1. Fuzzy Applications In Finance and Investment 1390 March School of Economic Sciences In the Name of God Dr. K.Pakizeh k.Dehghan Manshadi E.Jafarzade

2. Forecasting Demand Using Fuzzy Averaging 1

3. Experts estimates for annual demand for a new product. Five experts are asked to forecast the annual demand for a new product using Fuzzy Delphi technique which requires use of triangular numbers Ai = (a(i) 1 ; a(i) M; a(i) 2 ); i = 1; …..; 5. Here a(i) 1 is the smallest number of units to be produced, a(i) M is the most likely number of units, and a(i) 2 is the largest number of units. The experts opinions are shown on Table bellow: Forecasting Demand Using Fuzzy Averaging

4. The Defuzzied Average Forecasting Demand Using Fuzzy Averaging

5. 2 Fuzzy Zero-Based Budgeting

6. The fuzzy zero-based budgeting method uses triangular numbers to model fuzziness in budgeting. it is more realistic to use fuzzy data instead of crisp data. Consider a company with several decision centers, say A;B; and C. Assume that the decision makers agree on some preliminary budgets using a specified number of budget levels for each center depending on its importance. The budgets are expressed in terms of triangular fuzzy numbers obtained by certain procedure. The following possible budget levels were suggested: for the centerA;A 0 < A 1 < A 2 ; for the centerB;B 0 < B 1 ; for the centerC;C 0 < C 1 < C 2 : Fuzzy Zero-Based Budgeting normal minimal improved Fuzzy Zero-Based Budgeting

7. The total budget available to the company is limited but it is flexible and could be expressed by a right trapezoidal number L of the type shown in Fig. bellow with membership function: Total available budget. Fuzzy Zero-Based Budgeting

8. The decision makers follow a step by step budget allocation procedure according to the importance of each center in their opinion. where Fuzzy Zero-Based Budgeting

9. Example The limited available budget L given by and Fuzzy Zero-Based Budgeting

10. Cumulative budgets. NOTE: The budget of center B is at level 0 (smaller than normal ); the decision makers may consider the option to close this center and redistribute the money to the other two centers which are more important. Fuzzy Zero-Based Budgeting

11. 3 Fuzzy Valuation

12. Valuation is One of the most important aspect of Investment and Finance Problems. Although there are many methods in valuation, but most of them are based on calculation of present value of cash flows. In most cases its assumed that the discount rate is fixed and deterministic. But we know that such assumption can rarely be true. So one of the applicable method in order to consider a probabilistic discount rate is Fuzzy procedure. Here this procedure is introduced with an example. F i = cash flow in period I R= discount rate PV=ordinary present value (its with uncertainty) Fuzzy Valuation

13. Now we assume a fuzzy discount rate and rewrite the PV formula as bellow: Discount rate in period i ( triangular fuzzy number ) Example Year(i)0123 Amount invested3000500040002000 Fuzzy Valuation

14. Year(i)123 Fuzzy Discount Rate(%)(8,10,13)(9,12,15)(7,10,12) Fuzzy Valuation

15. 4 Portfolio Selection Based on the Fuzzy Decision Theory

16. with the membership function: Furthermore, the optimal decision is defined by the following non-fuzzy subset Portfolio Selection Based on the Fuzzy Decision Theory

17. An investor can construct a portfolio based on m potential market scenarios from an investment universe of n assets with and xmax i being the minimum and the maximum weight of the ith asset, respectively. Let Rik denote the return of the ith asset for the kth market scenario and let Rk(x) = n i=1 Rikxi denote the portfolio return for the kth scenario, at the end of the investment period. For each scenario, the investor may have a target range for the expected return, over the investment period. Denoting Rmin k and Rmax k as the minimum and the maximum expected returns, respectively, for the kth market scenario, and characterizing the degree of the investor’s satisfaction with portfolio x for the kth scenario as the following linear membership function: Portfolio Selection Based on the Fuzzy Decision Theory

18. portfolio selection model can be written as follow: Portfolio Selection Based on the Fuzzy Decision Theory

19. References: [1] Yong Fang and et.al;Fuzzy portfolio optimization;theory and methods; [2] George Bojadziev and Maria Bojadziev;Fuzzy Logic for Business,Finance, and Management; [3] Ludmila Dymowa;Soft computing in Economics and Finance; [4]Kaufman, Arnold &Madan M.Gupta,1991,Fuzzy mathematical models in engineering and management science,Elsevier Science Publications. References

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