<p>Essentially this project report is a discussion of mathematical modelling in pension funds, presenting sections from Cairns, A.J.D., Blake, D., Dowd, K., Stochastic lifestyling: Optimal dynamic asset allocation for defined contribution pension plans, Journal of Economic Dynamics and Control, Volume 30, Issue 2006, Pages 843-877, with added details and background material in order to demonstrate the mathematical methods. In the investigation of the management of the investment portfolio, we only use one risky asset together with a bond and cash as other assets in a&nbsp / continuous time framework. The particular model is very much designed according to the members&rsquo / preference and then the funds are invested by the fund manager in the financial market. At the end, we are going to show various simulations of these models. Our methods include stochastic control for utility maximisation among others. The optimisation problem entails the optimal&nbsp / investment portfolio to maximise a certain power utility function. We use MATLAB and MAPLE programming languages to generate results in the form of graphs and tables</p>

<p>Essentially this project report is a discussion of mathematical modelling in pension funds, presenting sections from Cairns, A.J.D., Blake, D., Dowd, K., Stochastic lifestyling: Optimal dynamic asset allocation for defined contribution pension plans, Journal of Economic Dynamics and Control, Volume 30, Issue 2006, Pages 843-877, with added details and background material in order to demonstrate the mathematical methods. In the investigation of the management of the investment portfolio, we only use one risky asset together with a bond and cash as other assets in a&nbsp / continuous time framework. The particular model is very much designed according to the members&rsquo / preference and then the funds are invested by the fund manager in the financial market. At the end, we are going to show various simulations of these models. Our methods include stochastic control for utility maximisation among others. The optimisation problem entails the optimal&nbsp / investment portfolio to maximise a certain power utility function. We use MATLAB and MAPLE programming languages to generate results in the form of graphs and tables</p>

Magister Scientiae - MSc / Essentially this project report is a discussion of mathematical modelling in pension funds, presenting sections from Cairns, A.J.D., Blake, D., Dowd, K., Stochastic lifestyling: Optimal dynamic asset allocation for defined contribution pension plans, Journal of Economic Dynamics and Control, Volume 30, Issue 2006, Pages 843-877, with added details and background material in order to demonstrate the mathematical methods. In the investigation of the management of the investment portfolio, we only use one risky asset together with a bond and cash as other assets in a continuous time framework. The particular model is very much designed according to the members’ preference and then the funds are invested by the fund manager in the financial market. At the end, we are going to show various simulations of these models. Our methods include stochastic control for utility maximisation among others. The optimisation problem entails the optimal investment portfolio to maximise a certain power utility function. We use MATLAB and MAPLE programming languages to generate results in the form of graphs and tables. / South Africa

臺閔地區65歲以上的老年人口於民國82年底占總人口之7.1%，達到聯合國所界定之「高齡化」水準，至民國94年底老年人口大幅增加為占總人口之9.74%，人口老化的趨勢使得退休後的生活保障更顯重要;年長者生活保障的主要來源是退休金，而我國已於民國94年7月1日由確定給付制轉變為確定提撥制(DC制)。基於醫學的快速成長，以及生活環境水準亦顯著提昇，採用現有的生命表預測未來死亡率可能會有極大的誤差，故本文參考許鳴遠(2006)的Reduction Factor模型，預測台灣未來的高齡人口死亡率，進而推計未來的各年齡人口數，再將改善的死亡率應用在確定提撥制的退休基金，並參考MacDonald and Cairns (2007)，假設退休基金投資在五種不同的投資標的，分別考慮20歲的個人與不同時間點20~55歲的各年齡人口，在不同限制函數下找尋確定提撥制退休基金的最適投資策略，並比較不同限制函數對依賴比造成的影響。

>Magister Scientiae - MSc / Pensions are essentially the only source of income for many retired workers. It is thus critical that the pension fund manager chooses the right type of plan for his/her workers.Every pension scheme follows its own set of rules when calculating the benefits of the fund’s members at retirement. Whichever plan the manager chooses for the members,he/she will have to invest their contributions in the financial market. The manager is therefore faced with the daunting task of selecting the most appropriate investment strat-egy as to maximize the returns from the financial assets. Due to the volatile nature of stock markets, some pension companies have attached minimum guarantees to pension contracts. These guarantees come at a price, but ensure that the member does not suffer a loss due to poorly performing equities.In this thesis we study four types of mathematical problems in pension fund management,of which three are essentially optimization problems. Firstly, following Blake [5], we show in a discrete time setting how to decompose a pension benefit into a combination of Euro-pean options. We also model the pension plan preferences of workers, sponsors and fund managers. We make a number of contributions additional to the paper by Blake [5]. In particular, we contribute graphic illustrations of the expected values of the pension fund assets, liabilities and the actuarial surplus processes. In more detail than in the original source, we derive the variance of the assets of a defined benefit pension plan. Secondly,we dedicate Chapter 6 to the problem of minimizing the cost of a minimum guarantee included in defined contribution (DC) pension contracts. Here we work in discrete time and consider multi-period guarantees similar to those in Hipp [25]. This entire chapter is original work. Using a standard optimization method, we propose a strategy that cal- culates an optimal sequence of guarantees that minimizes the sum of the squares of the present value of the total price of the guarantee. Graphic illustrations are included to in-dicate the minimum value and corresponding optimal sequence of guarantees. Thirdly, we derive an optimal investment strategy for a defined contribution fund with three financial assets in the presence of a minimum guarantee. We work in a continuous time setting and in particular contribute simulations of the dynamics of the short interest rate process and the assets in the financial market of Deelstra et al. [19]. We also derive an optimal investment strategy of the surplus process introduced in Deelstra et al. [19]. The results regarding the surplus are then converted to consider the actual investment portfolio per- taining to the wealth of the fund. We note that the aforementioned paper does not use optimal control theory. In order to illustrate the method of stochastic optimal control, we study a fourth problem by including a discussion of the paper by Devolder et al. [21] in Chapter 3. We enhance the work in the latter paper by including some simulations. The specific portfolio management strategies are applicable to banking as well (and is being pursued independently).

現行各國的退休金計畫逐漸地由確定給付制轉變為確定提撥制。這表示投資的風險由原本退休金計畫的發起者(雇主)轉移到了參與者(員工)的身上。為了減少每個確定提撥制計畫參與者的投資風險，本文中採用退休時所得替代率為預估的目標，藉由模擬與最適化的方法找到最適投資策略與最適提撥率。 能反映出時間性的隨機模型在精算科學的領域是日漸重要，本文試著藉由隨機性的變化來估計代替以往精算上各種假設下所求得的負債。本文藉由隨機模擬的方式，得到各種資產在市場上或者是經濟上的價值來建構相關投資標的之報酬率，並利用動態隨機規劃模型去改善財務上避險以及資產負債管理。此外，為了避免模擬分析時間過長的問題，本文採用了情境抽樣的方法去改善電腦模擬分析計算時的效率。 我們主要得到以下結論： (一)確定提撥制下的負債受薪資水準波動的影響，所以此時會持有較 多的指數連結型債券以反應薪資水準及通貨膨脹的影響。整體投 資的結果與Vigna & Haberman (2001) 文中的結果及實務上生命 週期型態(lifestyle)投資方式呈現相同的現象。 (二)考慮每期下跌風險(downside risk)時，期中的投資可能會偏向 於投資風險較高的股票。在每年觀察下跌風險的情況下其投資因 為必須考慮避免每一年的下跌風險，需要比每五年觀察下跌風險 的情況做風險較大的投資，以達到其目標。 (三)在本文的調整投資組合策略下，因為調整次數不多，所以在考慮 交易成本的情況，當交易成本很小時對於整體的最適化資產配置 與最適化提撥率的影響是很小的。在本文的調整投資組合策略 下，交易成本的影響只有在交易成本非常大的情況下才能看得出 來。 (四)均勻抽樣法抽出的400組情境幾乎可以完全的代替4000組情境， 其結果可以看出與未抽樣相同的生命週期型態(lifestyle)投資 方式。而隨機抽樣法的結果雖然也可看出趨勢，但準確性相對於 均勻抽樣法仍稍嫌不足，並不適合用來代替原先的4000組情境。 / A shift from defined-benefit pension plan towards defined-contribution pension plan is currently popular around the world. This means that a serious investment risk transfers from defined-benefit sponsors to the individual members of defined-contribution plans. In order to reduce the risk of individual DC member, we investigate the methodology of finding the optimal contribution rate and asset allocation to reach a certain target of the retirement replacement rate in this paper. Stochastic processes are getting more important to the field of actuarial science. Instead of trying to approximate liabilities by a single deterministic set of actuarial assumption, we seek to take account of market or economic valuation for both assets and liabilities using stochastic simulation. We applied dynamic stochastic programming models to improve financial hedging and asset liability management. Moreover, in order to avoid the problem of time-consuming, we use scenario sampling method to improve the efficiency of computer calculation. We draw four conclusions from our investigations： (1)We will hold more assets in indexed-linked bonds because the pension liability is highly related to the wage- index and inflation rate. The optimal investment strategy is very like the so called "lifestyle" investment strategy. (2)When we consider downside risk, we should hold more risky equities. The investment strategy is more risky when we consider downside risk every year than every 5 years. (3)Under our rebalancing strategy, if the transaction cost is small, the influence on the investment strategy and contribution rate is small. We can see the influence of the transaction cost in a situation that the transaction cost is very big only. (4)There are almost no different between uniform sampling scenarios and original simulation scenarios, so uniform sampling scenarios may replace the original simulation scenarios perfectly. And random sampling method is unsuitable to replace the original simulation scenarios.