通貨膨脹學習效果之動態投資組合

本研究探討長期投資人在面臨通貨膨脹風險時的最適投資決策。就長期投資者而言，諸如退休金規劃者等，通貨膨脹是無可避免卻又不易被數量化之風險，因為各國僅公布與之相關的消費者物價指數而沒有公布真實通貨膨脹數值，因此我們延伸Campbell和Viceira(2001)及Brennan和Xia(2002)的模型假設，以消費者物價指數的資訊來校正原先假定符合Vasicek模型之通貨膨脹動態過程。本研究之理論背景為：利用貝式過濾方法(Baysian Filtering Method)，將含有雜訊之消費者物價指數，透過後驗分配得出通貨膨脹動態過程。利用帄賭過程(Martingale Method)求解資產之公帄價格。再引進定值相對風險趨避(Constant Relative Risk Aversion，CRRA)的效用函數，求出最適投資組合下之期末累積財富、各期資產配置以及效用值。

本研究歸納數值結果如下： 一、投資期間越長，通貨膨脹學習效果越顯著。投資期間達25年以上時，有學習效果之累積財富為無學習效果時兩倍以上，25年為2.36倍；30年為2.18倍。此外，學習效果對投資人效用改善率於長期投資時也較顯著，投資10年效用改善率為35%，而投資30年則高達1289%，呈非線性成長。以上結果顯示：資產在市場上累積越久，受到通膨影響越明顯，更需要以學習方式動態調整資產配置進行通貨膨脹風險管理。

二、風險較趨避之投資人，CRRA參數值越大；於最適投資組合下之期末財富較少，因為風險較趨避投資人偏好低波動度資產組合。風險容忍度低之投資人較需要通貨膨脹之學習，否則效用減損過高，例如CRRA參數為1.5之投資人30年後效用減損65%，CRRA參數為4之投資人效用減損達96.5%。以上數據顯示：風險趨避投資人對風險關注程度較高，考慮學習效果時，較能根據目前通貨膨脹調整資產配置。

This study examines the optimal portfolio selection incorporating inflation risk learning adjustments for a long-term investor. For long-term investors, it is inevitable to face the uncertainty of inflation. On the other hand, quantifying inflation risk needs more effort since the government announced the information on Consumer Price Index (CPI) rather than the real inflation rates.

In order to measure the inflation rate in planning the long-term investment strategies, we extend the works in Campbell and Viceira (2001) and Brennan and Xia (2002) to construct a stochastic process of the inflation rate. The prior distribution of inflation rate process, which is not directly observable, is assumed to follow the diffusion process. Based on the information of CPI, we then employ the optimal linear filtering equations to estimate the posterior distribution of the inflation rate process. Through these mechanisms, the inflation rate process is closer to reality by learning from CPI. We also construct the optimal portfolio strategy through a Martingale formulation based on the wealth constraints. The optimal portfolio strategies are given in closed-form solutions.

Furthermore, the importance of learning about inflation risk is summarized through the numerical results. (1) When the investment interval is longer, the learning effect becomes more significant. If the investment horizon is longer than 25 years, the wealth accumulation under learning will be twice more than that without learning effect, e.g., the wealth accumulation is approximately 2.36, 2.18 folds at the end of 25, 30 years. Utility increase under learning also become larger for long-term investor, e.g., the utility values will improve 35% after considering learning ability on inflation from 10-year interval, improve 1289% from 30 years.

(2)When the CRRA parameter increases, the investor have lower risk tolerance; and their wealth accumulation become less due to the lower volatility portfolio. A conservative investor requires more learning ability given the inflation, otherwise their utility value will be reduced, e.g., the utility values will be reduced 35% when CRRA=1.5 after 30 years’ investment, 96,5% when CRRA=4.