| 摘要: | 本篇文章主要是探討退休基金的評價，主要是由確定提撥與確定給付制度下，員工於退休日，何者價值為高，則為其可請領之退休金金額。因其具有較佳的特性與較佳選擇權（maximum option）之意涵相同﹔再者，文中假定強制退休年齡為65歲，則如選擇權之到期日﹔對單一員工而言，其具有選擇是否提前退休，一如美式選擇權之提前履約。綜上，我們可將退休基金以選擇權評價模式，求出退休基金之價值。 Michael Sherris (1997)以評價選擇權之方式，計算退休基金的價值，本文即由Sherris的文章延伸，內容分為兩大部分。第一部份，Sherris 在1997年的論文中假設利率為固定，對於退休時服務年限動輒數十年，如此的假設並不合理，因此本文將使用隨機利率替代固定利率，因此評價的模型具有三個狀態變數；薪資成長率、基金資產報酬率與利率。此外，評價方法並將脫退率考量進去，如死亡、離職等等。由於評價模型屬於多狀態變數，並屬路徑相依的問題─與薪資成長有關，因此使用樹狀圖的方法沒有效率，我們將採行Longstaff and Schwartz 於2001年發表的“A simple least-squares approach“關徐美式選擇權的評價。第二部份主要是考量到景氣循環的效果，因此在原本基金資產報酬率預定的趨勢上，加上一些隨機跳躍，並觀察退休金之價值之變動。我們假設隨機跳躍的個數是服從Poisson 的隨機變數，跳上跳下的機率和幅度都是一致的。 模擬的結果，發現Sherris 高估了退休金大約34％的價值，原因應是利率使得折現因子變小，而使得退休基金價值較低。另外，我們亦做了與利率有關之參數的敏感度分析，發現利率與薪資成長率或利率與基金資產報酬的相關係數改變，並不能改變退休基金價值太多。反倒是利率的初始價值影響基金價格甚巨。利率支初始價值由0.1到0.05大約始退休金價值增加3.7％--4％。 另外我們亦可發現加入隨機的跳動亦不影響結果甚巨，我想最主要的原因及在於報酬率增加與減少的機率皆相同，跳動的幅度也相同。因此正的影響力與負的影響力剛好抵消，所以結果就顯示不出有極大的差異點。 This paper utilizes the contingent claim approach to value option features in a retirement benefit. Extending from Michael Sherris (1997), the contributions of this paper are two-fold. First, we assume the valuation model contains three state variables: salary growth rate, crediting rate, and interest rate, where Sherris assumed the interest rate to be constant. Since retirement benefits involve a long-time membership, the assumption made by Sherris is impractical. Our model is still able to take some decrements such as retirement, resignation, death, and disability into consideration. Second, because business cycles are often observed in the real world, we consider that the crediting rate has some random jumps in the life of a pension fund, and the number of jumps follows a Poisson random variable. A discrete lattice commonly used for valuing a financial option is unsatisfactory for multi-variances. A simulation is more efficient. In this paper we use the Longstaff and Schwartz’s simulation approach to calculate benefit values for a range of ages. The results show that, after considering the stochastic character of interest rates, the costs of benefit valued by Sherris are overstated by about an average of 34 percent. When we examine the sensitivity of benefit values to parameters, such asρfr, ρsr, σr, and r0, they have little effect to the benefit costs except for r0. As for changing r0 from 0.1 to 0.5, the costs of benefit rise about 4%. Considering random jumps in crediting rate will thus not affect the cost of retirement benefits too much. |
