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國科會研究計劃期中進度報告 前言
投資型保險商品在世界各地都很受歡迎，近年來在台灣，投資型保險商品亦 非常受歡迎，而變額年金(Variable Annuity) 與權益連結年金 (Equity Indexed Annuity) 是投資型保險商品中最重要的兩種類型，但是這類保險商品的評價模 型與相關風險管理的議題都十分複雜。本計劃希望利用變異數縮減技術來設計有 效率的蒙地卡羅評價方法。在這一年，我們計劃能夠設計權益連結年金(通常為 一般帳戶商品)的評價方法。 研究目的 近年來，敝人因為參與保險局與保發中心多項與投資型商品有關的計劃，了 解財務工程的方法十分適合投資型商品的評價與風險管理，特別是模疑方法(隨 機模型)。投資型商品主要有變額年金 (Variable Annuity(VA)) 與權益連結年金 (Equity Indexed Annuity(EIA)). 這兩大類保險商品的評價模型與相關風險管理的 議題都十分複雜。在學術界，有不少學者投入此方面的研究。而且，近年來這些 商品在台灣的銷售量與日俱增。業界與監理單位在此方面的人才都仍十分缺乏。 因此本計劃希望對學界、實務界都能有所貢獻。
In this project, we wish to study effective pricing methods for VA and EIA contracts. We will focus on EIA contracts in the first year and VA contracts in the second year. The steps are essential the same each year. VA contracts are separate account insurance and most EIA contracts are general account insurance (unless the contracts are registered). Hence VA contracts need different treatments.
An Equity Indexed Annuity contract provides the policyholder with a guaranteed minimum annual return and offers participation in the equity market. The returns to be credited are based on the index-linked return, which is tied to the performance of an equity price index such as the Standard and Poor’s 500, the participation rate and the guaranteed minimum return. The participation rate determines how much of the index increases will be used to compute the index-linked return. With the guaranteed minimum return, the downside risk of the equity market is limited. EIAs usually have a maturity ranging from one to ten years, with seven years being typical.
There exist a few variations of EIA contracts. The major contract types include point-to-point (PTP), ratchet (which comes in simple and compound versions), and high water mark; see, Hardy (2003) for more information. The ways to calculate the index-linked return is usually called indexation method or indexing. For PTP EIAs,
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the indexing is based on the growth of the two time points, which is the simplest indexation method. Under the ratchet contract design, the indexing is usually evaluated annually based on the index growth during each time period. With a compound ratchet (CR), the returns in each time period are compounded, while the returns of each period of a simple ratchet (SR) are summed arithmetically. High water mark EIAs have the feature similar to a lookback option. They use the maximum index level over the entire term of an annuity for calculating payoff at maturity.
VA contracts are separate account insurance and most EIA contracts are general account insurance (unless the contracts are registered). Hence VA contracts need different treatments.
文獻探討
Below are summaries of some important papers for valuation of equity-linked insurance products.
EIA
• Valuation of equity-indexed annuities under stochastic interest rates. X. S. Lin and Ken Seng Tan. 2003. North American Actuarial Journal 7(3): 72-91. A more general economic model is considered by assuming that the interest rates are stochastic and follow a diffusion process. A detailed numerical analysis is performed.
• Pricing equity-indexed annuities with path-dependent options. H. Lee. (2003). Insurance: Mathematics and Economics 33(3): 677- 690. This paper derive explicit pricing formulas for an up-and-in barrier EIA, an annual reset EIA with up-and-in barriers, a partial time lookback EIA, a partial lookback EIA with variable guarantee and a floating-strike lookback option by using the method of Esscher transforms.
• Pricing of Ratchet equity-indexed annuities under stochastic interest rates. M. Kijima, T. Wang. 2006. Insurance: Mathematics and Economics. This paper derives the closed-form solutions for both simple and compound Ratchet EIAs under stochastic interest rate. The short rate is assumed to follow extended Vasicek model (interest rates are unobservable and could be negative). An efficient Monte Carlo simulation scheme is also established. Future research topics: first, consider LIBOR market model (interest rates are positive and observable); second, the hedging of EIA contracts is important. Finally, the surrender risk should be considered.
• Ratchet equity indexed annuities. Mary Hardy. 2004. In 14th Annual International AFIR Colloquium. This paper derives a valuation formula for the tractable,
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compound version of the ratchet EIA. It presents a non-recombining trinomial lattice approach for valuing simple ratchet EIAs (which are not analytically tractable). It also compares the accuracy of the lattice method with Monte Carlo simulation for the simple ratchet EIA and find that the lattice approach achieves a very high degree of accuracy much more efficiently than the Monte Carlo approach. The underlying life of contract guarantee value (not analytically tractable) is estimated using Monte Carlo simulation; the value is shown to be very small compared with the ratchet part of the benefit. All the valuation in the paper has assumed a standard lognormal risk neutral distribution.
• Valuation of equity-linked insurance and annuity products with binomial models. P. Gaillardetz, X. Sheldon Lin. 2006. North American Actuarial Journal 7(3): 72-91. This paper introduces an age dependent, mortality risk-adjusted martingale probability measures for each of the term life, pure endowment, and endowment insurances using their corresponding premiums. Two different pricing approaches for equity-linked products are presented. The early withdrawal risk is also discussed.
• Valuation of Equity-Indexed Annuities when Interest Rates are Stochastic. H. K. Fung, L. K. Li. This paper carry out a simple simulation exercise to inspect the mispricing effect when true interest rates follow a stochastic model but EIAs are evaluated with deterministic interest rates inferred from the initial yield curve. It inspects the pricing differences between the use of the Black-Scholes (BS) model and the use of the Black-Scholes-Hull-White (BSHW) model that is considered in Lin and Tan (2003). The simulation results show that pricing biases are in general much larger for PTP contracts than for CARs.
研究方法 我們使用風險中立的評價方法去導出對權益連結保險商品評價函數或設計 評價演算法。這是一個標準化的方法，且經常用於評價類似的商品。我們希望能 推導出一個的評價函數，針對複雜的權益連結保險商品創新設計且有效的模擬演 算法。這些想法，它們的數學分析都相當複雜，因此敝人與史丹佛大學的 Professor Glynn 共同合作，來克服這個困難。 結果與討論 由於敝人國科會計畫擬用創新的蒙地卡羅法來評價年金商品，因此於97年12 月前往參加 Winter Simulation Conference 2008 (WSC08) 國際學術會議，在 此發表的學術論文 (特別是 analysis methodology 和Risk Analysis tracks)，
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Risk Analysis tracks 每年約有12-15篇學術論文發表，論文探討內容以蒙地卡 羅法在財務領域的創新應用為主。 另外，我們希望在選擇變異數縮減技術上能有突破性的想法，所以必須先對 評價問題有一定的了解，而且必須有不錯的模擬理論的背景。 為了讓這樣的想 法考慮更為周詳，敝人於98年2月前往史丹佛大學，與 Professor Glynn和其它 傑出的博士生進行短期研究，透過這樣面對面的直接討論，加速我們之間的合作， 並討論了一些不錯的想法。 在過去一年的研究中，我們了解了權益連結的保險商品契約的特性，也完成 部份針對權益連結的保險商品的程式庫。今年我計畫參加 ARIA 年會 (論文 "Valuation of Ratchet Equity Indexed Annuities with Quanto Features" 已 被接受) 。 另將繼續把權益連結的保險商品的程式庫完成並開始設計變額年金 的評價方法。
