前导班定量资深培训师

1、FRM一级培训项目iveysis讲师：么峥资深培训师地点： 么峥么峥称：FRM，浙江大学数学学士，浙江大学金融学。教授课程：数量分析，与产品，估值与风险模型工作背景：曾就职于交通风险管理部，负责全行新资本协议的实施工作；参与各新资本协议有关项目，包含信用风险初级法改造、市场风险验证、第二支柱建设等项目；跟进三定量测算与最新动态。现就职于某制总行风险部，负责模型验证工作。：96 2 -96 2-iveysis 20%Discrete and continuous probability distributionsPo ulation and sam le sisticsSistical infe

2、rence and hypothesis testingEstimating the parameters of distributionsGraphical represenion of sistical relationshipsLinear regreswith sin le and multi le re ressorsThe ordinary least squares (OLS) methodreting and using regresHypothesis testing andcoefficients ,the t-servalsistic, and other outputH

3、eteroskedasticity and multicollinearitySimulation methodsEstimating correlation and volatility: EWMA and GARVolatility term structuress96 3 -96 3-Readings foriveysis10. Michael Miller, Mathematics and Sistics for Finanl RiskManagement (Hoboken, NJ: John Wiley & Sons, 2012).Chapter 2 -ProbabilitiesCh

4、apter 3 -Basic SisticsChapter 4 -DistributionsChapter 5 -Hypothesis Testing &ervals11. James stock and Mark Watson,roduction to econometrics, Briefedition(ton: Pearson Education, 2008).Chapter 4 - Linear regreswith one regressorsingle regressor: Hypothesis Tests andChapter 5 - RegreservalsChapter 6

5、-Linear regreswiwith multiple regressorsChapter 7 - Hypothesis Tests and regreservalsultiple96 4 -96 4-Readings foriveysis12. Dessislava Pachamanova and FrFabozzi, Simulation andOptimization in Finance (Hoboken, NJ: John Wiley & Sons, 2010)Chapter 4 Simulation Ming13.John Hull, Options, Futures, and

6、 Other Derivatives, 8th Edition ork: Prentice Hall, 2012).(New YChapter 22 Estimating Volatilities and Correlations如何学好定量分析部分？抓住基本概念搞清来龙去脉忽略理论推导多做习题练习96 5 -96 5-FRMive OutlinePreparationThe Usage of FinanBasic conceptsl ComputerNumerical Characteristics of Random VariablesProbability Distributions (

7、Discrete & Continuous)PoEstimate andervalsHypothesis TestsLinear regresMonte Carlo Methods96 6 -96 6-FRMive OutlinePreparationTheerest rateEffective annual rateSummation and differentiationConcave and convexityPV、FV、PMT、I/YNPV、IRR96 7 -96 7-Theerest rateHow much is $100 after 3 years if y=10%?FV=？20

8、13$100How much is the present value of $100 three years later if y=10%?PV3=FV3/(1+y)3PV2=FV2/(1+y)2PV1=FV1/(1+y)FV11FV22FV33096 8 -96 8-Effective Annual Rate (EAR)Simple v.s. compoundingCompounding conventions are important ideterminingffeiannual rate (EAR), in order to compare securities with diffe

9、rent compounding periods, we must convert their yields to EARTeralized formula to compute the EARr nEAR=1+ n 1n : number of compounding periods per yearr : annual rate (quoted)Annually, semiannually, quarterly, monthly, continuously compoundingkly, daily,n , EAR erContinuously compounded rate196 9 -

10、96 9-Summation and differentiationSummation NoionsionThe Summatininn X . Xn2ii1i1i1XProperties of the Summation Operatorn k nki1 kXi k Xi( Xi Yi ) XiYi(a bXi ) na b Xi96-96 10-10Summation and differentiationf(x)y x) f ( x0f ( x0 )f ( x ) l lim0 x 0f (x0 )y0 y f (yf ( x x) f ( x )00 xxy f (x )00 x0 x

11、0 x9161-96x0 x)Summation and differentiationf ( x0 x) f ( x0 )f ( x ) lim0 x x) x 0f ( x0f ( x0 ) lim(x)2x 096-96 12-12PV、FV、PMT、I/YNPV, IRRPV, PMT, FV, I/YTCt(1 y)ty)Tt 1Example1: calculate the price of a bond. Coupon rate=5%, 10 years, annually compounding. What about semi-annual compounding?Examp

12、le2: the price of a bond is 99. Coupon rate=5%, 10 years, annually compounding.Calculate the yield of the bond.NPV, IRRExample3: a project is going to earn 10, 20, 20 million dollars IRR is 10%. Calculate the NPV.Example4: a project is going to earn 10, 20, 20 million dollars invests 35 million doll

13、ars. Calculate the IRR.hree years. Supe thehree years. A company96-96 13-13FRMive OutlinePreparationThe Usage of FinanBasic conceptsl ComputerNumerical Characteristics of Random VariablesProbability Distributions (Discrete & Continuous)PoEstimate andervalsHypothesis TestsLinear regresMonte Carlo Met

14、hods96-96 14-14指定金融计算器的使用BA II Plus/Profes常见al96-96 15-15计算器基础TI BAII PLUS计算器的基本设定主要功能按键: 都印在键上。如按右上方【ON/OFF】 键，表示开机关机。次要功能按键:按【2ND】切换键之后，显示写在按键上方的次要功能。如【2ND 】【ENTER】表示调用SET功能。小数位数的设置:默认为两位小数；更改设置时,依次按【2ND】【】，表示调用 FORMAT功能，出现DEC=2.00，若要改为四位小数，输入4，再按【ENTER】，出现DEC=4.0000。时一般最好设为4位小数。这样输入金额时可以万元计，结果的小数

15、点4可以精确到元。位，小数位数设置将保持有效，不会因退出或重新开机而改变，要重新设置FORMAT才会改变。数字重新输入按【CE/C】键。96-96 16-16计算器基础特殊计算功能操作负号功能（-2： 【2】【+|-】）括号的使用1/X 功能（1/2：【2】【1/X】）ex功能（e2：【2】【2nd】【ex】）yx功能（23 ：【2】【 yx】 【3】 【=】）（21/3：【2】【 yx】 【3】 【1/X】 【=】）nCr功能（C3：【10】 【2nd】【nCr】【3】 【=】）10nPr功能（ P3：【10】 【2nd】 【nPr】【3】 【=】）10阶乘功能（5！=120 【5】【2nd

16、】【 】 ）96-96 17-17货币时间价值时间价值 / 财务计算【CPT】【N】【I/Y】【PV】:计算( Compute )供款期数( Number of Payments )利率(erest Rate )现在价值供款( Premium / Payment )将来价值( Future Value )【PMT】【FV】【2ND】【 CLR TVM】: 清除全部货币时间值(All Clear )PV 、FV、 N 、I/Y、 PMT这五个货币时间价值功能键中会存有上次运算的结果，通过【OFF】或【CE/C】键无法清除其中数据。正确的清空方法是按【2ND】调用【CLR TVM 】。在计算器中输

17、入 I/Y 时，不需要加百分号，例如： I/Y 8%，直接输入 8 【I/Y】 即可。为表述简单，凡直接书写第二功能键，即表示先按2ND，然后按其所对应的主功能键。96-96 18-18货币时间价值运算规则：PV现值、FV终值、PMT年金、I/Y利率、N期数，运用财务计算器计算货币时间价值的五大变量。只要输入任何四个变量，可以求出剩下一个变量。例题一：由现值求终值投资100元，以累积率为10%，投资期限为10年，问这项投资10年后一共可？计算器按键依次为：10【N】；10【I/Y】；0【PMT】；-100【PV】；【CPT】【FV】。计算结果为：FV=259.3742例题二：由终值求现值面值1

18、00元的零息债券，到期收益率为6%，10年到期，该债券当前的价?格应该是计算器按键依次为：10【N】；6【I/Y】；0【PMT】；100【FV】；【CPT】【PV】。计算结果为PV=-55.839596-96 19-19NP求VIRN和RPVIRR例题B公司计划以100一台新机器，这家公司希望的投资回报率为10%，未来五年内公司预计现金流如下表所示，试求净现值和内部收益率。023451-100203020202096-96 20-20年数预计现金流120230320420520现金流现金流方法一96-96 21-21按键解释显示CF 2ND CLR WORK清除CF功能中的CF0=0.0000

19、100+/-ENTER期初投入CF0=-100.0000 20 ENTER第一期现金流C01=20.0000 30 ENTER第二期现金流C02=30.0000 20 ENTER第三期现金流C03=20.0000 20 ENTER第四期现金流C04=20.0000 20 ENTER第五期现金流C05=20.0000NPV 10 ENTER折现率10%I=10.0000计算NPVNPV=-15.9198IRR CPT计算IRRIRR=3.3675现金流现金流方法二96-96 22-22按键解释显示CF 2ND CLR WORK清除CF功能中的CF0=0.0000100+/-ENTER期初投入CF

20、0=-100.000 20 ENTER第一期现金流C01=20.0000 30 ENTER第二期现金流C02=30.0000 20 ENTER第三期现金流C03=20.0000 3 ENTER现金流20将会连续出现三次F03=3.0000NPV 10 ENTER折现率10%I=10.0000 CPT计算NPVNPV=-15.9198IRR CPT计算IRRIRR=3.3675统计运算统计运算例题已知某之前五年的收益率结果为：0%，5%，10%，15%，20%。求它的均值和方差。操作步骤：DATA功能96-96 23-23按键解释显示【2nd】【7】进入DATA功能X010.0000【2nd】【

21、CE/C】-【CLR WORK】清除DATA功能中的X010.00000【ENTER】第一个收益率X01=0.0000【】【】5【ENTER】第二个收益率X02=5.0000【】【】10【ENTER】第三个收益率X03=10.0000【】【】15【ENTER】第四个收益率X04=15.0000【】【】20【ENTER】第五个收益率X05=20.0000统计运算统计运算S功能另一只密切相关，五年对应收益率为1%，4%，10%与该，13%，21%。求回归直线。96-96 24-24按键解释显示【2nd】【DATA】进入DATA功能X010.0000【2nd】【CE/C】-【CLR WORK】清除D

22、ATA功能中的X010.00000【ENTER】【】1【ENTER】【】第一个收益率X01=0.0000 Y01=1.00005【ENTER】【】4【ENTER】【】第二个收益率X02=5.0000 Y01=4.000010【ENTER】【】10【ENTER】【】第三个收益率X03=10.0000 Y01=10.000015【ENTER】【】13【ENTER】【】第四个收益率X04=15.0000 Y01=13.000020【ENTER】【】21【ENTER】【】第五个收益率X05=20.0000 Y01=21.0000【2nd】【8】-【S】LIN表示线性关系LIN【】【】【】【】各统计与回

23、归指标结果n=5.0000 计算日期计算日期计算日期间隔功能例：2012年1月12日到2012年3月15日【2nd】【1】月.日年： 【1】【.】【1212】【ENTER】【】月.日年： 【3】【.】【1512】【ENTER】【】【CPT】96-96 25-25FRMive OutlinePreparationThe Usage of FinanBasic conceptsl ComputerNumerical Characteristics of Random VariablesProbability Distributions (Discrete & Continuous)PoEstima

24、te andervalsHypothesis TestsLinear regresMonte Carlo Methods96-96 26-26FRMive OutlineBasic conceptsRandom events, results and eventsPopulation and sampleRandom variablesProbability and probability calculationProbability density function and cumulative density function96-96 27-27Random events, result

25、s and eventsPhenomenonCertain phenomenonUncertain phenomenonRandom Experiment & Random Variables: is an uncertainty/number.Sam le S ace or Po ulationSample PoRandom Event: is a singlee or a set ofes.Mutually exclusive events: are events same time.Collectively exhaustive events: are those est cannot

26、both happen at thet include allsible96-96 28-28Random events, results and eventsRandom VariablesRandom variables are denoted by capital letters X, Y, Z, etcThe values taken by these variables are often denoted by small letters,x, y, z, etc. eDiscrete random variableContinuous random variableProbabil

27、ityProbability of an Event: The Classical or A Priori DefinitionP( A) number ofes favorable to Atotal number ofes96-96 29-29Probability and probability calculationProperties of ProbabilitiesThe probability of an event alwayss betn 0 and 1. Thus, the probabilityof event A, P(A), satisfies this relati

28、onship:0 P( A) 1If A,B,Care mutually exclusive events, the probabilityt any one ofthem will occur is equal to the sum of the probabilities of their individualoccurren.P( A B C .) P( A) P(B) P(C) .If A,B,C,are mutually exclusive and collectively exhaustive set of events,the sum of the probabilities o

29、f their individual occurrenis 1.P( A B C .) P( A) P(B) P(C) . 196-96 30-30Probability and probability calculationProperties of ProbabilitiesAddition rule：P( A B) P( A) P(B) P( AB)For every event A, there is an event A, called the complement of A:P A A AA P96-96 31-31Probability and probability calcu

30、lationUnconditional probability: P(A), P(B)Conditionalrobabilit : P A BWe want to find out the probabilityt the event A occurs knowingt the event B has already occurred. This probability is called theconditionalrobabilitof Aiven B.P( A|B) P( AB) ;P(B) 0P(B)The conditional probability of A, given B,

31、is equal to the ratio of theirjoprobability to the marginal probability of B. In like manner,P(B|A) P( AB) ;P( A) 0P( A)probability P(AB)=P(A) P(B|A)=P(B) P(A|B)Jo96-96 32-32Probability and probability calculationIndependent eventsThe occurrence of A has no influence on the occurrence of BB is indep

32、endent of AP(AB)=P(A)P(B) P(B|A) = P(B) P(A|B) = P(A)Three events A1, A2, A3 are independent if Ak ) P(Aj )P(Ak ), j k.P(Ajwhere j, k=1,2,3andP( A1 A2 A3 ) P( A1 )P( A2 )P( A3 )96-96 33-33Probability density function and cumulative density functionRandom Variables and Their Probability Distributions

33、Probability Distribution of a Discrete Random VariableProbability Mass Function (PMF) or Probability Function (PF)f ( X xi ) P( X xi ), i 1, 2, 3.Properties of the PMFFor example: Binomial n=3 p=0.5, x xifBinomial: n=3 p=.50 f (x ) 1xP(x)i01230.1250.3750.3750.1251.000f (x ) 1ix96-96 34-34P(x)0.40.30

34、.20.10.00123C1Probability density function and cumulative density functionProbability Distribution of a Continuous Random VariableProbability density function (PDF)x22 ) f (x)dxx1PA PDF has the following properties:The total area under the curve f(x) is 1P(x1Xx2)is the area under the curvebetPn x1 a

35、nd x2. P P P2 )3.22296-96 35-35Probability density function and cumulative density functionCumulative Distribution Function (CDF)F ( X ) P( X x)F(x)1F bP(a X b)=F(b) - F(a)F(a)0 xf(x)P(a X b) = Area underf(x) betn a and b= F(b) - F(a)xa0b96-96 36-36abFRMive OutlinePreparationThe Usage of FinanBasic

36、conceptsl ComputerNumerical Characteristics of Random VariablesProbability Distributions (Discrete & Continuous)PoEstimate andervalsHypothesis TestsLinear regresMonte Carlo Methods96-96 37-37FRMive OutlineNumerical Characteristics of Random VariablesExpecionsVariance, standard deviationCovarianceCor

37、relation coefficientSkewnesskurtosis96-96 38-38ExpecionsExpected Value: A Measure of Central Tendency themomentE( X ) x f (x) x1P(x1 ) x2 P(x2 ) xn P(xn )XE( X ) xf (x)dxProperties of Expected Value1. If b is a constant, E(b)=b 2. E(X+Y)=E(X)+E(Y)3. In general, E(XY) E(X)E(Y); If X and Y are indepen

38、dent random variables, then E(XY) =E(X)E(Y)4. E(X2) E(X)2If a is a constant, E(aX)=aE(X)If a and b are constants, then E(aX+b)=aE(X)+E(b)=aE(X)+b96-96 39-39Variance and standard deviationVariance: a Measure of DisperThe definition of variance the second momentVAR( X ) x E( X 22X )itive square root o

39、f VAR(X), x , is known as the standardThedeviation.To compute the variance, we use the following formula:VAR( X ) ( X )2 P( X )ixiXiVAR( X ) (x x )f (x)dx2VAR( X ) E( X 2 ) E( X )296-96 40-40Variance and standard deviationProperties of VarianceThe variance of a constant is zero. By definition, a con

40、stan variability.If X and Y are two independent random variables, thens noVAR(X+Y)=VAR (X)+VAR (Y)andVAR (X-Y)=VAR (X)+VAR (Y).b is a constant, then: VAR (X+b)=VAR (X)4. If a is constant, then: VAR (aX)=a2VAR (X).a and b are constant, then: VAR (aX+b)=a2VAR (X)6. If X and Y are independent random va

41、riables and a and b areconstants, then VARaX+bY =a2VARX +b2VARY7. For compuional convenience, we can get: VAR (X)=E(X2)-E(X)2 ,E( X 2 ) x2 f ( X )xt96-96 41-41CovarianceCovariancecov(X, Y)E(X - E(X)(Y - E(Y)E(XY) - E(X)E(Y)Covariance measures how one random variable moves wirandom variable.notherCov

42、ariance ranges from negative infinity toitive infinity.Properties of CovarianceIf X and Y are independent random variables, their covariance is zero.cov(X,Y)=cov(Y,X)3. cov(X, X) E(X-E(X)(X-E(X) 2 (X)cov(a+bX, c+dY) bd cov( X ,YIf X and Y are NOT independent, then:var( X Y ) var X var Y 2 cov X Y96-

43、96 42-42Correlation coefficientCorrelation coefficient cov(X,Y) XYxyProperties of Correlation coefficientCorrelation measures the linear relationship bet variables.n two randomCorrelation has no units, ranges from 1 to +1.If two variables are independent, their covariance is zero, therefore, the cor

44、relation coefficient will be zero. The converse, however, is NOT true.For example, Y=X2Varianof correlated Variables.var( X Y ) var( X ) var(Y ) 2 x y96-96 43-43Correlation coefficient96-96 44-44Correlation coefficientreionr = +1perfectitive correlation0 r +1itive linear correlationr = 0no linear co

45、rrelation1 r 3=30=00Exs kurtosisTails(amingFat tailnormalThailsame variation)96-96 46-46FRMive OutlinePreparationThe Usage of FinanBasic conceptsl ComputerNumerical Characteristics of Random VariablesProbability Distributions (Discrete & Continuous)PoEstimate andervalsHypothesis TestsLinear regresMo

46、nte Carlo Methods96-96 47-47FRMive OutlineProbability Distributions (Discrete & Continuous)BernoulliBinomialNormal distribution96-96 48-48Some Important Probability DistributionsBernoulli random variableP(Y=1)=pP(Y=0)=1-pBinomial random variablethe probability of x suces in n trails n x px)X )x( 1 p

47、nxp(P) x Jacob Bernoulli (1654-1705)数学家Expecions and varian96-96 49-49ExpecionVarianceBernoulli random variable (Y)pp(1-p)Binomial random variable (X)npnp(1-p)Some Important Probability DistributionsThe Cumulative Binomial Probability Tablex0.050.7740.9770.9991.0001.0001.0000.10.5900.9190.9911.0001.

48、0001.0000.20.3280.7370.9420.9931.0001.0000.30.1680.5280.8370.9690.9981.0000.40.0780.3370.6830.9130.9901.0000.50.0310.1880.5000.8130.9691.0000.60.0100.0870.3170.6630.9221.0000.70.0020.0310.1630.4720.8321.0000.80.0000.0070.0580.2630.6721.0000.90.0000.0000.0090.0810.4101.0000.950.0000.0000.0010.0230.22

49、61.000012345F (x) P( X x) P(i)all i xF(x1)P(X) = F(x)Deriving Individual Probabilities from Cumulative ProbabilitiesFor example:P(3) F (3) F (2) .813 .500 .31350-95906Some Important Probability DistributionsThe Binomial Distribution- Overviewp = 0.5Binomial Probability: n=4 p=0 5p = 0.1p = 0.3Binomi

50、al Probability: n=4 p=0 0.50.4n = 0.10.0012x34012x34Binomial Probability: n=10 p=0.1Binomial Probability: n=10 p=0 3Binomial Probability: n=10 p=0.50.4n = 0.001 2 3 4 5 6 7 108x9Binomial Probability: n=20 p=0.1Binomial Probability: n=20 p=0.3

51、Binomial Probability: n=20 p=0.50 20 20 2n = 0 00 00 016817 192016817 192016817 1920 xxxBinomial distributionse more symmetric as n increases and as= . .51- -95916P(x)P(x)P(x)P(x)P(x)P(x)P(x)P(x)P(x)0.10.00 1 2 3 4 5 6 7 108 9x0.10.00 1 2 3 4 5 6 7 108 9xBinomial Probabi

52、lity: n=4 p=01234xSome Important Probability DistributionsNormal DistributionAs n increases, teistribuaches Normal Distribution.n = 6Binomial Distribution: n=6, p=.5n = 10Binomial Distribution: n=10, p=.5n = 14Binomial Distribution: n=14, p=.50 30 30 30 20 20 0 0

53、0 000 00123x45612345x671809xNormal Distribution: = 0, =1 Normal Probability Density Function:0.4103 x f (x)exp 02220.100wheree 2 .7182818 3 . and.-505x52- -95926P(x)P(x)P(x)f(x)Some Important Probability DistributionsThe Shof the Normal Distribution Density FunctionThe normal curve is symmetrical Th

54、e two halves are identicalTheoretically, the curve extends to +Theoretically, the curve extends to -The mean, median, and mode are equal.Properties: N (, 2 ) , Fully described by its two parameters and 2.XBell-shd, Symmetrical distribution swness=0; kurtosis=3.A linear combination of two (or more) i

55、ndependent normally distribution random variables is normally distributed.53- -95936Some Important Probability DistributionsTheervalsy 68% of all observations fall y 90% of all observations fall y 95% of all observations fall y 99% of all observations fallApproxima Approxima ApproximaApproximahe he

56、heheerval erval 1.65 erval 1.96 erval 2.5854- -95946Some Important Probability DistributionsThe standard normal distributionN(0,1) or ZStandardization: if XN( , ), then Z X N(0,1)How to use Z-table?How we use the standard normal distribution to compute variousprobabilities?X 75 .9Example: X N (70，9)

57、 , compute the probability ofZ 75.9 70 1.96 , then compute P(Z 1.96) 1 0.975 0.025364.12 X 75.9Question 1: compute the probability ofQuestion 2: compute the probability of 64.12 X and X 75.955- -95956Some Important Probability DistributionsThe central limit theorem (CLT)Laplace: ifis a random sample

58、 from any population (i.e.,nprobability distribution) with mean tends to be normally distributeX2Xand, the sample meanX2t X nsample size increases indefiniX xy (technically, infiniy 30) N (0,1) xn xStandard Error (se) of mean X:nHowever, the populations standard deviation is almost never known.Inste

59、ad,use the standard deviation of the sample mean. 1 X X 2S 2xin 156- -9596Some Important Probability DistributionsThe Chi-Square( 2 ) Probability DistributionnX N (0,1), 2 2 (n)2iXii1(n 1)s2 (n 1)22057- -95976Some Important Probability DistributionsThe t Distribution (Students distribution)Z ( X X )

60、 N (0,Recallt,1) ,bothandare known.2x/nXXe we only know2xSupand estimateby its (sample) estimatorX(Xi - X )2, we obtain a new variable.2Sx =n 1t= X Xt(n1)S /nxX N (0,1);Y 2 (n);X且X,Y独立，t t(n)Y n58- -95986Some Important Probability DistributionsProperties of the t DistributionSymmetricThe mean of t d