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 / 1r

Length of smaller part of line = 1r
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
= br(ba)
b.
x
2
= a+r(ba)
2)
If f(x
1
) < f(x
2
)
a.
The new interval of uncertainty is [x1,b]
b.
The new lefthand endpoint is x3=br(bx1)
c.
The new righthand endpoint is x4=
x1+r(bx1)
3)
If f(x1) ≥f(x2)
a.
The new interval of uncertainty is [a,x2]
b.
The new lefthand endpoint is x3= x2r(x2a)
c.
The new righthand endpoint is x4=
a+r(x2a)
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
= (RevenuesCosts)(1Tax Rate)+ Tax Rate * Deprecation
Ex:
2 mutually exclusive proposals of = risk have been made for the purchase of a new
mach. Assume straightline 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
13 yrs
$3,500
$1,800
45 yrs
$3,000
$1,800
Project A:
Depr = Initial Val – Salvage Val / Useful Life = 85000/5 = $1700
Yrs 13: Taxable Inc = Earnings – Depr = 35001700  $1800
Tax = .4(1800) = $720
Cash Flow(yrs 13) = Earnings – Taxes = 3500720 = $2870
Yrs 45: Tax Inc = 30001700 = $1300
Tax = .4(1300) = $520
Cash Flow(45) = 3000520 = 2480
PV = 8500+2780(1.1)
1
+ 2780(1.1)
2
+…+ 2780(1.1)
5
= $1647.21
Project B:
Depr = 60001000/5 = $1000
Taxable Income(yrs 15) = 18001000 = $800
Tax = .4(800) = $320
Cash Flow(yrs 14) = Earnings – Tax = 1800320 = $1480
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 Spring '08
 Patton
 Depreciation, Net Present Value

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