本文提出模糊理論在現金流量折現問題的應用。以現金流量折現模式的基本架構為基礎，藉由模糊集合理論的導入，來探討財務管理領域中運用現金流量模式評價的相關議題。
第三章主要發展出模糊邏輯系統來延伸傳統現金流量折現模式，其中模糊現金流量及折現率同時被考慮在模式中。為了明確地構建出一個較符合實際的評價模型，模式中具不確定性的參數將被模糊化為三角模糊數來量化及評估公司或金融資產的內在真實價值。利用 符號距離方法解模糊化後，可證明本章所提出之模糊現金流量折現模式為傳統現金流量折現模式的一種延伸。
延續第三章之架構，第四章將現金流量折現模式轉換為永續年金模式，進一步分析不同現金流量形式的永續年金現值。相同地，不確定性與決策者評估態度等基本概念被同時用來解釋不精確現金流量、必要報酬率及成長率等參數，並同時模糊化為三角模糊數。結果提供決策者在評估一般永續年金及成長型永續年金的價值，同時亦說明模糊永續年金模式為傳統永續年金模式的一種延伸。
第五章以模糊現金流量折現模式為基礎，進一步發展出模糊資本預算模式，模式中涵蓋未來各期間之不確定現金流量及必要報酬率。藉由模糊資本預算模式的推導與模擬分析結果，說明此一模糊模式對財務管理者在分析資本預算時是較有實用價值的。
本文透過嚴謹及具體的數學分析，導入模糊集合理論，指出模糊現金流量折現模式為傳統評價模式的一種合理的延伸，在不失模式簡單易懂的本質下發展出較適合一般財務管理者使用之評價模型及資本預算模型。分析結果亦可解釋過去資料無法完全精確預測複雜財務環境中之不確定資訊現象。另外，本文所推導之定理皆以數值範例加以說明模糊觀點下各個模糊模式的義涵，並討論真實價值與資本預算決策是如何受到不確定現金流量水準、公司成長率及必要報酬率的影響。因此，本文之主要貢獻在於構建較實際的現金流量折現模式，並將其應用於財務管理領域的一些重要模型。

This thesis puts forward some applications of fuzzy theory in discounted cash flow (DCF) problems. On the basis of the basic frameworks of financial valuation models, the fuzzy set theory is introduced to deal with the related topics of DCF models such as perpetuity and capital budgeting. 
A fuzzy logic system has been developed and it can be concluded that it is some extensions of the classical DCF models. In order to explicitly construct a more appropriate valuation model, uncertain information will be fuzzified as triangular fuzzy numbers to quantify and evaluate the intrinsic value of a company or a financial asset. Using  signed distance method to defuzzify the fuzzy model, we find that the fuzzy discounted cash flow (FDCF) model proposed in this thesis is some extensions of classical (crisp) model and it is considered to be more suitable to capture the imprecise elements of valuation. 
Following the basic framework of Chapter 3, Chapter 4 stresses the DCF model on the analysis of perpetuity model, in which the impreciseness and the decision maker’s attitude toward the estimate of the uncertain parameters are simultaneously incorporated into the descriptions of different cash flow streams, required rate of return and growth rate. Similar to Chapter 3, the uncertain information will be fuzzified as triangular fuzzy numbers; therefore it would be more realistic for typical decision maker to analyze the present values of ordinary and growing perpetuities. We also find that the fuzzy perpetuity models are one extension of crisp perpetuity models.
In Chapter 5, we further develop a fuzzy capital budgeting model by extending the classical net present value (NPV) method that takes the vague future cash flows and required rates of returns in different time periods into account. The results are more useful and practical for financial manager to analyze the capital budgeting decision of firms by means of the derivations of fuzzy model with numerical simulation.
Through conscientious and concrete mathematical analyses, this thesis addressed that the FDCF model is one reasonable extension of the crisp models. In addition, numerical examples are also used to illustrate each theorem in this thesis. In summary, the main contributions of this thesis are to construct the easier understand and more realistic FDCF model and then apply it to extend some important valuation models in financial management without losing the essence of original models.

List of Table	IV
List of Figure	V
Chapter 1  Introduction	1
1.1  Motivation and objective 1
1.2  Organization of the dissertation 4
Chapter 2  Literature Review on Discounted Cash Flow Model 6
2.1  Historical review on discounted cash flow models 	6
2.2  Preliminaries for fuzzy operations 	9
2.3  The explanation of using triangular fuzzy numbers in fuzzification 	14
2.4  The explanation of using the  signed distance 15
Chapter 3  Valuation by Using a Fuzzy Discounted Cash Flow Model	19
3.1  Exordium  	19
3.2  Valuation by using the crisp DCF model  	20
3.3  Valuation by using the FDCF model  	23
3.4  Numerical examples  	29
3.5  Discussions  	33
3.5.1  The relation between Theorem 3.1 and crisp case  34
3.5.2  The results of using the  signed distance method to defuzzify with different levels 34
Chapter 4  Using Fuzzy Discounted Cash Flow Models in Evaluating the Present Value of Perpetuities 36
4.1  Exordium 	36
4.2  The crisp perpetuities with different types of cash flow streams  	37
4.2.1  The present value of an ordinary perpetuity  	37
4.2.2  The present value of an growing perpetuity  	38
4.3  The fuzzy perpetuities with different types of cash flow streams  	38
4.3.1  Fuzzification of crisp ordinary perpetuity  	39
4.3.2  Fuzzification of crisp growing perpetuity  	42
4.4  Numerical examples  	45
4.5  Discussions  	48
4.5.1  The relations among Theorems 4.1, 4.2 and the crisp ordinary perpetuity  48
4.5.2  The relations among Theorems 4.3, 4.4 and the crisp growing perpetuity  49
Chapter 5  Using Fuzzy Discounted Cash Flow Model in Capital Budgeting: A Fuzzy Net Present Value Criteria	51
5.1  Exordium  	51
5.2  NPV method in crisp  	52
5.3  Fuzzy net present value method  	53
5.3.1  Capital budgeting with FNPV method  	53
5.3.2  Defuzzification by using the  signed distance approach 56
5.4  Case study and numerical simulations  	57
5.4.1  The application of FNPV method : a project for constructing the students’ dormitory in Nan Hwa University  57
5.4.2  Simulation results   	59
5.5  Discussions 	61
5.5.1  The relations among Theorem 5.2 and crisp case  62
5.5.2  The relations of the estimated net cash inflow   and required rate of return in the fuzzy sense under different levels 63
Chapter 6  Conclusions 65
6.1  Results  	65
6.2  Future studies   	67
Bibliography	68
Appendix	72



List of Tables
3.1 The numerical comparisons of crisp case and fuzzy case for n=3 with
different λ levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 The numerical comparisons of crisp case and fuzzy case for n=2 with
different λ levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3 The numerical comparisons of crisp case and fuzzy case for n=1 with
different λ levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.1 The numerical comparisons of crisp and fuzzy present value of ordinary
perpetuity with different λ levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2 The numerical comparisons of crisp and fuzzy present value of growing
perpetuity with different λ levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47


List of Figures
2.1 cut − α )] ( ~ ), ( ~ [ α α U L D D and point ) ( ~ ) 1 ( ) ( ~ α λ α λ U L D D − + in
)] ( ~ ), ( ~ [ α α U L D D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
3.1 Infinite cash-flow stream and present value . . . . . . . . . . . . . . . . . . . . . . . . 20
5.1 Graph of the FNPV ( *
, 10 λ N ) and crisp NPV ( 10 N ) with different λ levels
and left-hand-side variations of net cash inflow t
ε . . . . . . . . . . . . . . . . . 59
5.2 Graph of the FNPV ( *
, 10 λ N ) and crisp NPV ( 10 N ) with different λ levels
and right-hand-side variations of net cash inflow t
β . . . . . . . . . . . . . . . . 60
5.3 Graph of the FNPV ( *
, 10 λ N ) and crisp NPV ( 10 N ) with different λ levels
and left-hand-side variations of required rate of return t
θ . . . . . . . . . . . . . 60
5.4 Graph of the FNPV ( *
, 10 λ N ) and crisp NPV ( 10 N ) with different λ levels
and right-hand-side variations of required rate of return t
ω . . . . . . . . . . . 61

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