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題名：以實質選擇權評估序列資本預算:多元常態積分數值求解與臨界值非線性求根法
作者：段昌文
作者(外文)：Chang-Wen Duan
校院名稱：淡江大學
系所名稱：財務金融學系
指導教授：林蒼祥
學位類別：博士
出版日期：2003
主題關鍵詞：正割法；非交易性資產；格狀法；類股利率；多階段複合式實質買權；臨界值；巒生證券；數值積分法；secant method；non-traded asset；lattice method；dividend-like yield；multi-stage compound real call options；critical value；twin security；numerical method
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本文應用複合式實質買權 (compound real call options) 模型，以評估台灣首家以 0.35μm 製程技術切入生產六千四百萬位元 (64 Megabytes, 64Mb) 動態隨機存取記憶體 (Dynamic Random Access Memory, DRAM) 之新設晶片製造廠 (chipmakers) 投資計劃案；模型之標的變數為全球單顆 DRAM 之平均銷售價格 (average sale price, ASP)，然而投資計劃屬非交易性資產 (non-traded asset)，我們以台灣半導體製造公司之上市股價成投資組合作為巒生證券 (twin security)；茂德上市時單位投資預算為四階段複合式實質買權最後一階段的履約價 (exercise price)，而由後期 (上市階段) 往前期 (設廠階段) 推導，於價平 (at-the-money) 條件下時所求得之臨界值 (critical value) 為第一階段至第三階段之隨機履約價。
根據 Trigeorgis (1993) 指出，應用實質選擇權理論以評價非交易性資產時，模型中將產生一類股利率 (dividend-like yield)，本文參考 Duan, Lin and Lee (2003) 對類股利率之估計方法，採用資本資產定價模式 (capital asset pricing model, CAPM) 與持有成本模式 (cost of carry model) 兩種模型以估計個案研究之類股利率，結果兩種方法之估計值差只有0.0224，且為負值。此負值之類股利率表現出 CAPM 模式中之相關係數小於一，隱含新廠之投資計畫具風險分散能力，符合 Schumpeter (1939) 之創新理論。
比較數值求解法與數值積分法中，發現正割法 (secant method) 結合 Genz (1999) 對多元常態積分的格狀法 (lattice method) 於求解非線性模型之根時速度最快。求解結果發現，臨界值為逐年遞減，顯示個案公司生產成本遞減，製程技術依循莫爾定律 (Moore’s law)；實質買權值估計果分別為每顆8.1613與9.8661美元，表示茂德於1996年之設廠時，以此買權值可取得後續計畫案之投資與否的權利；進一步根據廠商預期於上市階段之產能與發行股本，我們可估計於設廠至上市階段之預期股價，結果發現以持有成本估計之類股利率值以求算之每股實質買權值時，是接近標的公司之掛牌價。
在模擬分析中發現，投資預算愈高臨界值愈高；相關係數為正值時，波動性、無風險利率愈高臨界值愈高，相關係數為負值時，波動性愈高臨界值愈低，無風險利率愈高臨界值愈高，且標的物價值是不會影響各期投資與否之決策。
參數敏感度分析顯示，實質買權之值與無風險利率為負向關係，與波動性則不一定成正向關係，相異於金融買權理論，主因在於實質選擇權之標的物為非交易性資產的特性，使複合式實質買權公式中產生類股利率，此類股利率為相關係數之增函數，當相關係數大於零，而類股利率亦為正值，實質買權之值會隨波動性之增加而下降，表示新計劃案風險分散效果差；反之，相關係數小於零，類股利率亦小於零，實質買權值則隨波動性增加而上升，表示投資風險分散效果佳。
We employed the closed-form solution for multi-stage compound real call option to evaluate ProMos Technologies Inc. It was constructed in 1996 as the first plant to product new generation 64Mb DRAM chips manufactured on 0.35μm to 0.2μm in Taiwan. We cannot construct a synthetic real option using the DRAM foundry because the foundry is a non-traded asset. Therefore, we applied stock portfolio of the listing semiconductor firms as twin securities. The underling variable is the W.W. DRAM ASP. The unit investment budget is set to be the exercise price.
Trigeorgis (1993) noted if the object of investment is a non-traded asset, there exists a dividend-like yield (δ) in the real option pricing model. We referred to the method of Duan, Lin, and Lee (2003) for the estimation of dividend-like yield. The result of dividend-like yield estimation showed that yields obtained from CAPM and cost of carry model were close, differed by only 0.0224 and both values were negative, indicating the small correlation between the new DRAM foundry of ProMos and existing portfolio of representative chipmakers. It indicated that the new investment offered diversification advantage and consistence with the theory of innovation from Schumpeter (1939).
We used three numerical methods to simulate the approximation value of the multivariate normal integral and found that lattice method was the best method in terms of execution efficiency and the Monte Carlo method was the worst.
In the solving critical values, we found that the at-the-money critical value decreased by the year, which indicated that the manufacturing process of the ProMos followed the Moore’s law. The estimated values of real call options were US$8.1613 and US$9.8661 respectively. It expressed the intrinsic values of manufacturing each unit of DRAM by the firm when it is constructed. Finally we estimated the real call value by share of the ProMos based on the projected capacity and its outstanding shares at the time of IPO. The result showed that the value was close to the listing price in IPO with the dividend-like yield by the cost of carry model.
In the simulative analysis, we find that all critical values are influenced by the production cost for stage i and stages thereafter. The higher production cost, the higher critical value. We shows that volatility, risk-free rate and critical value move together when ρ value was large than zero, and the lower volatility, the higher risk-free rate, the higher critical value when ρ value was less than zero. But when the company goes IPO, its value does not affect the critical values in stage 1 and stages thereafter. Thus the underlying value at the time of IPO does not affect the investment decision made at each stage.
In volatility and interest rate sensitivity analysis, we found that the valuation result of an investment project derived from real call option differed from financial call option, mainly because an investment project a non-traded asset that is subjected to the influence of dividend-like yield. In the formula for dividend-like yield, there exists ρ; when the ρ value is large or positive, it indicated the high correlation of the new investment project with the current market status, hence poor diversification advantage of the project and higher investment risks. That means when σ increases, the value of decision flexibility brought by the new project declines, and hence the real call value drops. Conversely, when ρ is small or negative, the project offers more diversification advantage and the real call value rises with the increase of σ. Regardless the ρ value, when interest rate increases, the real call option value drops. In addition, when ρ is large under high interest rate market, that is when the project offers poor diversification advantage, its value is subjected to the influence of interest rate only and nearly totally unrelated to σ.
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