战略分析工具+分析方法-bainmath.ppt

1、Author: Collins Qian,Reviewer: Brian Bilello,bc,Bain Math,March 1998,Copyright 1998 Bain semi-log paper is logarithmic on one axis and linear on the other,Log Scale,Linear Scale,23,CU7112997ECA,Bain Math,Graphing - Logarithmic (3),The most useful feature of a log graph is that equal multiplicative c

2、hanges in data are represented by equal distances on the axes the 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, 10 It is convenient to use log scales to examine the rate of change between d

3、ata 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 used prices and costs over time Growth Share matrices ROS/RMS graphs,Line Shape of Data Plots,Explanation,A straight line,The data points are changing at the

4、 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.,24,CU7112997ECA,Bain Math,Agenda,Basic math,Financial math simple interest compound interest present value r

5、isk and return net present value internal rate of return bond and stock valuation Statistical math,25,CU7112997ECA,Bain Math,Simple Interest,Definition:,Simple interest is computed on a principal amount for a specified time period The formula for simple interest is:i = prt where, p = the principal r

6、 = the annual interest rate t = the number of years,Application:,Simple interest is used to calculate the return on certain types of investments Given: 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 m

7、onths? Answer:i = prt i = $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 computed on a principal amount and any accumulated interest. A bank that pays

8、 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 the amount, A

9、, you will receive at the end of period n is: A = p (1 + )ntwhere,p = the principal r = the annual interest rate n = the number of times compounding is done in a year t = the number of years,r n,Notes:,As the number of times compounding is done per year approaches infinity (as in continuous compound

10、ing), the amount, A, you will receive at the end of period n is calculated using the formula: A = pert The 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,CU7112997ECA,Bain Math,Compound Interest - Ap

11、plication,$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 invest $1,000 at an annual rate of

12、 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 earned (i = prt):i1

13、 = $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= $32.78 New 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 - Defin

14、itions (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 m

15、oney that will be received in the future is worth today The formula for present value is:PV = Where, C =the amount of money received in the future r = the annual rate of return n = the number of years is called the discount factor The present value PV of a stream of cash is then: PV = C0+ + + Where

16、C0 is the cash expected today, C1 is the cash 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:

17、 PV =,The present value PV 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:,30,CU7112997ECA,Bain Math,Present Value - Exercise (1),1)$10.00 today 2)$20.00 five years from today 3)A perpetu

18、ity of $1.50 4)A perpetuity of $1.00, growing at 5% 5)A six year annuity of $2.00 Assume 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

19、today receive a cash inflow at a later time,1)$10.00 today, PV = $10.00 2)$20.00 five years from today, For HP12C: 5 16 3)A perpetuity of $1.50, PV = = $9.38 4)A perpetuity of $1.00, growing at 5%, PV = = $9.09 5)A six year annuity of $2.00, PV = - =$7.37,$1.50 0.16,$1.00 0.16-0.05,The option with t

20、he highest present value is #1, receiving $10.00 today,$2.00 0.16,1 (1+ 0.16)5,$2.00 0.16,FV,i,PV,N,=(9.52)*,20,( ),( ),Answer:,32,CU7112997ECA,Bain Math,Risk and Return,Not all investments have the same risk investing in the U.S. stock market is more risky than investing in a U.S. government treasu

21、ry bill, but less risky than investing in the stock market of a developing country Most investors are risk averse - they avoid risk when they can do so without sacrificing return Risk averse investors demand a higher return on higher risk investments,A safe dollar is worth more than a risky one.,33,

22、CU7112997ECA,Bain Math,Net Present Value,Net present value (NPV) is the method 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 investment Cash outflows have negative present

23、values; cash inflows have positive present values The rate used to calculate 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 NP

24、V is positive In capital budgeting, the discount rate used is called the hurdle rate,Definition:,34,CU7112997ECA,Bain Math,Internal 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 f

25、lows generated by the investment) The investment is acceptable when the IRR is greater than the required rate of return, or hurdle rate Unfortunately, comparing IRRs and choosing the highest one sometimes does not lead to the correct answer. Therefore, IRRs should not be used to compare projects. pr

26、oject A can have a higher IRR but lower NPV than project B; that is, IRRs do NOT indicate the magnitude of an opportunity projects with cash flows that fluctuate between negative and positive more than once have multiple IRRs IRRs cannot be calculated for all negative cash flows,Definition:,35,CU711

27、2997ECA,Bain Math,NPV and IRR - Exercise,*You can use this abbreviated format since the other data has not changed from part a,Given:,An investment costing $2MM will produce cash flows of $700,000 in Year 1, $700,000 in Year 2, and $900,000 in Year 3. Calculate its net present value at discount rate

28、s of (a) 5%, (b) 10%, and (c) 15%. Also, (d) calculate the projects IRR.,Answer:,Using a 5% discount rate, NPV = -$2MM + + + = $79,041,$700,000 (1.05),$700,000 (1.05)2,$900,000 (1.05)3,For HP12c: Another easy way to calculate an IRR is to use the IRR function in Excel: = IRR (C1, C2, C3, Cn) where C

29、1 is the cash flow in Year 1, C2 is the cash flow in Year 2, etc. In this example, “ = IRR (-2,000,000, 700,000, 700,000, 900,000)” = 7.01%,f,CLX,f,i,CHS,CF0,2,000,000,g,700,000,CFj,g,700,000,CFj,g,900,000,CFj,g,5,NPV,=79,041,For HP12C:,Using 10% discount rate, NPV = -$2MM + + + = -$108,941,$700,000

30、 (1.10),$700,000 (1.10)2,$900,000 (1.10)3,For HP12C:,f,i,10,NPV,= -$108941,Using 15% discount rate, NPV = -$2MM + + + = -$270,239,$700,000 (1.15),$700,000 (1.15)2,$900,000 (1.15)3,For HP12C:,f,i,15,NPV,= -$270,239,f,IRR,= 7.01%,*,*,*,a),b),c),d),36,CU7112997ECA,Bain Math,Bond and Stock Valuation,The

31、 concept of Net Present Value can be applied to valuing bonds and stocks.,Bonds:,The market value, MV, today of a bond with a face value of f and a coupon rate of c maturing in n years when the market interest rate is r is:,Stocks:,The price today, p0 of a stock that pays dividends of dt in Year t a

32、nd that you sell in Year n for pn is, for someone expecting a rate of return of r:,fc 1+ r,fc (1+ r)2,fc (1+ r)n,f (1+ r)n,MV =,+,+ +,+,d1 1+ r,d2 (1+ r)2,dn (1+ r)n,pn (1+ r)n,p0 = +,+ +,+,37,CU7112997ECA,Bain Math,Agenda,Basic math Financial math Statistical math averages weighted averages linear

33、regression,38,CU7112997ECA,Bain Math,Averages,*Sometimes referred to as the arithmetic mean,Definition:,(Sum of data points),Exercise:,Answer:,39,CU7112997ECA,Bain Math,Using Averages,(Arithmetic) Mean:,The mean is an artificial statistic in that it need not coincide with any point in your data set

34、in the previous example, the mean of 9.3 is not in the original data set Extreme points have more effect than those closer to the middle,The median is affected by the number of, but not the value of, extreme points in the data set The median is least useful in small data sets (e.g., the median of 1.

35、3 and 10 is 3, which tells us little),The mode can be meaningless in a data set with several values that occur repeatedly,Most useful for data sets that have outliers,Most useful when you need a simple average that weights all of the data points equally,Most useful for data sets with one or few valu

36、es that occur repeatedly,Median:,Mode:,Characteristics,When to Use,40,CU7112997ECA,Bain Math,Weighted Averages,Definition:,Weighted average =,(Sum of each data point multiplied by its weight),(Sum of the weight),Weighted averages are appropriate for data sets in which the points to be averaged have

37、different levels of importance. Weighted averages adjust for the importance of different data points by using their weights.,Exercise:,A mattress manufacturer produces mattresses of all sizes (twin, queen, and king size). What is the average price of a mattress?,Given:,Answer:,Application:,Mattress

38、Size,Price,Monthly Volume,Twin,Queen,King,$99.00,$159.00,$199.00,80,120,30,The weighted average of mattress price is:,41,CU7112997ECA,Bain Math,Linear Regression - Definitions,Regressions are used to understand how one or more independent variables affect a dependent variable.,Dependent variable,Ind

39、ependent variables,y = a + b x1 + c x2 + + e,A linear regression can be expressed as an equation:,42,CU7112997ECA,Bain Math,Linear Regression - Key Statistics,R-squared:,Tells to what extent the independent variables “explain” the value of the dependent variable Generally a low R-squared means that

40、your independent variables do not explain the value of dependent variable well. Therefore, you may need to rethink your independent variables.,Tells whether the estimated coefficients are significantly different from zero If a coefficient of a particular independent variable is not significantly dif

41、ferent from zero, you may not conclude that the particular independent variable has an effect on the dependent variable,Tells the range of values in which an estimated coefficient is likely to fall (i.e., how much larger and smaller the true coefficient may be from the estimated coefficient),T-stati

42、stics should be = 2 or = -2 (for a 95% confidence level),R-squared can only be between 0 and 1 Usually an R-squared above 0.7 is considered high,It is typical to use two standard deviations (or a 95% confidence interval),T-statistics:,Standard deviation:,Explanation,Reference Points,Statistics,There

43、 are three key statistics that help explain regression output.,43,CU7112997ECA,Bain Math,Single Variable Linear Regression - Data,For a commercial bank client, a Bain AC has been asked to explain the relationship between the number of deposit accounts and the demand deposit account (DDA) balance. He

44、 has obtained the following data from thirty different branches of the bank:,Branch,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,DDA Balance ($000),2,240.0,2,210.2,1,368.4,1,840.0,1,884.2,2,016.2,1,890.8,1,345.0,1,448.2,1,505.0,1,525.8,1,603.4,1,884.2,1,528.2,1,384.2,Number of Accounts,900,906,500,741,789,88

45、9,874,510,529,420,679,872,924,607,452,Branch,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,DDA Balance ($000),1,790.4,1,865.8,2,046.2,2,645.6,1,481.6,1,641.8,1,360.6,1,680.0,1,756.4,2,324.2,1,250.2,1,165.8,1,450.0,1,885.2,1,842.4,Number of Accounts,729,794,844,1,005,621,703,679,718,785,938,572,508,62

46、9,801,876,44,CU7112997ECA,Bain Math,Linear Regression - Steps,Input the data into an excel spreadsheet Each variable should be put in its own column,Step 1:,Run the regression Choose “data analysis” from the “tools” bar, and then choose “regression” When the “regression” window appears, drag the dep

47、endent variable data (in this case, the “DDA balance”) into “Input Y range”, and the independent variable data (“number of accounts”) into “Input X range) To view the regression results graphically, check off the box for a line fit plot,Step 2:,Interpret the results,Step 3:,The three steps for doing

48、 regression analysis are:,45,CU7112997ECA,Bain Math,Single Variable Linear Regression - Output,SUMMARY OUTPUT,Regression Statistics,Multiple R,0.86823719,R Square,0.753835818,Adjusted R Square,0.74504424,Standard Error,174.6960575,Observations,30,ANOVA,df,SS,MS,F,Significance F,Regression,1,2616833.

49、837,2616833.837,85.74522395,5.10952E-10,Residual,28,854523.9497,30518.71249,Total,29,3471357.787,Coefficients,Standard Error,t Stat,P-value,Lower 95%,Upper 95%,Lower 95.0%,Upper 95.0%,Intercept,384.9724417,148.5732885,2.591128227,0.015023879,80.63351466,689.3113687,80.63351466,689.3113687,X Variable

50、,1.849629565,0.199746781,9.259871702,5.10952E-10,1.440466373,2.258792758,1.440466373,2.258792758,# of accounts,Excel generates the following regression output:,46,CU7112997ECA,Bain Math,Single Variable Linear Regression - Actual vs. Predicted Data,R = 0.75,0,0,The regression does a good job of predi

51、cting DDA Balance.,47,CU7112997ECA,Bain Math,Single Variable Linear Regression - Answer,R-squared = 0.75. This means that 75% of the DDA balance can be “explained by the number of accounts. The T-statistics on the x variable (i.e., number of accounts) is 9.3, which is significantly bigger than the r

52、eference value of “2”, indicating a strong relationship between the number of accounts and DDA balance The line fit plot shows how closely the predicted DDA balances tie with the actual numbers The estimated regression line is = DDA balance = $385,000 + ($1,850 x number of accounts) This means that

53、every incremental account brings in $1,850 of additional DDA balances,y-intercept,Coefficient for number of accounts,Using the data from the regression analysis, you can summarize the relationship between DDA balance and number of accounts.,48,CU7112997ECA,Bain Math,Multi-Variable Linear Regression

54、- Data,Note: This is different from the previous exercise in that it is a regression with multiple independent variables, (i.e., Horsepower and Engine Displacement),For an automobile client, a Bain AC has been asked to explain the drivers of miles per gallon for small automobile models. The initial

55、hypothesis is that Horsepower and Engine Displacement are the two most important explanatory variables. The following data has been gathered for 18 small automobile models in 1995:,Automobile,Miles per Gallon,Horsepower,Engine Displacement,1,27,130,112,2,28,81,90,3,30,93,135,4,28,113,97,5,31,90,114,

56、6,33,63,81,7,40,55,61,8,30,102,97,9,33,92,91,10,32,81,90,11,29,103,113,12,31,90,97,13,33,74,98,14,35,73,73,15,29,102,97,16,32,78,89,17,28,100,109,18,28,100,109,49,CU7112997ECA,Bain Math,Multi-Variable Linear Regression - Output,Regression Statistics,Multiple R,0.856946547,R Square,0.734357385,Adjusted R Square,0.69893837,Standard Error,1.750074559,Observations,18,ANOVA,df,