CFA 2018 Level 2 Quantitative

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Topic weight: Study Session 1-2 Ethics & Professional Standards 10 -15% Study Session 3 Quantitative Methods 5 -10% Study Session 4 Economics 5 -10% Study Session 5-6 Financial Reporting and Analysis 15 -20% Study Session 7-8 Corporate Finance 5 -15% Study Session 9-11 Equity Investment 15 -25% Study Session 12-13 Fixed Income 10 -20% Study Session 14 Derivatives 5 -15% Study Session 15 Alternative Investments 5 -10% Study Session 16-17 Portfolio Management 5 -10% Weights: 100%
  Quantitative Methods Level 2 -- 2017
  Instructor: Feng Brief Introduction
  Brief IntroductionContent:
  Ø
Reading 9: Correlation and Regression • Reading 10: Multiple Regression and Issues in
  Regression Analysis
Reading 11: Time-Series Analysis • Reading 12: Excerpt from “Probabilistic approaches:
  scenario analysis, decision trees, and simulations”
  Brief Introduction 考纲对比: Ø
  与2016年相比,2017年的考纲没有变化。Study session 3: Quantitative Methods for Valuation
  Brief Introduction 推荐阅读: Ø
  定量投资分析 • Richard A. DeFusco, Dennis W.
  Mcleavey, Jerald E. Pinto, David E. Runkle
  Brief Introduction 学习建议: Ø 本门课程逻辑递进关系很强, 要把每个知识点学懂了再继
  续往前学 ; Ø 听课与做题相结合,但并不建议“刷题”; Ø 最重要的,认真、仔细的听课。ISBN: 978-7-111-388机械工业出版社
  幸福就是,有人爱、有事做、 有所相信、有所期待! Correlation AnalysisTasks:
  Ø Calculate and interpret a sample covariance and a
  sample correlation coefficient;
  Ø Formulate a hypothesis test of population correlation
  coefficient;
  Ø Correlation Analysis Correlation Analysis
  Scatter plotsSample covariance
  Ø Ø
  A graph that shows the relationship between the A statistical measure of the degree to which two observations for two data series in two dimensions. variables move together, and capture the linear
  relationship between tow variables.
  South Korea n Australia X - X Y - Y
    i   i  i= 1 C o v (X ,Y )= U.K. n - 1 U.S.
  Ø Ranges of Cov(X,Y): -∞ < Cov(X,Y) < +∞.
  Switzerland ü
  Cov(X,Y) > 0: the two variables tend to move together;
  Japan ü
  Cov(X,Y) < 0: the two variables tend to move in opposite direction.
  Correlation Analysis Correlation Analysis
  Sample correlation coefficientSample correlation coefficient (Cont.) r = +1 r = -1
  Ø
  A measure of the direction and extent of linear
  (perfect positive linear (perfect negative linear association between two variables. correlation) correlation)
  C o v (X ,Y ) r = X Y s s X Y
  Ø Ranges of r : -1 < r < +1.
  XY
  XY Correlation AnalysisSample correlation coefficient (Cont.) 0 < r < 1Sample correlation coefficient (Cont.) r = 0
-1 &lt; r &lt; 0
  (positive linear correlation)
  (negative linear correlation) Correlation Analysis
  (no linear correlation) Correlation AnalysisSteps of hypothesis testing (Review of Level 1)Hypothesis testing of correlation
  Ø
  t-test: df=n-2;
  Ø
  Step 1: stating the hypotheses: relation to be tested;
  Ø
  1 - r Correlation Analysis
  critical 2 r n - 2 t =
  , or t &lt; - t
  critical
  Decision rule: reject H if t &gt; + t
  ü
  Two-tailed test;
  ü
  ü
  Step 3: specifying the significance level;
  : ρ≠0;
  a
  H : ρ=0, H
  Step 2: identifying the appropriate test statistic and its probability distribution;
  Test the correlation coefficient between two variables is equal to zero.
  Ø
  Step 6: making the statistical decision;
  Ø
  Step 5: collecting the data and calculating the test statistic;
  Ø
  Step 4: stating the decision rule;
  Ø
  üAnswer:Example:
  A analyst want to test the correlation between variable X and variable Y. The sample size is 20, and he find the covariance between X and Y is 16. The standard deviation of X is 4 and the standard deviation of Y is 8. With 5% significance level, test the significance of the correlation coefficient between X and Y.
  Since 2.45 is larger than 2.101, the null hypothesis can be rejectted, and we can say the correlation coefficient between X and Y is significantly different from zero.
  Correlation Analysis
  exists when in fact there is no relation (no economic explanation).
  Ø Spurious correlation: statistically significant correlation
  Correlation Analysis
  Ø Outlier: may result in false statistical significance of linear relationship.
  Correlation Analysis
  2 0 - 2 t= 0 .5 x = 2 .4 5 1 - 0 .2 5
  Ø
  Correlation Analysis
  The critical value of two-tailed t-test with df=18 and significance level of 5% is 2.101;
  Ø
  t-statistic:
  Ø
  Sample correlation coefficient r = 16/(4×8) = 0.5;
  Ø
  : ρ≠0;
  a
  H : ρ=0, H
  ØLimitation to correlation analysisLimitation to correlation analysis (Cont.)
  Correlation Analysis Summary
  Ø ☆☆Importance: Limitation to correlation analysis (Cont.)
  ØContent:
  Ø Nonlinear relationships: two variables can have a strong
  Covariance and correlation coefficient; • nonlinear relation and still have a very low correlation.
Hypothesis testing of correlation coefficient;
  Limitation of correlation analysis. •
  ØExam tips:
  这一部分是后面学习的基础,出题点比较多,出题形式也 • 比较灵活。 Simple Linear RegressionDependent variable (Y)
  Simple Linear Regression Ø
  The variable that you are seeking to explain;
  Ø
  Also referred to as explained variable or predicted
  Tasks: variable.
  Ø Describe the assumptions underlying linear regression;
  Ø Calculate and interpret the predicted value andIndependent variable (X)
  Ø
  confidence interval for the dependent variable; The variable(s) that you are using to explain changes in the dependent variable.
  Ø Interpret regression coefficients, formulate its
  Ø
  Also referred to as explanatory variable or predicting hypothesis testing, calculate and interpret its variable.Linear regressionSimple linear regression model
  Ø
  Simple Linear RegressionAssumptions of simple linear regression modelThe regression line (the line of best fit)
  The variance of the error term is the same for all observations (homoscedasticity):
  Ø
  The error term is uncorrelated (independent) across observations: E(ε
  i
  ε
  j
  )=0 for all i ≠ j;
  Ø Ordinary least squares (OLS) regression: chooses values
  2 2 i ε
  E (ε )= σ i= 1 , .... ,n
  The expected value of the error term is 0: E(ε)=0;
  for the intercept (estimated intercept coefficient, ) and slope (estimated slope coefficient, ), to minimize the sum of squared errors (SSE).
  ü Sum of squared errors (SSE): sum of squared vertical
  distances between the observations and the regression line.
  Ø
  ˆ ˆ ˆ i 1 i Y = b + b X Simple Linear Regression
  ˆb0 ˆb1
  Ø
  Ø
  Use linear regression model to explain the dependent variable using the independent variable(s).
  th
  Simple Linear Regression
  Ø
  where: Y
  i
  = i
  th
  observation of the dependent variable, Y;
  X
  i
  = i
  observation of the independent variable, X; b = intercept; b
  The independent variable (X) is not random;
  1
  = slope coefficient; ε
  i
  = error term for the i
  th
  observation (also referred to as residual of disturbance term).
  .... i Y = b + b X + ε 1 i i i= 1 , ,n Simple Linear Regression
  Ø
  The relationship between the dependent variable (Y) and the independent variable (X) is linear;
  ØEquation of regression line:The regression linePredicted value of dependent variable
  Simple Linear Regression c t : f s :
  C o v ˆ 1 b =  ˆ 0 b
  Simple Linear Regression
  Simple Linear Regression Y : ˆ X : p
  ˆ ˆ ˆ
  The values that are predicted by the regression equation, given an estimate of the independent variable. where: predicted value of the dependent variable; forecasted value of the independent variable. p 1 Y = b + b X
  Ø
  ˆ ˆ 1 b = Y - b X
  ˆb1 Simple Linear Regression X Y 2 X
  Ø
  ü Interpretation: the value of Y when X is equal to zero.
  Calculation:
  ü
  Estimated intercept coefficient ( )
  ü Interpretation: the sensitivity of Y to a change in X.
  Calculation:
  ü
  Estimated slope coefficient ( )
The change of Y for 1-unit change of X. ØSignificance test for a regression coefficientPredicted value of dependent variable (Cont.)
  H : b
  Test statistic: df=n-2;
  Ø
  The confidence interval for a predicted value of dependent variable is: where: two-tailed critical t-value with df=n-2; standard error of the prediction. c f c f c f
  Y t s or Y t s Y Y t s        ˆ ˆ ˆ ( ) - ( ) ( )
  b - b t = s ˆ ˆ
  Rejection of null hypothesis means the regression coefficient is significantly different from the hypothesized 1 1 1 b
  Ø
  ;
  critical
  , or t &lt; - t
  critical
  Decision rule: reject H if t &gt; + t
  Ø
  Ø
  1
  ≠ 0, which means to test whether an independent variable explains the variation in the dependent variable.
  1
  : b
  a
  = 0; H
  1
  Typically, H : b
  ü
  ≠ hypothesized value;
  1
  Ø
  a
  = hypothesized value; H
  : b Simple Linear Regression 1 1 1 1 c 1 c 1 1 c b b b b t s or b t s b b t s        ˆ ˆ ˆ ˆ ˆ ˆConfidence interval for a regression coefficientImportance:Content:
  ( ) - ( ) ( ) c t : 1 b s : ˆ Ø
  Ø
  The confidence interval for a regression coefficient is: where: two-tailed critical t-value with df=n-2; standard error of the regression coefficient.
  Summary ANOVA Analysis (1)
  ☆☆☆ Ø
Underlying consumptions of linear regression; Prediction of dependent variable; Interpretation of hypothesis testing for regression coefficient.
  If the confidence interval does not include zero, the null hypothesis (H : b
  Simple Linear Regression
  =0) is rejected, and the coefficient is said to be statistically significantly different from zero.
  ü
  Can be applied to significance test for a regression coefficient.
  Ø
  ØExam tips:
常考点1:underlying consumption,概念题;
  1
常考点2:predicted value of dependent variable,计算题。Analysis of variance (ANOVA)
  Ø
  A statistical procedure for dividing the total variability of a variable into components that can be attributed to different sources.Tasks:
  ü
  Total variation = explained variation + unexplained variation
  squares (RSS) + Sum of squared errors (SSE)
  Ø Describe limitations of regression analysis.
  , and F-statistics;
  2
  Ø Calculate and interpret SEE, R
  Ø Describe and interpret ANOVA;
Total sum of squares(SST) = Regression sum ofAnalysis of variance (Cont.)Analysis of variance (Cont.)
  Simple Linear RegressionAnalysis of variance (Cont.)Analysis of variance (Cont.)
  Total n-1 SST - Simple Linear Regression
  1 RSS MSR=SSR/1 Error (unexplained) n-2 SSE MSE=SSE/(n-2)
  Squares (MS) Regression (explained)
  df Sum of Squares (SS) Mean Sum of
  MSR: mean regression sum of squares;
  ü
  ANOVA table
  Ø
  ˆ
  Ø
  
  S S E = (Y -Y )
  Ø Sum of squared errors (SSE): measures the unexplained variation in the dependent variable.
  ˆ
  R SS= (Y-Y) 
   Simple Linear Regression n 2 i-1
  in the dependent variable that is explained by the independent variable. n 2 i i- 1 S S T = (Y - Y )
  Ø Regression sum of squares (RSS): measures the variation
  Ø Total sum of squares(SST): measures the total variation in the dependent variable.
  Simple Linear Regression
  A graphic explanation of the components of total variation:
Also known as the sum of squared residuals or the residual sum of squares. i n 2 i-1 Simple Linear Regression Simple Linear Regression
  Standard error of estimate (SEE)Coefficient of determination (R²)
  Ø
  The standard deviation of error terms in the regression. Ø The percentage of the total variation that is explained by
  S S E the regression. S E E = = M S E n - 2 2 E x p la in e d v a ria tio n R S S S S T -S S E
  R = = =
  Ø
  Measures the degree of variability of the actual Y-values T o ta l v a ria tio n S S T S S T relative to the estimated Y-values from a regression
  ü
  For simple linear regression, R² is equal to the squared equation; correlation coefficient: R² = r².
  ü
  Gauges the "fit" of the regression line. The smaller the SEE, the better the fit.
  Simple Linear Regression Simple Linear Regression
  F-statisticF-statistic (Cont.)
  Ø
  An F-statistic assesses how well the independent
  Ø
  For
  simple linear regression, the F-test duplicate the t-
  variable s , as a group , explains the variation in the test for the significance of the slop coefficient. dependent variable; or used to test whether at least one
  ü
  H : b = 0; H : b ≠ 0;
  1 a
  1 independent variable explains a significant portion of the
  R SS M SR
  1 F= ü variation of the dependent variable.
  df =1; df =n-2;
  numerator denominator = SSE M SE n -2 R S S
  M S R k F =
  ü M S E = SSE Decision rule: reject H if F &gt; F . c n -k -1
  ü
  ☆☆☆ Ø
  Simple Linear Regression ØImportance:Limitations of regression analysisContent:
  Ø
  Regression relations can change over time (parameter instability).
ANOVA; SEE, R
  If the regression assumptions are violated, hypothesis tests and predictions based on linear regression will not be valid.
  Multiple Regression
  2 , and F-statistic.
  ØExam tips:
  题都可能考。 Summary
  Ø
  To investment contexts, public knowledge of regression relationships may negate their future usefulness.
  Ø
常考点1:给出ANOVA表,计算某空白格; 常考点2:R 2 的calculation and interpretation,计算题和概念Multiple regression
  ØTasks:
  observation of the j
  Multiple Regression
  Ø Formulate a multiple regression and explain the
  assumptions of a multiple regression model;
  Ø Interpret estimated regression coefficients, formulate hypothesis tests for them and interpret the results.
  Y = b + b X + b X + ...+ b X + ε 2 2 i k k i i j i 1 1 i
  independent variable X
  th
  th
  Ø Calculate and interpret the predicted value for the
  = the i
  ji
  X
  observation of the dependent variable Y
  th
  = the i
  i
  Multiple linear regression model where: Y
  ü
  Regression analysis with more than one independent variable . Multiple Regression Multiple RegressionAssumptions of multiple linear regression Assumptions of multiple linear regression (Cont.)
  Ø Ø
  The relationship between the dependent variable and The variance of the error term is the same for all 2 2 the independent variables is linear; observations (homoscedasticity, 同方差性): ; E (ε )= σ i ε
  Ø Ø
  The independent variables are not random. Also, no The error term is uncorrelated across observations: E( ε ε
  i
  exact linear relation exists between two or more of the )=0 for all i≠j;
  j Ø
  independent variables; The error term is normally distributed.
  Ø
  The expected value of the error term, conditioned on the independent variables, is 0: E(ε | X , X , …, X ) = 0;
  1 2 k Multiple Regression
  Multiple RegressionHypothesis testing of regression coefficients Intercept term (b )
  ˆ ˆ Ø  
  Hypothesis: H : b j b j H a : b j b j
  Ø
  The value of the dependent variable when the
  ˆ ˆ   H : b j b j H a : b j b j independent variables are all equal to zero.
  ˆ ˆ   H : b j b j H a : b j b j b ˆ - b j jSlope coefficient (b ) j
  Ø t 
  Test statistic:
  s b j ˆ Ø
  The expected increase in the dependent variable for a 1-
  ü
  df = n-k-1, k = number of independent variables unit increase in that independent variable, holding the
  Ø
  Decision rule: reject H if other independent variables constant.
  ü
  t &gt; + t , or t &lt; - t ;
  c c Multiple Regression Multiple RegressionStatistical significance of independent variable Interpret the testing results
  ˆ ˆ Ø H : b  H : b 
  Ø
  Hypothesis: j a j Rejection of null hypothesis means the regression
  ˆ b j
  coefficient is different from/greater than/less than the
   Ø t
  Test statistic:
  s b ˆ j hypothesized value given a level of significance ( α).
  ü
  df = n-k-1, k = number of independent variables
  Ø
  For significance testing, rejection of null hypothesis
  Ø
  Decision rule: reject H if means the regression coefficient is different from zero, or
  ü
  t &gt; + t , or t &lt; - t ;
  critical critical
  the independent variable explains some variation of the
  ü p-value &lt; significance level ( α).
  dependent variable.
  Multiple Regression Multiple RegressionConfidence interval for a regression coefficient Predicting the dependent variable
  Ø
  The confidence interval for a regression coefficient is:
  Ø
  The regression equation can be used to predict the value
  ˆ ˆ ˆ        b ( t s ˆ ) or b - ( t s ˆ ) b b ( t s ˆ ) j c b j c b j j j 1 1 c b
  of the dependent variable based on assumed values of where: the independent variables.
  t :
  two-tailed critical t-value with df=n-k-1; c
  ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ Y = b + b X + b X + ...+ b X s : ˆ i 1 1 i 2 2 i k k i
  standard error of the regression coefficient. b 1
  Ø
  Can be applied to significance test for a regression where: coefficient.
  = predicted value of the dependent variable ˆY i
  ü
 &ems

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