推導資本投資之財務決策公式──使用下檔風險指標與模糊集合運算

Deriving the Financial Decision Making Formula for Capital Investments: Using Indicators of Downside Risk and Algorithms of Fuzzy Sets

企業成長與永續發展需要持續進行資本投資，投資決策包含財務考量與策略考量。本文針對資本投資決策之財務面可行性，以較少被使用的下檔風險指標取代普遍被使用的全距報酬波動度，詳述其於實務風險估計更合理有效的因果；同時結合模糊集合，應用於四個變數的估計與模糊運算，有系統的建構模糊淨現值與模糊下檔風險值，以提高估計的彈性。本文中投資案分為三種情況：單一投資案、不同存續期間但相同資金成本的互斥案、與不同存續期間且不同資金成本的互斥案。使用模糊數的α-截集與信賴區間，推導報酬的歸屬函數：模糊淨現值期望值、模糊淨現值期望值之等值年金、與模糊淨現值期望值之永續等值年金。經解模糊後，惟有正報酬的投資案可被接受、並進行後續風險評估。以低於報酬期望值的模糊半變異數為基礎，推導下檔標準差，以衡量投資案的絕對風險；再以下檔變異係數作為報酬與風險的綜合衡量指標。在沒有策略性考量下，優選投資案的財務決策規則為：正模糊淨現值期望值愈大愈好、下檔標準差愈小愈好、下檔變異係數愈小愈好。 Capital investments are required for business growth and perpetual development. They include the financial and strategic considerations. This work makes the less used downside risk indicators replace the commonly adopted range return volatility, describing the cause and effect of a more reasonable and effective practical risk estimation. The fuzzy sets are applied to the estimation and operation of four variables to systematically obtain fuzzy net present values and fuzzy downside risk measures in order to promote the flexibility of estimation. In this work the capital investments are composed of three conditions: a single project, mutually exclusive projects with different life spans but the same cost of capital, and mutually exclusive projects with different life spans and different costs of capital. α-cuts and confidence of intervals are used to derive the membership functions of returns: the expected fuzzy net present values, the fuzzy equivalent annuity of expected fuzzy net present values, and the equivalent annuity to infinity of expected fuzzy net present values. After dufuzzification, only the positive-return projects can be accepted and processed by the subsequent risk evaluation. Based on the fuzzy semi-variance under expected returns, the down-side standard deviations are obtained to measure the absolute risks of investment projects. The downside coefficients are further developed to be the returnrisk complex indicators. If there's no strategic consideration, the financial decisionmaking rules for priority project are: greater positive fuzzy expected net present value, smaller down-side standard deviation, and smaller coefficient of variation.