Engineering Economics List of Formulas 2021 PDF

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Prepared by Ron Mackinnon, University of British Columbia, © 2008. 7-Feb-EECE 450 — Engineering Economics — Formula SheetCost Indexes:Index valu eat timeBIndex valu eat timeA Cost at timeBCost at timeA=Power sizing:power-sizing exponentSize (capacity) of asset BSize (capacity) of asset A Cost ofasse...

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EECE 450 — Engineering Economics — Formula Sheet Ordinary Geometric Gradient Annuity:

Cost Indexes:

 1 − (1 + g) n (1 + i) − n  P= A1 ; i ≠ g i −g  

Cost at time A Index value at time A = Cost at time B Index value at time B

Power sizing: Cost of asset A  Size (capacity) of asset A  =  Cost of asset B  Size (capacity) of asset B  x = power - sizing exponent

x

Learning Curve: TN = Tinitial × N b log(learningcurve rate) b= log 2 TN = time to make N th unit Tinitial = time to make first unit N = number of finished units b = learning curve exponent

Simple Interest:

P=

nA1 ;i = g (1 + i)

 (1+ i) n − (1+ g ) n F = A1 i −g 

 ; i ≠ g 

F = nA1 (1 + i) n−1; i = g A1 = payment in first period (end) g = periodic rate of growth P ,F ,i ,n as above for compound interest

Simple Annuity Due:  1− (1+ i) − n  P= A  (1+ i ) i   n  (1+ i ) − 1 F = A  (1+ i ) i  

Interest earned on amount P : I = Pin Maturity value : F = P (1 + in ) i = interest rate per time period n = number of time periods

Nominal, Periodic, Effective Interest Rates:

Compound Interest:

i=

F = P(1 + i) n F = future value P = present value i = periodic interest rate n = number of periods

Ordinary Simple Annuity:

A = cash amount (beginning of period) P ,F ,i ,n as above for compound interest r m

(

)m

(1+ ieff ) = 1+ rm r = nominal interest rate per year m = number of compounding periods per year ieff = effective interest rate (compounded annually) i = periodic interest rate

1 − (1 +i )− n  P = A  i  

Equivalent Interest Rates:

 (1 + i) n −1  F = A  i   A = periodic payment (end of period) P ,F ,i ,n as above for compound interest

ip = interest rate for payment period

Ordinary Arithmetic Gradient Annuity: 1  n Aeq = G  −  n i (1 + i) −1 

(1+ i p ) p = (1 + ic ) c p = number of payment periods per year ic = interest rate for compounding period c = number of compounding periods per year

Ordinary General Annuity:  1− (1+ ip )− n  =  P A ip  

Aeq = equivalent periodic payment

 (1 + ip ) n −1   F = A ip   ip = interest rate for payment period

G = gradient amount (periodicincrement)

n = number of payment periods

P ,i ,n as above for compound interest

P , F , A as above for annuities

 (1+ i )n − in − 1 P =G 2 n   i (1 + i) 

Prepared by Ron Mackinnon, University of British Columbia, © 2008.

7-Feb-08

Perpetual Annuities: Ordinary : P =

A i

A A (1+ i ) = + A i i A Geometric Growth : P = ;i > g i− g P , A ,i , g as above for annuities Due : P =

Investment Criteria: CF1 CF2 CFn + + ... + n 1 2 (1+ r ) (1+ r ) (1 + r) NPV = net present value

NPV = CF0 +

NFV = CF0 (1 + r )n + CF1 (1 + r ) n −1 + ... + CFn NFV = net future value EACF = equivalent annual cash flow =

NPV −n 1 −(1 +r)   r 

CFj = cash flow at time j n = lifetime of investment r = MARR = minimum acceptable rate of return CF1 CF2 CFn 0 = CF0 + + + ... + (1 + i ) 1 (1 + i) 2 (1+ i) n i = IRR = internal rate of return PV(neg CFs, efin )× (1+ i′ ) n = FV(pos CFs, einv ) i ′ = MIRR= modified internal rate of return efin = financing rate of return einv = reinvestment rate of return PV(positive cash flows) Benefit - cost ratio, BCR = PV(negative cash flows)

Probability: E(X ) = Weighted average =

w1 S1 +  + w kS k w1 + + wk

wi = weight for Scenario i S i = value of X for Scenario i E(X ) = µ X = expected value of X =

 P(x j ) x j all j

Var(X ) = variance of X =

 P(x j)( x j − µX )2 all j

P( x j ) = Probability( X = x j )

Depreciation: B= initial (purchase) value or cost basis S= estimated salvage value after depreciable life dt= depreciation charge in year t N= number of years in depreciable life t

Book value at end of period t: BVt = B −

 di i =1

Straight-Line (SL): Annual charge: dt = (B – S)/N Book value at end of period t: BVt = B − t×d Prepared by Ron Mackinnon, University of British Columbia, © 2008.

Sum-of-Years’-Digits (SOYD): SOYD = N(N+1)/2 Annual charge: dt = (B − S)(N − t + 1)/SOYD Declining balance (DB): D= proportion of start of period BV that is depreciated Annual charge: dn = BD(1–D) n–1 n Book value at end of period n: BVn = B(1-D) Capital Cost Allowance (CCA): d= CCA rate UCC n= Undepreciated capital cost at end of period n Annual charge: CCA1 = B(d/2) for n = 1; CCAn = Bd(1–d/2)(1–d)n–2 for n ≥ 2 UCC at end of period n: UCCn = B(1–d/2)(1–d) n–1 BdT C  1 + i 2  PV(CCA tax shields gained) =     i + d  1+ i 

SdT C  1  PV(CCA tax shields lost) =     i + d  (1 + i )N  TC = firm's tax rate; i = discount rate

Investment Project Cash Flows: Taxable income = OR−OC−CCA−I Net profit = taxable income ×(1−T) Before-tax cash flow (BTCF) = I+CCA+taxable income After-tax cash flow (ATCF) = Net profit + CCA + I = (Taxable income)×(1−T) + CCA + I = (BTCF − I − CCA)(1 −T) + CCA + I = (OR − OC)(1 −T) + I(T) + CCA(T) Net cash flow from operations = ATCF – I – DIV = (OR − OC)(1−T) + I(T) + CCA(T) − I − DIV = (OR − OC − I)(1−T) + CCA(T) − DIV = Net profit + CCA − DIV OR= operating revenue; OC= operating cost I= interest expense; DIV = dividends; T= tax rate Net cash flow = Net cash flow from operations + New equity issued + New debt issued + Proceeds from asset disposal − Repurchase of equity − Repayment of debt (principal) − Purchase of assets  dT 1+ i 2  Net capital investment = B1 − C   i + d 1+ i 

 dT   1 Net salvage value = S 1− C    i + d  (1+ i) N

  

Inflation: (1+i) = (1+i′ )(1+f) i = i′ + f + ( i ′)(f) i= market interest rate; i′ = real interest rate f= inflation rate

Weighted Average Cost of Capital (WACC): WACC =

D E × (1 − TC )id + × ie V V

V = D+E D= market value of debt; E= market value of equity V= market value of firm id= cost of (rate of return on) debt after-tax cost of debt: idt = id (1–T) ie= cost of equity 7-Feb-08...