# 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
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
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): -∞ &lt; Cov(X,Y) &lt; +∞.
Switzerland ü
Cov(X,Y) &gt; 0: the two variables tend to move together;
Japan ü
Cov(X,Y) &lt; 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 &lt; r &lt; +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
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
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:

1

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

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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Cfa 2018 Level 2 Corporate Finance Cfa 2018 Level 2 Derivatives Cfa 2018 Level 2 Economics Cfa 2018 Level 2 Equity Quest Bank Cfa 2018 Level 2 Ethics Quest Bank