1、1CU7112997ECAAuthor: Collins QianReviewer: Brian Bilello bcBain MathMarch 1998Copyright 1998 Bain & Company, Inc. BOSCopyright 1998 Bain & Company, Inc. 2CU7112997ECABain MathAgenda Basic mathFinancial mathStatistical mathBOSCopyright 1998 Bain & Company, Inc. 3CU7112997ECABain MathAgenda Basic math
2、 ratioproportionpercentinflationforeign exchangegraphingFinancial mathStatistical mathBOSCopyright 1998 Bain & Company, Inc. 4CU7112997ECABain 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
3、 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 = $9.0 MM1.5 MM$?$6.0BOSCopyright 1998 Bain & Company, Inc. 5CU7112997ECABain MathProportion Definition:If the ratio of A t
4、o 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 project SG&A c
5、osts 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)?= $166MMBOSCopyright 1998 Bain & Company, Inc. 6CU7112997ECABain MathPercent Definition:A percentage (abbreviated “percent”) is a convenient w
6、ay 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 miles by flying to her client. In her second year, she logged 25,000 miles. What is the percentage increase in miles?Give
7、n:A percentage can be used to express the change in a number 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 0.25520BOSCopyright 1998 Bain & Company, Inc. 7CU71
8、12997ECABain MathInflation - DefinitionsIf 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., revenue growth, un
9、it 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(current) do
10、llars:3. Price deflatorPrice 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 take
11、n 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 each individual years dollar terms (i.e., inflation has not been taken out).Inflation is defined as the year-over-year decrease in the v
12、alue of a unit of currency.BOSCopyright 1998 Bain & Company, Inc. 8CU7112997ECABain 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 empl
13、oyees to use.Year1996=100*% ChangeYear1996=100*% Change197027.79 5.32 1996100.00 1.95 197129.23 5.18 1997101.97 1.97 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.
14、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 2007132.96 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 3.06 201
15、3156.05 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 3.97 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
16、years.BOSCopyright 1998 Bain & Company, Inc. 9CU7112997ECABain MathInflation - 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 fig
17、ures 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 or cost has changed over time due to accumulated experience, you must use real figuresNote :When to use real vs. Nom
18、inal 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 spreadsheets whether you are using real or nominal figures. If you are using real figures, you should also note what
19、you have chosen as the base year.BOSCopyright 1998 Bain & Company, Inc. 10CU7112997ECABain MathInflation - 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.
20、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$1,500CAGRBOSCopyright 1998 Bain & Company, Inc. 11CU7112997ECABain M
21、athInflation - 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$3,000Nominal dollarsReal (1992) dollarsIf you use real dollars, you can see what has happened to inflation-adjusted prices. The
22、y have fallen an average of 1.0% per year.BOSCopyright 1998 Bain & Company, Inc. 12CU7112997ECABain MathInflation - Exercise (1) Consider the following revenue stream in nominal dollars:Revenue ($ million)199020.5199125.3199227.4199331.2199436.8199545.5199651.0How do we calculate the revenue stream
23、in real dollars?BOSCopyright 1998 Bain & Company, Inc. 13CU7112997ECABain MathInflation - Exercise (2) Answer:Step 1: Choose a base year. 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) =
24、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)Step 4: Calculate the revenue stream in real (1990) dollars terms:1990:1991:1992:1993: = , X = 20.585.34 85.341994:1995:19
25、96:=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)199020.5199124.3199225.7199328.5199432.8199539.6199643.5 (1996) = 100.00BOSCopyright 1998
26、Bain & Company, Inc. 14CU7112997ECABain MathForeign Exchange - Definitions Investments employed in making payments between countries (e.g., paper currency, notes, checks, bills of exchange, and electronic notifications of international debits and credits)Price at which one countrys currency can be c
27、onverted 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 interest and inflation rates between curre
28、nciesForeign Exchange:Exchange Rate:BOSCopyright 1998 Bain & Company, Inc. 15CU7112997ECABain MathForeign Exchange Rates1) US$ equivalent = US dollars per 1 selected foreign currency unit2) Currency per US$ = selected foreign currency units per 1 US dollar The Wall Street Journal Tuesday, November 2
29、5, 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$5.81851.7384CountrySingapore (dollar)US$ Equiv.0.6289Currency per US$1.5900Fi
30、nancial publications, such as the Wall Street Journal, provide exchange rates. BOSCopyright 1998 Bain & Company, Inc. 16CU7112997ECABain MathForeign Exchange - Exercises Question 1:Answer:Question 2:Answer:Question 3:Answer: 650.28 US dollars = ? British poundsfrom table: 0.5914 = US$ 1.00 $650.28 x
31、 = 384.581490.50 Francs = ? US$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$0.5752 1 Mark 1.59 Singapore dollar US$
32、 1BOSCopyright 1998 Bain & Company, Inc. 17CU7112997ECABain 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 xBOSCopyright 1998 Bain & Company, Inc. 18CU7112997ECABai
33、n MathGraphing - Linear 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
34、 = = 15b = Y intercept = 400 dollars charged (when 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-versaBOSCopyrig
35、ht 1998 Bain & Company, Inc. 19CU7112997ECABain MathGraphing - Linear 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, an
36、d what is the cost of making 1,500 lamps?BOSCopyright 1998 Bain & Company, Inc. 20CU7112997ECABain 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 = produc
37、tion 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, 2,0
38、00)X = 900Y = 9,000yxThe cost of producing 1,500 lamps is $16,000BOSCopyright 1998 Bain & Company, Inc. 21CU7112997ECABain 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
39、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 log10 100=2 or log 100=2BOSCopyright 1998 Bain & Company, Inc. 22CU7112997ECABain MathGraphing - Logarithmic (2) For a log scale i
40、n 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 typically uses base 10Log-log paper is logarithmic on both axes; semi-log paper is logarithmic on one axis and linear on th
41、e otherLog ScaleLinear ScaleBOSCopyright 1998 Bain & Company, Inc. 23CU7112997ECABain MathGraphing - 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 dista
42、nce 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 in a seriesLog scales are often used for:Experience curve (a log/log scale is mandatory - natural logs (l
43、n 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 point to the nextCurving upwardThe rate of change is increasingCurving downwardThe rate of change is decr
44、easingIn many situations, it is convenient to use logarithms.BOSCopyright 1998 Bain & Company, Inc. 24CU7112997ECABain MathAgenda Basic mathFinancial mathsimple interestcompound interestpresent valuerisk and returnnet present valueinternal rate of returnbond and stock valuationStatistical mathBOSCop
45、yright 1998 Bain & Company, Inc. 25CU7112997ECABain MathSimple Interest Definition: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 inter
46、est 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 two months?Answer:i = prti = $5,000 x 0.06 x i = $50 2 12BOSCopyright 1998 Bain & C
47、ompany, Inc. 26CU7112997ECABain 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 interest. A bank that pays compound interest on a savings account computes
48、 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 the amount, A, you will receive at the end of period n is:A =
49、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 amount, A, you will receive at the end of per
50、iod 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 rateBOSCopyright 1998 Bain & Company, Inc. 27CU7112997ECABain MathCompound Interest - Application $1,000.00$30
