一級(jí)定量分析版

1、FRM一級(jí)培訓(xùn)項(xiàng)目Quantitative Analysis講師：么崢日期：2015.1.10 1.11地點(diǎn)： 上海北京深圳Quantitative Analysis 20%Ø Discrete and continuous probability distributions， Bayesian analysisØ Estimating the parameters of distributionsØ Population and sample statisticsØ Statistical inference and hypothesis testin

2、g¾Linear regression with single and multiple regressorsØ Time series analysis¾Correlations and copulasØ Estimating correlation and volatility using EWMA and GARCH modelsØ Volatility term structuresØ Simulation methods2-201Readings for Quantitative Analysis¾13. Mich

3、ael Miller, Mathematics and Statistics Management (Hoboken, NJ: John Wiley & SoChapter 2 -ProbabilitiesChapter 3 -Basic StatisticsChapter 4 -DistributionsFinancial Risk 013).Chapter 5 -Hypothesis Testing & Confidence Intervals¾14. James stock and Mark Watson, introductionedition (Boston

4、: Pearson Education, 2008).econometrics, Brief Chapter 4 - Linear regression with one regressor Chapter 5 - Regression with a single regressor: Hypothesis Tests and confidence intervals Chapter 6 -Linear regression with multiple regressors Chapter 7 - Hypothesis Tests and confidence intervals in mul

5、tiple regression3-201Readings for Quantitative Analysis¾15. John hull, Risk Management and Financial Institutions, 3rd Edition (Boston: Pearson Prentice hall, 2012). Chapter 11. Correlations and Copulas¾16. Francis x. Diebold, Elements of Forecasting, 4th Edition (Mason, Ohio: Cengage Lear

6、ning, 2006). Chapter 5. Modeling and Forecasting Trend (Section 5.4 onlySelecting Forecasting Models Using the Akaike andSchwarz Criteria) Chapter 7. Characterizing Cycles Chapter 8. Modeling Cycles: MA, AR, and ARMA Models¾17.John Hull, Options, Futures, and Other Derivatives, 9th Edition ork:

7、 Prentice Hall, 2014).(New YChapter 23 Estimating Volatilities and Correlations for Risk Management¾18. Dessislava Pachamanova and Frank Fabozzi, Simulation and Optimization in Finance (hoboken, NJ: John Wiley & Sons, 2010). Chapter 4. Simulation Modeling4-201¾如何學(xué)好定量分析部分？抓住基本概念搞清來(lái)龍去脈忽略

8、理論推導(dǎo)多做習(xí)題練習(xí)5-201FRM Quantitative Outline¾Probability¾Probability Distributions (Discrete & Continuous)¾Basic Statistics¾Hypothesis Testing and confidence intervals¾Linear regression with one regressor¾Regression with a single regressor: Hypothesis Tests and confidenc

9、e intervals¾Linear regression with multiple regressors¾Hypothesis Tests and confidence intervals in multiple regression¾Elements of Forecasting¾Correlations and Copulas¾Estimating Volatilities and Correlations ¾Simulation Modeling6-201Probability and Probability Distrib

10、utionsØ Phenomenon Certain phenomenon Uncertain phenomenØ Random Experiment & Random Variables: is an uncertain quantity/number.Ø Sample Space or PopulationØ Sample PointØ Random Event: is a single outcome or a set of outcomes. Mutually exclusive events: are events that

11、cannot both happen at the same time. Collectively exhaustive events: are those that include all possible outcomes7-201Probability and Probability DistributionsØ Venn DiagramsAAAB(a)(b)ABAB(c)(d)8-201Probability and Probability DistributionsAims: describe and distinguish between continuous and d

12、iscrete random variables.¾Random VariablesØ Random variables are denoted by capital letters X, Y, Z, etcØ The values taken by these variables are often denoted by small letters, x, y, z, etc. outcomeØ Discrete random variableØ Continuous random variableProbabilityØ Prob

13、ability of an Event: The Classical or A Priori DefinitionP( A) = number of outcomes favorable to A¾total number of outcomes9-201Probability and Probability DistributionsAims: calculate the probability of an event given a discrete probability function. Aims: distinguish between independent and m

14、utually exclusive events.¾Properties of Probabilities¾The probability of an event always lies between 0 and 1. Thus, the probability of event A, P(A), satisfies this relationship:0 £ P( A) £ 1If A,B,Care mutually exclusive events, the probability that any one of them will occur i

15、s equal to the sum of the probabilities of their individual occurrences.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 of their individual occurrences is 1.P( A + B + C +.) = P( A) + P(B) + P(C) + . =

16、 1¾¾10-201Probability and Probability DistributionsAims: define joint probability, describe a probability matrix, and calculate joint probabilities using probability matrices.Aims: define and calculate a conditional probability, and distinguish between conditional and unconditional probabi

17、lities.Ø Properties of ProbabilitiesØ Addition rule：P( A + B) = P( A) + P(B) - P( AB)Ø For every event A, there is an event A, called the complement of A:P( A + A ') = 1P( AA ') = 011-201Probability and Probability Distributions¾¾Unconditional probability: P(A), P(B)

18、 Conditional probability: P(A|B)Ø We want to find out the probability that the event A occurs knowing that the event B has already occurred. This probability is called the conditional probability of A given B.P( A|B) = P( AB) ;P(B) > 0P(B)Ø The conditional probability of A, given B, is

19、equal to the ratio of theirjoint probability to the maral probability of B. In like manner,¾Joint probability: P(AB)=P(A) P(B|A)=P(B) P(A|B)12-201P(B|A) =( AB) ;P( A) > 0P( A)Probability and Probability DistributionsØ Independent eventsØ The occurrence of A has no influence on the

20、occurrence of BØ B is independent of AP(AB)=P(A)P(B)Ù P(B|A) = P(B) Ù P(A|B) = P(A)Ø Three events A1, A2, A3 are independent ifÇ Ak ) = P(Aj )P(Ak ), j ¹ k.P(Ajwhere j, k=1,2,3andP( A1 Ç A2 Ç A3 ) = P( A1 )P( A2 )P( A3 )13-201Probability and Probability Distri

21、butionsØ Total Probability FormulaØ If an event A must result in one of the mutually exclusiveevents A1, A2 , A3 ,., An , thenP(A) = P(A1)P(A A1) + P(A2)P(A A2) +.+ P(An )P(A An )A1Ai Aj=F （ij）;n(1)A2AnA i=WU(2)Ai = 1314-201Probability and Probability DistributionsExampleX: Companys choice

22、 -A,B,CY: Whether company will default 0,12%default20%98%Not defaultA5%Conditional Probability:P(Y=1|X=A)=?defaultBCompanies50%95%Not defaultdefaultCUnconditional Probability:P(Y=1)=?10%30%Not default90%15-201Probability and Probability DistributionsAims: describe Bayes's Theorem and apply this

23、theorem in the calculation of conditional probabilities.Ø Bayes TheoremP( A | B) = P(B | A) ´ P( A)Prior ProbabilityP(B)Probability and Probability DistributionsExample:快速診斷儀:有病 0.1沒(méi)病 0.9現(xiàn)一個(gè)人診斷為有病，問(wèn)其真有病的概率？P(B|A)99%診斷有病10%P(A)有病1%診斷沒(méi)病P(B|A)人P(A)沒(méi)病診斷有病5%90%17-20195%診斷沒(méi)病機(jī)器說(shuō)有病機(jī)器說(shuō)沒(méi)病如果人真有病0.990

24、.01如果人真沒(méi)病0.050.95Probability and Probability DistributionsAims: define and distinguish between the probability density function, the cumulative distribution function, and the inverse cumulative distribution function, and calculate probabilities based on each of these functions.¾Random Variables

25、 and Their Probability Distributions¾Probability Distribution of a Discrete Random VariableProbability Mass Function (PMF) or Probability Function (PF)f ( X = xi ) = P( X = xi ), i = 1, 2, 3.For example: Binomial n=3 p=0.5Properties of the PMFf ( X = xi ) = 0, x ¹ xi0 £ f (xi ) £

26、 1Binomial: n=3 p=.50.4xP(x)0.301230.1250.3750.3750.1251.0000.20.1å f (x ) = 10.00123iC1x18-201P(x)Probability and Probability Distributions¾Probability Distribution of a Continuous Random VariableØ Probability density function (PDF)x2P(x1 < X < x2 ) = ò f (x)dxx1A PDF has

27、the following properties:1. The total area under the curve f(x) is 12. P(x1<X<x2)is the area under the curve between x1 and x2.P(x1 £ X £ x2 ) = P(x1 < X £ x2 ) = P(x1£ X < x2 ) = P(x1< X < x2 )3.19-201Probability and Probability DistributionsØ Cumulative D

28、istribution Function (CDF)F ( X ) = P( X £ x)F(x)1F(b)P(a £ X £ b)=F(b) - F(a)F(a)0xf(x)P(a £ X £ b) = Area underf(x) between a and b= F(b) - F(a)xa0b20-201abProbability and Probability DistributionsØ Properties of CDFF(-)=0 and F()=1, where F(-) and F() are the limits

29、of F(X) as x tends to - and ,respectively.F(X) is a non-decreasing function such that if x2>x1 then F(x2)F(x1)P(Xk)=1-F(k); that is, the probability that X assumes a value equal to or greater than k is 1 minus the probability that X takes a value below k.P(x1Xx2)=F(x2)-F(x1)21-201Probability and

30、Probability Distributions¾Multivariate probability density functionWe take X from 1 or 2 with the same probability. We take Y from 1,X with the same probability.XY12Total120.500.000.250.250.750.25Total0.500.501.00¾¾Definition: f(X,Y)=P(X=x and Y=y)Properties of the bivariate or joint

31、probability mass function (PMF)1.f(X,Y) 0 for all pairs of X and Y. This is because all probabilities are nonnegative.f(X,Y) =1.2.22-201Probability and Probability Distributions¾Statistical Independence: f(X,Y) = f(X)f(Y)Statistical Independence of two random variables X123f(Y)11/91/91/93/91/91

32、/91/93/91/91/91/93/93/93/93/9123Yf(X)¾¾EXAMPLE: FRM EXAM 2007 Q93The joint probability distribution of random variables X and Y is given by f(x,y)=kxy for x=1,2,3, y=1,2,3 and k is a positive constant, what is the probability that X+Y will exceed 5?A. 1/9B. 1/4C. 1/36D. Cannot be determine

33、d.24-201Characteristics of Probability Distributions¾Expected Value: A Measure of Central Tendency the first momentE( X ) = å x f (x) = x1P(x1 ) + x2 P(x2 ) +L+ xn P(xn )XE( X ) = ò xf (x)dxProperties of Expected Value¾1. If b is a constant, E(b)=b 2. E(X+Y)=E(X)+E(Y)3. In genera

34、l, E(XY) E(X)E(Y); If X and Y are independent random variables, then E(XY) =E(X)E(Y)4. E(X2) E(X)25. If a is a constant, E(aX)=aE(X)6. If a and b are constants, then E(aX+b)=aE(X)+E(b)=aE(X)+b25-201Characteristics of Probability DistributionsØ Expected Value of Multivariate Probability Distribu

35、tionØ In the a bivariate PMF, it can be shown thatE( XY ) = åå xyf ( XY )XY¾Example¾Continuing with our X,Y example, and applying above formula:E( XY ) = (1´1´ 0.50) + (1´ 2´ 0.00) + (2´1´ 0.25) + (2´ 2´ 0.25)=226-201Characteristics of

36、 Probability DistributionsØ Variance: a Measure of Dispersion the second momentØ The definition of varianceVAR( X ) = s x= E( X - m )22XØ The positive square root of VAR(X), s x , is known as the standard deviation.Ø To compute the variance, we use the following formula:VAR( X )

37、= å( X - m )2 P( X )ixiXiVAR( X ) = ò (x - mx )f (x)dx2VAR( X ) = E( X 2 ) -E( X )227-201Characteristics of Probability Distributions28-201Example:Given the following data, what is the best estimate of its expectation and its variance?ProbabilityValue 20%-1225%1015%1530%2510%30Characterist

38、ics of Probability DistributionsØ Properties of Variance1. The variance of a constant is zero. By definition, a constant has no variability.2. If X and Y are two independent random variables, thenVAR(X+Y)=VAR (X)+VAR (Y)andVAR (X-Y)=VAR (X)+VAR (Y)3. If b is a constant, then: VAR (X+b)=VAR (X)4

39、. If a is constant, then: VAR (aX)=a2VAR (X)5. If a and b are constant, then:6. If X and Y are independent rAR (aX+b)=a2VAR (X)om variables and a and b areconstants, then VAR (aX+bY)=a2VAR (X)+b2VAR (Y)7. For computational convenience, we can get: VAR (X)=E(X2)-E(X)2 ,E( X 2 ) = å x2 f ( X )xth

40、at29-201Characteristics of Probability DistributionsØ Coefficient of VariationCV = s x ´100mxExample: Coefficient of variationYou have just been presented with a report indicates that the mean monthly return on T-bills is 0.25% with a standard deviation of 0.36%, and the mean monthly retur

41、n for the S&P 500 is 1.09% with a standard deviation of 7.03%. Your unit manager has asked you to compute the CV for these two investments and to interpret your results.30-201Characteristics of Probability DistributionsØ Covariancecov(X, Y) = E(X - E(X)(Y - E(Y) = E(XY) - E(X)E(Y)Covariance

42、 measures how one random variable moves with another random variable.Covariance ranges from negative infinity to positive infinity.¾Properties of Covariance1. If X and Y are independent random variables, their covariance is zero.2. cov(X,Y)=cov(Y,X)3. cov(X, X) = E(X-E(X)(X-E(X) = s 2 (X)4. cov

43、(a+bX, c+dY) = bd cov( X ,Y )5. If X and Y areOT independent, then:var( X ± Y ) = var( X ) + var(Y ) ± 2 cov( X ,Y )31-201Characteristics of Probability Distributions¾¾EXAMPLE 1: FRM EXAM 2007 Q127Suppose that A and B are random variables, each follows a standard normal distribut

44、ion, and the covariance between A and B is 0.35, what is the variance of 3A+2B?A.14.47B. 17.20C. 9.20D. 15.10¾¾EXAMPLE 2: FRM EXAM 2002 Q70Given that x and y are random variables, and a, b, c and d are constant, which of the following definition is wrong?A. E(ax+by+c)=aE(x)+bE(y)+c, if x a

45、nd y are correlated.B. V(ax+by+c)=V(ax+by)+c, if x and y are correlated.C. cov(ax+by, cx+dy)= acV(x)+bdV(y) +(ad+bc)cov(x,y), if x and y are correlated.D. V(x-y)=V(x)+V(y)=V(x+y), if x and y are uncorrelated.32-201Characteristics of Probability DistributionsØ Correlation coefficient= cov(X,Y)rs

46、 sXYxy¾Properties of Correlation coefficientCorrelation measures the linear relationship between two random variables.Correlation has no units, ranges from 1 to +1.If two variables are independent, their covariance is zero, therefore, the correlation coefficient will be zero. The converse, howe

47、ver, is NOT true. For example, Y=X2Variances of correlated Variables.var( X ± Y ) = var( X ) + var(Y ) ± 2rs xs y33-201Interpretations of Correlation Coefficients34-201Correlation coefficientInterpretationr = +1perfect positive correlation0 < r < +1positive linear correlationr = 0no

48、linear correlation1 < r < 0negative linear correlationr = 1perfect negative correlationInterpretations of Correlation Coefficients¾EXAMPLE 1: Which of the following statements about correlation coefficient is false?A. It always ranges from -1 to +1B. A correlation coefficient of zero mean

49、s that two random variables are independent.C. It is a measure of linear relationship between two random variables.D. It can be calculated by scaling the covariance between two random variables.¾EXAMPLE 2: If X and Y are independent random variables,A. the covariance between the two variables i

50、s equal to zeroB. the correlation between the two variables is equal to -1C. the covariance and correlation between the two variables are both equal to 0D. the variables are perfectly correlatedE. for two (possibly dependent) random variables, X and Y, an upper bound on the covariance of X and Y is

51、135-201Characteristics of Probability DistributionsØ Chebyshevs InequalityØ 對(duì)任何一組觀測(cè)值，個(gè)體落于均值周?chē)?fù)k個(gè)標(biāo)準(zhǔn)差之內(nèi)的概率不小于1 - 1/k² ，對(duì)任意k>1。P(| X - m |£ ks ) ³ 1-1 / k 2 , k > 1122= 1 - 1 = 3 = 75%1 -244Standard deviations of the meanAt least132= 1 - 1 = 8 = 89%Lie within31 -9941421=

52、 15 = 94%1 -= 1 -161636-201Characteristics of Probability DistributionsØ Skewness-A measure of asymmetry of a PDF- the 3rd momentE ( X - m)3third moment about meanS =x=s3xcube of standard deviationNegative-SkewedSymmetricPositive-SkewedMeanMedianModeMean= Median= ModeModeMedianMean37-201Charact

53、eristics of Probability Distributions¾Kurtosis-A measure of tallness or flatness of a PDF-the 4th momentE( X - m)4s 4fourth moment about meanK =fourth power of standard deviationØ For a normal distribution the K value is 3, and such a PDF is calledmesokurticØ Excess kurtosis = kurtosi

54、s - 3Ø Leptokurtic vs. platykurtic38-201leptokurticmesokurticplatykurticKurtosis>3=3<3Excess kurtosis>0=0<0Tails(assuming same variation)Fat tailnormalThin tailCoskewness and Cokurtosis¾The third cross central moment is known as coskewness and the fourth cross central moment is

55、 known as cokurtosis.Example:Assume four series of fund returns (A B C D) where the mean, standard deviation, skew, kurtosis all the same, but only the order of returns is different.¾¾The two portfolios (A+B and C+D) havethe same mean and standarddeviation, but the skews of the portfolios

56、are different.39-201Coskewness and CokurtosisØ Scatterplots show the difference between (B versus A) and (D versus C):A and B: their best positive returns occur during the same time period, but their worst negative returns occur in different periods. This causes the distribution of points to be skewed toward the top-right of the chart.C and D: their worst negative returns occur in the same period, but their best positive returns occur in different periods. In the second chart, thepoints are skewed toward the bottom-left ofthe