© CFA Institute For candidate use only Not for distribution QUANTITATIVE METHODS CFAđ Program Curriculum 2022 ã LEVEL I ã VOLUME © CFA Institute For candidate use only Not for distribution © 2021, 2020, 2019, 2018, 2017, 2016, 2015, 2014, 2013, 2012, 2011, 2010, 2009, 2008, 2007, 2006 by CFA Institute All rights reserved This copyright covers material written expressly for this volume by the editor/s as well as the compilation itself It does not cover the individual selections herein that first appeared elsewhere Permission to reprint these has been obtained by CFA Institute for this edition only Further reproductions by any means, electronic or mechanical, including photocopying and recording, or by any information storage or retrieval systems, must be arranged with the individual copyright holders noted CFA®, Chartered Financial Analyst®, AIMR-PPS®, and GIPS® are just a few of the trademarks owned by CFA Institute To view a list of CFA Institute trademarks and the Guide for Use of CFA Institute Marks, please visit our website at www.cfainstitute.org This publication is designed to provide accurate and authoritative information in regard to the subject matter covered It is sold with the understanding that the publisher is not engaged in rendering legal, accounting, or other professional service If legal advice or other expert assistance is required, the services of a competent professional should be sought All trademarks, service marks, registered trademarks, and registered service marks are the property of their respective owners and are used herein for identification purposes only ISBN 978-1-950157-42-6 (paper) ISBN 978-1-950157-66-2 (ebk) 10 © CFA Institute For candidate use only Not for distribution CONTENTS How to Use the CFA Program Curriculum Background on the CBOK Organization of the Curriculum Features of the Curriculum Designing Your Personal Study Program CFA Institute Learning Ecosystem (LES) Prep Providers Feedback vii vii viii viii ix x xi xii Quantitative Methods Study Session Quantitative Methods (1) Reading The Time Value of Money Introduction Interest Rates Future Value of a Single Cash Flow (Lump Sum) Non-Annual Compounding (Future Value) Continuous Compounding, Stated and Effective Rates Stated and Effective Rates Future Value of a Series of Cash Flows, Future Value Annuities Equal Cash Flows—Ordinary Annuity Unequal Cash Flows Present Value of a Single Cash Flow (Lump Sum) Non-Annual Compounding (Present Value) Present Value of a Series of Equal Cash Flows (Annuities) and Unequal Cash Flows The Present Value of a Series of Equal Cash Flows The Present Value of a Series of Unequal Cash Flows Present Value of a Perpetuity and Present Values Indexed at Times other than t=0 Present Values Indexed at Times Other than t = 0 Solving for Interest Rates, Growth Rates, and Number of Periods Solving for Interest Rates and Growth Rates Solving for the Number of Periods Solving for Size of Annuity Payments (Combining Future Value and Present Value Annuities) Present Value and Future Value Equivalence, Additivity Principle The Cash Flow Additivity Principle Summary Practice Problems Solutions indicates an optional segment 5 13 15 16 17 17 19 20 22 23 24 28 29 30 32 32 35 36 39 41 42 44 49 ii © CFA Institute For candidate use only Not for distribution Contents Reading Organizing, Visualizing, and Describing Data Introduction Data Types Numerical versus Categorical Data Cross-Sectional versus Time-Series versus Panel Data Structured versus Unstructured Data Data Summarization Organizing Data for Quantitative Analysis Summarizing Data Using Frequency Distributions Summarizing Data Using a Contingency Table Data Visualization Histogram and Frequency Polygon Bar Chart Tree- Map Word Cloud Line Chart Scatter Plot Heat Map Guide to Selecting among Visualization Types Measures of Central Tendency The Arithmetic Mean The Median The Mode Other Concepts of Mean Quantiles Quartiles, Quintiles, Deciles, and Percentiles Quantiles in Investment Practice Measures of Dispersion The Range The Mean Absolute Deviation Sample Variance and Sample Standard Deviation Downside Deviation and Coefficient of Variation Coefficient of Variation The Shape of the Distributions The Shape of the Distributions: Kurtosis Correlation between Two Variables Properties of Correlation Limitations of Correlation Analysis Summary Practice Problems Solutions 63 64 64 65 67 68 72 72 75 81 86 86 88 91 92 93 95 99 100 103 103 107 109 110 120 120 126 126 126 127 128 131 135 136 139 142 143 146 149 154 166 Reading Probability Concepts Introduction, Probability Concepts, and Odds Ratios Probability, Expected Value, and Variance 175 176 176 indicates an optional segment Contents © CFA Institute For candidate use only Not for distribution iii Conditional and Joint Probability Expected Value (Mean), Variance, and Conditional Measures of Expected Value and Variance Expected Value, Variance, Standard Deviation, Covariances, and Correlations of Portfolio Returns Covariance Given a Joint Probability Function Bayes' Formula Bayes’ Formula Principles of Counting Summary Practice Problems Solutions 181 Study Session Quantitative Methods (2) 237 Reading Common Probability Distributions Introduction and Discrete Random Variables Discrete Random Variables Discrete and Continuous Uniform Distribution Continuous Uniform Distribution Binomial Distribution Normal Distribution The Normal Distribution Probabilities Using the Normal Distribution Standardizing a Random Variable Probabilities Using the Standard Normal Distribution Applications of the Normal Distribution Lognormal Distribution and Continuous Compounding The Lognormal Distribution Continuously Compounded Rates of Return Student’s t-, Chi-Square, and F-Distributions Student’s t-Distribution Chi-Square and F-Distribution Monte Carlo Simulation Summary Practice Problems Solutions 239 240 241 244 246 250 257 257 261 263 263 265 269 269 272 275 275 277 282 288 292 299 Reading Sampling and Estimation Introduction Sampling Methods Simple Random Sampling Stratified Random Sampling Cluster Sampling Non- Probability Sampling Sampling from Different Distributions Distribution of the Sample Mean and the Central Limit Theorem The Central Limit Theorem Standard Error of the Sample Mean 305 306 306 307 308 309 310 315 316 317 319 indicates an optional segment 192 199 205 208 208 214 220 224 230 iv © CFA Institute For candidate use only Not for distribution Contents Point Estimates of the Population Mean 322 Point Estimators 322 Confidence Intervals for the Population Mean and Selection of Sample Size 326 Selection of Sample Size 332 Resampling 334 Data Snooping Bias, Sample Selection Bias, Look-Ahead Bias, and Time- Period Bias 338 Data Snooping Bias 338 Sample Selection Bias 340 Look- Ahead Bias 342 Time- Period Bias 342 Summary 344 Practice Problems 347 Solutions 351 Reading Hypothesis Testing Introduction Why Hypothesis Testing? Implications from a Sampling Distribution The Process of Hypothesis Testing Stating the Hypotheses Two-Sided vs One-Sided Hypotheses Selecting the Appropriate Hypotheses Identify the Appropriate Test Statistic Test Statistics Identifying the Distribution of the Test Statistic Specify the Level of Significance State the Decision Rule Determining Critical Values Decision Rules and Confidence Intervals Collect the Data and Calculate the Test Statistic Make a Decision Make a Statistical Decision Make an Economic Decision Statistically Significant but Not Economically Significant? The Role of p-Values Multiple Tests and Interpreting Significance Tests Concerning a Single Mean Test Concerning Differences between Means with Independent Samples Test Concerning Differences between Means with Dependent Samples Testing Concerning Tests of Variances (Chi-Square Test) Tests of a Single Variance Test Concerning the Equality of Two Variances (F-Test) Parametric vs Nonparametric Tests Uses of Nonparametric Tests Nonparametric Inference: Summary Tests Concerning Correlation Parametric Test of a Correlation indicates an optional segment 357 358 358 359 360 361 361 362 363 363 364 364 366 367 368 369 370 370 370 370 371 374 377 381 384 388 388 391 396 397 397 398 399 Contents © CFA Institute For candidate use only Not for distribution Tests Concerning Correlation: The Spearman Rank Correlation Coefficient Test of Independence Using Contingency Table Data Summary Practice Problems Solutions Reading Introduction to Linear Regression Simple Linear Regression Estimating the Parameters of a Simple Linear Regression The Basics of Simple Linear Regression Estimating the Regression Line Interpreting the Regression Coefficients Cross-Sectional vs Time-Series Regressions Assumptions of the Simple Linear Regression Model Assumption 1: Linearity Assumption 2: Homoskedasticity Assumption 3: Independence Assumption 4: Normality Analysis of Variance Breaking down the Sum of Squares Total into Its Components Measures of Goodness of Fit ANOVA and Standard Error of Estimate in Simple Linear Regression Hypothesis Testing of Linear Regression Coefficients Hypothesis Tests of the Slope Coefficient Hypothesis Tests of the Intercept Hypothesis Tests of Slope When Independent Variable Is an Indicator Variable Test of Hypotheses: Level of Significance and p-Values Prediction Using Simple Linear Regression and Prediction Intervals Functional Forms for Simple Linear Regression The Log-Lin Model The Lin-Log Model The Log-Log Model Selecting the Correct Functional Form Summary Practice Problems Solutions v 401 404 409 412 422 431 431 434 434 435 438 440 443 443 445 447 448 450 450 451 453 455 455 459 459 461 463 467 468 469 470 472 474 477 490 Appendices 495 Glossary G-1 indicates an optional segment © CFA Institute For candidate use only Not for distribution © CFA Institute For candidate use only Not for distribution How to Use the CFA Program Curriculum Congratulations on your decision to enter the Chartered Financial Analyst (CFA®) Program This exciting and rewarding program of study reflects your desire to become a serious investment professional You are embarking on a program noted for its high ethical standards and the breadth of knowledge, skills, and abilities (competencies) it develops Your commitment should be educationally and professionally rewarding The credential you seek is respected around the world as a mark of accomplishment and dedication Each level of the program represents a distinct achievement in professional development Successful completion of the program is rewarded with membership in a prestigious global community of investment professionals CFA charterholders are dedicated to life-long learning and maintaining currency with the ever-changing dynamics of a challenging profession CFA Program enrollment represents the first step toward a career-long commitment to professional education The CFA exam measures your mastery of the core knowledge, skills, and abilities required to succeed as an investment professional These core competencies are the basis for the Candidate Body of Knowledge (CBOK™) The CBOK consists of four components: ■■ A broad outline that lists the major CFA Program topic areas (www.cfainstitute org/programs/cfa/curriculum/cbok); ■■ Topic area weights that indicate the relative exam weightings of the top-level topic areas (www.cfainstitute.org/programs/cfa/curriculum); ■■ Learning outcome statements (LOS) that advise candidates about the specific knowledge, skills, and abilities they should acquire from readings covering a topic area (LOS are provided in candidate study sessions and at the beginning of each reading); and ■■ CFA Program curriculum that candidates receive upon exam registration Therefore, the key to your success on the CFA exams is studying and understanding the CBOK The following sections provide background on the CBOK, the organization of the curriculum, features of the curriculum, and tips for designing an effective personal study program BACKGROUND ON THE CBOK CFA Program is grounded in the practice of the investment profession CFA Institute performs a continuous practice analysis with investment professionals around the world to determine the competencies that are relevant to the profession, beginning with the Global Body of Investment Knowledge (GBIK®) Regional expert panels and targeted surveys are conducted annually to verify and reinforce the continuous feedback about the GBIK The practice analysis process ultimately defines the CBOK The CBOK reflects the competencies that are generally accepted and applied by investment professionals These competencies are used in practice in a generalist context and are expected to be demonstrated by a recently qualified CFA charterholder © 2021 CFA Institute All rights reserved vii viii © CFA Institute For candidate use only Not for distribution How to Use the CFA Program Curriculum The CFA Institute staff—in conjunction with the Education Advisory Committee and Curriculum Level Advisors, who consist of practicing CFA charterholders—designs the CFA Program curriculum in order to deliver the CBOK to candidates The exams, also written by CFA charterholders, are designed to allow you to demonstrate your mastery of the CBOK as set forth in the CFA Program curriculum As you structure your personal study program, you should emphasize mastery of the CBOK and the practical application of that knowledge For more information on the practice analysis, CBOK, and development of the CFA Program curriculum, please visit www cfainstitute.org ORGANIZATION OF THE CURRICULUM The Level I CFA Program curriculum is organized into 10 topic areas Each topic area begins with a brief statement of the material and the depth of knowledge expected It is then divided into one or more study sessions These study sessions should form the basic structure of your reading and preparation Each study session includes a statement of its structure and objective and is further divided into assigned readings An outline illustrating the organization of these study sessions can be found at the front of each volume of the curriculum The readings are commissioned by CFA Institute and written by content experts, including investment professionals and university professors Each reading includes LOS and the core material to be studied, often a combination of text, exhibits, and in- text examples and questions End of Reading Questions (EORQs) followed by solutions help you understand and master the material The LOS indicate what you should be able to accomplish after studying the material The LOS, the core material, and the EORQs are dependent on each other, with the core material and EORQs providing context for understanding the scope of the LOS and enabling you to apply a principle or concept in a variety of scenarios The entire readings, including the EORQs, are the basis for all exam questions and are selected or developed specifically to teach the knowledge, skills, and abilities reflected in the CBOK You should use the LOS to guide and focus your study because each exam question is based on one or more LOS and the core material and practice problems associated with the LOS As a candidate, you are responsible for the entirety of the required material in a study session We encourage you to review the information about the LOS on our website (www cfainstitute.org/programs/cfa/curriculum/study-sessions), including the descriptions of LOS “command words” on the candidate resources page at www.cfainstitute.org FEATURES OF THE CURRICULUM End of Reading Questions/Solutions All End of Reading Questions (EORQs) as well as their solutions are part of the curriculum and are required material for the exam In addition to the in-text examples and questions, these EORQs help demonstrate practical applications and reinforce your understanding of the concepts presented Some of these EORQs are adapted from past CFA exams and/or may serve as a basis for exam questions (15) Start with the definition of variance for a three-asset portfolio and see how it decomposes into three variance terms and six covariance terms Dispensing with the derivation, the result is Equation 16: V2 R p E êô R p ER p ằ ẳ ^ E êơw1R1 w2 R2 w3 R3 E w1R1 w2 R2 w3 R3 º¼ ^ ` ` E >w1R1 w2 R2 w3 R3 w1ER1 w2 ER2 w3 ER3@ using Equatioon 13 w12 V2 R1 w1w2 Cov R1 , R2 w1w3Cov R1 , R3 w1w2Cov R1 , R2 w22 V2 R2 w2 w3Cov R2 , R3 w1w3Cov R1 , R3 w2 w3Cov R2 , R3 w32 V2 R3 (16) © CFA Institute For candidate use only Not for distribution Expected Value, Variance, Standard Deviation, Covariances, and Correlations of Portfolio Returns Noting that the order of variables in covariance does not matter, for example, Cov(R2,R1) = Cov(R1,R2), and that diagonal variance terms σ2(R1), σ2(R2), and σ2(R3) can be expressed as Cov(R1,R1), Cov(R2,R2), and Cov(R3,R3), respectively, the most 3 ... Solutions 63 64 64 65 67 68 72 72 75 81 86 86 88 91 92 93 95 99 10 0 10 3 10 3 10 7 10 9 11 0 12 0 12 0 12 6 12 6 12 6 12 7 12 8 13 1 13 5 13 6 13 9 14 2 14 3 14 6 14 9 15 4 16 6 Reading Probability Concepts Introduction,... follows: PV rs m rs m N mN FVN A $1, 000,000 6% 0.06 12 0.06 12 0.0050 12 1 12 interest periods r Ã Đ PV ? ?1 s mạ © mN 12 A $1, 000,000 1. 005 A $1, 000,000 1. 0 616 78 A $1, 0 61, 677. 81 If you had been paid 6 percent... rs m rs m N mN FVN $10 ,000 8% 0.08 0.08 0.02 4 2 interest periods r Ã Đ PV ? ?1 s mạ â mN $10 ,000 1. 02 $10 ,000 1. 1 716 59 $11 , 716 .59 At maturity, the CD will be worth $11 , 716 .59 The future value