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Test 2 Cheat Sheet - Golden Section Search(be able to go...

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Golden Section Search : (be able to go thru 1 iteration of the method to find the new interval of uncertainty) (max. f(x) s.t. such that a ≤ x ≤ b) Consider the line segment [0,1] that is divided into 2 parts: 0 --------------------------------- r -------------------- 1 The line segment is said to be divided into the Golden Section if: Length of the whole line / Length of larger part of line = Length of larger part of line / length of smaller part of line Find r that divides the line into the golden section. - Length of whole line = 1 Whole / Lg = 1/r - Length of larger part of line = r Lg / Sm = r / 1-r - Length of smaller part of line = 1-r Golden Ratio: r = -1 ± √5 / 2 = . 6180 Interval of Uncertainty – interval containing x that maximizes f(x) L k = length of the interval of uncertainty after k iterations I k = interval of uncertainty after k iterations 1) Initially evaluate f(x) at 2 points x 1 and x 2 on the interval [a,b] a. x 1 = b-r(b-a) b. x 2 = a+r(b-a) 2) If f(x 1 ) < f(x 2 ) a. The new interval of uncertainty is [x1,b] b. The new left-hand endpoint is x3=b-r(b-x1) c. The new right-hand endpoint is x4= x1+r(b-x1) 3) If f(x1) ≥f(x2) a. The new interval of uncertainty is [a,x2] b. The new left-hand endpoint is x3= x2-r(x2-a) c. The new right-hand endpoint is x4= a+r(x2-a) Ex: Max f(x) = x 2 + 2x s.t. -3 ≤ x ≤5. a = -3, b=5 x1= 5- r[5-(-3)] = 5 - .618(8)= .055728 x2= -3 + r[5-(-3)] = -3 + .618(8)=1.944271 f(x1) =(.055728)2+2(.055728)=.11456 f(x2) =(1.944271)2+2(1.944271)=7.6687 f(x2) > f(x1)=> new endpoints are (x1,5) Taxes & Depreciation: Devaluation Expense - Amount an asset decreases in market value during service life. Depreciation Expense - Decrease in value of an asset reported for income tax purposes (a. Straight Line Dep.: Dep. per yr = Total Deval Exp / Service Life) Ex: A piece of equip. is purchased for $135,000 & sold for $15,000 after 10 yrs. Find devaluation & deprecation expense. Total Deval Exp = $135,000-$15,000 = $120,000 Dep Exp = $120,000/10yr = $12,000/yr Determining Cost of Money: Debt Capital : $ borrowed to purchase an asset Equity C: $ belonging to a business thats used to purchase asset Factors Determining Cost of $: o Proportion of debt financing o Proportion of equity financing o Cost of debt capital (interest expense) o Cost of equity capital (opp. cost) Cash Flow After Taxes: Taxable Income = Revenues – Costs – Depreciation Tax = TaxRate * (Revenues – Costs – Deprecation) Cash Flow (after taxes) = Revenues – Costs – Tax Cash Flow = (Revenues-Costs)(1-Tax Rate)+ Tax Rate * Deprecation Ex: 2 mutually exclusive proposals of = risk have been made for the purchase of a new mach. Assume straight-line deprecation, a corporate tax rate of 40%, and 10% cost of capital. Determine the best project using present value analysis Project A Project B Net Investment $8,500 $6,000 Salvage Value $0 $1,000 Estimated Life 5 years 5 years Earnings b4 taxes & deprecation 1-3 yrs $3,500 $1,800 4-5 yrs $3,000 $1,800 Project A: Depr = Initial Val – Salvage Val / Useful Life = 8500-0/5 = $1700 Yrs 1-3: Taxable Inc = Earnings – Depr = 3500-1700 - $1800 Tax = .4(1800) = $720 Cash Flow(yrs 1-3) = Earnings – Taxes = 3500-720 = $2870 Yrs 4-5: Tax Inc = 3000-1700 = $1300 Tax = .4(1300) = $520 Cash Flow(4-5) = 3000-520 = 2480 PV = -8500+2780(1.1) -1 + 2780(1.1) -2 +…+ 2780(1.1) -5 = $1647.21 Project B: Depr = 6000-1000/5 = $1000 Taxable Income(yrs 1-5) = 1800-1000 = $800 Tax = .4(800) = $320 Cash Flow(yrs 1-4) = Earnings – Tax = 1800-320 = $1480
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