【文章內(nèi)容簡介】 100=2 or log 100=221CU7112997ECABOSCopyright169。 1998 Bain Company, Inc. Bain MathGraphing Logarithmic (2) For a log scale in base 10, as the linear scale values increase by ten times, the log values increase by 1.98765432101,000,000,000100,000,00010,000,0001,000,000100,00010,0001,000100101?Log paper typically uses base 10?Loglog paper is logarithmic on both axes。 semilog paper is logarithmic on one axis and linear on the otherLog Scale Linear Scale22CU7112997ECABOSCopyright169。 1998 Bain Company, Inc. Bain 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 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 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 used–prices and costs over time–Growth Share matrices–ROS/RMS graphsLine Shape of Data Plots ExplanationA straight line The data points are changing at the same rate from one point to the nextCurving upward The rate of change is increasingCurving downward The rate of change is decreasingIn many situations, it is convenient to use logarithms.23CU7112997ECABOSCopyright169。 1998 Bain Company, Inc. Bain MathAgenda ?Basic math?Financial math–simple interest–pound interest–present value–risk and return– present value–internal rate of return–bond and stock valuation?Statistical math24CU7112997ECABOSCopyright169。 1998 Bain Company, Inc. Bain MathSimple Interest Definition: Simple interest is 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 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 x i = $50 2 1225CU7112997ECABOSCopyright169。 1998 Bain Company, Inc. Bain MathCompound Interest “Money makes money. And the money that money makes, makes more money.” Benjamin FranklinDefinition: Compound interest is puted on a principal amount and any accumulated interest. A bank that pays pound interest on a savings account putes interest periodically (., daily or quarterly) and adds this interest to the original principal. The interest for the following period is puted by using the new principal (., the original principal plus interest).The formula for the amount, A, you will receive at the end of period n is:A = p (1 + )nt where, p = the principalr = the annual interest raten = the number of times pounding is done in a yeart = the number of yearsr nNotes: As the number of times pounding is done per year approaches infinity (as in continuous pounding), 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 pound rate26CU7112997ECABOSCopyright169。 1998 Bain Company, Inc. Bain MathCompound Interest Application $1, $ $1, $$1, $ $1,$ $1,$0$250$500$750$1,000$1,250Dollarsi1 i2i3 i4A1 A2 A3 A41st Quarter 2nd Quarter 3rd Quarter 4th QuarterGiven: What amount will you receive at the end of one year if you invest $1,000 at an annual rate of 12% pounded quarterly?Answer: A = p (1+ ) nt = $1,000 (1 + ) 4 = $1, n 4Detailed Answer:At the end of each quarter, interest is puted, and then added to the principal. This bees the new principal on which the next period’s interest is calculated.Interest earned (i = prt): i1 = $1, i2 = $1, i3 = $1, 14 = $1,= $ = $ = $ = $New principle A1 = $1,000+$30 A2 = $1,030+ A3 = $1,+ A4 = $1,+= $1,030 = $1, = $1, = $1,27CU7112997ECABOSCopyright169。 1998 Bain Company, Inc. Bain MathPresent Value 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 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 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)n28CU7112997ECABOSCopyright169。 1998 Bain Company, Inc. Bain MathPresent Value Definitions (2) The present value of a perpetuity (., 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 which pays a fixed sum, each year for a specific number of years from year 1 to year n is: Perpetuity:Growing perpetuity:Annuity:CrC rgPV = C r 1 (1+ r)nC r29CU7112997ECABOSCopyright169。 1998 Bain Company, Inc. Bain MathPresent Value Exercise (1) 1) $ today2) $ five years from today3) A perpetuity of $4) A perpetuity of $, growing