人口高齡化是許多國家必須面對的問題，為因應長壽風險，保障年長的退休民眾的經濟安全無虞，應當設計新的退休金籌措管道，以避免退休民眾須降低其退休後之生活水平或是延後退休時間。本研究以英澳無保險之反向房屋抵押貸款為基礎，分析壽險公司在台灣販售反向房屋抵押貸款時，屋主可以獲得之最高年金金額，並進一步考量當死亡率改善後對於屋主可以獲得之最高年金金額之影響。並分別以反向房屋抵押貸款最高年金金額所提供之所得替代率及其可支應每人每月生活開銷之比重，衡量反向房屋抵押貸款能為退休民眾增加多少退休金。最後，藉由分析反向房屋抵押貸款之存續期間與一般年金商品之存續期間的不同，進一步了解反向房屋抵押貸款與一般年金商品對於利率風險之敏感程度差異。

論文名稱：可轉換公司債存續期間之分析 校所組別：國立政治大學金融研究所 畢業時間：九十年度第二學期 提要別：碩士學位論文提要 研究生：陳嘉霖 指導教授：陳松男博士 論文提要及內容： 本研究在分析可轉債的存續期間，在存續期間的衡量上是採用有效存續期間法；而在可轉換公司債的評價上，假設股票價格服從幾何布朗寧運動，無風險利率的變動符合Hu1I-white利率模型，並且考量利率與股票報酬之間的相關性，建立可轉換公司債評價六元樹形圖。 本研究分別針對到期期限長短、價內外程度、股價波動度、利率波動度、股價與利率相關係數及票面利率等六項參數，作可轉換公司債存續期間的敏感度分析，研究結果為：1 加入贖回條款後，可轉債的存續期間高於未加任何條款下的可轉債存續期間。2 加入賣回條款後，可轉債的存續期間低於未加任何條款下的可轉債存續期間。3 加入贖回及賣回候款後，可轉債的存續期間會介於僅含贖回條款與僅含賣回條款的存續期間之中。4 距到期日愈長可轉債的存續期間愈高。5 愈價外的可轉債其存續期間愈高。6 股票波動度愈高，可轉債的存續期間愈低。7 利率波動度增加則可轉債的存續期間上升。8 股票價格與利率相關係數由正至負，可轉債的存續期間上升。9 若贖回權愈小，則票息上升會增加可轉債的存續期間。 關鍵字：可轉換公司債、存續期間、有效存續期間、六元樹、Hull-white、利率模型 / Title of Thesis: Anatomy of the Convertible Bond Duration Name of Institute: Graduate Institute of Money and Banking, NCCU Graduate Date: June, 2002 Name of Student: Chen, Chia-Lin Advisor: Dr. Chen, Son-Nan Abstract: This thesis uses effective duration method to anatomize the convertible bond duration. With the assumptions that stock price follows Geometric Brownian Motion and risk-free interest rate follows Hull and White model, we built a hexanomial tree to value the convertible bond. This thesis analyses the effects of the six parameters . They are maturity date, the ratio of the stock price versus the strike price, the correlation between stock return and interest rate, stock return volatility, interest rate volatility, and coupons. The conclusions include nine points. First, the value of convertible bond duration including call clauses is higher then pure convertible bond duration. Second, the value of convertible bond duration including put clauses is lower than pure convertible bond duration. Third, the value of convertible bond duration including both call and put clauses is between only including call or put clauses ones. Fourth, the longer the time to maturity is, the higher the convertible bond duration is. Fifth, the higher the ratio of the strike price versus the stock price is , the higher the convertible bond duration is. Sixth, the higher the stock volatility is , the lower the convertible bond duration is. Seventh, the higher the interest rate volatility is , the higher the convertible bond duration is. Eighth, the value of the correlation between stock return and interest rate increases from a negative value to a positive one, then the convertible bond duration increases. Ninth, if the value of call right is very small , the convertible bond duration will increase by the increasing of the coupon . Keywords: Convertible Bond, Duration, Effective Duration, Hexanomial Tree, Hull and White Interest Rate Model

本研究運用的投資組合理論(Portfolio Theory)與免疫理論(Immunization Theory)建構資產負債管理模型，希望在於免除利率風險下，能夠極大化勞保基金的投資報酬率。本研究探討勞保老年給付年金制實行後，勞保基金在資產負債管理之下最適資產配置。我們以勞保局編印之「勞工保險統計年報」中勞保基金民國81年到91年實際投資的資料及勞保局委託研究之精算報告對於老年給付年金制實行後未來勞保基金的給付預測值，在不同年金選擇率以及不同的費率與控管年限下，根據勞保基金資產與負債的存續期間，建議勞保基金最適的投資組合，並計算資產負債管理成本，研究結果發現： 1 年金選擇率為100％及80%時，勞保費率提高至8.3%僅能確保未來30年與40年勞保基金不會因為利率變動而導致基金破產甚至無力清償，但考慮年限為50年時，國內市場無法找到存續期間可以配合的投資工具，無法規避利率風險。年金選擇率為50%時，由於未來各年之勞保的給付獲得舒緩，使得資產配置所需的存續期間也降低，故當勞保費率提高 至8%即可確保勞保基金未來50年可以規避利率風險的危機，且在國內市場上可以找到投資工具配合。 2. 要使勞保基金免於利率風險的考慮年限越長，其投資組合的重心應該從現行的銀行存款移轉到債券及股票與受益憑證。 3. 進行資產負債管理是需要成本的，若以資產負債管理前後效率前緣下的投資報酬率的差異為資產負債管理成本，在年金選擇率100%時資產負債管理平均成本為0.3695%；選擇率80%時平均成本為0.434%；年金選擇率為50%時資產負債管理平均成本為0.384%，研究結果顯示資產負債管理平均成本都低於0.5%以下，故建議勞保基金應盡早進行資產負債管理以因應老年給付年金化後利率風險對於勞保基金財務上的衝擊。 / This paper investigates the Asset-Liability Management for Labor Insurance Fund. We utilize Immunization Theory and Portfolio Theory selection model to immunize the surplus of Labor Insurance Funds against interest-rate fluctuations and to maximize expected return of Labor Insurance Funds simultaneously. In addition, we use the data from Labor Insurance Funds from 1992 to 2002 to demonstrate the implementation of our model. We calculate the optimal asset allocation and the ALM cost under different lump-sum/annuity selection ratio、time horizon and contribution rates. The empirical results from this study show that： 1. Assuming 100% and 80% participants choice annuity, to prevent the insolvency of Labor Insurance Fund from interest-rate fluctuations in 30 and 40 years, the Labor Insurance premium must increase to 8.3%. Assuming 50% participants choice annuity, to prevent the insolvency of Labor Insurance Fund from interest-rate fluctuations in 50 years, the Labor Insurance premium must increase to 8%. 2. To prolong the period over which the Labor Insurance Funds can immunize its surplus against interest-rate fluctuations, a large proportion of the investment asset should be allocate from bank deposit to bond and stock. 3. ALM needs cost. Assuming 100% participants choice annuity, the average ALM cost is 0.3695%.Assuming 80% participants choice annuity, the average ALM cost is 0.434%.Assuming 50% participants choice annuity, the average ALM cost is 0.384%. We find the average ALM cost is very small under any lump-sum/annuity selection ratio. Therefore, we suggest Bureau of Labor Insurance should start to implement ALM as soon as possible to avoid the affect of interest-rate fluctuations.

本研究以免疫理論（Immunization Theory）與投資組合理論（Portfolio Theory）來架構退休基金的資產負債管理模型，希望在免除利率風險的情況下，極大化退休基金的投資報酬率。本文以退撫基金86年到88年實際投資的實證資料及其對未來給付支出的預測模擬值，在不同考慮年限與提撥率下，建議其最適投資組合，並計算出資產負債管理的成本。最後，再以84年到88年市場平均資料的實證結果，支持並驗證我們以退撫基金內部資料所做的實證結果。研究發現： 1、資產負債管理的成本相當少，因此退撫基金應該儘早進行資產負債管理。若以資產負債管理前後效率前緣下投資報酬率的差異為資產負債管理的成本，以退撫基金內部資料的研究顯示，資產負債管理的平均成本為0.133﹪；以市場平均資料的研究顯示，資產負債管理的平均成本為0.234﹪。 2、在進行資產負債管理的分析後，退撫基金的薪資提撥率應提高至14.84﹪，才能確保未來的30年內，退撫基金不會因為利率變動而導致基金破產甚至無力清償。 3、要使退撫基金免於利率風險的年限愈長，其投資重心必須從短期票券和債券移到債券與股票或受益憑證上。 / This paper investigates the Asset-Liability Management for pension fund. We utilize Immunization Theory and Portfolio Theory selection model to immunize the surplus of pension funds against interest-rate fluctuations and to maximize expected return of pension fund simultaneously. In addition, we use the public trading data of the investment market in Taiwan from 1995 to1999 and the data from the Taiwan Public Employees Retirement System（TPERS）from1997 to 1999 to demonstrate the implementation of our model. We calculate the optimal asset allocation and the ALM cost under different time horizons and contribution rates. The empirical results from this study show that: 1、 The ALM cost is very small. Therefore, we suggest TPERS should start to implement ALM as soon as possible. Given the investment performance of TPERS, We find the ALM cost is 0.133﹪. Given the performance of the investment market, the ALM cost is 0.234﹪. 2、 The TPERS must increase its contribution rate to 14.84﹪ in order to make sure that the TPERS will not be insolvent as a result of interest-rate fluctuations in 30 years. 3、 To prolong the period over which the TPERS can immunize its surplus against interest-rate fluctuations, a large proportion of the investment asset should be allocate from commercial paper and bond to bond and stock.

退休基金面臨的風險來自資產與負債兩方面，負債的風險為實際給付大於預期給付的不確定性，資產面的風險為實際價值小於預期價值的不確定性。造成這兩個風險的因素有很多，其中利率風險是引起資產與負債不確定性的重要因素。因為非預期的利率變動會使資產與負債價值發生變動，而影響退休基金的給付能力。如果資產與負債之間能維持一適當關係，退休基金則可免於利率變動所帶來的不利益。本研究以存續期間作為退休基金資產負債管理的方法，債券為投資工具進行退休基金免疫策略的模擬，結果發現：在資產與負債名目現金流量不受利率變動影響的假設下，不論市場利率如何變動，在傳統的免疫策略下，投資組合必能創造出預期的報酬率；但當上述假設不存在時，免疫狀態將不復存在。為防止退休基金給付能力不足，除應每年確實進行精算評估外，應於市場利率、投資報酬以及薪資上漲率發生變動後，立刻衡量資產與負債現金流量受此變動所引起的改變，並計算變動後資產與負債現值的差額，若發現資產現值小於負債現值時，基金管理人可買賣期貨以延長或縮短投資組合的平均存續期間，促使資產與負債對利率之敏感性不同，好讓投資組合能於利率變動後自動補足此差額。或甚於平均存續期間的計算公式中，將此種變動狀況考慮進去，使免疫策略於資產與負債的名目現金流量因利率變動而改變時仍能發揮免疫效果。

本研究探討國內債券市場如何決定債券殖利率之風險貼水及交易成本貼水，其中風險貼水可分為時間及信用風險貼水兩部份，屬於前者之重要變數有存續期間(Duration)與凸性(Convexity)，屬於後者則有信用評等(Credit Ranking)與銀行擔保效果，而交易成本分析是在控制風險貼水因素後，看稅賦效果是否會影響投資者之必要報酬率。此外，本研究亦對殖利率曲線作分析，討論長短期資金市場是否存在明顯互動關係。首先對存續期間及凸性之特性作研究，檢測在既有到期期間變數下，加入存續期間及凸性對於債券殖利率邊際解釋能力的影響，冀描述國內投資者之訂價行為。其次就債券信用評等與銀行擔保效果作分析，公司債可能有不同信用等級，面對不等級公司債券，投資人或會要求不同程度之違約風險貼水，本文以實證對此作探討。又公司債因擔保與否，區分為擔保公司債及無擔保公司債，因此本研究關心的第二組變數是發行人為政府抑民間機構暨公司債之信用評等與銀行擔保效果。本研究亦對債券交易成本作探討，就證券交易稅而言，政府公債免徵交易稅而公司債券買賣須課徵千分之一的交易稅，因此我們想要知道存在公債與公司債之間，因稅賦差異造成交易成本不同，是否會影響到投資者的意願，故第三項變數為以證交稅為主之交易成本。最後對市場資金供需情形作分析，同時探討長短期資金市場是否存在互動關係，故第四項變數為全面資金供需情況。本研究藉檢測總體經濟資金供需變數對債券殖利率之影響，同時也可檢驗國內長短期資金市場之區隔程度。由於國內債市尚淺，仍舊有很大的發展空間，因此在可預期的未來，國內債券市場勢必會受到應有的重視。但由於早期國內債市之不發達及不受重視，使得有關債券資料的保存，特別是公司債券部份十分缺乏，連帶相關文獻亦寥寥可數，故本文以國內債券市場資料做實證研究、分析，冀望能有邊際之貢獻。

近年來台灣的市場利率持續下滑，可能造成保險公司無法實現對被保險人之高預定利率的保單給付承諾，利率風險已經成為壽險公司是否能繼續經營之重要關鍵。壽險公司如未能衡量利率變動風險而貿然推出保單，將對財務造成極大的負擔，不但會影響公司清償能力，亦會對社會造成衝擊，因此壽險公司應先衡量現在及未來將面臨多大的利率風險，做適當資產負債管理，以避免將來失卻清償能力。 　　本研究利用財務上平均存續期間（duration）觀念與Barney（1997）所提出之K值來衡量利率風險，以民國87-89年財務報表資料，實證研究利率風險會對那些經營指標產生影響，以喚起業界對於利率風險之重視。研究結果發現： 　　1.利率風險對於壽險公司之投資報酬率、股東權益報酬率有顯著的影響，並且呈負相關。 　　2.利率風險對於壽險公司之流動比率無顯著相關；與負債比率有顯著之正相關。 　　3.利率風險對於新契約保費成長率、保單繼續率無顯著影響，顯示國內並無明顯之逆中介情況。 　　4.壽險公司可藉由投資較長期之公債、公司債及減少保單貸款、不動產投資與固定資產項目之利率敏感度，以增加壽險公司之獲利性。 / The interest rates have been decreasing recently. Under this circumstance, it might be difficult for insurance companies to gain sufficient investment returns to fulfill the commitment of insurance policies. The interest-rate risk has become one of the critical factors for the solvency of life insurance companies. Therefore, life insurance companies should evaluate the impact of interest-rate risk and perform asset-liability management to prevent insolvency. 　　This study applies the concept of duration and K value (Barney 1997) to measure interest-rate risk and its impact on the operations of life insurance companies in Taiwan. The empirical analysis is conducted based on the financial data of life insurance companies in Taiwan during the period of 1998-2000. The empirical findings are listed as follows: 　　1.Interest-rate risk has a significantly negative impact on both investment return and ROE.. 　　2.Interest-rate risk does not have significant impact on current ratio of the life insurance companies, but it is positively related to debt ratio. 　　3.Interest-rate risk does not have significant impact on either new contract growth rate or policy renewal rate, which indicates that the process of disintermediation does not happen in life insurance industry in Taiwan. 　　4.By investing in the long-term government bonds and corporate bonds and reducing the interest-rate sensitivity of policy loans、investment on real estates and fixed assets , life insurance companies may be able to increase their profits.

本論文是由三篇關於金融市場的實證研究組合而成。第一篇以權益存續期間為主題，主要是利用迴歸模型估計台灣上市產業指數的實證權益存續期間，以探討股票報酬率的利率敏感度。迴歸模型中控制了三個重要的股票風險因子─市場因子、規模因子與價值因子。但其中，我們改以正交市場因子代替市場因子，以避免因為利率變動與市場報酬間存在共線性，而造成權益存續期間有可能錯估的問題。此外，基於權益存續期間具有會隨時間改變的動態特性，本文亦對各產業指數最近一次結構性變化的發生時點進行偵測，並據以推估最近期的權益存續期間。實證結果顯示：除了鋼鐵業的權益存續期間不顯著之外，其他所有產業指數皆具有負的權益存續期間，表示其報酬率與利率變動呈現出正向關係。在程度上，則以營建類指的利率敏感度最大，汽車類指最小。 第二篇應用了Diebold and Yilmaz (2009)的外溢指標分析台灣上市產業指數間的連動性，其優點是可以瞭解到產業間相互影響的方向以及程度。實證結果顯示：台灣上市產業指數間的外溢程度頗高，並以營建業為最主要的影響者，而相反地，鋼鐵業則是主要的被影響者。外溢指標具有隨時間改變的動態特性，而且透過動態外溢指標可觀察到次貸風暴蔓延的嚴重性。 第三篇應用了Goyal, Perignon and Villa (2008)所提出的多群組因素分析法，檢測美國總人口死亡率的共同因子個數。該方法最大的優點是能夠有效地辨識出真正的共同因子，避免了一般因素分析容易將解釋能力高的群組內獨特因子誤認為共同因子的缺點。根據檢測結果顯示，美國總人口死亡率的共同因子共有兩個，而且第二個因子的重要性隨時間愈來愈明顯。 / This thesis consists of three empirical essays about financial markets. The first essay analyzes the sensitivity of stock returns to changes in interest rates by estimating empirical equity duration of 18 industrial indices in the Taiwan Stock Exchange. In the regression models, we also control for the market excess return and the Fama-French mimicking returns for size and book-to-market factors. To avoid the effects of the multicolinearity between the market excess return and the interest rate changes, we replace the market excess return by the orthogonalized market factor. In addition, considering the time-varying pattern of empirical equity duration, we further adopt the reversed ordered Cusum test proposed by Pesaran and Timmermann (2002) to identify the most recent break of the regression relationship, and then extract the post-break data to re-estimate the up-to-date empirical equity duration. The result shows that except the Steel index, all industrial indices exhibit significantly negative equity durations, indicating a positive relationship between industrial index returns and interest rate changes in Taiwan. Among them, the Construction index has the largest interest rate sensitivity, while the lowest one is for the Automobile index. The second essay focuses on the nature of financial market interdependence, both in terms of returns and returns volatilities. Being capable of identifying the direction and magnitude of linkages among financial markets, the spillover index proposed by Diebold and Yilmaz (2009) is used to measure return and volatility spillovers between the top eight industrial indices based on market value in the Taiwan Stock Exchange. We find that for both returns and volatilities, the spillover effects among industrial indices in Taiwan are substantial. In particular, the Construction index is the major transmitter of shocks to other industries, and the Steel index, in contrast, suffers the most shocks from others. The spillover index fluctuates over time and indeed detects the severity of subprime mortgage crisis. The third essay adopts the multigroup factor analysis proposed by Goyal, Perignon and Villa (2008) to estimate the number of common pervasive factors for annual age-specific mortality for the entire U.S. populations. While the standard principal component analysis easily treats any group-specific factor as pervasive one due to its high contribution to total system variance, this methodology is able to estimate the space spanned by common and group-specific pervasive factors and recognize the true common factors. Empirical result shows that there are only two common pervasive factors governing the death rates in the United States; in particular, the importance of the second factor increases over time.

關於Merton (1969) 最適投資組合策略問題，所考慮之投資情境為：一個將其財富資金安排配置於風險性資產(各類證券)與無風險短期現金部位之投資人，在給定此投資人心目中財富效用函數之前提下，希望事先決定出投資組合之最適投資權重(策略)，藉此達成在投資期滿時極大化財富效用之期望值。基於Merton (1974) 公司價值觀點，具有違約風險之證券(公司債與股票)乃是公司價值之衍生性商品，無法以傳統資產配置對股票與債券部位採取現貨方式處理最適投資策略，在此必需同時結合財務工程處理衍生性金融商品計價與避險之技術來解決。本研究利用Kron & Kraft (2003) 彈性求解法來針對市場是否有投資限制、債券提前違約、到期違約及利率隨機與否等假設，基於不同投資組合情境分析來最適投資部位策略。本研貢獻和究創新突破之處在於特別探討公司違約時，債券投資人不再享有全部公司殘值之求償權，此時股東亦享有部份比例之求償權，違約後之公司殘值將由債券投資人與股東兩者比例共分之特殊情境下，對數型態財富效用之投資人對於提前違約風險之接受度高於到期違約風險，若一般情境(股東無任何求償權)則為相反。此外亦特別提供最適成長投資組合之動態避險策略封閉解，藉以提供投資人面臨企業違約風險時應制定之投資決策與動態調整，使本研究臻至週延與實用。 / Under the Merton (1969) optimal portfolio problem, we only consider the specific investor, whose wealth utility follows the type of logarithm function; wants to maximize the expected value of the terminal wealth utility through determine the optimal investment strategy in advance. He divides his wealth into the riskless asset and risky assets such as the money market account and the various-risky securities issued by the corporate. Based on the Merton firm value framework (1974), the defaultable securities, such as the corporate bonds and stocks, are the derivatives instruments of the firm value. It will be inappropriate if we deal with this optimal portfolio problem under the original methods. Therefore, we need to handle this optimal asset allocation problem through the pricing, valuation and hedging techniques from the financial engineering simultaneously. This study apply the elasticity approach to portfolio optimization (EAPO, Kraft ,2003) to solve the optimal portfolio strategy under various scenarios, such as the market contains the investment constrain or not, intermediate default risks, mature default risk, interest rate risky under the stochastic process. The innovation and contribution of this paper are especially breaking the common setting and analysis the optimal-growth-portfolio strategy under the special scenario. In the common setting, as soon as the default event occurs, the residual firm value will be claimed by the corporate bondholders with fully proportion and the stockholder cannot share any residual value. Oppositely, the stockholder will be able to share the residual firm value proportionally with the corporate bondholder together under the so-called special scenario. We found that the investor would have higher acceptance of the premature default risk than the mature default risk in the special scenario. This phenomenon will be reversed under the common scenario. Furthermore, in order to make this study more completely and useful, we do not only illustrate the optimal investment strategy but also provide the closed-formed solution of the dynamic hedge strategy of the risky position, composed by the defaultable securities. This could help the optimal-growth-portfolio-oriented investor to make investment decision while they face the firm value downward decreasing.

摘要 衡量壽險公司利率風險的初步作法是估算保單的存續期間。保單的存續期間因現金流量模式的不同與一般債券的存續期間有很大的差異。壽險保單未來不只會產生現金流出 （給付和費用），還會有現金流入（保費），其淨現金流量因而有可能變號，所以壽險保單的存續期問可能小於○，也可能大於到期日，甚至還可能因為準備金接近○的關係而有很大的數值。此外，保單的存續期間不太受死亡率下降的影響，解約率升高通常會使存續期問的數值降低，而佣金的平準化則會使原本正的存續期問變小。從壽險公司銷售保單組合可能的結果（平穩、成長、衰退等三種型態）來分析負債存續期間，由其結果可知一個新創立或成長型的壽險公司其所面臨利率風險的衝擊要大於一個處於平穩型或衰退型的壽險公司。 關鍵字：存續期間、利率風險、淨現金流量、壽險保單、準備金 / Abstract Estimating the duration of the life insurance policy is the first step in measuring the interest rate risk of the life insurance company. Life insurance policy's duration is quite different from bond's due to the difference in the pattern of cash flows. Life insurance policies generate not only cash outflows as payments to policyholders from insurance companies but also cash inflows as premiums from policyholders to insurers. Furthermore, the net cash flow usually turns from inflow to outflow as time goes by. The duration of the life insurance policy therefore could be negative or longer than the maturity of the policy. It could even be huge if its reserve is close to zero. Besides, the mortality rate does not have a significant impact on policy duration; early surrenders of policies would reduce policy duration in general; and leveling commission rate would make positive duration smaller. Findings concluding from analyzing the likely results, referring to the steadying, growing and declining modes, of insurance portfolios offered by life insurance companies for analyzing their liability duration, indicate that the interest rate risk exposure by a start-up life insurance company or a growing life insurance company is greater than a life insurance company that is at a steadying or declining phase. Keywords: duration, interest rate risk, net cash flow, life insurance policies, reserve.