某咨询分析方法bainmath

1、某咨询分析方法bainmathBain MathAgenda Basic mathFinancial mathStatistical math2Bain MathAgenda Basic math ratioproportionpercentinflationforeign exchangegraphingFinancial mathStatistical math3Bain MathRatio Definition:Application:Note:The ratio of A to B is written or A:BABA ratio can be used to calculate

2、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 = $9.0 MM1.5 MM$?$6.04Bain MathProportion Definition:If the ratio of A to B

3、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&

4、;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)?= $166MM5Bain MathPercent Definition:A percentage (abbreviated “percent”) is a convenient way to express a ratio. Literally, p

5、ercentage 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?Given:A percentage can be used to expre

6、ss 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 0.255206Bain MathInflation - DefinitionsIf an item cost $1.00 in 1997 and cost $1.03 i

7、n 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, unit cost growth), it is necessary to subtract out the effects of inflation. Inflationar

8、y 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) dollars:3. Price deflatorPrice deflator (current year) Price deflator (base year)Inflati

9、on 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 taken out). Any year can be chosen as the base year, but all dollar figures must be stated

10、 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 value of a unit of currency.7Bain Math Inflation - U.S. Price Deflators *1996 is the ba

11、se 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.Year1996=100*% ChangeYear1996=100*% Change197027.79 5.32 1996100.00 1.95 197129.23 5.18 1997101.97 1.97 197230.46 4.23 19981

12、04.48 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 2007132.96 2.82 198263.97

13、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 3.97 2017173.83 2.71

14、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.8Bain MathInflation - Real vs. Nominal Figures To understand how a company has performed over time (e.g., in terms of revenue, cost

15、s, 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 or c

16、ost 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 sprea

17、dsheets 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.9Bain MathInflation - Example (1) (1970 -1992)Adjusting for inflation is critical for any analysis looking at prices over time. In nominal dollars, GEs was

18、her 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 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$1,500CAG

19、R10Bain 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 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 pr

20、ices. They have fallen an average of 1.0% per year.11Bain 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?12Bain MathInflation - Exe

21、rcise (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) = 95.67(1995) = 98.08Step 3: Use the formula to calculate real dollars:Price deflator (current

22、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 85.341994:1995:1996:=20.5 X = , X = 24.388.72 85.3425.3 X = , X = 25.791.16 85.3427.4 X = , X = 28.593.54 85.3

23、431.2 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) = 100.0013Bain MathForeign Exchange - Definitions Investments employed in making payments between countries (e.g., pape

24、r currency, notes, checks, bills of exchange, and electronic notifications of international 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

25、other currencies. In efficient international markets, exchange rates will adjust to compensate for differences in interest and inflation rates between currenciesForeign Exchange:Exchange Rate:14Bain MathForeign Exchange Rates1) US$ equivalent = US dollars per 1 selected foreign currency unit2) Curre

26、ncy per US$ = selected foreign currency units per 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.

27、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. 15Bain MathForeign Exchange - Exercises Question 1:Answer:Question 2:Answer:Question 3:Answer: 650.28 US dollars = ?

28、 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 US$1$

29、0.1719 1 Franc$0.5752 1 Mark 1.59 Singapore dollar US$ 116Bain 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 x17Bain MathGraphing - Linear Exercise #1 Formula for

30、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 (wh

31、en 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-versa18Bain MathGraphing - Linear Exercise #2 (1) A lamp manufa

32、cturer 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?19Bain MathGraphing - Linear Exercise #2 (2) 08,00016,000P

33、roduction 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 + bb = y-mxb = 2,000 - 10 (100)b = 1,00

34、0 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 $16,00020Bain MathGraphing - Logarithmic (1) Log:A “log” or logarithm of given number is def

35、ined 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 log10 100=2 or log 100=221Bain

36、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 typically uses base 10Log-log paper is logarithmic on both axes; semi-log p

37、aper is logarithmic on one axis and linear on the otherLog ScaleLinear Scale22Bain 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 distance

38、 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 (ln o

39、r 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 decreas

40、ingIn many situations, it is convenient to use logarithms.23Bain MathAgenda Basic mathFinancial mathsimple interestcompound interestpresent valuerisk and returnnet present valueinternal rate of returnbond and stock valuationStatistical math24Bain MathSimple Interest Definition:Simple interest is com

41、puted 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 calculate the return on certain types of investmentsGiven: A person invests $5,000 in a sa

42、vings 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 1225Bain MathCompound Interest “Money makes money. And the money that money makes, makes more money.”- Benjamin FranklinDefinition:C

43、ompound 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 original principal. The interest for the following period is computed by us

44、ing 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 done in a yeart = the number of yearsr nNotes:As the nu

45、mber 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 interest rate that would generate the same amount

46、of interest as would the compound rate26Bain 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 will you receive at the end of one year

47、 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 periods interest is calc

48、ulated.Interest earned (i = prt):i1 = $1,000 x0.12x0.25i2 = $1,030 x0.12x0.25i3 = $1,060.90 x0.12x0.2514 = $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= $1,125.5127Bain MathPresent Value - Definition

49、s (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 money that

50、 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

51、 expected today, C1 is the cash expected in one year, etc. 1 (1+r)nC (1+r)nC1 1+rC2 (1+r)2Cn (1+r)n28Bain 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 P

52、V of an annuity, an investment 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 r29Bain MathPresent Value - Exercise (1) 1)$10.00 today2)$20.00 five years from today3)A perpetuity of $1.504)A p

53、erpetuity of $1.00, growing at 5%5)A six year annuity of $2.00Assume you can invest at 16% per yearWhich of the following would you prefer to receive? 30Bain MathPresent Value - Exercise (2) *The present value is negative because this is the cash outflow required today receive a cash inflow at a lat

54、er time1)$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 year annuity of $2.00, PV = - =$7.37 $1.50 0.16The option with the highest present value is #1, receiving $10.00 today$2.0

55、0 0.16 1 (1+ 0.16)5 $2.00 0.16FViPVN=(9.52)*20( )( ) PV = = $9.52$20.00 (1+0.16)5Answer:31Bain MathRisk and Return Not all investments have the same riskinvesting in the U.S. stock market is more risky than investing in a U.S. government treasury bill, but less risky than investing in the stock mark

56、et of a developing countryMost investors are risk averse - they avoid risk when they can do so without sacrificing returnRisk averse investors demand a higher return on higher risk investmentsA safe dollar is worth more than a risky one.32Bain MathNet Present Value Net present value (NPV) is the met

57、hod used in evaluating investments whereby the present value of all case outflows required for the investment are added to the present value of all cash inflows generated by the investmentCash outflows have negative present values; cash inflows have positive present valuesThe rate used to calculate

58、the present values is the discount rate. The discount rate is the required rate of return, or the opportunity cost of capital (i.e., the return you are giving up to pursue this project)An investment is acceptable if the NPV is positiveIn capital budgeting, the discount rate used is called the hurdle

59、 rateDefinition:33Bain MathInternal Rate of Return The internal rate of return (IRR) is the discount rate for which the net present value is zero (i.e., the cost of the investment equals the future cash flows generated by the investment)The investment is acceptable when the IRR is greater than the r

60、equired rate of return, or hurdle rateUnfortunately, comparing IRRs and choosing the highest one sometimes does not lead to the correct answer. Therefore, IRRs should not be used to compare ject A can have a higher IRR but lower NPV than project B; that is, IRRs do NOT indicate the magnitude of an opportu