1、Author:Collins QianReviewer:Brian Bilello bcBain MathMarch 1998Bain MathAgendaBasic mathFinancial mathStatistical mathBain MathAgendaBasic math ratioproportionpercentinflationforeign exchangegraphingFinancial mathStatistical mathBain MathRatioDefinition:Application:Note:The ratio of A to B is writte
2、n or A:BABA ratio can be used to calculate price per unit (),given the total revenue and total unitsPrice Unittotal revenue=Given:=Answer:Price Unit$9MM 1.5MMThe math for ratios is simple.Identifying a relevant unit can be challengingtotal units=price/unit=$9.0 MM1.5 MM$?$6.0Bain MathProportionDefin
3、ition:If the ratio of A to B is equal to the ratio of C to D,then A and 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 pr
4、oject SG&A costs in 1999,given revenue in 1996,SG&A costs in 1996,and revenue in 1999(assuming SG&A and revenue in 1999 are proportional to SG&A and revenue in 1996)?=$166MMBain MathPercentDefinition:A percentage(abbreviated“percent”)is a convenient way to express a ratio.Literally,percentage means“
5、per 100.”Application:In 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 numbe
6、r from one time period 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 0.25520Bain MathInflation-DefinitionsIf an item cost$1.00 in 1997 and cost$1.03 in 1998,inflation was 3%from 1997 to 1998.The it
7、em 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 because inflation does not crea
8、te 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 figure(current year)Dollar
9、 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 years that are stated in ea
10、ch 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.Bain Math Inflation-U.S.Price DeflatorsYear1996=100*%ChangeYear1996=100*%Change197027.79 5.32 1996100.00 1.95 197129.23 5.18 1997101.97 1.97
11、197230.46 4.23 1998104.48 2.46 197332.18 5.64 1999107.10 2.51 197435.07 8.99 2000109.80 2.52 197538.36 9.37 2001112.51 2.47 197640.61 5.86 2002115.41 2.58 197743.23 6.45 2003118.58 2.75 197846.37 7.26 2004122.02 2.90 197950.35 8.58 2005125.65 2.97 198055.00 9.25 2006129.31 2.92 198160.18 9.41 200713
12、2.96 2.82 198263.97 6.30 2008136.57 2.71 198366.68 4.24 2009140.26 2.70 198469.21 3.79 2010144.06 2.71 198571.59 3.43 2011147.89 2.65 198673.46 2.62 2012151.90 2.72 198775.71 3.06 2013156.05 2.73 198878.48 3.65 2014160.29 2.72 198981.79 4.22 2015164.73 2.76 199085.34 4.34 2016169.25 2.75 199188.72 3
13、.97 2017173.83 2.71 199291.16 2.75 2018178.53 2.70 199393.54 2.62 2019183.33 2.69 199495.67 2.28 2020188.31 2.71 199598.08 2.51 A deflator table lists price deflators for a number of years.Bain MathInflation-Real vs.Nominal FiguresTo understand how a company has performed over time(e.g.,in terms of
14、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 use nominal figures.For an experience curve,where you want to understand how price
15、 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 situation and the clients preference.It is important to specify on slides and spreadsh
16、eets 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.Bain MathInflation-Example(1)(1970-1992)Adjusting for inflation is critical for any analysis looking at prices over time.In nominal dollars,GEs washer prices ha
17、ve 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 dollars4.5%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$1,500CAGRBain MathInflat
18、ion-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 91 92$0$500$1,000$1,500$2,000$2,500$3,000Nominal dollarsReal(1992)dollarsIf you use real dollars,you can see what has happened to inflation-adjusted prices.They have fallen an
19、average of 1.0%per year.Bain MathInflation-Exercise(1)Consider the following revenue stream in nominal dollars:Revenue($million)199020.5199125.3199227.4199331.2199436.8199545.5199651.0How do we calculate the revenue stream in real dollars?Bain MathInflation-Exercise(2)Answer:Step 1:Choose a base yea
20、r.For this example,we will 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)Do
21、llar figure(base year)Step 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.
22、0 XRevenue($Million)199020.5199124.3199225.7199328.5199432.8199539.6199643.5(1996)=100.00Bain MathForeign Exchange-DefinitionsInvestments employed in making payments between countries(e.g.,paper currency,notes,checks,bills of exchange,and electronic notifications of international debits and credits)
23、Price at which one countrys currency can be converted into anothersThe interest and inflation rates of a given currency determine the value of holding money in that currency relative to in other currencies.In efficient international markets,exchange rates will adjust to compensate for differences in
24、 interest and inflation rates between currenciesForeign Exchange:Exchange Rate:Bain MathForeign Exchange RatesThe Wall Street Journal Tuesday,November 25,1997Currency TradingMonday,November 24,1997Exchange RatesCountryArgentina(Peso)Britain(Pound)US$Equiv.11.00011.6910Currency per US$20.99990.5914Co
25、untryFrance(Franc)Germany(Mark)US$Equiv.0.17190.5752Currency per US$5.81851.7384CountrySingapore(dollar)US$Equiv.0.6289Currency per US$1.5900Financial publications,such as the Wall Street Journal,provide exchange rates.Bain MathForeign Exchange-ExercisesQuestion 1:Answer:Question 2:Answer:Question 3
26、:Answer:650.28 US dollars=?British poundsfrom 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 dollarsfrom table:$0.5752=1 Mark 1.59 Singapore dollar=US$1 1,000 German Marks x x =914.57 Singapore dollars 0.5914
27、US$1$0.1719 1 Franc$0.5752 1 Mark 1.59 Singapore dollar US$1Bain MathGraphing-LinearX0Y(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 xBain MathGraphing-Linear Exercise#1Formula for line:y=mx+bIn this exercise
28、,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(when people=0)A caterer charges$400.00 for setting up a p
29、arty,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-versaBain MathGraphing-Linear Exercise#2(1)A lamp manufacturer has collected a set of production data as follows:Number of lamps P
30、roduced/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?Bain MathGraphing-Linear Exercise#2(2)08,00016,000Production Cost/Day05001,0001,500Produced/Day(1,500,16,000)(1,000,11,000)Formula for
31、 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+bb=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
32、,2,000)X=900Y=9,000yxThe cost of producing 1,500 lamps is$16,000Bain MathGraphing-Logarithmic(1)Log: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=XMath
33、ematically,ifWhere,Y=given number10=base X=power(or log)For example:100=102 can be written as log10 100=2 or log 100=2Bain 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
34、,000,000100,00010,0001,000100101Log paper typically uses base 10Log-log paper is logarithmic on both axes;semi-log paper is logarithmic on one axis and linear on the otherLog ScaleLinear ScaleBain MathGraphing-Logarithmic(3)The most useful feature of a log graph is that equal multiplicative changes
35、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 points
36、 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 poin
37、t to the nextCurving upwardThe rate of change is increasingCurving downwardThe rate of change is decreasingIn many situations,it is convenient to use logarithms.Bain MathAgendaBasic mathFinancial mathsimple interestcompound interestpresent valuerisk and returnnet present valueinternal rate of return
38、bond and stock valuationStatistical mathBain MathSimple InterestDefinition:Simple interest is computed on a principal amount for a specified time periodThe formula for simple interest is:i=prtwhere,p=the principalr=the annual interest ratet=the number of yearsApplication:Simple interest is used to c
39、alculate 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 two months?Answer:i=prti=$5,000 x 0.06 x i=$50 2 12Bain MathCompound Interest“Money makes money.And th
40、e money that money makes,makes more money.”-Benjamin FranklinDefinition:Compound interest is computed on a principal amount and any accumulated interest.A bank that pays compound interest on a savings account computes interest periodically(e.g.,daily or quarterly)and adds this interest to the origin
41、al principal.The interest for the following period is computed by using the new principal(i.e.,the original principal plus interest).The formula for the 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 do
42、ne 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 amount,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 interes
43、t rate that would generate the same amount of interest as would the compound rateBain MathCompound Interest-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 w
44、ill you receive at the end of one year if you invest$1,000 at an annual rate of 12%compounded quarterly?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 peri
45、ods interest is calculated.Interest earned(i=prt):i1 =$1,000 x0.12x0.25i2 =$1,030 x0.12x0.25i3 =$1,060.90 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=$1,125.51Bain MathPresent Va
46、lue-Definitions(1)Time Value of Money:At different points 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
47、money that will be received in the future is worth todayThe 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 expe
48、cted today,C1 is the cash expected in one year,etc.1 (1+r)nC (1+r)nC1 1+rC2 (1+r)2Cn (1+r)nBain MathPresent Value-Definitions(2)The present value of a perpetuity(i.e.,an infinite cash stream)of is:PV=A perpetuity growing at rate of g has present value of:PV=The present value PV of an annuity,an inve
49、stment which pays a fixed sum,each year for a specific number of years from year 1 to year n is:Perpetuity:Growing perpetuity:Annuity:C rC r-gPV=C r-1 (1+r)nC rBain MathPresent Value-Exercise(1)1)$10.00 today2)$20.00 five years from today3)A perpetuity of$1.504)A perpetuity of$1.00,growing at 5%5)A
50、six year annuity of$2.00Assume you can invest at 16%per yearWhich of the following would you prefer to receive?Bain MathPresent Value-Exercise(2)1)$10.00 today,PV=$10.002)$20.00 five years from today,For HP12C:5 163)A perpetuity of$1.50,PV=$9.384)A perpetuity of$1.00,growing at 5%,PV=$9.095)A six ye
