EECE 450 — Engineering Economics — Formula Sheet

1 Prepared by Ron Mackinnon, University of British Columbia, 2008. 7-Feb-08 eece 450 Engineering Economics Formula SheetCost Indexes: B at time eIndex valuA at time eIndex valuB at timeCost A at timeCost = Power sizing: exponent sizing-powerBasset of (capacity) SizeAasset of (capacity) SizeBasset ofCost Aasset ofCost = =xx Learning Curve: exponent curve learningunits finished ofnumber unitfirst make totimeunitth make totime2lograte) curve learninglog(initialinitial===== =bNTNTbNTTNbN Simple Interest: periods timeofnumber period per time rateinterest )1( :alueMaturity v :amount on earnedInterest ==+==niinPFPinIP Compound Interest.

2 Periods ofnumber rateinterest periodicluepresent va valuefuture)1(====+=niPFiPFn Ordinary Simple Annuity: interest compoundfor above as ,,,period) of (endpayment periodic1)1()1(1niFPAiiAFiiAPnn= += + = Ordinary Arithmetic Gradient Annuity: interest compoundfor above as ,,increment) (periodicamount gradient payment periodic equivalent)1(1)1(1)1(12niPGAiiiniGPiniGA eqnnneq== + += + =Ordinary Geometric Gradient Annuity: interest compoundfor above as ,,,growth of rate periodic(end) periodfirst in payment ;)1(;)1()1(;)1(;)1()1(1111111niFPgAgiinA FgigigiAFgiinAPgigiigAPnnnnn===+= + +==+= ++ = Simple Annuity Due: interest compoundfor above as ,,,period) of (beginningamount cash )1(1)1()1()1(1niFPAiiiAFiiiAPnn=+ +=+ + = Nominal, Periodic, Effective Interest Rates: ()rateinterest periodicannually) d(compounde rateinterest effectiveyearper periods gcompoundin ofnumber yearper rateinterest nominal1)1(effeff====+=+=iimrimrimmr Equivalent Interest Rates: yearper periods gcompoundin ofnumber period gcompoundinfor rateinterest yearper periodspayment ofnumber periodpayment for rateinterest )1()1(====+=+cipiiicpccpp Ordinary General Annuity.

3 Annuitiesfor above as ,,periodspayment ofnumber periodpayment for rateinterest 1)1()1(1 AFPniiiAFiiAPppnppnp== += + = Prepared by Ron Mackinnon, University of British Columbia, 2008. 7-Feb-08 Perpetual Annuities: annuitiesfor above as ,,,; :Growth Geometric)1( :Due :OrdinarygiAPgigiAPAiAiiAPiAP> =+=+== Investment Criteria: flows) cash ePV(negativflows) cash ePV(positiv BCRratio,cost -Benefitreturn of ratent reinvestmeereturn of rate financingereturn of rate internal modifiedMIRR)eCFs, FV(pos)1()eCFs, PV(negreturn of rate internalIRR)1(..)1()1(0return of rate acceptable minimumMARR investment of lifetime at timeflow cashNPVflow cash annual equivalentEACF valuefuturenet )1()1(NFVluepresent vanet NPV)1(.

4 1()1(NPVinvfininvfin22110)1(111022110=== == = + ==+++++++===== ===+++++==+++++++= + iiiiCFiCFiCFCFrnjCFCFrCFrCFrCFrCFrCFCF nnnjrrnnnnnn Probability: )y(Probabilit)(P))((P of variance)(Var)(P of valueexpected)E( Scenariofor of value Scenariofor weight average Weighted)E(2 all all111jjXjjjjjjXiikkkxXxxxXXxxXXiXSiwwwS wSwX== =======++++== LL 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 Book value at end of period t: BVt = B =tiid1 Straight-Line (SL): Annual charge: dt = (B S)/N Book value at end of period t: BVt = B t dSum-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 Book value at end of period n: BVn = B(1-D)n Capital Cost Allowance (CCA): d= CCA rate UCCn= 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 ++ +=iidiBdTC121gained) shields PV(CCA tax ()ratediscount rate.)

5 Tax sfirm'11lost) shields PV(CCA tax== + +=iTidiSdTCNC 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 +++ =iididTBC1211investment capitalNet () + + =NCididTS111 valuesalvageNet 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): ()EDViVEiTVDedC+= + =1 WACC 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