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1、Author:Collins Qian,Reviewer:Brian Bilello,bc,Bain Math,March 1998,Copyright 1998 Bain&Company,Inc.,2,CU7112997ECA,Bain Math,Agenda,Basic math,Financial math,Statistical math,3,CU7112997ECA,Bain Math,Agenda,Basic math ratioproportionpercentinflationforeign exchangegraphing,Financial math,Statistical
2、 math,4,CU7112997ECA,Bain Math,Ratio,Definition:,Application:,Note:,The ratio of A to B is written or A:B,AB,A ratio can be used to calculate price per unit(),given the total revenue and total units,Price Unit,total revenue=,Given:,=,Answer:,Price Unit,$9MM 1.5MM,The math for ratios is simple.Identi
3、fying a relevant unit can be challenging,total units=,price/unit=,$9.0 MM,1.5 MM,$?,$6.0,5,CU7112997ECA,Bain Math,Proportion,Definition:,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 C,AB,CD,Revenue=SG&A=,Giv
4、en:,$135MM$83MM,$270MM$?,1996,1999,Answer:,$135MM$270MM,$83MM$?,135MM x?=83MM x 270MM,83MMx270MM 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 revenue in 1999(assuming SG&A and revenue in 1999 are proportional to SG&A and rev
5、enue in 1996),?=,=$166MM,6,CU7112997ECA,Bain Math,Percent,Definition:,A percentage(abbreviated“percent”)is a convenient way to express a ratio.Literally,percentage means“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 mile
6、s 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 to the next,%change=-1,new value-original value original value,new valueoriginal value,The ratio of 5
7、to 20 is or 0.25,7,CU7112997ECA,Bain Math,Inflation-Definitions,If an item cost$1.00 in 1997 and cost$1.03 in 1998,inflation was 3%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.,reven
8、ue growth,unit cost growth),it is necessary to subtract out the effects of inflation.Inflationary growth is not“real”growth because 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(cur
9、rent)dollars:,3.Price deflator,Price deflator(current year)Price deflator(base year),Inflation between current year and base year,=,Dollar figure(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 t
10、aken out).Any year can be chosen as the base year,but all dollar figures must be stated in the same base year,Dollar figures for a number of years 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
11、value of a unit of currency.,8,CU7112997ECA,Bain Math,Inflation-U.S.Price Deflators,*1996 is the base yearNote:These 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 pric
12、e deflators for a number of years.,9,CU7112997ECA,Bain Math,Inflation-Real vs.Nominal 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 ann
13、ual 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 or cost has changed over time due to accumulated experience,you must use real figures,Note:,When to use real vs.Nominal figures:,Whether
14、 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 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 bas
15、e year.,10,CU7112997ECA,Bain Math,Inflation-Example(1),(1970-1992),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 t
16、he price increase was due to inflation.,CAGR,11,CU7112997ECA,Bain Math,Inflation-Example(2),Price of a GE Washer,CAGR,(1970-1992),If you use real dollars,you can see what has happened to inflation-adjusted prices.They have fallen an average of 1.0%per year.,12,CU7112997ECA,Bain Math,Inflation-Exerci
17、se(1),Consider the following revenue stream in nominal dollars:,Revenue($million),199020.5,199125.3,199227.4,199331.2,199436.8,199545.5,199651.0,How do we calculate the revenue stream in real dollars?,13,CU7112997ECA,Bain Math,Inflation-Exercise(2),Answer:,Step 1:Choose a base year.For this example,
18、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)=98.08,Step 3:Use the formula to calculate real dollars:,Price deflator(current year)Dollar figure(current year),Price deflator(base year),Dollar figure(bas
19、e year),Step 4:Calculate the revenue stream in real(1990)dollars terms:,1990:1991:1992:1993:,=,X=20.5,85.34 85.34,1994:1995:1996:,=,20.5 X,=,X=24.3,88.72 85.34,25.3 X,=,X=25.7,91.16 85.34,27.4 X,=,X=28.5,93.54 85.34,31.2 X,=,X=32.8,95.67 85.34,36.8 X,=,X=39.6,98.08 85.34,45.5 X,=,X=43.5,100.00 85.34
20、,51.0 X,Revenue($Million),199020.5,199124.3,199225.7,199328.5,199432.8,199539.6,199643.5,(1996)=100.00,14,CU7112997ECA,Bain Math,Foreign Exchange-Definitions,Investments employed in making payments between countries(e.g.,paper currency,notes,checks,bills of exchange,and electronic notifications of i
21、nternational debits and credits)Price at which one countrys currency can be converted into anothersThe interest and inflation rates of a given currency determine the value of holding money in that currency relative to in other currencies.In efficient international markets,exchange rates will adjust
22、to compensate for differences in interest and inflation rates between currencies,Foreign Exchange:,Exchange Rate:,15,CU7112997ECA,Bain Math,Foreign Exchange Rates,1)US$equivalent=US dollars per 1 selected foreign currency unit2)Currency per US$=selected foreign currency units per 1 US dollar,The Wal
23、l Street Journal Tuesday,November 25,1997,Currency Trading,Monday,November 24,1997Exchange Rates,Country,Argentina(Peso),Britain(Pound),US$Equiv.1,1.0001,1.6910,Currency per US$2,0.9999,0.5914,Country,France(Franc),Germany(Mark),US$Equiv.,0.1719,0.5752,Currency per US$,5.8185,1.7384,Country,Singapor
24、e(dollar),US$Equiv.,0.6289,Currency per US$,1.5900,Financial publications,such as the Wall Street Journal,provide exchange rates.,16,CU7112997ECA,Bain Math,Foreign Exchange-Exercises,Question 1:,Answer:,Question 2:,Answer:,Question 3:,Answer:,650.28 US dollars=?British poundsfrom table:0.5914=US$1.0
25、0$650.28 x=384.581490.50 Francs=?US$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 US$1,$0.1719 1 Franc,$0.5752 1 Mark,1.59 Singapore dollar US$1,17,CU7
26、112997ECA,Bain Math,Graphing-Linear,X,0,Y,(X1,Y1),(X2,Y2),b,X,Y,The formula for a line is:y=mx+bWhere,m=slope=,y2-y1 x2-x1,b=the y intercept=the y coordinate when the x coordinate is“0”,y x,18,CU7112997ECA,Bain Math,Graphing-Linear Exercise#1,Formula for line:y=mx+bIn this exercise,y=15x+400,where,0
27、,200,400,600,800,1,000,1,200,1,400,1,600,1,800,$2,000,Dollars changing,0,50,100,People,(100,1900),(50,1150),The caterer would charge$1900 for a 100 person party.,X axis=peopleY axis=dollars chargedm=slope=15,b=Y intercept=400 dollars charged(when people=0),A caterer charges$400.00 for setting up a p
28、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-versa,19,CU7112997ECA,Bain Math,Graphing-Linear Exercise#2(1),A lamp manufacturer has collected a set of production data as follo
29、ws:,Number of lamps Produced/Day,Production Cost/Day,1008509009501,000,$2,000$9,500$10,000$10,500$11,000,What is the daily fixed cost of production,and what is the cost of making 1,500 lamps?,20,CU7112997ECA,Bain Math,Graphing-Linear Exercise#2(2),0,8,000,16,000,Production Cost/Day,0,500,1,000,1,500
30、,Produced/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+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=
31、15,000+1,000y=16,000,11,000-2,000 1,000-100,9,000 900,(100,2,000),X=900,Y=9,000,The cost of producing 1,500 lamps is$16,000,21,CU7112997ECA,Bain Math,Graphing-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 numb
32、er,Unless otherwise stated,the base is assumed to be 10,Y=10 x,then log10 Y=X,Mathematically,if,Where,Y=given number,10=base,X=power(or log),For example:100=102 can be written as log10 100=2 or log 100=2,22,CU7112997ECA,Bain Math,Graphing-Logarithmic(2),For a log scale in base 10,as the linear scale
33、 values increase by ten times,the log values increase by 1.,9876543210,1,000,000,000100,000,00010,000,0001,000,000100,00010,0001,000100101,Log paper typically uses base 10Log-log paper is logarithmic on both axes;semi-log paper is logarithmic on one axis and linear on the other,Log Scale,Linear Scal
34、e,23,CU7112997ECA,Bain Math,Graphing-Logarithmic(3),The most useful feature of a log graph is that equal multiplicative changes 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 c
35、hange in both sets of numbers is the same,10It is convenient to use log scales to examine the rate of change between data points in a series,Log 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 ma
36、tricesROS/RMS graphs,Line Shape of Data Plots,Explanation,A straight line,The data points are changing at the same rate from one point to the next,Curving upward,The rate of change is increasing,Curving downward,The rate of change is decreasing,In many situations,it is convenient to use logarithms.,
37、24,CU7112997ECA,Bain Math,Agenda,Basic math,Financial mathsimple interestcompound interestpresent valuerisk and returnnet present valueinternal rate of returnbond and stock valuationStatistical math,25,CU7112997ECA,Bain Math,Simple Interest,Definition:,Simple interest is computed on a principal amou
38、nt for a specified time periodThe formula for simple interest is:i=prtwhere,p=the principalr=the annual interest ratet=the number of years,Application:,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 a
39、n 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 12,26,CU7112997ECA,Bain Math,Compound Interest,“Money makes money.And the money that money makes,makes more money.”-Benjamin Franklin,Definition:,Compound interest is comput
40、ed 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 original principal.The interest for the following period is computed by using the new principal(i.e.,the
41、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 done in a yeart=the number of years,r n,Notes:,As the number of times compounding is done per year ap
42、proaches 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 interest rate that would generate the same amount of interest as would the compound rate,27,CU7112997EC
43、A,Bain Math,Compound 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,$1,250,Dollars,i1,i2,i3,i4,A1,A2,A3,A4,1st Quarter,2nd Quarter,3rd Quarter,4th Quarter,Given:,What amount will you receive at the end of one year if you in
44、vest$1,000 at an annual rate of 12%compounded quarterly?,Answer:,A=p(1+)nt=$1,000(1+)4=$1,125.51,r n,0.12 4,Detailed 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 earne
45、d(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=$1,125.51,28,CU7112997ECA,Bain Math,Present Value-Definitions(1),Time Val
46、ue 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 money that will be rece
47、ived 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 expected today,C1 is the ca
48、sh expected in one year,etc.,1(1+r)n,C1 1+r,C2(1+r)2,Cn(1+r)n,29,CU7112997ECA,Bain Math,Present 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 investment wh
49、ich pays a fixed sum,each year for a specific number of years from year 1 to year n is:,Perpetuity:,Growing perpetuity:,Annuity:,30,CU7112997ECA,Bain Math,Present 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 six year annu
50、ity of$2.00Assume you can invest at 16%per year,Which of the following would you prefer to receive?,31,CU7112997ECA,Bain Math,Present Value-Exercise(2),*The present value is negative because this is the cash outflow required today receive a cash inflow at a later time,1)$10.00 today,PV=$10.002)$20.0
