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1、Author:Collins QianReviewer:Brian Bilello bcBain MathMarch 1998Copyright 1998 Bain&Company,Inc.1CU7112997ECABOSCopyright 1998 Bain&Company,Inc.Bain MathAgenda Basic mathFinancial mathStatistical math2CU7112997ECABOSCopyright 1998 Bain&Company,Inc.Bain MathAgenda Basic math ratioproportionpercentinfl
2、ationforeign exchangegraphingFinancial mathStatistical math3CU7112997ECABOSCopyright 1998 Bain&Company,Inc.Bain MathRatio Definition:Application:Note:The ratio of A to B is written or A:BABA ratio can be used to calculate price per unit (),given the total revenue and total unitsPrice Unittotal reven
3、ue=Given:=Answer:Price Unit$9MM 1.5MMThe math for ratios is simple.Identifying a relevant unit can be challengingtotal units=price/unit=$9.0 MM1.5 MM$?$6.04CU7112997ECABOSCopyright 1998 Bain&Company,Inc.Bain MathProportion Definition:If the ratio of A to B is equal to the ratio of C to D,then A and
4、B are proportional to C and D.Application:=It follows that A x D=B x CABCDRevenue=SG&A =Given:$135MM$83MM$270MM$?19961999Answer:$135MM$270MM$83MM$?135MM x?=83MM x 270MM83MMx270MM 135MM=The concept of proportion can be used to project SG&A costs in 1999,given revenue in 1996,SG&A costs in 1996,and re
5、venue in 1999(assuming SG&A and revenue in 1999 are proportional to SG&A and revenue in 1996)?=$166MM5CU7112997ECABOSCopyright 1998 Bain&Company,Inc.Bain MathPercent Definition:A percentage(abbreviated“percent”)is a convenient way to express a ratio.Literally,percentage means“per 100.”Application:In
6、 percentage terms,0.25=25 per 100 or 25%In her first year at Bain,an AC logged 7,000 frequent flier miles by flying to her client.In her second year,she logged 25,000 miles.What is the percentage increase in miles?Given:A percentage can be used to express the change in a number from one time period
7、to the nextAnswer:-1=3.57-1=2.57=257%25,000 7,000%change=-1 new value-original value original valuenew valueoriginal valueThe ratio of 5 to 20 is or 0.255206CU7112997ECABOSCopyright 1998 Bain&Company,Inc.Bain MathInflation-DefinitionsIf an item cost$1.00 in 1997 and cost$1.03 in 1998,inflation was 3
8、%from 1997 to 1998.The item is not intrinsically more valuable in 1998-the dollar is less valuableWhen calculating the“real”growth of a dollar figure over time(e.g.,revenue growth,unit cost growth),it is necessary to subtract out the effects of inflation.Inflationary growth is not“real”growth becaus
9、e inflation does not create intrinsic value.Definition:A price deflator is a measure of inflation over time.Related Terminology:1.Real(constant)dollars:2.Nominal(current)dollars:3.Price deflatorPrice deflator(current year)Price deflator(base year)Inflation between current year and base year=Dollar f
10、igure(current year)Dollar figure(base year)=Dollar figures for a number of years that are stated in a chosen“base”years dollar terms(i.e.,inflation has been taken out).Any year can be chosen as the base year,but all dollar figures must be stated in the same base yearDollar figures for a number of ye
11、ars that are stated in each individual years dollar terms(i.e.,inflation has not been taken out).Inflation is defined as the year-over-year decrease in the value of a unit of currency.7CU7112997ECABOSCopyright 1998 Bain&Company,Inc.Bain Math Inflation-U.S.Price Deflators*1996 is the base yearNote:Th
12、ese are the U.S.Price Deflators which WEFA Group has forecasted through the year 2020.The library has purchased this time series for all Bain employees to use.A deflator table lists price deflators for a number of years.8CU7112997ECABOSCopyright 1998 Bain&Company,Inc.Bain MathInflation-Real vs.Nomin
13、al Figures To understand how a company has performed over time(e.g.,in terms of revenue,costs,or profit),it is necessary to remove inflation,(i.e.use real figures).Since most companies use nominal figures in their annual reports,if you are showing the clients revenue over time,it is preferable to us
14、e nominal figures.For an experience curve,where you want to understand how price or cost has changed over time due to accumulated experience,you must use real figuresNote:When to use real vs.Nominal figures:Whether you should use real(constant)figures or nominal(current)figures depends on the situat
15、ion and the clients preference.It is important to specify on slides and spreadsheets whether you are using real or nominal figures.If you are using real figures,you should also note what you have chosen as the base year.9CU7112997ECABOSCopyright 1998 Bain&Company,Inc.Bain MathInflation-Example(1)(19
16、70-1992)Adjusting for inflation is critical for any analysis looking at prices over time.In nominal dollars,GEs washer prices have increased by an average of 4.5%since 1970.When you use nominal dollars,it is impossible to tell how much of the price increase was due to inflation.$2,00072Nominal dolla
17、rs4.5%Price of a GE Washer19707173 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92$0$500$1,000$1,500CAGR10CU7112997ECABOSCopyright 1998 Bain&Company,Inc.Bain MathInflation-Example(2)Price of a GE Washer CAGR(1970-1992)(1.0%)4.5%197071 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90
18、 91 92$0$500$1,000$1,500$2,000$2,500$3,000Nominal dollarsReal(1992)dollarsIf you use real dollars,you can see what has happened to inflation-adjusted prices.They have fallen an average of 1.0%per year.11CU7112997ECABOSCopyright 1998 Bain&Company,Inc.Bain MathInflation-Exercise(1)Consider the followi
19、ng revenue stream in nominal dollars:Revenue($million)199020.5199125.3199227.4199331.2199436.8199545.5199651.0How do we calculate the revenue stream in real dollars?12CU7112997ECABOSCopyright 1998 Bain&Company,Inc.Bain MathInflation-Exercise(2)Answer:Step 1:Choose a base year.For this example,we wil
20、l use 1990Step 2:Find deflators for all years (from the deflator table):(1990)=85.34(1991)=88.72(1992)=91.16(1993)=93.54(1994)=95.67(1995)=98.08Step 3:Use the formula to calculate real dollars:Price deflator(current year)Dollar figure(current year)Price deflator(base year)Dollar figure(base year)Ste
21、p 4:Calculate the revenue stream in real(1990)dollars terms:1990:1991:1992:1993:=,X=20.585.34 85.341994:1995:1996:=20.5 X =,X=24.388.72 85.3425.3 X =,X=25.791.16 85.3427.4 X =,X=28.593.54 85.3431.2 X =,X=32.895.67 85.3436.8 X =,X=39.698.08 85.3445.5 X =,X=43.5100.00 85.3451.0 XRevenue($Million)19902
22、0.5199124.3199225.7199328.5199432.8199539.6199643.5(1996)=100.0013CU7112997ECABOSCopyright 1998 Bain&Company,Inc.Bain MathForeign Exchange-Definitions Investments employed in making payments between countries(e.g.,paper currency,notes,checks,bills of exchange,and electronic notifications of internat
23、ional debits and credits)Price at which one countrys currency can be converted into anothersThe interest and inflation rates of a given currency determine the value of holding money in that currency relative to in other currencies.In efficient international markets,exchange rates will adjust to comp
24、ensate for differences in interest and inflation rates between currencies Foreign Exchange:Exchange Rate:14CU7112997ECABOSCopyright 1998 Bain&Company,Inc.Bain MathForeign Exchange Rates1)US$equivalent=US dollars per 1 selected foreign currency unit2)Currency per US$=selected foreign currency units p
25、er 1 US dollar The Wall Street Journal Tuesday,November 25,1997Currency TradingMonday,November 24,1997Exchange RatesCountryArgentina(Peso)Britain(Pound)US$Equiv.11.00011.6910Currency per US$20.99990.5914CountryFrance(Franc)Germany(Mark)US$Equiv.0.17190.5752Currency per US$5.81851.7384CountrySingapor
26、e(dollar)US$Equiv.0.6289Currency per US$1.5900Financial publications,such as the Wall Street Journal,provide exchange rates.15CU7112997ECABOSCopyright 1998 Bain&Company,Inc.Bain MathForeign Exchange-Exercises Question 1:Answer:Question 2:Answer:Question 3:Answer:650.28 US dollars=?British poundsfrom
27、 table:0.5914 =US$1.00$650.28 x =384.581490.50 Francs=?US$from table:$0.1719=1 Franc 1490.50 Franc x =$256.221,000 German Marks=?Singapore dollars from table:$0.5752=1 Mark 1.59 Singapore dollar=US$1 1,000 German Marks x x =914.57 Singapore dollars 0.5914 US$1$0.1719 1 Franc$0.5752 1 Mark 1.59 Singa
28、pore dollar US$116CU7112997ECABOSCopyright 1998 Bain&Company,Inc.Bain MathGraphing-Linear X0Y(X1,Y1)(X2,Y2)bXYThe formula for a line is:y=mx+bWhere,m=slope=y2-y1 x2-x1b=the y intercept=the y coordinate when the x coordinate is“0”y x17CU7112997ECABOSCopyright 1998 Bain&Company,Inc.Bain MathGraphing-L
29、inear Exercise#1 Formula for line:y=mx+bIn this exercise,y=15x+400,where,02004006008001,0001,2001,4001,6001,800$2,000Dollars changing050100People(100,1900)(50,1150)The caterer would charge$1900 for a 100 person party.yxX axis=peopleY axis=dollars chargedm=slope=15b=Y intercept =400 dollars charged(w
30、hen people=0)A caterer charges$400.00 for setting up a party,plus$15.00 for each person.How much would the caterer charge for a 100 person party?Using this formula,you can solve for dollars charged(y),given people(x),and vice-versa18CU7112997ECABOSCopyright 1998 Bain&Company,Inc.Bain MathGraphing-Li
31、near Exercise#2(1)A lamp manufacturer has collected a set of production data as follows:Number of lamps Produced/DayProduction Cost/Day1008509009501,000$2,000$9,500$10,000$10,500$11,000What is the daily fixed cost of production,and what is the cost of making 1,500 lamps?19CU7112997ECABOSCopyright 19
32、98 Bain&Company,Inc.Bain MathGraphing-Linear Exercise#2(2)08,00016,000Production Cost/Day05001,0001,500Produced/Day(1,500,16,000)(1,000,11,000)Formula for line:y=mx+bX axis=#of lamps produced/day Y axis=production cost/dayM=slope=10b=Y intercept=production cost(i.e.,the fixed cost)when lamps=0y=mx+b
33、b=y-mxb=2,000-10(100)b=1,000 The fixed cost is$1,000y=10 x+1,000For 1,500 lamps:y=10(1,500)+1,000y=15,000+1,000y=16,00011,000-2,000 1,000-1009,000 900(100,2,000)X=900Y=9,000yxThe cost of producing 1,500 lamps is$16,00020CU7112997ECABOSCopyright 1998 Bain&Company,Inc.Bain MathGraphing-Logarithmic(1)L
34、og:A“log”or logarithm of given number is defined as the power to which a base number must be raised to equal that given numberUnless otherwise stated,the base is assumed to be 10Y=10 x,then log10 Y=XMathematically,ifWhere,Y=given number10=base X=power(or log)For example:100=102 can be written as log
35、10 100=2 or log 100=221CU7112997ECABOSCopyright 1998 Bain&Company,Inc.Bain MathGraphing-Logarithmic(2)For a log scale in base 10,as the linear scale values increase by ten times,the log values increase by 1.98765432101,000,000,000100,000,00010,000,0001,000,000100,00010,0001,000100101Log paper typica
36、lly uses base 10Log-log paper is logarithmic on both axes;semi-log paper is logarithmic on one axis and linear on the otherLog ScaleLinear Scale22CU7112997ECABOSCopyright 1998 Bain&Company,Inc.Bain MathGraphing-Logarithmic(3)The most useful feature of a log graph is that equal multiplicative changes
37、 in data are represented by equal distances on the axesthe distance between 10 and 100 is equal to the distance between 1,000,000 and 10,000,000 because the multiplicative change in both sets of numbers is the same,10It is convenient to use log scales to examine the rate of change between data point
38、s in a seriesLog scales are often used for:Experience curve(a log/log scale is mandatory-natural logs(ln or loge)are typically usedprices and costs over timeGrowth Share matricesROS/RMS graphsLine Shape of Data PlotsExplanationA straight lineThe data points are changing at the same rate from one poi
39、nt to the nextCurving upwardThe rate of change is increasingCurving downwardThe rate of change is decreasingIn many situations,it is convenient to use logarithms.23CU7112997ECABOSCopyright 1998 Bain&Company,Inc.Bain MathAgenda Basic mathFinancial mathsimple interestcompound interestpresent valuerisk
40、 and returnnet present valueinternal rate of returnbond and stock valuationStatistical math24CU7112997ECABOSCopyright 1998 Bain&Company,Inc.Bain MathSimple Interest Definition:Simple interest is computed on a principal amount for a specified time periodThe formula for simple interest is:i=prtwhere,p
41、=the principalr=the annual interest ratet=the number of yearsApplication:Simple interest is used to calculate the return on certain types of investmentsGiven:A person invests$5,000 in a savings account for two months at an annual interest rate of 6%.How much interest will she receive at the end of t
42、wo months?Answer:i=prti=$5,000 x 0.06 x i=$50 2 1225CU7112997ECABOSCopyright 1998 Bain&Company,Inc.Bain MathCompound Interest“Money makes money.And the money that money makes,makes more money.”-Benjamin FranklinDefinition:Compound interest is computed on a principal amount and any accumulated intere
43、st.A bank that pays compound interest on a savings account computes interest periodically(e.g.,daily or quarterly)and adds this interest to the original principal.The interest for the following period is computed by using the new principal(i.e.,the original principal plus interest).The formula for t
44、he amount,A,you will receive at the end of period n is:A=p(1+)ntwhere,p=the principalr=the annual interest raten=the number of times compounding is done in a yeart=the number of yearsr nNotes:As the number of times compounding is done per year approaches infinity(as in continuous compounding),the am
45、ount,A,you will receive at the end of period n is calculated using the formula:A=pertThe effective annual interest rate(or yield)is the simple interest rate that would generate the same amount of interest as would the compound rate26CU7112997ECABOSCopyright 1998 Bain&Company,Inc.Bain MathCompound In
46、terest-Application$1,000.00$30.00$1,030.00$30.90$1,060.90$31.83$1,092.73$32.78$1,125.51$0$250$500$750$1,000$1,250Dollarsi1i2i3i4A1A2A3A41st Quarter2nd Quarter3rd Quarter4th QuarterGiven:What amount will you receive at the end of one year if you invest$1,000 at an annual rate of 12%compounded quarter
47、ly?Answer:A=p(1+)nt=$1,000(1+)4=$1,125.51r n0.12 4Detailed Answer:At the end of each quarter,interest is computed,and then added to the principal.This becomes the new principal on which the next periods interest is calculated.Interest earned(i=prt):i1 =$1,000 x0.12x0.25i2 =$1,030 x0.12x0.25i3 =$1,06
48、0.90 x0.12x0.2514 =$1,092.73x0.12x0.25=$30.00=$30.90=$31.83=$32.78New principleA1=$1,000+$30A2=$1,030+30.90A3=$1,060.90+31.83A4=$1,092.73+32.78=$1,030=$1,060.90=$1,092.73=$1,125.5127CU7112997ECABOSCopyright 1998 Bain&Company,Inc.Bain MathPresent Value-Definitions(1)Time Value of Money:At different p
49、oints in time,a given dollar amount of money has different values.One dollar received today is worth more than one dollar received tomorrow,because money can be invested with some return.Present Value:Present value allows you to determine how much money that will be received in the future is worth t
50、odayThe formula for present value is:PV=Where,C=the amount of money received in the futurer=the annual rate of returnn=the number of years is called the discount factorThe present value PV of a stream of cash is then:PV=C0+Where C0 is the cash expected today,C1 is the cash expected in one year,etc.1