short rate to estimate the parameters of each model. Finally we use these models to construct the yield curve of Taiwan Government Bonds market and to tell which model has the best fitting bond prices performance. Our metric of performance for these models is RMSE (Root mean squared Price Prediction Error). The main contribution of this study is to construct the yield curve of TGB market and it is useful to price derivatives and fixed income securities and I introduce two stochastic interest rates models, which first appear in Taiwan’s research. I also show how to solve the PDE for a bond price and it is a useful practice for someone who wants to construct his/her own model.

政府為推動台灣成為亞太金融中心，逐漸放寬許多金融管制，因此，規避匯率風險將是台灣落實金融自由化與國際化的重要課題。 過去探討外幣選擇權訂價模式的文獻通常在利率固定的假設下進行研究，本研究將HJM利率模型應用於評價外幣選擇權，考慮國內外利率皆為隨機性，歐式與美式外幣選擇權的訂價。本文運用風險中立評價法，推導出與Grabbe(1983)類似的歐式外幣選擇權封閉解，並採用Amin and Bodurtha(1995)的模型設定，以間斷時間的HJM模型為基礎，運用模擬的方法決定美式外幣買權的價格，進而改變各參數的設定，進行敏感度分析。模擬結果顯示長天期的美式外幣買權對遠期利率波動度的敏感度較短天期大。本文呈現另一種外幣選擇權的評價模式，後續的研究可考慮將本文所採用的方法應用於外匯期貨選擇權、交換選擇權等衍生性金融商品的評價上。 第一章 緒論 第一節 研究背景與動機 1 第二節 研究目的 2 第三節 研究架構 3 第二章 相關文獻探討 第一節 歐式外幣選擇權之固定利率模式 4 第二節 歐式外幣選擇權之隨機利率模式 8 第三節 美式外幣選擇權評價模式 13 第三章 外幣選擇權定價模式 第一節 隨機利率下歐式外幣選擇權訂價理論 16 第二節 隨機利率下美式外幣選擇權訂價模式 26 第四章 模擬結果分析 33 第五章 結論與建議 43 附錄一 45 附錄二 46 附錄三 47 附錄四 49 附錄五、美式外幣選擇權電腦模擬程式 50 參考文獻 53

當保險契約被發行時，保險公司必須被要求盡可能的具備承擔未來不可知的風險的能力。本文將死亡風險、利率風險、脫退風險以及流動性風險引入，並針對生死合險契約進行盈餘分析。在此以 Vasicek (1977) 所提出之隨機利率模型、根據被保險人理性行為作為基礎之脫退模型以及引入簡化後的 Longstaff、Mithal與Nies (2005)流動性風險債券價格來描述各種風險。根據上述模型假設下計算保費及準備金，遂以蒙地卡羅模擬法量化源於各種風險之盈餘。最後，本文計算保險公司之盈餘對各風險參數之敏感度分析，並計算各期破產與發生流動性問題之可能性。 / Once insurance contracts are issued, the insurers should be capable to deal with the unknown conditions in the future as possible. In this paper, we analyze the impact of mortality, interest rate, surrender and liquidity risks on the surplus of endowment contract. We model the interest rate risk by Vasicek model, the surrender rate based on the rational behavior of policyholders and introduce the discounted price of zero coupon bonds as the liquidity risk. Under such assumptions, we compute the premium and reserve, demonstrate the simulated insurance surplus, and finally exhibit the statistics of the surplus from different sources. The simulated results show the sensitivity of the surplus to the parameters of the risks. At the same time, we also show the probabilities of insolvency and illiquidity of the insurer before the maturity date of the contract due to the fluctuating surrender rate and liquidity risk resulting from the stochastic interest rate.

本論文推導了四種隨機利率下匯率連動選擇權評價模型及其避險比率，其依序為匯率連動選擇權、匯率連動交換選擇權、後定選擇權與匯率連動遠期契約，並比較上述選擇權在隨機利率下與固定利率下評價模型與避險比率之差異。在固定利率下的評價公式與避險比率，其折現因子為固定利率，然而在隨機利率下的評價公式，是以零息債券折現，因此能反映未來利率波動。若發行券商預期未來利率有大幅波動或選擇權的到期日較長時，應使用隨機利率下的評價公式，方能得到較合理的價格。

在WTO成立，各國國際化程度日益提高的同時，企業與個人進行跨國投資的情形也愈來愈普遍，跨國投資除了要考慮標的資產之報酬與波動性之外，尚須考量匯率變動所產生之風險與不確定性。當某一國外資產具有正向預期報酬率的同時，實現後的報酬率卻又不一定為正，正是因為匯率波動所產生的影響。又，傳統財務理論告訴我們，藉由增加投資組合中所有非完全正相關的資產個數可以有效的降低投資組合的非系統風險，因此投資人在進行投資時往往採用建構投資組合的方式取代持有少數資產的型態。然而，在建構跨通貨避險投資組合時，若是對於投資組合中的各項資產與外幣分別進行避險（分別利用衍生性商品避險），往往是費時、費力又不具有效率。因此，對於整個投資組合進行避險反而是一個比較好的方法，當投資組合價值發生變動時，可以即時對於各項資產部位與外幣分別做調整，遠較於對個別資產進行避險來的方便、快速且有效。 / In most cases, investment is made of building a portfolio rather than single asset. Therefore, it is necessary to develop techniques of valuing portfolio derivatives. Moreover, we consider a cross-currency portfolio that account for currency and interest rate risk. As interest rate is stochastic, we use Heath-Jarrow Morton (HJM) Approach to describe its dynamics. Applying Vorst (1992); Geman, Karoui and Rochet(1995), we derive the approximated close-form of the cross-currency portfolio option. In HJM Approach, it is difficult to acquire hedge ratios of options. We apply another method to build a hedging portfolio. Then, we perform numerical simulations to test its hedging efficiency and sensitivity with respect to different variables.

本研究主要是針對投資型年金之資產負債管理作探討，其中是就規避利率風險對於資產負債管理上的影響以及分析資產配置最適化作為研究的架構，而所利用的研究方法乃是取決於建構利率隨機模型並輔以免疫理論與Markowitz投資組合理論，以期在規避利率風險的同時，亦能將資產配置達至最佳化。 首先，為實際模擬出符合現實經濟環境變動下的隨機利率期間模型，本研究利用C.I.R利率期間結構模型來建構年金保單期間的利率結構，並且由於投資型年金之保單價值的累積特性，因此本研究同時亦建構出連接保單價值的投資資產之報酬率型態，進而模擬出各期之現金流量以及各項投資資產的存續期間；再者，藉由Markowitz投資組合理論，以在免疫條件之限制下進行最適資產配置之評估。 最後，以某知名的保險公司所推出的投資型年金商品作為本研究之實證對象，透過模擬之方法，將研究模型中之各項參數予以評估，並且根據上述之研究過程將免疫理論與投資組合理論相連接，以檢視投資型年金商品在規避利率風險的狀態下，其最適之資產配置比例是否與現行法令之規範相牴觸，而能給予適時之建議。另外，由本實證結果可知，經由本研究的分析流程，可以有效地給予年金管理者規劃出年金資產負債管理時的最適投資組合比例，並且在增加外國投資資產時，更能有效的增加年金資產之報酬，同時也不影響保險法對於投資資產的比例與總金額之限制。再者，對於探討規避利率風險前後之資產組合之資產報酬之變化時，可以進一步了解到，當年金管理者在運用免疫策略來規避利率風險時，其所面對的風險成本之多寡，以作為制定避險決策時的依據。 / This research explores the asset-liability management (ALM) for the Investment-Link-Annuity. Two aspects investigated in this research are the interest rate risk and the optimal asset allocation. Moreover, the major issue investigated here is the trade-off between the optimal investment return and the hedge of interest rate risk. We refer this trade-off as ALM cost. By using stochastic interest rate model, Immunization theory and Portfolio Selection Model, we construct an ALM model to achieve the optimal asset allocation given on hedging the interest rate risk under the immunization strategies for the insurance company. First, we utilize the public trading data for investment market in Taiwan and in USA from 1985 to 2000 and the investment-link annuity product of a well-know insurance company in Taiwan to simulate the cash flow and demonstrate the implementation of our model. By analyzing different simulations under various scenarios, the empirical results are as the followings: 1.The ALM cost for immunization strategies is very small, and is estimated to be about 1% to 2%. Therefore, we suggest that insurance companies should start to undertake the asset liability management as soon as possible. 2.If relaxing the investment restrictions of Insurance Law or allowing insurance company to invest in foreign investment market, the overall investment return will be increased and the ALM cost will be reduced effectively.