以模擬退火法建構資本資產模型的獲利風險最佳化研究－以 0050 指數型 50 檔股票為例

Markowitz 所提出之投資組合選擇問題須面臨計算工作繁重且高度複雜的最佳化組合問題，如何在所選擇投資之資產分配最佳資金權重，使所建構之投資組合符合效用前緣線。本研究以股票市場為例，期望在股票市場中尋找風險最小且報酬率最大之資產投資組合並給予較高的資金權重。基於此理由，選擇夏普比率 (Sharpe Ratio) 當作投資組合的選擇策略，夏普比率是利用報酬率除以標準差去衡量承受風險的單位報酬率之大小。其次，本研究再利用霍普菲爾類神經網路 (Hopfield-Tank Neural Network) 結合波茲曼機器 (Boltzmann Machine) 優良的權重學習能力求解最佳投資組合問題。在退火程序中，溫度很高時，系統往高能量方向移動和往低能量方向移動的機率就愈來愈大；當溫度降低時，波茲曼機器則應用神經元狀態改變所造成的能量差 ΔΕ 概念，根據 ΔΕ 和溫度值給定的機率，當溫度高時不管 ΔΕ > 0 或 ΔΕ < 0，即不管能量是往上升或往下降，狀態改變的接受率大約相同，當溫度愈來愈低時，會使得 ΔΕ < 0 狀態被接受的機率愈來愈大。因此，波茲曼機器擁有跳脫局部最佳解 (Local Optimum) ，往全域最佳解 (Global Optimum) 方向移動的能力。本研究期望以模擬退火法建構夏普比率最大化並符合效益前緣的股票投資組合。

關鍵字

模擬退火法；波茲曼機器；夏普比率；風險最佳化；股票

並列摘要

The investment portfolio selection problem proposed by Markowitz has to face the intensive and highly complex optimal portfolio problem. How to allocate the best capital weight to the selected investment assets in order to make the constructed investment portfolio meets the efficient frontier is the problem. This thesis takes the stock market as an example, hoping to find the asset portfolio with the lowest risk and the highest return rate in the stock market and give it a higher capital weight. For this reason, Sharpe Ratio is chosen as the investment portfolio selection strategy which uses the rate of return divided by the standard deviation to measure the rate of return per unit of risk. Secondly, this research reuses Hopfield-Tank Neural Network and excellent weight learning ability of Boltzmann Machine to solve the optimal portfolio problem. During the process, if the temperature is high, the probability of the system moving to high energy or low energy becomes greater; if the temperature decreases, Boltzmann Machine applies the concept of energy difference ΔΕ caused by the change of neuron state. According to the probability given by ΔΕ and temperature, when the temperature is high, regardless of ΔΕ > 0 or ΔΕ < 0, that is, regardless of whether the energy is rising or falling, the accepted probability of the changing state is about the same. When the temperature is getting lower, the accepted probability of ΔΕ<0 is getting higher. Therefore, Boltzmann Machine has the ability to escape the local optimum and move towards the global optimum. This research expects to construct a stock portfolio that maximizes Sharpe Ratio and meets the efficient frontier with Simulated Annealing Method.