2017 Financial Risk Manager (FRM) Exam Part I Quantitative Analysis

269 Pages·2017·17.442 MB·English

by  GARP

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PEARSON ALWAYS LEARNING
2017 Financial Risk
Manager (FRM®)
Exam Part I
Quantitative Analysis
Seventh Custom Edition for the
Global Association of Risk Professionals
@GARP
Global Association
of Risk Professionals
Excerpts taken from:
Introductoi n to Econometrics, Brief Edition by James H. Stock and Mark W. Watson
Financial Risk Manager (FRM) Psff I: Anslysis, Seventh Edition by Global Astisocia on of Riek Profes8ionals.
2017 Quanlilative
Copyright O 2017 by Peal"90n Education, Inc. All Rights ReseM!d. Pearson Custom Edition.
(cid:125)(cid:2771)(cid:4036)(cid:2377)(cid:4647)(cid:88)(cid:88)(cid:88)(cid:15)(cid:78)(cid:89)(cid:76)(cid:66)(cid:80)(cid:91)(cid:74)(cid:15)(cid:68)(cid:80)(cid:78)(cid:126) (cid:50)(cid:50)(cid:18)(cid:17)(cid:23)(cid:21)(cid:22)(cid:21)(cid:25)(cid:21)(cid:19) (cid:4613)(cid:4156)(cid:3636)(cid:1794)(cid:36)(cid:39)(cid:34)(cid:1)(cid:39)(cid:51)(cid:46)(cid:3206)(cid:1265)(cid:1754)(cid:3170)(cid:3434)(cid:3028)(cid:12)(cid:2179)(cid:4204)
Excerpts taken from:
Introduction Econometrics, Brief Edition
to
by lames H. Stock and Mark W. Watson
Copyright © 2008 by Pearson Education, Inc.
Published by Addison Wesley
Boston, Massachusetts 02116
Copyright© 2017, 2016, 2015, 2014, 2013, 2012, 2011 by Pearson Education, Inc.
All rights reserved.
Pearson Custom Edition.
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Grateful acknowledgment I• made the followlng murce• for permlHlon reprint materlal
to to
copyrighted or controlled by them:
Chapters 2, 3, 4, 6, and 7 by Michael Miller, reprinted from Mathematics and Statistics for Finandal Risk
Management, Second Edition (2013), by permission of John Wiley Sons, Inc.
&
Chapters 10 and 11 by John Hull, reprinted from Risk Management and Finandal Institutions, Fourth Edition,
by permission of John Wiley Sons, Inc.
&
Chapters 5, 6, 7, and 8 by Francis X. Diebold, reprinted from Elements of Forecasting, Fourth Edition (2006),
by permission of the author.
Chapter 13 "Simulation Methods," by Chris Brooks, reprinted from Introductory Econometrics for Finance,
Third Edition (2014), by permission of cam bridge University Press.
Leaming Objectives provided by the Global Association of Risk Professionals.
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ISBN 10: 1-323-57707-6
ISBN 13: 978-1-323-57707-3
Financial Risk Manager (FRM) Psff I: Anslysis, Seventh Edition by Global Astisocia on of Riek Profes8ionals.
2017 Quanlilative
Copyright O 2017 by Peal"90n Education, Inc. All Rights ReseM!d. Pearson Custom Edition.
(cid:125)(cid:2771)(cid:4036)(cid:2377)(cid:4647)(cid:88)(cid:88)(cid:88)(cid:15)(cid:78)(cid:89)(cid:76)(cid:66)(cid:80)(cid:91)(cid:74)(cid:15)(cid:68)(cid:80)(cid:78)(cid:126) (cid:50)(cid:50)(cid:18)(cid:17)(cid:23)(cid:21)(cid:22)(cid:21)(cid:25)(cid:21)(cid:19) (cid:4613)(cid:4156)(cid:3636)(cid:1794)(cid:36)(cid:39)(cid:34)(cid:1)(cid:39)(cid:51)(cid:46)(cid:3206)(cid:1265)(cid:1754)(cid:3170)(cid:3434)(cid:3028)(cid:12)(cid:2179)(cid:4204)
Variance and Standard Deviation 17
CHAPTER PROBABILITIES
1 3
Standardized Variables 18
Discrete Random Varlables 4 Covariance 19
Continuous Random Varlables 4 Correlation 19
Probability Density Functions 4
Application: Portfolio Variance
Cumulative Distribution Functions 5
and Hedging 20
Inverse Cumulative Distribution
Functions 6 Moments 21
7
Mutually Excluslve Events Skewness 21
7
Independent Events Kurtosis 23
Probablllty Matrices 8 Coskewness and Cokurtosls 24
Condltlonal Probablllty 8 Best Linear Unbiased
Estimator (BLUE) 26
CHAPTER2 STATISTICS
llAslC 11
CHAPTER DISTRIBUTIONS
I 29
Averages 12
Population and Sample Data Parametric Distributions 30
12
Discrete Random Variables
13 Uniform Distribution 30
Continuous Random Variables
13
Bernoul Ii Distribution 31
Expectations 14
Binomial Distribution 31
Ill
Financial Risk Manager (FRM) Psff I: Anslysis, Seventh Edition by Global Astisocia on of Riek Profes8ionals.
2017 Quanlilative
Copyright O 2017 by Peal"90n Education, Inc. All Rights ReseM!d. Pearson Custom Edition.
(cid:125)(cid:2771)(cid:4036)(cid:2377)(cid:4647)(cid:88)(cid:88)(cid:88)(cid:15)(cid:78)(cid:89)(cid:76)(cid:66)(cid:80)(cid:91)(cid:74)(cid:15)(cid:68)(cid:80)(cid:78)(cid:126)(cid:1)(cid:1)(cid:1)(cid:50)(cid:50)(cid:18)(cid:17)(cid:23)(cid:21)(cid:22)(cid:21)(cid:25)(cid:21)(cid:19)(cid:1)(cid:1)(cid:1)(cid:4613)(cid:4156)(cid:3636)(cid:1794)(cid:36)(cid:39)(cid:34)(cid:1)(cid:39)(cid:51)(cid:46)(cid:3206)(cid:1265)(cid:1754)(cid:3170)(cid:3434)(cid:3028)(cid:12)(cid:2179)(cid:4204)
33 59
Poisson Distribution Confidence Intervals
34 60
Normal Distribution Hypothesis Testing
Which Way to Test? 60
36
Lognormal Distribution
One Tail or Two? 6
1
36
Central Limit Theorem The Confidence Level Returns 6
1
62
Appllcatlon: Monte Chebyshev•s lnequallty
Carlo Slmulatlons
62
1: Appllcatlon: VaR
Part Creating Normal
38 Backtesting 64
Random Varlables
Subadditivity 65
39
Chi-Squared Distribution
Expected Shortfall 66
39
Student•s Distribution
t
40 CHAPTER LINEAR REGRESSION
F-Distribution 6
WITH ONE REGRESSOR
41 69
Triangular Distribution
42
Beta Distribution
70
The Linear Regression Model
42
Mixture Distributions
Estimating the Coefficients
72
of the Linear Regression Model
CHAPTER4 BAYESIAN ANALYSIS 47 The Ordinary Least
Squares Estimator 73
OLS Estimates of the Relationship
Overview 48 Between Test Scores and the
Student-Teacher Ratio 74
48
Bayes' Theorem
Why Use the OLS Estimator? 75
Bayes versus Frequentlsts 51 76
Measures of Fit
52 TheR2 76
Many-State Problems
The Standard Error of the Regression 77
Application to the Test Score Data 77
CHAPTER5 HvPoTHESIS TESTING 78
The Least Squares Assumptions
AND CoNFIDENCE
Assumption The Conditional
#1:
INTERVALS 57 Distribution of Given X, Has
u,
a Mean of Zero 78
58 Assumption #2: (X1, Y), i = 1, . . ., n
Sample Mean Revisited Are Independently and
Identically Distributed 79
59
Sample Variance Revisited
Iv
•
Contents
Financial Risk Manager (FRM) Psff I: Anslysis, Seventh Edition by Global Astisocia on of Riek Profes8ionals.
2017 Quanlilative
Copyright O 2017 by Peal"90n Education, Inc. All Rights ReseM!d. Pearson Custom Edition.
(cid:125)(cid:2771)(cid:4036)(cid:2377)(cid:4647)(cid:88)(cid:88)(cid:88)(cid:15)(cid:78)(cid:89)(cid:76)(cid:66)(cid:80)(cid:91)(cid:74)(cid:15)(cid:68)(cid:80)(cid:78)(cid:126)(cid:1)(cid:1)(cid:1)(cid:50)(cid:50)(cid:18)(cid:17)(cid:23)(cid:21)(cid:22)(cid:21)(cid:25)(cid:21)(cid:19)(cid:1)(cid:1)(cid:1)(cid:4613)(cid:4156)(cid:3636)(cid:1794)(cid:36)(cid:39)(cid:34)(cid:1)(cid:39)(cid:51)(cid:46)(cid:3206)(cid:1265)(cid:1754)(cid:3170)(cid:3434)(cid:3028)(cid:12)(cid:2179)(cid:4204)
Assumption #3: Large Outliers Mathematical Implications
Are Unlikely 80 of Homoskedasticity 95
Use of the Least Squares What Does This Mean in Practice? 96
Assumptions 81
The Theoretlcal Foundations
Sampllng Distribution of Ordinary Least Squares 97
81
of the OLS Estimators
Linear Conditionally Unbiased
The Sampling Distribution Estimators and the Gauss-Markov
of the OLS Estimators 81 Theorem 98
Regression Estimators Other
Conclusion 83
than OLS 98
Summary 83
Using the t-Statlstlc
84 In Regression When
Appendix A
99
the Sample Size Is Small
The California Test Score Data Set 84
The t-Statistic and the Student
Appendix B 84 Distribution 99
t
Derivation of the OLS Estimators 84 Use of the Student Distribution
t
in Practice 100
100
Concluslon
CHAPTER 7 REGRESSION WITH
A
SINGLE REGRESSOR 87 Summary 101
101
Appendix
Testing Hypotheses about The Gauss-Markov Conditions and
One of the Regression a Proof of the Gauss-Markov
Coefficients 88 Theorem 101
Two-Sided Hypotheses Concerning 131 88 The Gauss-Markov Conditions 101
The Sample Average Is the Efficient
One-Sided Hypotheses Concerning 131 90
Linear Estimator of E(Y) 102
Testing Hypotheses about
the Intercept (J0 91
CHAPTER 8 LINEAR REGRESSION
Confidence Intervals
for a Regression Coefficient 91 WITH MULTIPLE
REGRESSORS
Regression When X 105
93
Is a Binary Variable
Interpretation of the Regression 106
Omitted Varlable Blas
Coefficients 93
Definition of Omitted Variable Bias 106
Heteroskedasticity A Formula for Omitted Variable Bias 107
94
and Homoskedasticity Addressing Omitted Variable
What Are Heteroskedasticity Bias by Dividing the Data
and Homoskedasticity? 94 into Groups 108
Contents • v
Financial Risk Manager (FRM) Psff I: Anslysis, Seventh Edition by Global Astisocia on of Riek Profes8ionals.
2017 Quanlilative
Copyright O 2017 by Peal"90n Education, Inc. All Rights ReseM!d. Pearson Custom Edition.
(cid:125)(cid:2771)(cid:4036)(cid:2377)(cid:4647)(cid:88)(cid:88)(cid:88)(cid:15)(cid:78)(cid:89)(cid:76)(cid:66)(cid:80)(cid:91)(cid:74)(cid:15)(cid:68)(cid:80)(cid:78)(cid:126) (cid:50)(cid:50)(cid:18)(cid:17)(cid:23)(cid:21)(cid:22)(cid:21)(cid:25)(cid:21)(cid:19) (cid:4613)(cid:4156)(cid:3636)(cid:1794)(cid:36)(cid:39)(cid:34)(cid:1)(cid:39)(cid:51)(cid:46)(cid:3206)(cid:1265)(cid:1754)(cid:3170)(cid:3434)(cid:3028)(cid:12)(cid:2179)(cid:4204)
110
The Multlple Regression Model
CHAPTER
9 HYP011tESIS TESTS
The Population Regression Line 110
AND CONFIDENCE
The Population Multiple
INTERVALS IN MULTIPLE
Regression Model 110
REGRESSION 123
The OLS Estimator in Multiple
112
Regression
The OLS Estimator 112 Hypothesis Tests and Confidence
Application to Test Scores Intervals for a Single Coefficient 124
and the Student-Teacher Ratio 112
Standard Errors for the OLS
Estimators 124
Measures of Flt In Multiple
113 Hypothesis Tests for a Single
Regression
Coefficient 124
The Standard Error
Confidence Intervals
of the Regression CSER) 113
for a Single Coefficient 125
TheR2 114
Application to Test Scores
The "Adjusted R2" 114
and the Student-Teacher Ratio 125
Application to Test Scores 114
126
Tests of Joint Hypotheses
The Least Squares Assumptions
Testing Hypotheses on Two
115
In Multlple Regression
or More Coefficients 126
Assumption The Conditional The F-Statistic 128
#1:
Distribution of Given
u1 Application to Test Scores
x,,, x21 x,,, Has a Mean of Zero 115
• • • • • and the Student-Teacher Ratio 129
Assumption #2: CXv• X21, , x,,,, Y,),
• • • The Homoskedasticity-Only
1, Are i.i.d. 115
i = ... , n F-Statistic 129
Assumption #3: Large Outliers
Are Unlikely 115 Testing Single Restrictions
130
Assumption #4: No Perfect Involving Multlple Coefficients
Multicollinearity 115
Confidence Sets for Multiple
The Distribution of the OLS Coefficients 131
Estimators In Multiple
116 Model Specification
Regression
132
for Multlple Regression
117
Multlcoll lnearlty
Omitted Variable Bias
Examples of Perfect Multlcolllnearlty 117 In Multlple Regression 132
Imperfect Multicollinearity 118 Model Specification in Theory
and in Practice 133
119
Conclusion
Interpreting theR2 and the
119 AdjustedR2 in Practice 133
Summary
vi • Contents
2017 Financial Risk Manager (FRM) Psff I: Anslysis, Seventh Edition by Global Astisocia on of Riek Profes8ionals.
Quanlilative
Copyright O 2017 by Peal"90n Education, Inc. All Rights ReseM!d. Pearson Custom Edition.
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Analysls of the Test Score
CHAPTER CHARACTERIZING
134 12
Data Set
CYCLES
171
137
Concluslon
137
Summary
Covariance Stationary
172
138 Time Series
Appendix
The Bonferroni Test White Noise 175
of a Joint Hypothesis 138
178
The Lag Operator
Wold's Theorem, the General
CHAPTER MODEUNGAND
10
Linear Process, and Ratlonal
FORECASTING
178
Distributed Lags
TREND
141 Wold's Theorem 178
Theorem 179
142 The General Linear Process 179
Modellng Trend
Rational Distributed Lags 180
147
Estimating Trend Models
Estimation and Inference
148
Forecasting Trend for the Mean, Autocorrelation,
and Partlal Autocorrelatlon
Selectlng Forecasting Models
180
Functions
Using the Akalke and Schwarz
148 Sample Mean 180
Criteria
Sample Autocorrelations 180
Application: Forecasting Retail
Sample Partial Autocorrelations 182
151
Sales
Appllcatlon: Characterizing
Canadian Employment
182
CHAPTER MoDEUNGAND Dynamics
11
FORECASTING
SEASONALITY
161 CHAPTER MODELING CvcLES:
13
MA AR, AND
,
The Nature and Sources ARMA MODELS
187
162
of Seasonaility
Modeling Seasonaility 163 Moving Average (MA) Models 188
164 The MA(1) Process 188
Forecasting Seasonal Series
The MA(q) Process 191
Application: Forecasting Housing
192
164 Autoregressive (AR) Models
Starts
The AR(l) Process 192
The AR(p) Process 194
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Contents • vii
Psff
2017 Financial Risk Manager (FRM) I: Anslysis, Seventh Edition by Global Astisocia on of Riek Profes8ionals.
Quanlilative
Copyright O 2017 by Peal"90n Education, Inc. All Rights ReseM!d. Pearson Custom Edition.
(cid:125)(cid:2771)(cid:4036)(cid:2377)(cid:4647)(cid:88)(cid:88)(cid:88)(cid:15)(cid:78)(cid:89)(cid:76)(cid:66)(cid:80)(cid:91)(cid:74)(cid:15)(cid:68)(cid:80)(cid:78)(cid:126) (cid:50)(cid:50)(cid:18)(cid:17)(cid:23)(cid:21)(cid:22)(cid:21)(cid:25)(cid:21)(cid:19) (cid:4613)(cid:4156)(cid:3636)(cid:1794)(cid:36)(cid:39)(cid:34)(cid:1)(cid:39)(cid:51)(cid:46)(cid:3206)(cid:1265)(cid:1754)(cid:3170)(cid:3434)(cid:3028)(cid:12)(cid:2179)(cid:4204)
Autoregressive Moving Average
CHAPTER 15 CORRELATIONS
196
(ARMA) Models
AND COPULAS 225
Appllcatlon: Specifying and
Estimating Models for
Employment Forecasting 197 Definition of Correlation 226
Correlation vs. Dependence 226
227
Monitoring Correlatlon
CHAPTER VOLATILITY 207
14
EWMA 227
GAR CH 228
208
Definition of Volatlllty Consistency Condition
Variance Rate 208 for Covariances 228
Business Days vs. Calendar Days 208
Multlvarlate Normal
lmplled Volatllltles 209 Distributions 229
The VIX Index 209 Generating Random Samples
from Normal Distributions 229
Are Dally Percentage Changes
Factor Models 229
210
In Flnanclal Varlables Normal?
230
Copulas
212
The Power Law
Expressing the Approach
213 Algebraically 232
Monitoring Dally Volatlllty
Other Copulas 232
Weighting Schemes 214
Tail Dependence 232
The Exponentlally Weighted Multivariate Copulas 233
214
Moving Average Model A Factor Copula Model 233
215
The GARCH(1,1) Model Application to Loan Portfolios:
The Weights 216 Vasicek•s Model 234
216 Proof of Vasicek's Result 235
Choosing Between the Models
Estimating PD and 235
p
216
Maximum Likelihood Methods Alternatives to the Gaussian Copula 236
Estimating a Constant Variance 217
236
Summary
Estimating EWMA or GARCH(l,1) 217
How Good Is the Model? 220
CHAPTER 16 SIMULATION
Using GARCH(111) to Forecast
Future Volatll lty 220 METHODS 239
Volatility Term Structures 222
Impact of Volatility Changes 222
240
Motivations
223
Summary
240
Monte Carlo Simulations
viii • Contents
2017 Financial Risk Manager (FRM) Psff I: Anslysis, Seventh Edition by Global Astisocia on of Riek Profes8ionals.
Quanlilative
Copyright O 2017 by Peal"90n Education, Inc. All Rights ReseM!d. Pearson Custom Edition.
(cid:125)(cid:2771)(cid:4036)(cid:2377)(cid:4647)(cid:88)(cid:88)(cid:88)(cid:15)(cid:78)(cid:89)(cid:76)(cid:66)(cid:80)(cid:91)(cid:74)(cid:15)(cid:68)(cid:80)(cid:78)(cid:126) (cid:50)(cid:50)(cid:18)(cid:17)(cid:23)(cid:21)(cid:22)(cid:21)(cid:25)(cid:21)(cid:19) (cid:4613)(cid:4156)(cid:3636)(cid:1794)(cid:36)(cid:39)(cid:34)(cid:1)(cid:39)(cid:51)(cid:46)(cid:3206)(cid:1265)(cid:1754)(cid:3170)(cid:3434)(cid:3028)(cid:12)(cid:2179)(cid:4204)
241
Variance Reduction Techniques An Example of Monte Carlo
Antithetic Variates 241 Simulation in Econometrics:
Control Variates 242 Deriving a Set of Critical Values
247
for a Dickey-Fuller Test
Random Number Re-Usage
across Experiments 243
An Example of How to Slmulate
Bootstrapping 243 the Price of a Flnanclal Option 247
An Example of Bootstrapping in a Simulating the Price of a Financial
Regression Context 244 Option Using a Fat-Tailed Underlying
Process 248
Situations Where the Bootstrap
Will Ineffective 245 Simulating the Price an Asian Option 248
Be of
Random Number Generation 245 Appendix Table 1 251
253
Disadvantages of the Simulation Index
Approach to Econometric or
246
Financial Problem Solving
Contents • Ix
2017 Financial Risk Manager (FRM) Psff I: Anslysis, Seventh Edition by Global Astisocia on of Riek Profes8ionals.
Quanlilative
Copyright O 2017 by Peal"90n Education, Inc. All Rights ReseM!d. Pearson Custom Edition.