死亡風險的自然避險與商品設計 - 政大學術集成

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(1)國立政治大學風險管理與保險學系博士學位論文. 死亡風險的自然避險與商品設計 Natural Hedging of Mortality Risks 治 and Product Design. 立. 政. 大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. 指導教授：蔡政憲博士研究生：黃芳文撰. 中華民國一○四年十二月.

(2) 謝辭能夠完成博士學位，我非常感謝周遭每一位協助過我的人，不管是幫助我解決課業學習上所遭遇到的問題，或是在我心情上down到谷底時，給我一句鼓勵的話，在這過程中我得到許多人的幫助，我由衷的感謝許多人。首先要感謝的是我的指導老師蔡政憲老師，從在保發中心擔任研究員期間，從您那裡學到很多有關風險與財務領域的知識，感謝您不吝於給我所需要幫助。更感謝是在進入政大博士班後可以投入在您的門下，您的教導把我過去不甚了解的觀念理論釐清。您總是非常有耐心的教導，總是包容大過責備，您也從不催促. 政治大助學生的好老師。能夠做為老師的學生，是上帝給我的恩典。立. 我的進度，我可以依照自己的步調去完成進度。謝謝老師，您是一位極盡所能幫. ‧ 國. 學. 感謝口試委員張士傑老師、謝明華老師、喬治華老師、楊曉文老師。由於有了您們寶貴的意見，使我的論文不僅在主軸設定上，且在內容上可以更加充實且. ‧. 更完整，在文章陳述的順序上提供給我很好的意見，增加了論文的流暢度，也更. y. sit. al. er. io. 導。. Nat. 能夠突顯論文所要呈現的重點。謝謝老師們針對我的論文提出您專業的意見與指. v. n. 在此也要感謝主任許永明老師，謝謝許老師幫助我在論文寫作以及期刊投稿. Ch. engchi. i n U. 的過程中，有許多的學習。使我從撰寫保險實務的主觀論述，可以慢慢的掌握到客觀佐證下學術論文的論述與邏輯。就算是我所熟知的風險資本領域，許老師仍可協助我強化了文章的深度與嚴謹度，這使我完成我的第一篇投上TSSCI期刊 (經濟論文)的學術論文，這給了我莫大的鼓勵與學習的動力。謝謝陳彩稚老師，做為我們博士班導師的您，在學業活動、生活輔導以及最重要的畢業所要求的各項規範等，幫助我們獲得許多諮詢管道。藉著許多和您互動的機會，從中了解到老師您對學生們的期待以及對教育的堅持，您總是給我們鼓勵也期許我們在學術上有更好的表現。授課上，您在保險理論的課程也給我許多保險與經濟理論的教導，謝謝陳老師。.

(3) 在學業修習各項專業課程的過程中，除蔡老師、許老師、陳老師，還有郭維裕老師教授的財務經濟、王儷玲老師教授的長壽風險、鄭士卿老師教授的保險產業，甚至是賴志仁老師與黃泓智老師的保險專題。每位老師都有老師的專業領域，使我學習到許多不同的學術領域，且每位老師的學術專業也深深吸引我學習的興趣，謝謝老師們的付出，不僅提升了我學習的深度與廣度，也使我的視野更廣闊。感謝文彬，你總是成為我各種疑難雜症諮詢的對象，你提供給我的資訊，讓我得以按部就班完成學業每個必經之路；謝謝凌玉助教、薏臻助教以及椀婷助教在系上各項事務上的協助；謝謝議謙、尚穎、可倫、彥志、景元、志宏、瑞祥，. 政治大增加了許多有趣的元素，也祝福尚在邁向博士學位路途的學弟妹可以順利達到那立在研究室裡總是可以找到你們協助與討論，有你們在研究室，這段學習的日子裡. 屬於你們最光榮的最後一哩路。. ‧ 國. 學. 接下來，還要感謝常常為我禱告的教會小組姊妹們，每每遇到無法克服的難. ‧. 關時，總是需要姊妹們特別為我代禱，記得有一門課要交一篇期末報告，我訂了. y. Nat. 題目，但腦袋是一片空白，不知道該如何下筆？我請小組姊妹為此事幫我禱告，. er. io. sit. 但自己心裡自忖不可能有人幫我寫啊，正在傷透腦筋，心裡狐疑禱告有用嗎？癱軟坐在研究室椅子上，眼睛呆滯地直視前方，驀然書架上有一本中譯簡體字的書. al. n. v i n 吸引我的目光，我打開書本，書上正好有我需要的資訊與簡略的程式，於是我有 Ch engchi U. 了開始可以完成這份期末報告，霎時間我心中充滿感激。我從不曾留意到這本書，卻早在兩年前它就放在我的書架上，是過去的學長所留下的，原來上帝兩年前就早已預備好了我所需用的。感謝上帝、也謝謝鳳品姐、燕慧以及小組姊妹們，給我屬靈的家，你們的禱告紀念使我走在上帝蒙福的道路上，一路蒙福到底。最后要感謝我的家人，謝謝先生及孩子體諒我放棄職場工作專心念書，給我足夠的空間與時間，滿足我對於知識的追求，對我百般的忍讓與支持。謝謝爸爸、哥哥、嫂嫂給我實際的支持與不時的關懷。我將把這份對每個人的感謝永遠銘記在心。.

(4) 摘要對必須維持長期清償能力的壽險公司來說，如何做好死亡率的風險管理是極為基本的。過去的文獻上提出，利用保險公司銷售的壽險商品(如：終身壽險)，因具有死亡風險，對具有長壽風險的商品(如：年金險)可產生避險效果。這種商品組合的自然避險方式是顯而易見的，但可能因為僵化的銷售市場與誘因導向的銷售方式等因素，商品組合的自然避險是較為難以執行的。我們提出的是，把自然避險策略植入商品設計中，可以把避險效果內含於商品內容裡面。關鍵在於讓. 政治大. 這張保單在死亡事件發生的時間點所產生的影響可以被抵銷，我們技巧性地選取. 立. 給付成長參數，即“應給付多少”因子，可以決定死亡給付的金額現值，而利率. ‧ 國. 學. 因子δ洽可反映給付的時間價值。本論文集提供壽險商品與年金商品的理論推導、. 計之中。. ‧. 圖解說明以及數值分析，一一闡述我們的想法以及如何將自然避險融入於商品設. y. Nat. io. sit. 在第一篇中，我們利用精算的方法推導，得到利用壽險商品設計範疇內可以. n. al. er. 達到最佳化化的自然避險策略。在第二篇的論文中，依據第一篇自然避險策略的. Ch. i n U. v. 理論基礎下的商品設計，我們除更進一步探討在死亡率與利率條件不確定的因素. engchi. 下，對於以最佳化避險策略的商品，可產生的影響，我們以實際的數據分析，得到即使對於未來的死亡率與利率不確定的情況下，我們所提出的依理論條件下的最佳化避險策略所設計的商品，仍能使商品所產生的死亡率風險極小化，甚至接近於無死亡率風險。除以壽險商品為主要設計外，在第三篇論文中，我們也呈現以年金商品為架構的商品設計。. 關鍵字: 死亡風險、長壽風險、自然避險.

(5) Abstract How to manage mortality rate risks is essential to the long-term solvency of life insurance companies.. The literatures proposed to hedge the products subject to the. longevity risk (such as annuities) by using the products subject to the mortality risk (e.g., whole life insurance) sold by an insurer.. Such natural hedging is intuitive but. may be difficult to implement due to the rigid sales market and incentive issues. We propose to embed natural hedging into product design so that the hedging may occur within a product. The key is to offset the impact of mortality on the timing of death. 政治大 growth rate of the death benefit, 立 the factor of “how much to pay” while δ would that in turn determines the present value of the death benefit by cleverly choose the. ‧ 國. This article provides theoretical derivations,. 學. reflect the time value of payment.. graphical illustrations, and numerical analyses of both life insurance products and. ‧. annuity products to illustrate the idea of embedding natural hedging into product. sit. y. Nat. design.. n. al. er. io. In the first essay, we use actuarial methods to come up with the theoretical. i n U. v. derivation of the optimal natural hedging strategy and we can easily embed it into. Ch. engchi. insurance product design. In the second essay, furthermore, we evaluate the impact of uncertainty passed on by future mortality rate and interest rate against the product design based on the optimal natural hedging strategy. We use the numerical illustration and obtain the minimum risk or nearly no risk at all under the optimal strategy. We develop not only on the life insurance product design, in the third essay, we also progress the product design of annuity products.. Keywords: mortality risk; longevity risk; natural hedging.

(6) Contents Essay I : The Optimal Strategy in Insurance Product Design. 11. 1. Introduction .............................................................................................................. 12 2. Literature Review..................................................................................................... 16 3. Theoretical Development ......................................................................................... 21 3.1 Idea Scratching .................................................................................................. 21 3.2 Formal Development ......................................................................................... 22. 政治大. 4. Numerical Illustrations............................................................................................. 34. 立. 4.1 The optimal strategy with γ=δ ........................................................................... 34. ‧ 國. 學. 4.2 The secondary strategy with 0<γ<δ ................................................................... 41. ‧. 5. Practice and Conclusion ........................................................................................... 42. sit. y. Nat. Essay II : The Uncertainty to Optimal Strategy in Life Insurance. n. al. er. io. Product design. Ch. i n U. 45. v. 1. Introduction .............................................................................................................. 46. engchi. 2. Theoretical Development ......................................................................................... 49 3. Models...................................................................................................................... 53 3.1 Model for Mortality Rate................................................................................... 53 3.2 Model for Interest Rate ...................................................................................... 56 4. Numerical Illustrations for Uncertainty ................................................................... 57 4.1 The optimal strategy with γ(t) =δ(t)................................................................... 57 4.2 The secondary strategy with 0<γ(t)<δ(t) ........................................................... 64 5. Conclusion ............................................................................................................... 67.

(7) Essay III: The Natural Hedging Strategy for Annuity Product in Product Design. 68. 1. Introduction .............................................................................................................. 69 2. Theoretical Development for Annuity Products ...................................................... 73 3. Numerical Illustrations for Annuity Products .......................................................... 78 4. Conclusion ............................................................................................................... 80. 政治大 Appendix ..................................................................................................................... 84 立 References ................................................................................................................... 81. ‧ 國. 學. A1. The deriving process of the equation (15) ........................................................ 84 A2. The close form of differential equation of equation (13).................................. 85. ‧. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v.

(8) List of Figures Figure 1 Cashflows of the Popular Increasing Whole Life Insurance, with δ = 2% and γ = 3% during the premium payment period and 6% afterwards. ......... 14 Figure 2 The boundary reserve with positive (δ – γ) on (μx+t, tVx) plane, with δ =2.5% and γ=2.3% in the case. ................................................................................ 26 Figure 3 The boundary reserve with negative (δ – γ) on (μx+t, tVx) plane, with δ =2.5% and γ=2.75% in the case. .................................................................. 26 Figure 4 The boundary reserve with (δ – γ) = 0 on (μx+t, tVx) plane, with δ =2.5% and. 政治大. γ=2.5% in the case. ...................................................................................... 28. 立. Figure 5 The boundary reserve with positive (δ – γ) on (x+t, tVx) plane, with δ =2.5%. ‧ 國. 學. and γ=2.3% in the case. ................................................................................ 29. ‧. Figure 6 The boundary reserve with negative (δ – γ) on (x+t, tVx) plane, with δ =2.5% and γ=2.75% in the case. .............................................................................. 30. y. Nat. io. sit. Figure 7 The boundary reserve with δ – γ = 0 on (x+t , tVx) plane, with δ =2.5% and. n. al. er. γ=2.5% in the case. ...................................................................................... 33. Ch. i n U. v. Figure 8 The boundary reserve of optimal strategy on (μx+t, tVx) plane, with δ =2.5%. engchi. and γ=2.5% in the case................................................................................. 51 Figure 9 The boundary reserve of optimal strategy on (x+t , tVx) plane , with δ =2.5% and γ=2.5% in the case................................................................................. 52 Figure 10 The boundary reserve of an annuity product with positive γ on (μx+t, tVx) plane, with δ =2.5% and γ=2.5% in the case ............................................... 75 Figure 11 The boundary reserve of an annuity with positive γ on (x+t, tVx) plane, with δ =2.5% and γ=2.5% in the case .................................................................. 77.

(9) List of Tables Table 1. Background Information for Cash Flow Testing of a Popular Compound Increasing Rate Whole Life Insurance Product ........................................... 14. Table 2. Basic Assumptions for the New Form of the Whole Life Insurance Product ...................................................................................................................... 35. Table 3. The Liability at the End of the 5th Policy Year of Illustrated Insurance Product for Different Mortality Bases ......................................................... 36. Table 4. The Realized Values of the Force of Interest Rate for the First Five Policy. 政治大. Years. ............................................................................................................ 37. 立. The Liability at the End of the 5th Policy Year of Product Design 2 for. 學. ‧ 國. Table 5. Different Mortality Bases ............................................................................ 38 The Realized Values of the Force of Interest Rate for the First Five Policy. ‧. Table 6. Years............................................................................................................. 40. y. Nat. The Liability at the End of the 5th Policy Year of Product 3 for Different. io. sit. Table 7. n. al. er. Mortality Bases ............................................................................................ 40 Table 8. Ch. i n U. v. The Liability at the End of the 5th Policy Year of Illustrated Insurance. engchi. Product for Different Mortality Bases ......................................................... 41 Table 9. The Parameters of the LC model ................................................................. 55. Table 10 The Parameters of CIR Model ..................................................................... 57 Table 11 Basic Assumptions for the New Form of the Whole Life Insurance Product ...................................................................................................................... 58 Table 12 The Liability at the End of the 5th Policy Year of Illustrated Insurance Product with Stochastic Interest Rate Model for Different Mortality Bases ...................................................................................................................... 60.

(10) Table 13 The Liability at the End of the 5th Policy Year of Illustrated Insurance Product with Stochastic Interest Rate and Stochastic Mortality Rate for Different Mortality Bases ............................................................................ 61 Table 14 The Realized Values of the Force of Interest Rate for the First Five Policy Years............................................................................................................. 63 Table 15 The Liability at the End of the 5th Policy Year of Product Design 2 for Different Mortality Bases ............................................................................ 63 Table 16 The Liability at the End of the 5th Policy Year of Illustrated Insurance. 政治大 Different Mortality Bases ............................................................................ 65 立 Product with Stochastic Interest Rate and Stochastic Mortality Rate for. ‧ 國. 學. Table 17 Indication of Increasing Rate to a Target Profit Rate of 5% in Mortality in a Compound Increasing Rate Whole Life Insurance Product ........................ 67. ‧. Table 18 The Basic Information of the Illustrated Annuity Product........................... 78. Nat. sit. y. Table 19 The Liability at the End of the 5th Policy Year of Illustrated Annuity. n. al. er. io. Product for Different Mortality Bases ......................................................... 79. Ch. engchi. i n U. v.

(11) Essay I :. 治政 The Optimal Strategy in 大 Insurance 立 ‧. ‧ 國. 學. Product Design. n. er. io. sit. y. Nat. al. Ch. engchi. 11. i n U. v.

(12) 1. Introduction In 2003, the increasing whole life insurance product was sold prosperously in Taiwan.. The increasing whole life insurance product was developed in early stage in. the history of Taiwan life insurance industry.. The kind with death benefit. compounded by high increasing rate stood out in the market with popularity. Since the product was designed against the currency inflation in increasing its death benefit by compound increasing rate, a lot of consumers brought this kind of compound increasing rate whole life insurance products at the time.. 政治大 From the point of view of the life insurance companies, the kind of compound 立. the selling technique of salespersons.. 學. ‧ 國. increasing rate whole life insurance products can catch the eye of consumers through. The life insurance company may quickly. ‧. collect the premium income that makes income statement look great.. The most. sit. y. Nat. important thing is that the company may take advantage of the collecting cash and For a life. io. er. financial leverage technique to optimize their investment strategy.. insurance sells an insurance product is not just a product sold, the main reason for. al. n. v i n them is to increase its premiumC income that let them U h e n g c h i obtain the right on the usage of. money from the insurance customers. This is not a news from the life insurance industry and easily to understand why the life insurance companies try very hard in product design in order to attract customers.. A branch company1 of global life. insurance group in Taiwan developed and sold first this kind of compound increasing rate whole life insurance products in 2003.. As they wish, the growth of premium. income of the life insurance company climbs sharply.. 1. The Aegon Life Insurance Company in Taiwan was the primary company that developed the first product of the kind of compound increasing rate whole life insurance products in 2003 12.

(13) At the same time, there are two to three life insurance companies selling the kind of compound increasing rate whole life insurance products in the market.. On the. other hand, most companies in the market state in a cautious way though the product may carry in a lot of premium income. They did not follow the trend to sell the kind of compound increasing rate whole life insurance products because they did not know the risk very well with the kind of the products.. They cannot explain why both of. the payoff for the buyers and the commission for the salespersons of products are better than other kinds of insurance products and then who is the loser in this business.. 政治大 is the risk?” or “Is the risk underestimated?” Unfortunately, the actuaries of those 立. Most people would ask “What kind of risk of the product is exposed to?” “How much. companies selling the compound increasing rate whole life insurance hardly know the. ‧ 國. 學. risk and don’t know how to manage such kinds of life insurance products.. ‧. In 2004, Taiwan Insurance Institute initiates a project on the study of this kind of. y. They took an example of this kind of compound. sit. Nat. increasing whole life products.. er. io. increasing rate whole life insurance products in analysis that may reveal some exotic. n. a l The background informationi vis in Table 1 and the cash C h in the Figure 1.U nThe researchers at the time flow testing of the product is shown engchi. phenomenon of the product.. discovered that the severity of insolvency existed in the kind of products.. 13.

(14) Table 1. Background Information for Cash Flow Testing of a Popular Compound Increasing Rate Whole Life Insurance Product. Age of insured. 35. Gender. Male. Face amount. 200,000. The initial value of force of interest rate (δ). 2%. Death benefit. 200,000 compounded by 3% and 6%. Benefit period. Whole life. Method of paying premium. 20-year period installment premium. Investment Return. 5%. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. Figure 1 Cash flows of the Popular Increasing Whole Life Insurance, with δ = 2% and γ = 3% during the premium payment period and 6% afterwards.. We can see the cash flows of the product are downward sharply on the latter age of the policy year. That is to say that the risk is exposed to the stage of older age of the policyholder.. If the policy lasts longer, the risk of this policy is getting severer.. This is unusual to a life insurance product. 14. Normally, the risk of a life insurance.

(15) product happens in the early policy years of the life insurance contracts.. That is why. the underwriting of life insurance is very important in order to prevent the early payout of death benefit of life insurance.. But the risk profile of the kind of. compound increasing rate life insurance seems not like that of a typical life insurance product.. It is getting curious for all the actuaries in solving the puzzle in the life. insurance industry. First of all, we focus on what the risk for the kind of insurance product is.. If it. is not exposed to the mortality risk, then we want to find out what risk it should be. 政治大 rate is important element 立 in related to the risk navigation and try to come out a. and how much it is exposed to.. And, moreover, we try to prove that the increasing. ‧. ‧ 國. alternatives.. 學. strategy in management the risk for the kind of the product in the different. io. than preset interest rate of the product.. We can see it is reasonable as it corresponds. n. al. sit. It is the product with compound increasing rate higher. er. Nat. whole life insurance product.. y. In this article we find out the mystery of this kind of compound increasing rate. i n U. v. to the theoretical results that we shows in the section 2 “Theoretical Development”:. Ch. engchi. when the (δ – γ) < 0, the product is exposed to the longevity risk.. It shows that the. company did not evaluate the product properly so as to collect not enough premiums and had to depend on a good return in the investment to cover the shortage in cash, the insolvency of the product would happen in decades later. And the illustration is presented under the condition that the company did not distribute the earnings during the whole policy year.. We can see the kind of products without properly evaluated. that lay the company in dangerous financial situation. We know the risk of the kind of the product and we can go further to deliver an optimal strategy in the product design according to its risk appetite. 15.

(16) In “Literature Review”,. The remainder of this article is organized as follows.. we study most papers of the subjects of risk mitigation and risk management in the In the section “Theoretical Development”, we deduce and. previous researches.. elaborate our theoretical development for a natural hedging strategy using the Laplace transform that we generate a specified life insurance.. In the next section “Numerical. Analysis” we use an existing model with U.S. mortality experience and demonstrate how to implement our proposed strategies to assess the natural hedging effect with numerical examples in the section.. We extend our analysis to the cases of annuity. 政治大. product in the next section, and then we conclude in the last section.. 學. ‧ 國. 立. 2. Literature Review. ‧. The risk emerging from the changes of mortality rate is a critical problem to the When a quick spread of deadly contagious disease causes a. sit. y. Nat. insurance providers.. n. al. er. io. great deal of casualty the mortality rate would increase and the mortality risk rises up.. i n U. v. In general timing, the mortality rate is decreasing gradually due to progressive medical technology nowadays.. CThat h emakes h i risk rise up. n g clongevity. Both mortality. risk and longevity risk are mortality rate risks caused by the dynamic changes of mortality rate. Whether it is mortality risk or longevity risk, the mitigation of mortality rate risk has a lot of discussion in literatures.. The longevity risk catches more attention. owing to its persistent trend. As the exceeding living becomes a normal case to everyone in the world, it is evolving into a systemic risk.. On the other hand, the. dynamic pattern of mortality risk is focusing on a sudden mortality rate shock. Those risks are not only a risk to the product providers but also a costly problem to 16.

(17) economic society.. The technique of risk mitigation in both risks is an important. issue to risk management. Considering how to mitigate mortality rate risk, many studies start with the longevity risk.. Transferring longevity risk externally with the financial vehicles of. capital market is the earliest suggestion in literatures to solve the problem.. Blake. and Burrows (2001) propose a solution accordingly through capital market by issuing survivor bonds to mitigate the longevity risk exposed to living benefit providers. Other prior studies also provide mitigation solutions through capital market, including. 政治大 Blake, Cairns, and Dawson, 立2006), mortality swaps (Lin and Cox, 2007), mortality. survivor bonds (Denuit, Devolder, and Goderniaux, 2007), survivor swaps (e.g. Dowd,. ‧ 國. 學. securitization (e.g. Dowd, 2003; Lin and Cox, 2005; Cairns, Blake, and Dowd, 2006a; Blake, Cairns, and Dowd, 2006; Cox, Lin, and Wang, 2006;; Blake, Cairns, Dowd,. ‧. and MacMinn, 2006).. These studies suggest mitigating longevity risk by. y. Transferring the risk. sit. Nat. transferring the risk to the investors of capital market.. al. n. policy.. er. io. externally may be a possible way but may not diminish the primary risk rooted in the. i n U. v. These methods are also involved with uncertainty in market environment. Ch. and transaction cost to the providers.. engchi. Natural hedging strategy is an alternative.. Instead of transferring risk outside. the initiative risk bearing company, some studies take steps to alleviate risk by way of inner mitigation within the company.. Natural hedging strategy makes risk. mitigation executed in the insurance company internally.. A whole life insurance. product is exposed to mortality risk and an annuity product is exposed to longevity risk. Companies can take advantage to diminish the threat of mortality rate risk through selling life insurance products and annuity products simultaneously.. 17.

(18) The main stream of the natural hedging strategy in studies is to optimize the proportion of product portfolio that helps maximize the hedging effect.. Cox and Lin. (2007) create a product portfolio with a life insurance product and an annuity product that shows the existence of natural hedging effect between life insurance product and annuity product.. Wang et al. (2010) soon propose an immunization model to achieve. the optimal life insurance to annuity ratio to mitigate longevity risks.. They address. how the natural hedging strategy is applicable in insurance industry.. They evaluate. the hedging effect to the reserves of a product portfolio and utilize matching of. 政治大 product by the impact of mortality rate changes. Mortality duration/convexity is the 立. durations and convexities to generate the optimal ratio of life insurance to annuity. most common used technique in determining optimal natural hedging strategy.. In. ‧ 國. 學. most recent studies Tsai and Chung (2013), Lin and Tsai (2013) adopt the. changed. proportionally. constantly.. They. extend. the. y. or. sit. mortality. Nat. the. ‧. duration/convexity matching to the prices of life insurance and annuity product with. io. er. duration/convexity matching application in determining the weights of two or three products in a product portfolio that can obtain the maximum effect of mitigation.. n. al. Ch. engchi. i n U. v. In other studies of mitigating mortality rate risk in a product portfolio, we see that Tsai, Wang, and Tzeng (2010) are taking the higher-order moments of the mortality risk distribution into consideration.. They suggest the Conditional. Value-at-Risk Minimization (CVaRM) approach in natural hedging strategy.. The. CVaRM approach can generate a narrower quantile of loss distribution compared to immunization model (Wang, Huang, and Hong, 2013).. They also show CVaRM. approach can be better off on risk reduction to immunization model (Wang et al., 2010) due to the missing consideration of the variance of the product portfolio. Unlike previous studies, Wang, Huang, and Hong (2013) employ an 18.

(19) insurance-policy-data set from the insurance company that allows them to build up an experienced mortality rate for life insurance and annuity product separately.. They. begin with a portfolio of zero coupon bonds, life insurance policies, and annuity policies to determine the value of the portfolio.. And they construct a feasible. objective function that makes it possible to catch the mispricing effect and the variance effect of the entire company portfolio.. Through the experienced data of the. company, they may carry out the objective function to control the variance and mispricing effects of longevity risk at the same time and address a natural hedging. 政治大 Through the natural hedge 立 strategy, the hedging effect helps to diminish or offset. strategy practically.. All those. immunization models in prior literatures are in common on three aspects.. First, the. ‧. ‧ 國. 學. the risk with appreciate allocation of life insurance and annuity policies.. essential portfolio of natural hedging strategy is constructed with the life insurance Second, the optimal weight of life insurance to annuity. sit. y. Nat. and annuity policies.. n. al. cannot mitigate risk efficiently.. er. io. policies determined by durations/convexities matching process, which is a static ratio,. i n U. v. Third, companies may have obstacle in executing. Ch. engchi. selected groups selling practically due to market environment and the life insuranceannuity product portfolio existing the cohort effect in different generations. We find a way of natural hedging strategy within a policy that we can /mitigate the mortality rate risk through delicate product design.. We are able to optimize the. risk mitigation in a simply death benefit protection product and have not to build up a weighted portfolio of life insurance and annuity.. Every insurance claim contains at. least two key factors of “when to pay” and “how much to pay” at the events written in policy but normally people overlook “how much to pay” can be a variable related to risk exposure.. We may set the death benefit as a function of face amount and vary 19.

(20) along with time in the content of product design. We attempt to utilize both the timing of benefit and the amount of benefit to be variables in determining the risk of the product.. If we design the product in which. the risk resulting from “when to pay” is more/less than the expected and that from “how much to pay” is less/more than the expected accordingly, we may immunize the risk within the product.. The expected value is determined by the present value of the. policy cash flows and the cash flows are decided by the future lifetime distribution based on mortality assumption.. 政治大. The time value is the bridge between “when to pay” and “how much to pay”.. 立. can be represented the time value of death benefit.. 學. ‧ 國. The risk rooting in “when to pay” controlled by the factor, the force of interest rate, If the death benefit is paid earlier. than expected timing the company faces losses of the time value.. In this case, we. ‧. may envisage that the amount of benefit is paid less in response to the loss of time. y. Nat. Thus, we introduce a new factor, the force of amount, in determining the. sit. value.. n. al. er. io. amount of death benefit and it is also engaged to the risk rooting in “how much to pay”.. i n U. v. We thus are able to create the product that when the benefit is paid earlier/later. Ch. engchi. than expected timing its amount of benefit is paid less/more accordingly. We implement our strategies by assessment of risk in product design as illustrated products in this article, in which we present the optimal strategy of perfect mitigation and the secondary strategy in diminishing risk.. Our strategies through. risk immunized product facilitate the marketing and management and also make mortality risk mitigation feasible to carry on.. Following our strategies in product. design, the insurance product has a lot more possibility in engaging to financial instrument without considering the mortality rate risk. 20. Our finding can also provide.

(21) a further research or re-design of previous studies that ignore the existence of a mortality rate risk in an insurance product valuation or risk management consideration. As far as we know, there are no such approaches in the field of the natural hedging literature yet.. 3. Theoretical Development 3.1 Idea Scratching. 政治大 We first analyze a traditional whole life insurance product in the framework of 立. The net single premium for the whole life insurance with 1- unit. 學. ‧ 國. Laplace transform2.. face amount expressed by the notations of Bowers et al. (1997) can also be expressed. ‧. in Laplace transform format as:. y. (1). sit. . Nat.  s 0 e b( s) s p x  x  s ds  L b( s)  f x( s)   x   ,. n. al. er. io. where δ is the force of interest, b(s) is the death benefit at time s per 1- unit face. Ch. i n U. v. amount, fx(s) is the probability density function of the future lifetime random variable. engchi. S at age x: spxx+s, in which μx+s is the force of mortality at age of x+s and spx denotes the probability of a person at age x who will survive s years. Note that b(s)=1 for traditional whole life insurance, which means that the death benefit is fixed at the face amount of the policy.. In such a case, equation (1) implies. that the net single premium is a function of the force of interest δ with respect to fx(s). Should mortality rise unexpectedly, the collected premium has insufficient time to. 2. The Laplace transform L  f (t )    e t f (t )dt is an integral transform widely used in mathematics with 0. many applications in physics and engineering. It turns convolution into multiplication, the latter being easier to solve because of its algebraic form. 21.

(22) accumulate to the death benefit; on the other hand, the policyholders as a whole pay too much if mortality improves more than expected. and insured subject to the mortality risk.. The product makes the insurer. Can we mitigate such risk through product. design? One design is to make the death benefit an increasing function of death time so that the accumulated value of the premium can match the death benefit no matter how mortality varies.. The Laplace transform format sheds light on a possible solution:. changing the parameter of (.) as. 政治大.  x      0 e  se s s p x  x  s ds  L e s f x( s) . . 立. (2). ‧ 國. 學. The product implied by equation (2) is an increasing whole life insurance policy. The parameter γ. ‧. Its death benefit increases continuously at the annual rate of .. Appropriate choices on (, δ), such as  δ,. n. al. sit. p x  x  s ds  1 .. io. s. er. . notice that x(0) = 0. We. y. Nat. controls how much to pay while δ would reflect the time value of payment.. i n U. v. can make the present value of the death benefit insensitive to the timing of death that in turn is affected by mortality.. Ch. engchi. Such design can thus mitigate the mortality risk.. 3.2 Formal Development We elaborate the above idea that we further investigate the impact of risks on the expected reserve of the appropriately calibrated whole life insurance product. mortality pressure increases when the policyholder gains his age.. The. We try to find a. way that may eliminate the difference of reserves by the impact of ages (with assumption that the mortality rate is a function of age). That also means the calibrated 22.

(23) whole life insurance product may be insensible to the ages.. Thus, when a new. policy of the calibrated product is sold now (i.e., at time 0) to a customer at age x with face amount 1, the expected reserve at time t of the policy is as in equation (3): . t.  t  s  s. V x  0 e e e. s. p x t  x t  s ds .. (3). Referring to the derivation in Bowers et al (1997, chapter 4), we may decompose t. Vx. t. into two parts:. V x  E  e.  ( t  S )  S. 政治大.   E  e (t  S )e  S | S  h  Pr  S  h   E  e (t  S )e  S | S  h  Pr  S  h  , e. 立. ‧ 國. 學. where Pr  S  h   h q x t and Pr  S  h   h p x t .. (4). .. ‧. Since the conditional p.d.f. of S given S ≤ h is 0sh. n. Ch. E  e (t  S )e  S | S  h   0 e (t  s )e  s h. elsewhere. er. io. al. sit. y. Nat.  f  s  s p x t  x t  s   f  s | S  h   F  h q h x t  0 . ,. engchi s. p x t  x t  s h. q x t. i n U. v. ds (5). and. E  e (t  S )e  S | S  h   e h E  e (t  S h )e  ( S h ) | ( S  h)  0  e he  h E  e (t  S )e  S | ( S  h)  0   e he  h t V x  h .. (6). Substituting equations (5) and (6) into (4) yields. V x  0 e h. t.  ( t  s )  s s. e. p x t  x t  s q h x t.  h  h ds h q xt  e e t V x h h p x t. 23. (7) ..

(24) Multiplying both sides of equation (7) by -1, adding t V xh , and then dividing by h, we obtain. 1  e e V x  h  t V x 1 h  (t  s )  s  0 e e s p x t  x t  s ds  t V x  h h h h.  h  h. t. h. p x t . (8) .. The limits of the two items on the right-hand side of equation (8) are as follows:. 1 h  (t  s )  s  s p x t  x t  s ds   0 e (t  s )e  s s p x t  x t  s ds |s 0 e e  s 0 h 0 h s  e (t  s )e  s s p x t  x t  s|s 0  e t  x t ,. lim. and. h 0. e h. s. p x t . ‧ 國. 1  e. 政治大. 立.  1  e s e  s s p x t  |s 0  s  t V x  e s e  s s p x t  x t  s|s 0  (   )e se  s s p x t|s 0 .  tV x. 學. lim t V x  h.  s  s. ‧.  t V x   x t  (   ) . Nat. sit. n. er. io. .. al. Ch. The above derivation means that. . (10). y. The limit of equation (8) is thus e  t  x  t  t V x   x  t    . (9). engchi. i n U. (11). v.  t . t V x  e  x  t  t V x   x  t      x. (12). Rearranging equation (12), we get.  t t V x  e  x t x . tV x   x t     Note that. t e  x t. (13) is always positive since  x t  0 .. following boundary conditions of the expected reserves:. 24. Then we have the.

(25) 1. when μx+t+δ–γ> 0 and  t V x ≥ 0, t V x . t e  x t ,  x t    . 2. when μx+t+δ–γ< 0 and  t V x ≥ 0, t V x  x. t e  x t ,  x t    . x. 3. when μx+t+δ–γ> 0and  t V x ≤ 0, t V x  x. t e  x t , and  x t    . 4. when μx+t+δ–γ< 0and  t V x ≤ 0, t V x . t e  x t .  x t    . x. 立. The line of boundary reserve is. 政治大. ‧ 國. 學. t e  x t tV x=  x t     .. (14). Nat. y. ‧. The above equation of the boundary reserve can be written3 as:. sit. n. al. (15). er. io.   xt       tV x  e t   e t     .. Ch. engchi. i n U. v. 3.2.1 The Analyses on the (μx+t, tVx) plane. As Figures 2 and 3 display, Equation (15) represent hyperbolas centered at (–δ+γ, eγt) on the (μx+t, tVx) plane. = –δ+γ and tVx = eγt.. The two asymptotes of the hyperbola are given by μx+t. The hyperbola with positive (δ – γ) lays in the 1st and 3rd. quadrants coordinated with respect to the center while the hyperbola with negative (δ – γ) lays in the 2nd and 4th quadrants as shown by Figure 2 and Figure 3 respectively.. 3. The hyperbola in Figure 2 exhibit positive slopes while that in Figure 3. Please see the appendix for the detail deducing process 25.

(26) have negative slopes.. Condition 2. Condition 1. 立. 政治大. ‧ 國. 學. Figure 2 The boundary reserve with positive (δ – γ) on (μx+t, tVx) plane, with δ =2.5% and γ=2.3% in the case. Note: We are aware that negative μx+t is not reasonable but retain them to show a complete. ‧. hyperbola.. n. er. io. sit. y. Nat. al. Ch. i n U. v. e n g c Condition hi 3. Condition 4. Figure 3. The boundary reserve with negative (δ – γ) on (μx+t, tVx) plane, with δ =2.5% and γ=2.75% in the case.. 26.

(27) In Figure 2, the curve on the down-right of the center matches the criteria of the condition 1.. Assuming that μx+t is an increasing function4 of age x, with criterion.  ≥ 0 in condition 1, we may deduce that the t V x is increasing along with μx+t, tV x x. and with criterion μx+t+δ–γ> 0, the lower bound of expected reserve in condition 1 should be on the down-right in respect to the center of hyperbola. in Figure 2 matches the criteria of the condition 2.. The up-left curve. With criteria of μx+t+δ–γ< 0 and.  ≥ 0, note that the condition 2 shown on the down-left part of the hyperbola tV x x. does not exist in reality since the force of mortality μx+t should not be negative.. 政治大 Following the same deduction, we identify the hyperbola with negative (δ – γ) in 立. Figure 3, of which the up-right curve represents the upper bound of the expected. ‧ 國. 學. reserve in the condition 3 and the down-left curve represents the lower bound of the. and 4 is decreasing along with μx+t.. The expected reserve may be more/less than or. sit. y. ‧. The feature of boundary reserve in condition 3. Nat. expected reserve in the condition 4.. io. al. er. equal to the lower/upper bound of expected reserve in practice.. v. n. When narrowing down the possibility of (δ – γ) in the four conditions we find out. Ch. engchi. i n U. the boundary reserve is equal to the expected reserve if (δ – γ) = 0.. When (δ – γ) is. equal to 0, the lower/upper bound of the expected reserve in Figure 2 is folded together with the upper/lower bound of that in Figure 3.. The feature of the special. case with (δ – γ) = 0 turns into a horizontal line as shown in Figure 4 which is exactly one of the asymptotes of the hyperbola tVx = eγt.. 4.  d t V x   x t  tV x  x d  x t x. 27.

(28) Figure 4. 政治大. The boundary reserve with (δ – γ) = 0 on (μx+t, tVx) plane, with δ =2.5% and γ=2.5% in the case.. 3.2.2 Further Analyses on the (x+t, tVx) plane. 學. ‧ 國. 立. ‧. Mortality rate risk exists because we evaluate risk according to ages.. If. In fact, neither can we observe. io. er. the companies may not have mortality rate risk.. sit. y. Nat. insurance companies price life insurance products according to real aging situation,. aging of individuals directly, nor can we price the mortality risk by each individual’s. al. n. v i n What we can C observe in nature is individual age and mortality rate hengchi U. aging situation.. with respect to age statistically.. Then, the force of mortality is quantified to express. aging with respect to age in methodology.. Since we price insurance product based. on ages, not aging, mortality rate risk emerges accordingly.. Thus, acquiring the. information on (x+t, t V x ) plane is important in determining the mortality rate risk of the specified product along with the changes of mortality rate. Next, we intend to demonstrate the relationship of expected reserve and age by the feature of the boundary reserve t V x . t e  x t coordinated on the (x+t , tVx) plane.  x t    . 28.

(29) With the information on the (μx+t, tVx) plane in last section, we can transform the t e  x t feature of the boundary reserve t V x  with respect to the force of  x t    . mortality into the feature of that with respect to age by assuming that the force of mortality μx+t is a monotonic increasing function of age x+t.. We then can illustrate. features of the boundary reserve according to age x+t on the (x+t , tVx) plane and explore the relationship of expected reserve and age. In case of the positive (δ – γ), we transform the feature of the boundary reserve on. 政治大 Also, in case 立 of the negative (δ – γ), we transform the feature of the. the (μx+t, tVx) plane in Figure 2 into the feature on the (x+t , tVx) plane as shown in Figure 5.. ‧ 國. Figure 6.. 學. boundary reserve in Figure 3 into the feature on the (x+t , tVx) plane as shown in The negative values of μx+t on the (μx+t, tVx) plane cannot exhibit on the. ‧. (x+t , tVx) plane for each age is of positive force of mortality.. n. er. io. sit. y. Nat. al. Ch. i n U. v. i e n g c hCondition 1: Upper bound. Figure 5. The boundary reserve with positive (δ – γ) on (x+t, tVx) plane, with δ =2.5% and γ=2.3% in the case.. Note that the illustration is using the Makeham model . x t.  9.566 10 4 + 5.162 10 5 1.09369 x t ,. which is cited from Melnikov and Romaniuk (2006) and the original data is based on the mortality rates from 1959 to 1999 in American (Pollard, 1973) 29.

(30) Condition 3: Upper bound. Condition 4: Lower bound. Figure 6. 政治大. The boundary reserve with negative (δ – γ) on (x+t, tVx) plane, with δ =2.5% and γ=2.75% in the case.. 立. Note that the illustration is using the Makeham model . x t.  9.566 10 4 + 5.162 10 5 1.09369 x t ,. which is cited from Melnikov and Romaniuk (2006) and the original data is based on the mortality. ‧ 國. 學. rates from 1959 to 1999 in American (Pollard, 1973).. The feature of the boundary reserve in Figure 5 shows the ages The case in condition 2. io. er. with positive value of μx+t which is the case in condition 1.. sit. y. Nat. the (x+t ,tVx) plane.. ‧. We demonstrate the four conditions of equation (13) in corresponding features on. does not show out on the (x+t ,tVx) plane for the value of μx+t is negative.. n. al. Ch. n U engchi. iv. In Figure 6,. the feature of boundary reserve on the up-right part is the case in condition 3 that follows the criterion of μx+t+δ – γ>0; while the other on the down-left part is the case in condition 4 that follows the criterion of μx+t+δ – γ<0 on the (x+t , tVx) plane. The feature of the boundary reserve in Figure 5 appears faster increasing than that in 1st quadrant of Figure 2 along with the horizontal axis.. As the horizontal axis of. μx+t is changed into that of the age x+t, the horizontal axis is re-scaled evenly at the measurement of age on the (x+t , tVx) plane.. Figure 5 is more than that. The slope  t V x of elderly ages in x.  d t V x   x t d tV x  in Figure 2 since t V x  and  x  t d  xt x d  x t x x 30.

(31) of the elderly age is much higher than that of the young ages. In Figure 5, the boundary reserve of specified product with positive (δ – γ), appears that the risk rooting in “when to pay” is more than that rooting in “how much to pay”.. As for our death benefit product, the risk rooting in “when to pay”. represents mortality risk because when the realized mortality rate rises up, it means that more than expected people of cohorts are paid earlier than expected timing.. The. insurance companies get loss by the time value of extra death benefit claim due to The risk rooting in “how much to pay” in our design is. increase of mortality rate.. 政治大 The立 risk rooting in “how much to pay” is a form of longevity. that we set the increment of death benefit each time that is only provided to the survivors at the time.. ‧ 國. 學. risk. The mortality risk exposure is more than longevity exposure in Figure 5.. In. general, most life insurance products are grouped in this type of feature in Figure 5 as. ‧. the mortality risk is more than the longevity risk.. y. Nat. io. The age of 33 years old is a critical age and the boundary reserve of all the. n. al. er. plane.. sit. The feature in Figure 6 is seemingly but not exact hyperbola on the (x+t , tVx). ages younger than age 33 are unbounded.. Ch. i n U. v. In this case, this means the maximum. engchi. reserve is unlimited and the company selling this product younger than age 33 may be responsible for more liability than he can afford.. The kind of compound increasing. rate whole life insurance products may lead the insurance company into dangerous situation.. We have to be cautious in selling this product to the younger ages.. The boundary reserve in the case of negative (δ – γ) is decreasing along with age more than 33 years old.. When (δ – γ) is negative, we may say that the risk rooting in. “how much to pay” in our design is more than that rooting in “when to pay”.. That is. to say the longevity risk of the case is more than mortality risk following the similar 31.

(32) elaboration in the case of positive (δ – γ) above. rare.. This type of insurance product is. But it ever shortly appeared in Taiwan insurance market.. designed an increasing whole life insurance with γ > δ.. The companies. For instance, the death. benefit is based on face amount compounded by 4% annually and annual interest rate is 2.5%. As for the special case on the (μx+t, tVx) plane in Figure 4, we transform the horizontal axis to age x+t and display the case of (δ – γ)=0 on (x+t, tVx) plane as shown in Figure 7. expected reserve.. The upper bound is overlapped with the lower bound of the. 政治大 The立 difference of features between Figure 4 and Figure 7 is the. The feature of the boundary reserve is a horizontal line that is the. same as in Figure 4.. ‧ 國. 學. measurement of horizontal axis that cannot alter the shape of horizontal line.. The. function of the special horizontal line tVx = eγt on (x+t, tVx) plane is the same as that on. ‧. the (μx+t, tVx) plane since the case is irrelevant to the variable of horizontal axis.. In. y. sit. n. al. er. We then come out an only solution that fulfills the criteria of the four. io. reserve.. Nat. this special case, the feature is not just the boundary reverse but also the expected. i n U. v. conditions of equation (13) represented the relationship between expected reserve and age.. Ch. engchi. 32.

(33) Upper bound=Lower bound. Figure 7. 政治大. The boundary reserve with δ – γ = 0 on (x+t , tVx) plane, with δ =2.5% and γ=2.5% in the case.. 立. x t.  9.566 10 4 + 5.162 10 5 1.09369 x t ,. 學. ‧ 國. Note that the illustration is using the Makeham model . which is cited from Melnikov and Romaniuk (2006) and the original data is based on the mortality rates from 1959 to 1999 in American (Pollard, 1973). ‧ y. sit. io. expected reserve is irrelevant to age.. We may make use of this property on risk. er. Nat. The expected reserve is a horizontal line on (x+t, tVx) plane reveals that the. n. aWhen v product with (δ – γ)=0 , i l C we design such a specified n U the risk rooting in “when to pay” ishexactly equal to the e n g c h i risk rooting in “how much to mitigation within product.. 5. pay”.. The mortality risk is mitigated totally by the longevity risk for each age in the. case.. Furthermore, the expected reserve is level to all ages and to the force of. mortality of each age.. The expected reserve of the policy will not be affected by the. Please note that the indication of the insurance product with criterion of (δ – γ) = 0 is based on that premium is paid completely. That is single premium paid for the product. For an installment premium paid for the product, we may take each time of installment premium as a single premium each time. With each single premium, we then design its benefit accordingly. Then, combining all installment premiums is also combining their benefit of each single premium of the policy. We may see the installment premium paid of the product as combining a lot of single premium paid products. If we want to have the optimal strategy for the installment premium paid in design of the kind of compound increasing rate whole life insurance product, we only need to optimize in design of the single premium paid for the product. 5. 33.

(34) changes of mortality rate and will be equal to the realized reserve.. As the realized. reserve remains the same as the expected reserve with the changes of mortality rate, the risk is immunized within the policy.. 4. Numerical Illustrations We provide two strategies of risk mitigation: the optimal strategy and the secondary strategy in product design whether is appropriate to the condition of insurance providers. insurance products.. 政治大 The secondary 立 strategy is used when the optimal strategy is not. The optimal strategy can minimize the risk emerged from. ‧ 國. 學. available or for other risk managements purpose.. ‧. 4.1 The optimal strategy with γ=δ. sit. y. Nat. To optimize the strategy of natural hedging in product design, we apply our. al. n. much as possible.. er. io. theoretical derivation to keep the γ in consistent with the force of interest rate δ as. i n U. v. For the timing respective to the most update information of the. Ch. engchi. force of interest rate δ, in determining the force of amount γ, we can separate into two different timing categories in product design, one is pre-determinate and the other is post-determinate.. The pre-determined category is that we design the product and. determine the force of amount before we get the information of the in-time interest rate in real time.. Thus, the post-determined category is the product design in. determining the force of amount γ, which is determined immediately after we know correspondent the interest rate in real time.. We also provide the effects according to. stochastic interest rate model and stochastic mortality rate model. 4.1.1 The Pre-determined Category 34.

(35) The typical product in this category is increasing whole life insurance with a general increasing amount on death benefit. The basic assumptions are set up in Table 2.. We assume that the face amount is. US$100,000 for the specified increasing whole life insurance and the premium is paid by single premium.. Assume that the company only sells the specified increasing. whole life insurance product with given value of the force of interest rate δ.. The. natural hedging strategy depends on the product design of the force of amount γ. The γ can be any given value in a reasonable risk mitigation strategy. Table 2. 政治大 25, 45. Basic Assumptions for the New Form of the Whole Life Insurance Product. 立. Age of insured. ‧ 國. Face amount. Male. 學. Gender. 100,000 4%. Death benefit. 100,000 compounded by γ(t). sit. Single premium. io. n. er. Method of paying premium. al. y. Whole life. Nat. Benefit period. ‧. The initial value of force of interest rate (δ). Ch. i n U. v. We first investigate the case with the value of γ equal to the value of δ.. engchi. pricing mortality rate is based on the LC model.. Our. We take the 5th policy year as an. example to examine the mortality rate risk which is the difference of the realized reserve and the expected reserve.. The realized reserve is evaluated by 20% up shock. or 20% down shock of the mortality rate.. Keeping the setting of δ and γ to satisfy. the equation (δ – γ) = 0 in all time, we illustrate three different effects of the designed product as the following situations. With γ = δ = constant number and fixed in all time Product design 1: traditional increasing whole life insurance. 35.

(36) Let γ = δ = constant = 4% through the whole policy year.. The death benefit is. compound at γ continuously and it is indicated as the following equation Ft = F0 exp(γt), where F0 is the face amount, t is the policy year.. The outcome is shown in the Table. 3. Table 3. The Liability at the End of the 5th Policy Year of Illustrated Insurance Product for Different Mortality Bases γ(=4%) = δ(=4%) (1). Basis. Shock. Shock. 25. 100,000. 122,140. 122,140. 122,140. 45. 100,000. 122,140. 122,140. 122,140. .. /(1). /(1). Reserve. Reserve. Changed. Changed. 0%. 0%. 0%. 0%. ‧. ‧ 國. Premium. (4)=[(2)-(1)] (5)=[(3)-(1)]. 學. Age. 立. 政治 (3)大 20% Up 20% Down (2). y. Nat. n. al. er. The result is consistent with the case of (δ – γ) = 0 in our theoretical. io. mortality rate.. sit. As we can see in Table 3, the reserve remains unchanged after the shock of. analysis.. Ch. i n U. v. Thus, the mortality rate risk of the specified product is none since the risk. engchi. exposure does not increase or decrease caused by the 20% shock at the 5th policy year. The specified whole life insurance product appears no mortality risk or longevity risk in all ages. The product design 1 is a traditional increasing whole life insurance.. If the. interest rate keeps the same as the force of amount γ, the product fulfills the criterion δ= γ and there is no mortality rate risk at all.. While in reality, the interest rate is not. fixed along with the whole policy years, the product may be exposed to risks. 4.1.2 The Post-determined Category 36.

(37) Product Design 2: The type of product is an interest rate variable whole life insurance. In order to keep the criterion of γ(t) = δ(t) during the same time period, t =1, 2…etc.. We decide the γ(t) immediately after we obtain a new δ(t) in the market each We let γ(t) as close to δ(t) as possible.. time.. The δ(t) is piecewise continuously. along with time t and so is the γ(t).We assume δ(1) = δ, γ(1) = δ(1) for the first policy year.. From the second policy year on, we declare a new interest rate at the. beginning of each policy year that generates a new force of interest rate δ(t). = γt = δ(t) for all t.. Let γ(t). We obtain the indication of the death benefit for this kind of. policies is Ft  F0exp( 0  (s)ds) .. 立. The interest rate is up-and-down at each policy year.. 學. ‧ 國. t. 政治大 We assume the realized. ‧. interest rate for the 2nd to 5th policy year, then the values of the force of interest rate. sit. n. al. er. io. Table 4. y. Nat. for the first five policy years are as shown in Table 4.. i n U. v. The Realized Values of the Force of Interest Rate for the First Five Policy Years.. Ch. engchi. Policy Year. 1. 2. 3. 4. 5. Interest rate A. δ=4%. δ(2)=4.25%. δ(3)=4.50%. δ(4)=4.50%. δ(5)=4.75%. Interest rate B. δ=4%. δ(2)=3.75%. δ(3)=3.75%. δ(4)=3.50%. δ(5)=3%. The outcome of product design 2 is shown in Table 5.. We can see that the. values of reserve are not changed by the shock of mortality rate in Table 5.. The. mortality risk and longevity risk indicated in column (4) and column (5), respectively, are shown no risk by the changes of mortality rate because the total mortality rate risk 37.

(38) is immunized within the policy.. Table 5. The Liability at the End of the 5th Policy Year of Product Design 2 for Different Mortality Bases γ(t) = δ(t) (1). Case. Age Premium 100,000. 45. 100,000. 25. 100,000. 45. 100,000. 20% Up Shock. (4)=[(2)-(1)] (5)=[(3)-(1)] /(1). /(1). 20% Down. Reserve. Reserve. Shock. Changed. Changed. 0%. 0%. 0%. 0%. 0%. 0%. 124,608 治 124,608 政 124,608 124,608 124,608 大立 119,722 119,722 119,722 124,608. ‧ 國. A. B. (3). 119,722. 119,722. 119,722. 學. 25. Basis. (2). 0%. 0%. ‧. The product includes a level benefit whole insurance and an extra insurance. y. Nat. sit. benefit (i.e. dividend or increment of death benefit) that depends on the declaration of. al. Our product design is based on the. n. er. The interest rate variable life insurance in the United. io. interest rate each policy year.. States is one of the kinds in this product group.. Ch. engchi. i n U. v. interest rate variable life insurance and keeping the criterion of γ(t)=δ(t).. The. difference from that the interest rate variable insurance product provides dividend to policyholders, the product 2 provides the increment of death benefit.. Each. increment of death benefit in our product should be the same as the dividend of interest rate variable insurance product generated by the declared interest rate.. As. long as the product meets the criterion of γ(t)=δ(t), whether the extra benefit is called either the increment of death benefit in our design or the dividend in the content of interest rate variable life insurance, it does not matter the achievement of risk mitigation.. In our product design here, we take the extra death benefit increased by 38.

(39) each declared interest rate and provide no dividend.. As to fulfill the criteria, the. design of interest rate variable life insurance should take death benefit and dividends into account on assessment of risk.. Since the interest rate is declared in related to. market interest rate, the type of product declines the threat of interest rate risk comparing to the product with fixed interest rate. Product Design 3: The interest rate variable increasing whole life insurance. The idea of this product is similar to Product design 2 that the decision of γ(t) is soon after the newest δ(t) we can get in the market.. Let γ(t) = γ0 +Δγt= δ(t), during. initial value of the force of amount γ 政Set the治大. the same time period, t =1, 2…etc. 2% < δ = 4% at time 0.. 0. Let δ(1) = δ, γ(1) = γ 立. 0. =. +Δγ1 = δ(1) in the first policy year.. ‧ 國. 學. From the 2nd policy year on, declare a new interest rate in each following policy year that generate a new force of interest rate δ(t).. Nat. y. ‧. benefit is. As γ(t) = γ0 + Δγt = δ(t), the death. Ft  F0exp( 0 t )  F0exp( 0  (s)ds) .. n. er. io. al. sit. t. i n U. v. The product is a combination of product 1 and product 2.. Ch. engchi. The product contains. a fixed increment death benefit that makes this part of product looks like a traditional increasing whole life and a variant increment death benefit that makes this part of product seems like an interest rate variable life insurance.. The variant increment. death benefit is like product design 2 determined by the declaration of interest rate. We assume the realized interest rate for the 2nd to 5th policy year.. The values of the. force of interest rate for the first five policy years are as shown in Table 6.. 39.

(40) Table 6. The Realized Values of the Force of Interest Rate for the First Five Policy Years. Policy Year. 1. 2. 3. 4. 5. Interest rate C. δ=4%. δ(2)=4%. δ(3)=4.50%. δ(4)=4.50%. δ(5)=4.50%. Δγ1=2%. Δγ2=2.25%. Δγ3=2.50%. Δγ4=2.50%. Δγ5=2.75%. δ=4%. δ(2)=3.50%. δ(3)=3.50%. δ(4)=3.50%. δ(5)=3.25%. Δγ1=2%. Δγ2=1.75%. Δγ3=1.75%. Δγ4=1.50%. Δγ5=1%. Interest rate D. The outcome of product 3 is shown in Table 7.. We can also see that the values. The mortality 政治大 risk and longevity risk indicated 立 in column (4) and column (5), respectively, are. of reserve are not changed by the shock of mortality rate in Table 7.. ‧ 國. 學. shown no risk by the changes of mortality rate because the total mortality rate risk is immunized within the policy.. ‧ y. Nat. The Liability at the End of the 5th Policy Year of Product 3 for Different. sit. Table 7. n. al (1). Case. Age Premium. γ(t) = γ0 + Δγt = δ(t). Ch. e n(2)g c h i. er. io. Mortality Bases. i n U. (3). v. (4)=[(2)-(1)] (5)=[(3)-(1)] /(1). /(1). 20% Up. 20% Down. Reserve. Reserve. Basis. Shock. Shock. Changed. Changed. 25. 100,000. 123,986. 123,986. 123,986. 0%. 0%. 45. 100,000. 123,986. 123,986. 123,986. 0%. 0%. 25. 100,000. 119,423. 119,423. 119,423. 0%. 0%. 45. 100,000. 119,423. 119,423. 119,423. 0%. 0%. C. D. 40.

(41) 4.2 The secondary strategy with 0<γ<δ6 When the strategy is with γ<δ, our objective is to diminish the mortality rate risk, As we elaborate in this article, the risk rooting in “when to. not to immunize the risk.. pay” of a death benefit life insurance is mortality risk along with the changes of mortality rate, and the risk rooting in “how much to pay” of a death benefit life insurance is longevity risk.. The outcome of the changes of the mortality rate with 20% shock is in. Table 8.. 立. 學. The Liability at the End of the 5th Policy Year of Illustrated Insurance Product for Different Mortality Bases. Basis. io 37,589. 45. 53,154. 20% Up Shock. al. (4)=[(2)-(1)] (5)=[(3)-(1)] /(1). /(1). 20%Down. Reserve. Reserve. Shock. Changed. Changed. Panel A: γ (=2%) <δ (=4%). n. 25. (3). y. Premium. (2). sit. Nat. Age. (1). ‧. ‧ 國. Table 8. 政治大. er. example.. We take a death benefit whole life insurance as an. i n C 63,758 h 66,020 i U e n g c h61,018 45,368. 47,287. 43,104. v 4.230%. -4.990%. 3.548%. -4.297%. Panel B: γ (=0%) <δ (=4%). 25. 15,811. 18,612. 20,153. 16,869. 8.280%. -9.350%. 45. 30,192. 35,216. 37,608. 32,420. 6.792%. -7.940%. The product in Panel A with δ, 4% and γ, 2% is compare to that in Panel B with δ, 4% and γ, 0%.. The product in Panel A is exposed to both the mortality risk and the. With product design of γ>δ, the life insurance product becomes exposed to the longevity risk only. That is not normal in a pure death benefit life insurance product. In the normal situation, insurance companies may not design such products to confuse themselves in identification of the longevity risk resulting from a pure death benefit life insurance.. 6. 41.

(42) longevity risk and the longevity risk is less than the mortality risk. Panel B is exposed to mortality risk only.. The product in. When the mortality rate is changed by. 20% shock, the expected reserve of both products is changed.. With increase of. reserve as indicated in column 4 with respect to 20% up shock of mortality rate, product in Panel A is exposed to risk of 3% ~ 4% more than that in basis mortality rate.. And the product in Panel B is exposed to that of 6 ~ 8% more than that in basis. mortality rate by the changes of mortality rate.. The secondary strategy is to diminish. the risk within the policy by creating longevity risk in a life insurance product to. 政治大 upon the demand of risk exposure. 立. lower the exposure of mortality risk and the strategy helps us to design such products. ‧ 國. 學 ‧. 5. Practice and Conclusion. sit. y. Nat. The evolution of the kind of compound increasing rate whole life insurances may. al. er. io. generally be divided into two stages. The first stage is selling those products that the. v. n. force of amount (i.e. increasing rate of the death benefit, γ) is much bigger than the. Ch. i n U. engchi. force of interest rate (i.e. δ) of the policy from 2004~2009. And the second stage is selling the products that the force of amount (i.e. γ) is mostly equal to the force of interest rate (i.e. δ) of the policy from 2009 till now. In 2009, the Financial Supervisory Authority of Taiwan mandated that increasing rate designed in the product is forbidden higher than the interest rate of the product. Thus, the life companies did not present the increasing whole life product with overly high increasing rate in death benefit ever since.. Furthermore, the companies have. known the compound increasing whole life insurance product better than before.. 42.

(43) From 2009 until now is the second stage of the product, almost every insurance company in the market designs and sells such compound increasing whole life insurance product.. In the perspective of the insurance companies, the product is. quick money collecting product that they have soon accumulated the premium income by selling such compound increasing whole life following the mandate of authority that meets the criterion of (δ – γ) = 0, which is optimal strategy in risk mitigation. The product is a tool in accumulating richness for the companies, needless in complicated underwriting process but charging the highest penalty in the condition of. 政治大 From the viewpoint of立 a buyer of the kind of insurance products, the compound. policy withdrawal.. ‧ 國. 學. increasing whole life insurance product is an insurance product that may resist the currency inflation and is convenient in saving without renewal, like a long-term bond. ‧. accumulated the coupons until the day of death.. The best of all, the death benefit The. sit. y. Nat. income is tax-free without upper limit at amount of death benefit in Taiwan.. n. al. er. io. drawback of the product is the penalty charge of the policy withdrawal that is 25% of. i n U. v. the policy value and it is way too much in charge for an insurance policy.. Ch. engchi. After the theoretical development and numerical analysis in the article, we are surely that the compound increasing rate whole life insurance product developing in the first stage is exposed to the longevity risk which is systemic risk along with the lifelong of the population.. While in the second stage, the compound increasing rate. whole life insurance product is limit in the criterion of (δ ≧ γ) which means the increasing rate should be less than the interest rate of the product.. Under the. condition of (δ ≧ γ), the compound increasing whole insurance is exposed to the mortality risk that is more controllable than longevity risk.. 43.

(44) Besides that we solve the mystery for the compound increasing rate whole life insurance product, we develop the risk mitigation in product design.. We discover. that natural hedging strategy through product design is an important step that we can immunize/mitigate the mortality rate risk within a policy.. The key point is that we. should take the risk rooting in “when to pay” and the risk rooting in “how much to pay” into consideration in risk mitigation technique.. We introduce a factor γ, the. force of amount (i.e. the variable of increasing rate in death benefit), as a risk factor rooting in “how much to pay” in the product design.. 政治大 mortality rate risk is immunized. 立 We deduce the optimal natural hedging strategy We utilize the γ to design a death benefit protection life insurance that the. ‧ 國. 學. within a policy with the criterion of settings on γ as to let γ = δ.. When the γ is equal. to δ, the specified product appears no risk with the changes of mortality rate.. The. ‧. mortality rate risk is immunized in such a kind of products with death benefit When the γ is not equal to δ, the strategy is used to diminish. n. al. If δ > γ, the death benefit protection. er. io. the risk rather than to immunize the risk.. sit. y. Nat. protection in all ages.. i n U. v. product is exposed to mortality risk rather than longevity risk with the changes of. Ch. engchi. mortality rate and the life insurance product with settings of γ is less exposed to mortality risk than the one without settings of γ. Following the optimal strategy in the product design, the insurance product has a lot more possibility in engaging to financial instrument without considering the mortality rate risk.. Our finding can also provide a further research or re-design of. previous studies that ignore the existence of a mortality rate risk in an insurance product valuation or risk management consideration.. 44.

(45) 立. 政治大 Essay II :. ‧ 國. 學. The Uncertainty to Optimal Strategy. ‧. in Life Insurance Product design. n. er. io. sit. y. Nat. al. Ch. engchi. 45. i n U. v.