品職教育2017年cfa一級框架圖quantitative何旋版

1、Qutative MethodsCFA一級框架圖utativeMethods講師：何 旋FrameworkTime Value CalculationR6 The Time Value of MoneyR7 Discounted Cash Flow ApplicationsProbability &Descriptive StatisticsR8 Statistical Concepts and Market ReturnsR9 Probability ConceptsR10 Common Probability DistributionsInferential statisticsR

2、11 Sampling and EstimationR12 Hypothesis TestingR13 Technical Analysising 6THE TIME VALUE OF MONEYInterest rateEARTVM計算FV=PV(1+r/m)mAnnuity： ordinary annuities annuity dueperpetuity計算ær ömær ömç1+¸= 1+ EAR Ò limç1+¸= er = 1+ EARèm ønÒ¥

3、 èm ø性質(zhì)The greater the compounding frequency,ü the greater the EAR will be in comparison to the stated rateü the greater the difference between EAR and the stated rate含義 Required rate of return Nominal risk-rate = real risk-rate + expected inflation rate Discount rate Required in

4、terest rate on a security = nominal risk-rate + default Opportucostrisk premium + liquidity risk premium + maturity risk premiuming 7DISCOUNTED CASH FLOW APPLICATIONSNPV & IRR 計算CFCFCFNCFNPV = CF0 +1 1 +2 2 + . +NN = åt t(1+ r )(1+ r )(1+ r )t =0 (1+ r )CFCFCFNCFNPV = 0 = CF0 +11 +22 + . +N

5、 N = ått(1+ IRR)(1+ IRR)(1+ IRR)t =0 (1+ IRR)Decision Rule Independent Projects： Accept it if NPV>0 Accept it if IRR>r (required rate of return) Mutually Exclusive Projects： NPV method: Choose the one with higher NPV IRR method: Choose the one with higher IRR NPV and IRR methods maywith e

6、ach other，以NPV為準IRR的特點ü When NPV= 0, the discount rate.ü IRR method assumes the projects cash flows will be reinvested at the IRR.ü Multiple solutions Problem of the IRR calculation (sign changes)率計算TWRR &MWRR 找到每一期的HPR；n期HPR的幾何平均找到每一期的現(xiàn)金流；MWRR計算IRR計算ü 會受到現(xiàn)金流改變的影響，所以客戶的決策對性質(zhì)&

7、#252;受到現(xiàn)金流流入流出的影響；MWRR會造成影響；ü 衡量基金經(jīng)理業(yè)績更準確ü如果客戶投資的越來越多，相當于給后期的率一個更高的權(quán)重TWRRMWRR定義幾何平均，相當于給每個率相同的權(quán)重以現(xiàn)金流作為權(quán)重，現(xiàn)金流越多的一期，率給的權(quán)重越大r= (F - P0 ) ´ 360BDFt單利年化r= HPY ´ 360HPY = P1 - P0 + CF1MMtP0復利年化365BEYEAY = (1+ HPY ) t-1(1+)2 =1+EAR2率折價率ing 8STATISTICAL CONCEPTS AND MARKET RETURNDescript

8、ive statistic：描述一組數(shù)據(jù)的基本特征Types of measurement scales性質(zhì)只能求modeMode、median沒有絕對零點Most refined中心位置Frequency Relative frequency Cumulative frequency Cumulative Relative FrequencyMean計算：arithmetic mean、weighted mean、geometric mean、harmonic mean注意：geometric mean=(1+HPR1)(1+HPRn) 1/n結(jié)論：harmonic mean<= ge

9、ometric mean<=arithmetic meanQules定義：Quartile /tile/Deciles/Percentile計算：Ly = (n+1)y/100性質(zhì)：比如The third quartile > meanNominal scalesOrdinal scalesInterval scalesRatio scales含義定性區(qū)分排序(>, <)(>, <, +, -)(>, <, +, -, *, /)離散程度Chebyshev inequality計算：1. regardless of the shape of th

10、e distribution;2. the proportion of the values that lie within k standard deviations of the mean is at least 1 1/k2Sharp ratio= RP -RfP性質(zhì)：越大越好計算Range =um value minimum valueNå Xi - XMAD = i =1nåNn ( X - X )2( X - m)2åiiFor population: s 2 = i=1For sample: s2 = i =1Nn -1性質(zhì)MAD < CVCV

11、= sx ´100%Xrelative dispersion,of scale (orof unit)Skewness 掌握性質(zhì)kurtosis 掌握性質(zhì)t分布：和z分布比，低峰肥尾。所以t分布更加分散一些，更大。Leptokurticü Sample kurtosis>3；Excess kurtosis>0ü 相同，尖峰肥尾(more frequent extremely large deviations from themean than a normal distribution.)PlatykurticSample kurtosis<3；

12、Excess kurtosis<0Normal distributionSample kurtosis=3；Excess kurtosis=0Positive skewedü Mode<median<meanü right fat tailü frequent small losses and a few extreme gains (mean0時)ü 投資者更加prefer positive skewnessNegative skewedü Mode>media>meanü left fat tail&

13、#252; frequent small gains and a few extreme losses. (mean0時)ing 9PROBABILITY CONCEPTS概率和(event)Properties of Probabilityü 0 P(E) 1ü P(E1)+ P(E2)+ P(En)=1 E1En: Mutually exclusive and Exhaustive概率Empirical probability: 分析過去，得到將來Priori probability: 分析過去，得到過去的推理Subjective probability:EventMu

14、tually exclusiveP(AB)=P(A|B)=P(B|A)=0Exhaustive eventsinclude all possible outcomesIndependenceü P(AB)=P(A)×P(B)ü If exclusive, must not independenceü Independence à=0; 反之不對概率計算Covariance & CorrelationüCorrelation measures the linear relationship between tworandom v

15、ariablesCorrelation has no units, ranges from 1 to +1, standardization of covarianceIf =0, this doesnt indicates independence.=COV(X,Y)rüCorrelationXYVar(X)Var(Y)ü計算性質(zhì)ü how one random variable moves with another random variable CovarianceCOV = E(X-E(X)(Y-E(Y) ü The covariance of

16、X with itself is equal to the variance of Xü Covariance ranges from negative infito positive infiOddsOdds for an event：P(E)/(1-P(E)Oddsan event：(1-P(E)/P(E)Joint probabilityP(AB)=P(A|B)×P(B)= P(B|A)×P(A)Addition ruleat least one of two events will occur P(A or B)=P(A)+P(B)-P(AB)BayesF

17、ormula計算，用二叉樹圖形，不用記公式80%股票上漲60%20%股票下跌好股票上漲10%差40%股票下跌90%排列組合了解Multiplication rulen1×n2××nkFactorialn！Labeling (or Multinomial)n!n1!´n2 !´K´ nk !Combinationn Cr (用計算器算)Permutationn Pr (用計算器算)注意：把非條件概率畫在第一支P( A | B) = P(B | A) * P( A)P(B)ing 10COMMON PROBABILITY DISTRIBU

18、TIONSProbability DistributionContinuousthe number of outcomes is infiniteP (x)=0 even though x can occurDiscrete Probability DistributionP(Y=1)=p，P(Y=0)=1-pExpectation=p, Variance=p(1-p)Bernoulli random variableBinomial random variablep(x) = P(X = x) = C px (1- p)n-xnxExpectation=np, Variance=np(1-p

19、)類型性質(zhì)&計算Discrete uniform例：X=1,2,3,4,5, p(x)=0.2注意：Cumulative probability function F(x)=P(X<=x)定義性質(zhì)Discretethe number of outcomes is countedmeasurable and positive probabilityContinuous Probability DistributionüüüüXN(µ , ²); 取值區(qū)間：(-, +)Symmetrical distribution: sk

20、ewness=0; kurtosis=3A linear combination of normally distributed is also normally distributed.The tails get thin and go to zero but extend infinitely.Propertiesm -s , m + s m -1.65s , m +1.65s m -1.96s , m +1.96s m - 2.58s , m + 2.58s 68% confidence interval is90% confidence interval is 95% confiden

21、ce interval is 99% confidence interval isNormaldistributionconfidenceintervalsZ = X - mStandard,Z分布sE(RP ) - RL /s P ;safety-first Ratioize SFR <=> MinimizeP (Rp< RL)üüüLognormaldistributionIf lnX is normal, then X is lognormal.Lognormal àthe price of asset; normal à

22、;the return of asset Right skewed; Bounded from below by zero (取值不能小于0)類型性質(zhì)&計算Uniform取值區(qū)間：(a, b)DistributionP(x1 £ X £1) /(b - a)Monte Carlo simulation & Historical simulation了解Monte Carlo simulationü based on their assumed distributions, to produce a distribution of possible

23、security values；ü It is fairly complex and will assume a parameter distribution；ü It is not an analytic method but a statistical one.Historical simulationü Selected historical data to generate a distribution;ü the past can not indicate the future;ü historical simulation cann

24、ot address the sort of “what if ” questionsthat Monte Carlo simulation can.ing 11,12難點SAMPLING AND ESTIMATION, HYPOTHESIS TESTINGFramework1.Sampling2.EstimationParameter( , )Statistics( X , Sx)3. Hypothesis testSamplePopulation1. Sampling抽樣ü Simple random samplingü Stratified random sampli

25、ng: to separate the population into smaller groupsDataü Time-series data: data taken over a period of timeü Cross-sectional data: data taken at a single point of timeSample statistic特點ü sampling error of the mean= sample mean- population meanü The sample statistic itself is a ran

26、dom variableü Central Limit Theory：n>=30 à sample mean N(µ , ²/n)ü Standard error =s /n or s /n2. EstimationDesirablepropertiesü Unbiasedness: expected value of the estimator is equal to parameter that estimateü Efficiency: dispersion is smallerü Consistenc

27、y: the accuracy increases as n increases.Estimation1. Point estimate:ss2. Confidence interval estimate x ± za 2or x ± ta 2nn選擇哪一個分布？ü 方差已知用z，方差未知用t，非正態(tài)總體小樣本不可估計；ü 如果n>=30，都可以用z.T分布ü Symmetricalü Degrees ofdom (df): n-1ü Less peaked than a normal distribution (“f

28、atter tails”)ü As the degrees ofdom gets larger, the shape of t-distribution approaches standard normal distribution估計的bias1. Data-mining bias2. Sample selection bias3. Survivorship bias4. Look-ahead bias5. Time-period bias3. Hypothesis test步驟：檢驗1. 提出假設Two-tailedOne-tailedH0 is what we want to

29、reject2. 計算test statisticTest Statistic = X - m0s /n3. 畫分布找到critical value2.5%95%2.5%-1.961.9695%5%1.645Reject H0Fail to Reject H0Reject H0Fail to Reject H0Reject H04.Reject H0 if |test statistic|>critical value à* is significantly different from *Fail to reject H0 if |test statistic|<cri

30、tical value à*is not significantly different from *H0 : m ³ m0 Ha : m < m0 or, H0 : m ³ m0 Ha : m < m0H0 : m = m0Ha : m ¹ m0P valueType I error and Type II errorIncorrect DecisionType errorDo not reject HCorrect Decision01. Type I error à Type error 2. Increase the Sam

31、ple Size àType I error & Type error Reject HIncorrect DecisionCorrect Decision0Significance level =P (Type I error)Power of test = 1- P (Type error)DecisionTrue conditionH0 H0 P value < àreject H0其他檢驗Nonparametric tests了解Parametric testsspecific to population parametersNonparametric

32、 testsNonparametric tests are used:ü The assumptions that support a parametric test are not met.ü When data are ranks (ordinal measurement scale) rather than values.ü The hypothesis does not involve the parameters of the distribution, such as testing whether a variable is normally dis

33、tributed.Test typeAssumptionsH0Test-statisticMeanIndependent populations12=0tPaired comparisons test =0t = d sddVarianceNormally distributed population²= ²c 20Two independent populations1²=2²F testing 13TECHNICAL ANALYSIS了解Principles1. Prices are determined by the interaction of supply and demand.2. Only participants who actually trade affect p