章節(jié)4數(shù)量和計(jì)算器的使用前導(dǎo)

1、CFA一級培訓(xùn)項(xiàng)目Quantitative Methods單晨瑋金程教育資深培訓(xùn)師地點(diǎn)： 上海 北京 Topic Weightings in CFA Level I2-62Session NO.ContentWeightingsStudy Session 1Ethics & Professional Standards15Study Session 2-3Quantitative Analysis12Study Session 4-6Economics10Study Session 7-10Financial Reporting and Analysis20Study Session

2、11Corporate Finance7Study Session 12Portfolio Management and Wealth Planning7Study Session 13-14Equity Investment10Study Session 15-16Fixed Income10Study Session 17Derivatives5Study Session 18Alternative Investments3Quantitative MethodsØ Time Value Calculationl R5 The Time Value of Moneyl R6 Di

3、scounted Cash Flow ApplicationsØ Probability & Statisticsl R7 Statistical Concepts and Market Returnsl R8 Probability Conceptsl R9 Common Probability DistributionsØ Inferential statisticsl R10 Sampling and Estimationl R11 Hypothesis Testing3-62 Time Value4-62Time Value of MoneyØRe

4、quired rate of return isl affected by the supply and demand of funds in the market;l the return that investors and savers require to get them to willingly lend their funds;l usually for particular investment.Discount rate isl the interest rate we use to discount payments to be made in the future.l u

5、sually used interchangeably with the interest rate.ØØOpportucost isl also understood as a form of interest rate. It is the value thatinvestors forgo by choosing a particular course of action.5-62Time Value of MoneyØ Decompose required rate of return:l Nominal risk-free rate = real ris

6、k-free rate + expected inflation ratel Required interest rate on a security= nominal risk-free rate + default risk premium + liquidity risk premium + maturity risk premium6-62Time Value of MoneyEAR calculation:m1+ EAR = æ1+rö= erçm ÷èøl 那么如果是semi, m=2; 如果是quarterly, m=4

7、l 如果是連續(xù)復(fù)利，公式則變?yōu)镋AR = eannual int-1Ø 定性（EAR和計(jì)息次數(shù)有關(guān)）l The greater the compounding frequency, the greater the EAR will bein comparison to the stated rate7-62mEAR=(1+periodic rate)-1Time Value of MoneyØFuture value (FV): Amount to which investment grows after one or morecompounding periods.

8、16;Present value (PV): Current value of some future cash flowØAnnuities: is a stream of equal cash flows that occurs at equal intervals over agiven periodØ 內(nèi)容：l N = number of periodsl I/Y = interest rate per periodl PV = present valuel PMT = amount of each periodic paymentl FV= future valu

9、e8-62Time Value of MoneyAn example of ordinary annuities（后付年金）:Example: Whats the FV of an ordinary annuity that pays 100 peryear at the end of each of the next 3 years, given the discount rate is 10%Solutions: enter relevant data for calculate.N=3, I/Y=10, PMT=-100, PV=0, CPTFV=3319-62Time Value of

10、 MoneyØAbout an annuity due（先付年金）l Definition: an annuity where the annuity payments occur at thebeginning of each compounding period.l Calculation:ü Measure 1: put the calculator in the BGN mode and inputrelevant data.ü Measure 2: treat as an ordinary annuity and simply multiple ther

11、esulting PV by (1+I/Y)10-62Example: Time Value of Money1.A company plans to borrow $50,000 for five years. The companys bank will lend the money at a rate of 9% and requires that the loan be paid off in five equal end-of-year payments. Calculate the amount of the payment that the company must make i

12、n order to fully amortize this loan in five years.Answer:N=5, I/Y=9, PV=50,000, FV=0; CPT: PMT=-12,854.62ØØ2.Using the loan described in the preceding example, determine the paymentamount if the bank requires the company to make quarterly payments.Answer:N=5×4=20, I/Y=9/4=2.25, PV=50,

13、000, FV=0; CPT: PMT=-3,132.10ØØ11-62Example: 房貸月供問題Ø張女士買了一套價(jià)值100萬的房子，首付比例30%，她從70萬，貸？款的年利率為6.2%，期限為20年。她每月月末需向還款Ø利用TVM功能：ØN=20*12=240, I/Y=6.2/12, PV=700,000, FV=0ØCPT PMT=-5096.12Ø還完第一以后，還有多少本金余額沒有還？第一中本金還了多少？利息還了多少？Ø利用AMORT功能：（2ND PV）ØP1=1，P2=1，BAL=698,

14、520.5485，PRN=-1,479.4515，INT=-3,616.666712-62Example: 養(yǎng)老問題Ø張女士今年60歲，她從今天開始即將退休。如果從退休開始每年年初都要支取10萬塊錢的的話，假設(shè)回報(bào)率為4%，張女士現(xiàn)在來支持她未來20年的生活？需要準(zhǔn)備ØBGN模式：(2ND BGN, 2ND SET, 2ND QUIT)Ø利用TVM功能：ØN=20, I/Y=4, PMT=100,000FV=0ØCPT PV=-1,413,393.9413-62Example: 教育金問題Ø張女士的女兒今年要上大學(xué)，她現(xiàn)在開始每年年初

15、都要給女兒支付5萬塊錢的教育金，一直持續(xù)到女兒大學(xué)畢業(yè)，為期4年，如果現(xiàn)在的市場利率為4%，張女士現(xiàn)在需要為女兒準(zhǔn)備多少教育金？ØBGN模式：(2ND BGN, 2ND SET, 2ND QUIT)Ø利用TVM功能：ØN=4, I/Y=4, PMT=50,000,FV=0ØCPT PV=-188,754.5514-62Time Value of MoneyØAbout perpetuityDefinition: A perpetuity is a financial instruments that pays a fixed amount of

16、 money at set intervals over an infinite period of time.lAAAAAAA0PV1 = A/(1+r) PV2 = A/(1+r)2 PV3 = A/(1+r)3 PV4 = A/(1+r)4etc.1234567etc.AAAAAPV =+(1+ r)PV = A +(1)(2)1+ r(1+ r)2(1+ r)31+ r(1+ r)2r ´ PV = A Þ PV = A(2) - (1) r15-62Discounted Cash Flow ApplicationsIRR（Internal Rate of Retu

17、rn）Ø When NPV= 0, the discount rate.16-62CFCFCFNCFNPV = 0 = CF0 +1+2+.+N= åt(1+ IRR)1(1+ IRR)2(1+ IRR)N(1+ IRR)tt =0CFCFCFNCFNPV = CF0 +1+2+ .+N= åt(1+ r )1(1+ r )2(1+ r )N(1+ r )tt =0Example: Discounted Cash Flow ApplicationsØ 張女士花了10萬塊錢買了一份，期限為5年。這份產(chǎn)品在這5年里每年年末分別可以給張女士帶來的2萬、3萬、4

18、萬。如果市場利率為5%，張女士為：3萬、2萬、這份凈賺？內(nèi)部率是多少？17-62Example: Discounted Cash Flow Applications18-62按鍵解釋顯示CF 2ND CLR WORK清除CF功能中的記憶CF0=0.000010+/-ENTER期初投入CF0=-10.0000 3 ENTER第一期現(xiàn)金流C01=3.0000 2 ENTER第二期現(xiàn)金流C02=2.0000 2 ENTER第三期現(xiàn)金流C03=2.0000 3 ENTER第四期現(xiàn)金流C04=3.0000 4 ENTER第五期現(xiàn)金流C05=4.0000NPV 5 ENTER折現(xiàn)率5%I=5.0000 C

19、PT計(jì)算NPVNPV=2.0011IRR CPT計(jì)算IRRIRR=11.5156Discounted Cash Flow ApplicationsHPRØ Define: the holding period return is simply the percentage change in the value of an investment over the period it is hold.Ø Calculate:19-62HPR = P1 - P0 + CF1P0Example: Discounted Cash Flow ApplicationsØA s

20、tock is purchased for $30 and is sold for $33 six months later, during which time it paid $0.50 in dividends. Calculate the holding periodreturn.ØAnswer:HPR = P1- P0 + CF1= 33 - 30 + 0.5 = 11.67%P03020-62 Statistical Concepts21-62Statistical ConceptsØ Descriptive statisticsl Summarize the

21、important characteristics of large data sets.Ø Inferential statisticsl Make forecasts, estimates, or judgments about a large set ofdata on the basis of the statistical characteristics of a smaller set (a sample)22-62Statistical ConceptsØ A measure used to describe a characteristic of a pop

22、ulation isreferred to as a parameter.Ø In the same manner that a parameter may be used to describe acharacteristic of a population, a sample statistic is used to measure a characteristic of a sample.23-62Statistical ConceptsØMeasures of central tendency: mode, median, mean24-62Statistical

23、ConceptsThe arithmetic mean：The weighted mean：The geometric mean：The harmonic mean：25-62XH =nnå(1/ Xi )i =1NG = N X1X2i )1/ Ni =1nXW = åwi Xi = (w1 X1 + w2 X 2 + wn Xn )i =1N å XiX = i =1nStatistical ConceptsØØStandard deviationXYZ Corp. Annual Stock Prices1.Assuming that th

24、e distribution of XYZ stock returns is a population, what is the population variance?A. 6.8%2B. 7.7%2C. 80.2%22.Assuming that the distribution of XYZ stock returns is a sample, the sample variance iscloset to?A. 5.0%2B. 72.4%2C. 96.3%226-6220032004200520062007200822%5%-7%11%2%11%Nå( X - m)2iFor

25、 population: s 2 = i=1NNå Xi - XMAD = i =1nn å( X - X )2iFor sample: s2 = i=1n -1Range =um value minimum valueStatistical ConceptsØCoefficient of variation measures the amount of dispersion in a distribution relative to the distributions mean. (relative dispersion)ØThe sharp rati

26、o measures excess return per unit of risk.27-62Sharp ratio= RP -RfPCV= sxXStatistical ConceptsMean=Median=ModeMode<Median<MeanMean<Median<ModeSymmetricalPositive (right) skewNegative (left) skewPositive skewed：Mode<median<mean, having a right fat tailNegative skewed：Mode>media&g

27、t;mean, having a left fat tail28-62Statistical ConceptsØLeptokurtic vs. platykurticl It deals with whether or not a distribution is more or less “peaked”than a normal distribution29-62leptokurticNormal distributionplatykurticSample kurtosis>3=3<3Statistical ConceptsLeptokurticNDiFat tailA

28、 leptokurtic return distribution has more frequent extremely large deviations from the mean than a normal distribution.30-62ormalstribution Probability Concepts31-6232-6233-6234-62Probability ConceptsØ Basic Conceptsl Random variable is uncertain quantity/number.l Outcome is an observed value o

29、f a random variable.l EventüüMutually exclusive eventscan not both happen at the same time.Exhaustive eventsinclude all possible outcomes.Ø Two Defining Properties of Probabilityl 0 P(E) 1l P(E1)+ P(E2)+ P(En)=135-62Probability ConceptsØ Unconditional Probability (marginal probab

30、ility): P(A)Ø Conditional probability : P(A|B)36-62Probability ConceptsØJoint probability : P(AB)l Multiplication rule:ü P(AB)=P(A|B)×P(B)= P(B|A)×P(A)l If A and B are mutually exclusive events, then: P(AB)=P(A|B)=P(B|A)=0Probability that at least one of two events will occu

31、r:l Addition rule:ü P(A or B)=P(A)+P(B)-P(AB)l If A and B are mutually exclusive events, then: P(A or B)=P(A)+P(B)Ø37-62Probability ConceptsØ The occurrence of A has no influence of on the occurrence of Bl P(A|B)=P(A) or P(B|A)=P(B)l P(AB)=P(A)×P(B)l P(A or B)=P(A)+P(B)-P(AB)

32、6; Independence and Mutually Exclusive are quite differentl If exclusive, must not independence;l Cause exclusive means if A occur, B can not occur, A influents B.ü P(AB)=P(A)×P(B)38-62Probability ConceptsE(X ) = åP(Xi )XiØ Expected value:E(X) = åxi*P(xi ) = x1 * P(x1 ) + x2

33、 * P(x2 ) +L+ xn * P(xn )N= å P ( X - EX )2s 2s=s2iii=139-62 Common Probability Distributions40-62Common Probability DistributionsØProbability Distributionl Describe the probabilities of all the possible outcomes for a randomvariable.Discrete and continuous random variablesl Discrete rando

34、m variables: the number of possible outcomes can be counted, and for each possible outcome, there is a measurable and positive probability.l Continuous variables: the number of possible outcomes is infinite, even if lower and upper bounds exist.ü P (x)=0 even though x can occur.ü P (x1<

35、X<x2)Ø41-62Common Probability DistributionsØProbability function:p(x)=P(X=x)l For discrete random variablesl 0 p(x) 1l p(x)=1Probability density function (p.d.f) : f(x)l For continuous random variable commonlyØ42-62Common Probability DistributionsØDiscrete uniforml A discrete

36、uniform random variable is one for which the probabilitiesfor all possible outcomes for a discrete random variable are equal.l For example, consider the discrete uniform probability distributiondefined as X=1,2,3,4,5, p(x)=0.2.ü Here, the probability for each outcome is equal to 0.2 i.e.,p(1)=p

37、(2)=p(3)=p(4)=p(5)=0.2.43-62Common Probability DistributionsØContinuous Uniform Distribution- is defined over a range that spans between some lower limit, a, and upper limit,b, which serve as the parameters of the distribution.ØProperties of Continuous uniform distributionl For all a x1<

38、; x2 bl P (X<a or X>b) = 0P(x1 £ X £1) /(b - a)l44-62Common Probability DistributionsØ 正態(tài)分布的前世今生45-62Common Probability DistributionsØ The shape of the density functionf(x)xØ Properties:l XN(µ , ²)l Symmetrical distribution: skewness=0; kurtosis=3l The tails

39、get thin and go to zero but extend infinitely, asympotic (漸近）46-62Common Probability DistributionsØ The confidence intervalsl 68% confidence interval isl 90% confidence interval isl 95% confidence interval isl 99% confidence interval isProbabilitym -s , m +s m -1.65s , m +1.65s m -1.96s , m +1.

40、96s m - 2.58s , m + 2.58s U-1.96U-1U-2.58U-2.58U+1.96U+1u 68%95%99%47-62Common Probability DistributionsØ Standard normal distributionl N(0,1) or ZZ = X - ml Standardization: if XN(µ , ²), then N(0,1)sl Z-tableØ F(-z)=1-F(z)Ø P(Z>z) = 1 F(z)48-62Common Probability Distrib

41、utions49-62Common Probability DistributionsØDefinition: If lnX is normal, then X is lognormal, which is used to describe the price of assetRight skewedØØBounded from below by zero, so it is useful for ming asset Prices50-620246810 Sampling and Estimation51-62Sampling and Estimation

42、16;Sampling and estimationl Simple random samplingl Stratified random sampling: to separate the population into smallergroups based on one or more distinguishing characteristics. Stratumand cells=M*NØSampling error: sampling error of the mean= sample mean- populationmean52-62Sampling and Estima

43、tionØTime-series datal consist of observations taken over a period of time at specific andequally spaced time intervals.ØCross-sectional datal a sample of observations taken at a single point in time.53-62Time-series dataCross-sectional dataa collection of data recorded over a period of ti

44、mea collection of data taken at a single point of time.Sampling and EstimationØ Point estimate: the statistic, computed from sample information, whichis used to estimate the population parameterØ Interval estimate: a range of values constructed from sample data so theparameter occurs withi

45、n that range at a specified probability.54-62Sampling and EstimationØData-mining biasl Refers to results where the statistical significance of the pattern is overestimated because the results were found through data mining.Sample selection biasl Some data is systematically excluded from the ana

46、lysis, usually because of the lack of availability.Survivorship biasl Usually derives from sample selection for only the existing portfolio areincludedLook-ahead biasl Occurs when a study tests a relationship using sample data that was not aavailable on the test date.Time-period biasl Time period ov

47、er which the data is gathered is either too short or too long. If the time period is too short, research results may reflect phenomena specific to that time period, or perhaps even data mining.ØØØØ55-62 Hypothesis Testing56-62Hypothesis TestingØ Hypothesis testingl The steps

48、 of hypothesis testingStep 1Step 2Step 3Step 5Step 457-62RejectFormulate adecision ruleTake a sample, arrive at decisionDo not rejectSelect a level of significanceIdentify the teststatisticState null and alternative hypotheses Technical Analysis58-62Technical AnalysisØ Principles:l Prices are d

49、etermined by the interaction of supply and demand.l Only participants who actually trade affect prices, and better- informed participants tend to trade in greater volume.l Price and volume reflect the collective behavior of buyers and sellers.Ø Assumptions:l Market prices reflect both rational and irrational investor behavior.ü Investor behavior is reflected in trends and patterns that trend to repeat and can