某咨詢(xún)－戰(zhàn)略方法之一

Author:CollinsQianReviewer:BrianBilello

bcBainMathMarch1998Copyright?1998Bain&Company,Inc.1CU7112997ECABainMathAgenda

BasicmathFinancialmathStatisticalmath2CU7112997ECABainMathAgenda

BasicmathratioproportionpercentinflationforeignexchangegraphingFinancialmathStatisticalmath3CU7112997ECABainMathRatio

Definition:Application:Note:TheratioofAtoBiswrittenorA:BABAratiocanbeusedtocalculatepriceperunit(),giventhetotalrevenueandtotalunitsPriceUnittotalrevenue=

Given:

==Answer:PriceUnit$9MM1.5MMThemathforratiosissimple.Identifyingarelevantunitcanbechallengingtotalunits=

price/unit=

$9.0MM1.5MM$?$6.04CU7112997ECABainMathProportion

Definition:IftheratioofAtoBisequaltotheratioofCtoD,thenAandBareproportionaltoCandD.Application:

=

ItfollowsthatAxD=BxCABCDRevenue=SG&A=Given:$135MM$83MM$270MM$?19961999Answer:$135MM

$270MM$83MM$?135MMx?=83MMx270MM83MMx270MM135MM=TheconceptofproportioncanbeusedtoprojectSG&Acostsin1999,givenrevenuein1996,SG&Acostsin1996,andrevenuein1999(assumingSG&Aandrevenuein1999areproportionaltoSG&Aandrevenuein1996)?==$166MM5CU7112997ECABainMathPercent

Definition:Apercentage(abbreviated“percent”)isaconvenientwaytoexpressaratio.Literally,percentagemeans“per100.”Application:Inpercentageterms,0.25=25per100or25%InherfirstyearatBain,anAClogged7,000frequentfliermilesbyflyingtoherclient.Inhersecondyear,shelogged25,000miles.Whatisthepercentageincreaseinmiles?Given:ApercentagecanbeusedtoexpressthechangeinanumberfromonetimeperiodtothenextAnswer:

-1=3.57-1=2.57=257%25,0007,000%change==-1

newvalue-originalvalueoriginalvaluenewvalueoriginalvalueTheratioof5to20isor0.255206CU7112997ECABainMathInflation-DefinitionsIfanitemcost$1.00in1997andcost$1.03in1998,inflationwas3%from1997to1998.Theitemisnotintrinsicallymorevaluablein1998-thedollarislessvaluableWhencalculatingthe“real”growthofadollarfigureovertime(e.g.,revenuegrowth,unitcostgrowth),itisnecessarytosubtractouttheeffectsofinflation.Inflationarygrowthisnot“real”growthbecauseinflationdoesnotcreateintrinsicvalue.Definition:Apricedeflatorisameasureofinflationovertime.RelatedTerminology:1.Real(constant) dollars:2.Nominal (current)dollars:3.PricedeflatorPricedeflator(currentyear)Pricedeflator(baseyear)Inflationbetweencurrentyearandbaseyear=Dollarfigure(currentyear)Dollarfigure(baseyear)=Dollarfiguresforanumberofyearsthatarestatedinachosen“base”year’sdollarterms(i.e.,inflationhasbeentakenout).Anyyearcanbechosenasthebaseyear,butalldollarfiguresmustbestatedinthesamebaseyearDollarfiguresforanumberofyearsthatarestatedineachindividualyear’sdollarterms(i.e.,inflationhasnotbeentakenout).Inflationisdefinedastheyear-over-yeardecreaseinthevalueofaunitofcurrency.7CU7112997ECABainMath

Inflation-U.S.PriceDeflators*1996isthebaseyearNote:ThesearetheU.S.PriceDeflatorswhichWEFAGrouphasforecastedthroughtheyear2020.ThelibraryhaspurchasedthistimeseriesforallBainemployeestouse.Adeflatortablelistspricedeflatorsforanumberofyears.8CU7112997ECABainMathInflation-Realvs.NominalFigures

Tounderstandhowacompanyhasperformedovertime(e.g.,intermsofrevenue,costs,orprofit),itisnecessarytoremoveinflation,(i.e.userealfigures).Sincemostcompaniesusenominalfiguresintheirannualreports,ifyouareshowingtheclient’srevenueovertime,itispreferabletousenominalfigures.Foranexperiencecurve,whereyouwanttounderstandhowpriceorcosthaschangedovertimeduetoaccumulatedexperience,youmustuserealfiguresNote:Whentouserealvs.Nominalfigures:Whetheryoushouldusereal(constant)figuresornominal(current)figuresdependsonthesituationandtheclient’spreference.Itisimportanttospecifyonslidesandspreadsheetswhetheryouareusingrealornominalfigures.Ifyouareusingrealfigures,youshouldalsonotewhatyouhavechosenasthebaseyear.9CU7112997ECABainMathInflation-Example(1)

(1970-1992)Adjustingforinflationiscriticalforanyanalysislookingatpricesovertime.Innominaldollars,GE’swasherpriceshaveincreasedbyanaverageof4.5%since1970.Whenyouusenominaldollars,itisimpossibletotellhowmuchofthepriceincreasewasduetoinflation.$2,00072Nominaldollars4.5%PriceofaGEWasher1970717374757677787980818283848586878889909192$0$500$1,000$1,500CAGR10CU7112997ECABainMathInflation-Example(2)

PriceofaGEWasherCAGR(1970-1992)(1.0%)4.5%197071727374757677787980818283848586878889909192$0$500$1,000$1,500$2,000$2,500$3,000NominaldollarsReal(1992)dollarsIfyouuserealdollars,youcanseewhathashappenedtoinflation-adjustedprices.Theyhavefallenanaverageof1.0%peryear.11CU7112997ECABainMathInflation-Exercise(1)

Considerthefollowingrevenuestreaminnominaldollars:Revenue($million)199020.5199125.3199227.4199331.2199436.8199545.5199651.0Howdowecalculatetherevenuestreaminrealdollars?12CU7112997ECABainMathInflation-Exercise(2)

Answer:Step1:Chooseabaseyear.Forthisexample,wewilluse1990Step2:Finddeflatorsforallyears(fromthedeflatortable):(1990)=85.34(1991)=88.72(1992)=91.16(1993)=93.54(1994)=95.67(1995)=98.08Step3:Usetheformulatocalculaterealdollars:Pricedeflator(currentyear)

Dollarfigure(currentyear)Pricedeflator(baseyear)Dollarfigure(baseyear)Step4:Calculatetherevenuestreaminreal(1990)dollarsterms:1990:1991:1992:1993:

=,X=20.585.3485.341994:1995:1996:=20.5X

=,X=24.388.7285.3425.3X

=,X=25.791.1685.3427.4X

=,X=28.593.5485.3431.2X

=,X=32.895.6785.3436.8X

=,X=39.698.0885.3445.5X

=,X=43.5100.0085.3451.0XRevenue($Million)199020.5199124.3199225.7199328.5199432.8199539.6199643.5

(1996)=100.0013CU7112997ECABainMathForeignExchange-Definitions

Investmentsemployedinmakingpaymentsbetweencountries(e.g.,papercurrency,notes,checks,billsofexchange,andelectronicnotificationsofinternationaldebitsandcredits)Priceatwhichonecountry’scurrencycanbeconvertedintoanother’sTheinterestandinflationratesofagivencurrencydeterminethevalueofholdingmoneyinthatcurrencyrelativetoinothercurrencies.Inefficientinternationalmarkets,exchangerateswilladjusttocompensatefordifferencesininterestandinflationratesbetweencurrencies

ForeignExchange:ExchangeRate:14CU7112997ECABainMathForeignExchangeRates1)US$equivalent=USdollarsper1selectedforeigncurrencyunit2)CurrencyperUS$=selectedforeigncurrencyunitsper1USdollarTheWallStreetJournalTuesday,November25,1997CurrencyTradingMonday,November24,1997ExchangeRatesCountryArgentina(Peso)Britain(Pound)US$Equiv.11.00011.6910Currency

perUS$20.99990.5914CountryFrance(Franc)Germany(Mark)US$Equiv.0.17190.5752CurrencyperUS$5.81851.7384CountrySingapore(dollar)US$Equiv.0.6289CurrencyperUS$1.5900Financialpublications,suchastheWallStreetJournal,provideexchangerates.15CU7112997ECABainMathForeignExchange-Exercises

Question1:Answer:Question2:Answer:Question3:Answer:

650.28USdollars=?Britishpoundsfromtable:0.5914￡=US$1.00 $650.28x=384.58￡1490.50Francs=?US$

fromtable:$0.1719=1Franc 1490.50Francx=$256.221,000GermanMarks=?Singaporedollars

fromtable:$0.5752=1Mark 1.59Singaporedollar=US$1 1,000GermanMarksxx=914.57Singaporedollars0.5914￡US$1$0.17191Franc$0.57521Mark1.59SingaporedollarUS$116CU7112997ECABainMathGraphing-Linear

X0Y(X1,Y1)(X2,Y2)bXYTheformulaforalineis:y=mx+bWhere,m=slope==y2-y1x2-x1b=theyintercept=theycoordinatewhenthexcoordinateis“0”y

x17CU7112997ECABainMathGraphing-LinearExercise#1

Formulaforline:y=mx+bInthisexercise,y=15x+400,where,02004006008001,0001,2001,4001,6001,800$2,000Dollarschanging050100People(100,1900)(50,1150)Thecatererwouldcharge$1900fora100personparty.yxXaxis=peopleYaxis=dollarschargedm=slope==15b=Yintercept=400dollarscharged(whenpeople=0)Acaterercharges$400.00forsettingupaparty,plus$15.00foreachperson.Howmuchwouldthecatererchargefora100personparty?Usingthisformula,youcansolvefordollarscharged(y),givenpeople(x),andvice-versa18CU7112997ECABainMathGraphing-LinearExercise#2(1)

Alampmanufacturerhascollectedasetofproductiondataasfollows:NumberoflampsProduced/DayProductionCost/Day1008509009501,000$2,000$9,500$10,000$10,500$11,000Whatisthedailyfixedcostofproduction,andwhatisthecostofmaking1,500lamps?19CU7112997ECABai三nM三athGra三phi三ng三-L三ine三ar三Exe三rci三se三#2三(2)08,0三0016,三000Pro三duc三tio三nC三ost三/Da三y05001,0三001,5三00Pro三duc三ed/三Day(1,三500三,1三6,0三00)(1,三000三,1三1,0三00)For三mul三af三or三lin三e:三y=三mx三+三bXa三xis三=三#o三fl三amp三sp三rod三uce三d/d三ayYa三xis三=三pro三duc三tio三nc三ost三/da三yM=三sl三ope三=三=三=三=三10b=三Y三int三erc三ept三=三pro三duc三tio三nc三ost三(i三.e.三,t三he三fix三ed三cos三t)三whe三nl三amp三s=三0y=三mx三+三bb=三y-三mxb=三2,三000三-三10三(10三0)b=三1,三000The三fi三xed三co三st三is三$1,三000y=三10三x+三1,三000For三1,三500三la三mps三:y=三10三(1三,50三0)三+1三,00三0y=三15三,00三0+三1,三000y=三16三,00三011,三000三-2,三0001,0三00三-1三009,0三00900(10三0,三2,0三00)X=三90三0Y=三9,三000yxThe三co三st三of三pro三duc三ing三1,三500三la三mps三is三$1三6,0三0020CU7三112三997三ECABai三nM三athGra三phi三ng三-L三oga三rit三hmi三c(三1)Log三:A“三log三”o三rl三oga三rit三hm三of三giv三en三num三ber三is三de三fin三ed三as三the三po三wer三to三wh三ich三a三bas三en三umb三er三mus三tb三er三ais三ed三to三equ三al三tha三tg三ive三nn三umb三erUnl三ess三ot三her三wis三es三tat三ed,三th三eb三ase三is三as三sum三ed三to三be三10Y=三10x,t三hen三lo三g10Y=三XMat三hem三ati三cal三ly,三ifWhe三re,三Y三=g三ive三nn三umb三er10三=b三aseX=三po三wer三(o三rl三og)For三ex三amp三le:三1三00=三102can三be三wr三itt三en三as三log10100三=2三or三log三10三0=221CU7三112三997三ECABai三nM三athGra三phi三ng三-L三oga三rit三hmi三c(三2)For三a三log三sc三ale三in三ba三se三10,三as三th三el三ine三ar三sca三le三val三ues三in三cre三ase三by三te三nt三ime三s,三the三lo三gv三alu三es三inc三rea三se三by三1.98765432101,0三00,三000三,00三0100三,00三0,0三0010,三000三,00三01,0三00,三000100三,00三010,三0001,0三00100101Log三pa三per三ty三pic三all三yu三ses三ba三se三10Log三-lo三gp三ape三ri三sl三oga三rit三hmi三co三nb三oth三ax三es;三se三mi-三log三pa三per三is三lo三gar三ith三mic三on三on三ea三xis三an三dl三ine三ar三on三the三ot三herLog三Sc三aleLin三ear三Sc三ale22CU7三112三997三ECABai三nM三athGra三phi三ng三-L三oga三rit三hmi三c(三3)The三mo三st三use三ful三fe三atu三re三of三al三og三gra三ph三is三tha三te三qua三lm三ult三ipl三ica三tiv三ec三han三ges三in三da三ta三are三re三pre三sen三ted三by三eq三ual三di三sta三nce三so三nt三he三axe三sthe三di三sta三nce三be三twe三en三10三and三10三0i三se三qua三lt三ot三he三dis三tan三ce三bet三wee三n1三,00三0,0三00三and三10三,00三0,0三00三bec三aus三et三he三mul三tip三lic三ati三ve三cha三nge三in三bo三th三set三so三fn三umb三ers三is三th三es三ame三,1三0It三is三con三ven三ien三tt三ou三se三log三sc三ale三st三oe三xam三ine三th三er三ate三of三ch三ang三eb三etw三een三da三ta三poi三nts三in三a三ser三iesLog三sc三ale三sa三re三oft三en三use三df三or:Exp三eri三enc三ec三urv三e(三al三og/三log三sc三ale三is三ma三nda三tor三y-三na三tur三al三log三s(三ln三or三loge)a三re三typ三ica三lly三us三edpri三ces三an三dc三ost三so三ver三ti三meGro三wth三Sh三are三ma三tri三cesROS三/RM三Sg三rap三hsLin三eS三hap三eo三fD三ata三Pl三otsExp三lan三ati三onAs三tra三igh三tl三ineThe三da三ta三poi三nts三ar三ec三han三gin三ga三tt三he三sam三er三ate三fr三om三one三po三int三to三th三en三extCur三vin三gu三pwa三rdThe三ra三te三of三cha三nge三is三in三cre三asi三ngCur三vin三gd三own三war三dThe三ra三te三of三cha三nge三is三de三cre三asi三ngIn三man三ys三itu三ati三ons三,i三ti三sc三onv三eni三ent三to三us三el三oga三rit三hms三.23CU7三112三997三ECABai三nM三athAge三ndaBas三ic三mat三hFin三anc三ial三ma三thsim三ple三in三ter三estcom三pou三nd三int三ere三stpre三sen三tv三alu三eris三ka三nd三ret三urnnet三pr三ese三nt三val三ueint三ern三al三rat三eo三fr三etu三rnbon三da三nd三sto三ck三val三uat三ionSta三tis三tic三al三mat三h24CU7三112三997三ECABai三nM三athSim三ple三In三ter三estDef三ini三tio三n:Sim三ple三in三ter三est三is三co三mpu三ted三on三a三pri三nci三pal三am三oun三tf三or三as三pec三ifi三ed三tim三ep三eri三odThe三fo三rmu三la三for三si三mpl三ei三nte三res三ti三s:三i=三pr三twhe三re,p=三th三ep三rin三cip三alr=三th三ea三nnu三al三int三ere三st三rat三et=三th三en三umb三er三of三yea三rsApp三lic三ati三on:Sim三ple三in三ter三est三is三us三ed三to三cal三cul三ate三th三er三etu三rn三on三cer三tai三nt三ype三so三fi三nve三stm三ent三sGiv三en:三Ap三ers三on三inv三est三s$三5,0三00三in三as三avi三ngs三ac三cou三nt三for三tw三om三ont三hs三at三an三ann三ual三in三ter三est三ra三te三of三6%.三How三mu三ch三int三ere三st三wil三ls三he三rec三eiv三ea三tt三he三end三of三tw三om三ont三hs?Ans三wer: i三=三prti=三$5三,00三0x三0.三06三xi=三$5三021225CU7三112三997三ECABai三nM三athCom三pou三nd三Int三ere三st“Mon三ey三mak三es三mon三ey.三A三nd三the三mo三ney三th三at三mon三ey三mak三es,三ma三kes三mo三re三mon三ey.三”-B三enj三ami三nF三ran三kli三nDef三ini三tio三n:Com三pou三nd三int三ere三st三is三com三put三ed三on三ap三rin三cip三al三amo三unt三an三da三ny三acc三umu三lat三ed三int三ere三st.三Ab三ank三th三at三pay三sc三omp三oun三di三nte三res三to三na三sa三vin三gs三acc三oun三tc三omp三ute三si三nte三res三tp三eri三odi三cal三ly三(e.三g.,三da三ily三or三qu三art三erl三y)三and三ad三ds三thi三si三nte三res三tt三ot三he三ori三gin三al三pri三nci三pal三.三Th三ei三nte三res三tf三or三the三fo三llo三win三gp三eri三od三is三com三put三ed三by三usi三ng三the三ne三wp三rin三cip三al三(i.三e.,三th三eo三rig三ina三lp三rin三cip三al三plu三si三nte三res三t).The三fo三rmu三la三for三th三ea三mou三nt,三A,三yo三uw三ill三re三cei三ve三at三the三en三do三fp三eri三od三ni三s:A=三p三(1三+三)ntwhe三re,三p三=t三he三pri三nci三palr=三th三ea三nnu三al三int三ere三st三rat三en=三th三en三umb三er三of三tim三es三com三pou三ndi三ng三is三don三ei三na三ye三art=三th三en三umb三er三of三yea三rsrnNot三es:As三the三nu三mbe三ro三ft三ime三sc三omp三oun三din三gi三sd三one三pe三ry三ear三ap三pro三ach三es三inf三ini三ty三(as三in三co三nti三nuo三us三com三pou三ndi三ng)三,t三he三amo三unt三,A三,y三ou三wil三lr三ece三ive三at三th三ee三nd三of三per三iod三n三is三cal三cul三ate三du三sin三gt三he三for三mul三a:A=三pertThe三ef三fec三tiv三ea三nnu三al三int三ere三st三rat三e(三or三yie三ld)三is三th三es三imp三le三int三ere三st三rat三et三hat三wo三uld三ge三ner三ate三th三es三ame三am三oun三to三fi三nte三res三ta三sw三oul三dt三he三com三pou三nd三rat三e26CU7三112三997三ECABai三nM三athCom三pou三nd三Int三ere三st三-A三ppl三ica三tio三n$1,三000三.00$30三.00$1,三030三.00$30三.90$1,三060三.90$31三.83$1,三092三.73$32三.78$1,三125三.51$0$25三0$50三0$75三0$1,三000$1,三250Dol三lar三si1i2i3i4A1A2A3A41stQua三rte三r2ndQua三rte三r3rdQua三rte三r4thQua三rte三rGiv三en:Wha三ta三mou三nt三wil三ly三ou三rec三eiv三ea三tt三he三end三of三on三ey三ear三if三yo三ui三nve三st三$1,三000三at三an三an三nua三lr三ate三of三12三%c三omp三oun三ded三qu三art三erl三y?Ans三wer三:A=三p三(1+三)nt=$三1,0三00三(1三+三)4=$三1,1三25.三51rn0.1三24Det三ail三edAns三wer三:At三the三en三do三fe三ach三qu三art三er,三in三ter三est三is三co三mpu三ted三,a三nd三the三na三dde三dt三ot三he三pri三nci三pal三.三Thi三sb三eco三mes三th三en三ew三pri三nci三pal三on三wh三ich三th三en三ext三pe三rio三d’s三in三ter三est三is三ca三lcu三lat三ed.Int三ere三st三ear三ned三(i三=三prt三):三i1=$三1,0三00x三0.1三2x0三.25三i2=$三1,0三30x三0.1三2x0三.25三i3=$三1,0三60.三90x三0.1三2x0三.25三14=$三1,0三92.三73x三0.1三2x0三.25=$三30.三00三=三$30三.90三=三$3三1.8三3三=$三32.三78New三pr三inc三ipl三e A1=$三1,0三00+三$30三A2=$三1,0三30+三30.三90三A3=$三1,0三60.三90+三31.三83三A4=$三1,0三92.三73+三32.三78=$三1,0三30三=三$1,三060三.90三=三$1三,09三2.7三3三=$三1,1三25.三5127CU7三112三997三ECABai三nM三athPre三sen三tV三alu三e-三De三fin三iti三ons三(1三)Tim三eV三alu三eo三fM三one三y:At三dif三fer三ent三po三int三si三nt三ime三,a三gi三ven三do三lla三ra三mou三nt三of三mon三ey三has三di三ffe三ren三tv三alu三es.One三do三lla三rr三ece三ive三dt三oda三yi三sw三ort三hm三ore三th三an三one三do三lla三rr三ece三ive三dt三omo三rro三w,三bec三aus三em三one三yc三an三be三inv三est三ed三wit三hs三ome三re三tur三n.Pre三sen三tV三alu三e:Pre三sen三tv三alu三ea三llo三ws三you三to三de三ter三min三eh三ow三muc三hm三one三yt三hat三wi三ll三be三rec三eiv三ed三in三the三fu三tur三ei三sw三ort三ht三oda三yThe三fo三rmu三la三for三pr三ese三nt三val三ue三is:三PV三=Whe三re,三C三=三the三am三oun三to三fm三one三yr三ece三ive三di三nt三he三fut三urer=三t三he三ann三ual三ra三te三of三ret三urnn=三t三he三num三ber三of三ye三arsis三cal三led三th三ed三isc三oun三tf三act三orThe三pr三ese三nt三val三ue三PV三of三as三tre三am三of三cas三hi三st三hen三:三PV三=C0+三+三+Whe三re三C0is三the三ca三sh三exp三ect三ed三tod三ay,三C1is三the三ca三sh三exp三ect三ed三in三one三ye三ar,三et三c.1(1+三r)nC

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