CFA® Quantitative Methods
Summary, Syllabus, Topics, and Sample Questions (L1, L2, L3)
CFA Institute’s Quantitative Methods 2023 curriculum introduces analytical tools and concepts such as the Time Value of Money (TVM), and probability concepts. TVM is a foundational principle in the quantitative part of the CFA exam that cascades into many other areas of the curriculum, such as equity investments, fixed income valuation, and corporate issuers. TVM explores the relationship between time and value and how that relationship affects valuing assets or determining rates of return.
CFA Quantitative Methods has a significant role in the Level 1 exam and a slightly smaller role in the Level 2 exam. On the Level 3 exam, these foundational concepts are not tested directly. However, the material has a significant overlap with other topics in the curriculum.
What to Expect in CFA Level 1 Quantitative Methods?
CFA Level 1 Quantitative Methods curriculum covers the understanding and interpretation of a normal distribution and how it relates to quantifying risk. At 812% of the exam, Quantitative Methods is given less weight than Ethics and Financial Statement analysis but is weighted more heavily than most topics. Candidates will also become familiar with data visualization, probability distributions, sampling and estimation, hypothesis testing, and regression analysis.
Exam Weighting
The CFA Quantitative Methods topic has a weight of 812%, meaning that approximately 1521 (ca. 10%) of the 180 CFA Level 1 exam questions focus on this topic.
Topic Weight  No. of Learning Modules  No. of Formulas  No. of Questions 

812%  7  ca. 100  ca. 18 
Level 1 Quantitative Methods 2023 Syllabus, Readings, and Changes
The CFA Level 1 Quantitative Methods syllabus spans 7 learning modules and contains 81 LOS. The CFA Level 1 curriculum includes 73 total learning modules for 2023. The first seven readings (17) center on Quantitative Methods (9.6% of the total curriculum).
The weight of the Quantitative Methods section was 12% in 2018 but dropped to 10% in 2020 through 2021. Since 2022, weight has fluctuated between 812%. There are no changes in CFA Level 1 curriculum for Quantitative methods.
2018  2019  2020  2021  2022  2023 

12%  10%  10%  10%  812%  812% 
Time Value of Money
We can express one of the core concepts of the time value of money through an ageold proverb—a bird in the hand is worth two in the bush. How does this relate to finance? Money now is worth more than money later: time imposes risk, which has a cost that detracts from value. Therefore, financial analysts must understand how to factor time into valuation.
This reading covers how to calculate
 the future value of a cash flow or a series of cash flows or
 the present value of a cash flow or series of cash flows happening in the future
Candidates are introduced to terminology and concepts that will curate a sense of economic intuition for material covered in later readings.
Organizing, Visualizing, and Describing Data
Never have financial analysts had such an abundance of data and data analysis tools as they have today. The analyst must learn to separate the wheat from the chaff to take advantage of this abundance rather than be smothered by it.
 This reading provides the foundation for understanding the more sophisticated Quantitative Methods concepts confronted later in the CFA curriculum.
 Candidates will learn core data types and how to organize, summarize, and visualize them.
 Candidates will also become familiar with the tools that allow analysts to transform raw data into useful information.
Probability Concepts
Financial analysts use tools based on probability concepts that allow them to evaluate data used to make decisions consistently and logically.
 The reading will discuss fundamental probability concepts and tools and apply them to a broad range of investment problems.
 Candidates will learn about independence, expectation, and variability, and acquire tools that will help them navigate a financial environment.
Common Probability Distributions
Decisions are made about the future, which is inevitably uncertain. Analysts must understand concepts of probability. Probability distributions provide the means to assess the possible outcomes of a random variable.
 The reading introduces seven probability distributions used extensively in investment analysis (the uniform, binomial, normal, lognormal, Student’s t, chisquare, and Fdistributions).
 Candidates will also learn about the Monte Carlo simulation; a model used to estimate possible outcomes that are influenced by random variables.
Sampling and Estimation
Samples are clues to the truth about a population. Financial analysts use sample data to assess past performance and forecast future performance.
 The reading introduces the process of obtaining samples and how to employ the mean as a measure of core tendencies of random variables.
 Candidates will learn the proper applications of the Central Limit Theorem and the implications of probability distributions.
Hypothesis Testing
Analysts have to sift through an avalanche of data to assess the investment environment and develop hypotheses. To test these hypotheses, analysts will employ statistical inference, allowing them to make judgments about populations based on smaller sample sizes.
 The reading discusses the three quantities commonly used in investments (mean, variance, and correlation) via a hypothesistesting framework.
Introduction to Linear Regression
The process of determining relationships between variables is an important tool in the analyst’s toolkit. One of these tools is regression analysis.
 The reading explains the assumptions underlying the simple linear regression model and the roles of independent variables within that model.
 Candidates will also learn to formulate various hypotheses using this tool.
CFA Quantitative Methods Level 1 Sample Questions and Answers
The sample questions are typical of the probing multiplechoice questions on the L1 exam. During the exam, you have about 90 seconds to read and answer each question, carefully designed to test knowledge from the CFA Curriculum. UWorld’s question bank is built to expose you to examlike questions and illustrate and explain the concepts tested thoroughly.
An analyst determines that the probabilities of two events are as follows:
Event  Description  Probability 

A  Price of oil increases  0.33 
B  S&P 500 Index increases  0.49 
Assuming the events are independent but not mutually exclusive, the probability that at least one of the two events will occur is closest to:
 0.16
 0.66
 0.82
The probability that at least one of the two events ( A or B ) will occur is the same as the probability that:
 Only A ( one event ) will occur, or
 Only B ( one event ) will occur, or
 Both A and B ( two events ) will occur.
The addition rule for probability states that for events that are not mutually exclusive:
P ( A or B ) = P (A ) + P (B ) − P (AB )
Note that the probability that both events occur, which is possible since the events are not mutually exclusive, must be subtracted to avoid double counting that probability.
In this scenario, P (A ) and P (B ) are given, but the probability of both A and B [ P ( AB ) ] is not given. However, for independent events, P (AB ) can be calculated according to the multiplication rule for probability as P (A ) P (B ). Substituting P (A ) P (B ) for P (AB ) into the addition rule above:
P (A or B ) = P (A ) + P (B ) − P (A ) P (B )
= 0.33 + 0.49 − (0.33 × 0.49)
= 0.6583 ≈ 0.66, or 66%
(Choice A) 0.1617 is the result of multiplying P (A ) and P (B ). This is P (AB ), the probability that both A and B will occur, not the probability of A or B.
(Choice C) 0.8200 is the result of treating A and B as mutually exclusive and not subtracting the probability that both events will occur, P (AB ). This has the effect of doublecounting the probability of A and B both occurring and so overestimates the probability of A or B occurring.
Things to remember:
The addition rule for probability states that for two events that are not mutually exclusive, such as A and B in this scenario, P (A or B ) = P (A ) + P (B ) − P (AB ). For independent events, the multiplication rule for probability states that P (AB ) = P (A ) P (B ).
A researcher formulates a null hypothesis stating that a country’s average inflation is 2%. After testing a sample, the researcher fails to reject the null hypothesis when the actual average inflation is not 2%. This most likely results in a:
 Type I error.
 Type II error.
 correct decision.
Actual Situation  

Decision from statistical test  Null (H_{0}) is true  Null (H_{0}) is false 
Do not reject null (H_{0})  Correct decision  Type II error 
Reject null (H_{0}), accept alternative (H_{a})  Type I error  Correct decision 
Hypothesis testing involves a decision regarding whether to reject a null hypothesis (H_{0}) based on statistical tests. In this question, the null hypothesis states that the country’s average inflation is 2% (H_{0}: μ = 2%). Conversely, the alternative hypothesis states that the country’s average inflation is not 2% (H_{a}: μ ≠ 2%).
There are two types of errors that can result from the decision to reject or to not reject the null hypothesis:
 Type I error: Rejecting the null hypothesis when it is true
 Type II error: Failing to reject the null hypothesis when it is false
Here, the null hypothesis is false since actual average inflation is not 2%. Failing to reject the false null hypothesis in this case results in a Type II error.
(Choice A) A Type I error results from rejecting a null hypothesis when it is true. An example of a Type I error would be the analyst rejecting the null hypothesis (H_{0}: μ = 2%) when the actual average inflation is 2%.
(Choice C) Given the actual situation that the average inflation is not 2% (ie, the null hypothesis is false), the correct decision is to reject the null hypothesis (H_{0}: μ = 2%).
Things to remember:
A Type I error results from rejecting the null hypothesis when it is true. A Type II error results from failing to reject the null hypothesis when it is false.
If a cumulative distribution function of a random variable indicates that P(X < a) is equal to P(X ≤ a), then the distribution is most likely :
 normal.
 discrete.
 binomial.
The value of a cumulative distribution function (CDF) for a random variable X is the probability that the variable takes on values less than or equal to a given value a: P (X ≤ a ). The CDF’s values range from 0 to 1 since there is a 100% chance that the random variable takes on values less or equal to its greatest value and the value of the distribution is “cumulative.”
A normal distribution is a continuous distribution. There are an uncountable number of possible outcomes in a continuous distribution, so the probability of a continuous random variable being equal to any one point, P (X = a ), is zero. Then:
P (X ≤ a )  = P (X < a ) + P (X = a ) 
= P (X < a ) + 0  
= P (X < a ) 
This is illustrated in the image shown above. The CDF of a continuous distribution (red line in the left graph) is smooth.
 For this continuous distribution, P (X < a ) = 0.80, while P (X ≤ a ) = 0.80, and thus P (X < a ) = P (X ≤ a ).
However, a discrete probability distribution involves a countable number of outcomes, so its CDF resembles a stairstep pattern (blue line in the graph on the right).
 For the discrete distribution, P (X < a ) = 0.60, while P (X ≤ a ) = 0.80, and thus P (X < a ) ≠ P (X ≤ a ).
(Choice B) In a discrete distribution, P (X < a ) ≠ P (X ≤ a ).
(Choice C) The binomial distribution is a type of discrete distribution, so P (X < a ) ≠ P (X ≤ a ).
Things to remember:
There are an uncountable number of possible outcomes in a continuous distribution, so the probability of a continuous random variable being equal to any one point, P (X = a ), is zero. Thus, P (X < a ) = P (X ≤ a ) for all continuous distributions, such as the normal distribution.
What to Expect in CFA Level 2 Quantitative Methods?
CFA Level 2 Quantitative Methods builds on the material covered in Level 1 while emphasizing hypothesis testing. At 510% of the exam, Quantitative Methods is one of the less heavily weighted topics.
Candidates will become familiar with tools used to identify relationships among variables and examine fintech, machine learning, and sentiment analysis as they relate to developing an investment hypothesis.
Exam Weighting
The CFA Quantitative Methods topic has a weight of 510% of the total exam content, so that approximately 48 of the 88 CFA Level 2 exam questions, or 12 of the 22 item sets focus on this topic.
Topic Weight  No. of Learning modules  No. of Formulas  No. of Questions 

510%  7  ca. 50  ca. 48 
Level 2 Quantitative Methods 2023 Syllabus, Readings, and Changes
The CFA Level 2 Quantitative Methods syllabus spans 5 learning modules and contains 50 LOS. The CFA Level 2 curriculum includes 49 total learning modules for 2023. Quantitative Methods holds 14.3% of the total curriculum. The weight of the Quantitative Methods section has consistently remained at 510% since 2018.
2018  2019  2020  2021  2022 

510%  510%  510%  510%  510% 
Readingrelated updates for 2023 include:
 Updated: Basics of Multiple Regression and Underlying Assumptions, Evaluating Regression Model Fit and Interpreting Results, Model Misspecification, Extensions of Multiple Regression
 Removed: Introduction to Linear Regression
Basics of Multiple Regression and Underlying Assumptions
Financial analysts typically work with sophisticated statistical models that involve more than one independent variable. For example, analysts may want to assess particular macroeconomic variables behind the demand for an individual company’s products or services. To make such assessments, analysts employ multiple linear regression (linear regression with more than one independent variable) to make such assessments.
 The reading introduces the core principles and models of multiple regression models and the foundational assumptions applied to and adjusted for realworld situations.
 Candidates will learn to diagnose an assumption violation and to adjust to these violations.
 The reading also dives into the role of logistic regression in machine learning for Big Data analysis.
Evaluating Regression Model Fit and Interpreting Results
The reading helps understanding:
 How well a multiple regression model explains the dependent variable by analyzing ANOVA table results and measures of goodness of fit
 The hypotheses on the significance of two or more coefficients in a multiple regression model and interpret the results of the joint hypothesis tests
 The Calculation and interpretation of a predicted value for the dependent variable, given the estimated regression model and assumed values for the independent variable
Model Misspecification
The reading helps understanding:
 How model misspecification affects the results of a regression analysis and how to avoid common forms of misspecification
 The types of heteroskedasticity and how it affects statistical inference
 Serial correlation and how it affects statistical inference
 Multicollinearity and how it affects regression analysis
Extensions of Multiple Regression
Financial analysts typically work with sophisticated statistical models that involve more than one independent variable. For example, analysts may want to assess particular macroeconomic variables behind the demand for an individual company’s products or services. To make such assessments, analysts employ multiple linear regression (linear regression with more than one independent variable) to make such assessments. Four learning modules delve into the complexities of multiple regression, including:
 The core principles and models of multiple regression models and the foundational assumptions applied to and adjusted for realworld situations.
 The diagnosis of an assumption violation and relevant adjustments.
 The role of logistic regression in machine learning for Big Data analysis.
TimeSeries Analysis
A set of these progressive observations is known as a time series, for example, a company’s quarterly sales over three years.
 The reading explores the two fundamental uses of timeseries models: understanding the past and forecasting the future of a time series.
 Candidates will learn to estimate timeseries models and how these models explain the changes in a time series over time.
Machine Learning
Since their introduction in the 1990s, machine learning techniques have become an integral tool in the tool kits of investment firms. Machine learning (ML) aids analysts in discovering new sources of value and efficiently executing trades.
 The reading provides an overview of machine learning and essential machine learning algorithms applied to investing.
 Candidates will learn about unsupervised machine learning algorithms, neural networks, deep learning nets, and how to choose an appropriate ML algorithm for the task.
Big Data Projects
Big data is an umbrella term that refers to data generated by organizations (business, financial markets, governments), individuals (credit cards, social media), sensors, and the Internet of Things. The true impact of big data on financial analysis is yet to be fully understood, but it has already become an integral part of analysts’ toolkits. Data can aid analysts in developing their hypotheses, forecasting trends in asset prices, detecting anomalies, etc.
 Candidates will learn about the concepts that allow analysts to make predictions using structured and unstructured data.
 The reading provides a realworld ‘big data’ project case study that uses sentiment analysis to assess stock movements.
CFA Quantitative Methods Level 2 Sample Questions and Answers
The sample questions here are typical of the L2 exam’s complexity and depth: formatted as item sets, with a vignette to deliver a scenario that tests the CFA L2 Curriculum. (On the actual exam, each vignette applies to four questions; we’ve thrown in a couple extra to get a bit more learning in). And be sure to review the illustrated explanations we’ve provided for each question: UWorld’s question bank is designed to expose you to examlike questions and explain the concepts tested thoroughly.
Passage
Jae Park, CFA, is a manager of a hedge fund that bases its security selection on advanced quantitative analysis. For several open job positions with the fund, Park is looking to hire people with scientific and research backgrounds. Using multiple regression, she would like to evaluate the relationship between the expected salary of the candidates based on their years of experience (EXP), number of published research papers (PRP), and amount of grant funding received in their career (GF). The results of that regression are shown below, along with sample critical values. Park wishes to test the results at a 5% significance level (α = 0.05).
Park also notices that each candidate attended one of five universities. She is considering how to add a variable for university attended to the regression model and believes dummy variables are the best way to capture this.
Finally, Park suspects that her regression in its current form may violate regression assumptions. Her concern is that her model might have an artificially large R^{2} and tstatistics that are understated.
Based on the data in Exhibit 1, the regression is most likely a good predictor of projected salary since:
 it has a high R ^{2}
 the F – statistic has a low p – value.
 most coefficients are statistically significant.
Exhibit 1 Selected Regression Output and ANOVA Data  

Coefficient  Standard Error  t – Statistic  
Intercept  94.222  11.785  7.995  
EXp  5.080  1.116  4.550  
PRP  0.820  1.873  0.438  
GF  0.212  0.136  1.552  
ANOVA Data  df 
Sum of Squares (SS) 
Mean SS  F  Significance F 
Regression (k )  3  30,430.34  10,143.40  18.643  0 
Residual (n – k – 1 )  22  11,969.66  544.08  
Total  25  42,400.00  
Observations  26  
R ^{2}  0.718  
Standard error  23.325 
Hypothesis testing verifies whether a regression is a good predictor of the dependent variable. Testing can be performed on coefficients or the overall regression.
The Fstatistic is used to test whether the overall regression (ie, the combination of all slope coefficients) is statistically significant. The null hypothesis (H_{0}) states that the model has no statistically significant coefficients, while the alternative hypothesis (H_{a}) states that at least one coefficient is significant.
The ANOVA table provides the F – statistic’s pvalue ( labeled “significance F ” ). H_{0} is rejected if the p – value is less than the level of significance (α). In this scenario, the p – value is less than α (0 < 0.05), so H_{0} is rejected and the regression is statistically significant. In other words, the regression is a good predictor of the dependent variable.
(Choice A) R ^{2} gauges how closely the data fit the regression line. Adding independent variables will increase R ^{2}, even if those variables are only slightly correlated with the dependent variable. Furthermore, R ^{2} does not measure statistical significance. Models may have multiple coefficients without statistical significance and still have a high R ^{2}. The model’s predictive power depends mainly on the statistical significance of the regression and the coefficients.
(Choice C) A t – test can determine if a slope coefficient is statistically significant. However, the model’s overall fit cannot be determined from t – tests due to possible interactions among the independent variables. Individual coefficients can be statistically significant, but the overall regression is not.
Things to remember:
The F – statistic is used to test how well the regression explains the dependent variable. The null hypothesis H_{0} states that none of the coefficients are statistically significant, and it can be rejected if the p – value of the F – statistic is less than α. Rejecting H_{0} indicates that the model is a statistically significant predictor of the dependent variable.
According to Exhibit 1, the most appropriate interpretation of the coefficients is that a higher expected salary will result from:
 more published papers and less grant funding.
 more published papers and more grant funding.
 fewer published papers and more grant funding.
Exhibit 1 Selected Regression Output and ANOVA Data  

Coefficient  Standard Error  t – Statistic  
Intercept  94.222  11.785  7.995  
EXp  5.080  1.116  4.550  
PRP  0.820  1.873  0.438  
GF  0.212  0.136  1.552  
ANOVA Data  df 
Sum of Squares (SS) 
Mean SS  F  Significance F 
Regression (k )  3  30,430.34  10,143.40  18.643  0 
Residual (n – k – 1 )  22  11,969.66  544.08  
Total  25  42,400.00  
Observations  26  
R ^{2}  0.718  
Standard error  23.325 
The predicted value of the dependent variable can be derived using the relationship expressed in the regression equation. The slope coefficients of the equation explain the change in the dependent variable, given a oneunit change to one of the independent variables.
The sign of the coefficient indicates the direction of the change in the dependent variable. If the sign of the coefficient is:
 positive, then the independent and dependent variables will move in the same direction.
 negative, then the independent and dependent variables will move in opposite directions.
In this scenario, the PRP and GF coefficients determine how changes to those variables affect the expected salary:
 The PRP coefficient is negative, so a higher PRP variable results in a lower projected salary (Choice A).
 The GF coefficient is positive, so a higher GF variable results in a higher projected salary (Choice B).
Thus, a higher salary would be expected from fewer published papers and more grant funding.
Things to remember:
The sign of a slope coefficient indicates how a change in an independent variable affects the dependent variable. If a coefficient has a positive sign, then the independent and dependent variables move in the same direction. If a coefficient has a negative sign, then the independent and dependent variables move in opposite directions.
Based on Exhibits 1 and 2, which slope coefficient is most likely to be statistically significant?
 GF
 EXP
 PRP
Exhibit 1 Selected Regression Output and ANOVA Data  

Coefficient  Standard Error  t – Statistic  
Intercept  94.222  11.785  7.995  
EXP  5.080  1.116  4.550  
PRP  0.820  1.873  0.438  
GF  0.212  0.136  1.552  
ANOVA Data  df 
Sum of Squares (SS) 
Mean SS  F  Significance F 
Regression (k )  3  30,430.34  10,143.40  18.643  0 
Residual (n – k – 1 )  22  11,969.66  544.08  
Total  25  42,400.00  
Observations  26  
R ^{2}  0.718  
Standard error  23.325 
Exhibit 2 Sample Values from t Distribution Table  

Significance – Twotailed  0.100  0.050  
Significance – Onetailed  0.050  0.025  
df  
21  1.7207  2.0796  
22  1.7171  2.0739  
23  1.7139  2.0687  
24  1.7109  2.0639  
25  1.7081  2.0595 
In a regression model, the slope coefficients describe how a change in an independent variable affects the dependent variable. A t – test is used to verify whether a coefficient meaningfully describes the relationship between an independent and dependent variable.
The test seeks to determine whether a coefficient is statistically different from zero. The null hypothesis (H_{0}) states that the coefficient equals zero, while the alternative hypothesis (H_{a}) states that the coefficient is not equal to zero. The test is also twotailed, as shown in the image above.
The test compares the t – statistic (t ) to the critical value (t_{c }), and the null is rejected if the absolute value of the t – statistic is greater than the critical value (ie, reject H_{0} if t > t_{c }). Rejection of H_{0} means that the coefficient is statistically different from zero (ie, statistically significant).
The steps for the t – test on each of the coefficients are as follows:
Steps  Calculations 

Solve for the residuals’ degrees of freedom 
df = n – k – 1 = 26 – 3 – 1 = 22 
Solve for the critical t – value  Critical value for 2tailed test, df = 22, and 5% significance = 2.0739 
Compare each t – statistic to the critical value; reject H0 if t  > t_{c} 
EXP: 4.550 > 2.0739; reject H_{0} PRP: 0.438  < 2.0739; cannot reject H_{0} GF: 1.552  < 2.0739; cannot reject H_{0} 
The null hypothesis is rejected only for EXP. Therefore, EXP is the only statistically significant coefficient; PRP and GF are not statistically significant (Choices A and C).
Things to remember:
The t – test is used to assess whether a slope coefficient is statistically significant. A t – test assumes a null hypothesis where the coefficient equals zero. If the null hypothesis is rejected, then the coefficient is not equal to zero and is therefore statistically significant.
What to Expect in CFA Level 3 Quantitative Methods?
The CFA Institute does not provide a standalone Level 3 Quantitative Methods curriculum. However, the foundational knowledge in the Level 1 and Level 2 curriculum is implicit at CFA Level 3.
Study Tips for CFA Quantitative Methods

Create a Solid Foundation
Because they will come up regularly as you move through the curriculum, Quantitative Methods subjects are fundamental knowledge you need to study closely. Candidates can expect to refer back to these readings for refreshers throughout the exam process.

Lots of Practice and Repetition
CFA Institute suggests the importance of practice, particularly the CFA quant section. UWorld’s Qbank provides over 425 individual questions and explanations (from Level 1 and Level 2) to help you gain fluency with quantitative concepts and practice. Answering as many questions as possible can significantly aid in your grasp of the concepts and the exam's testing formats.

Start your studies (early) with Quantitative Methods
The order of topics is a common question with most candidates. It is a good idea to start with Quantitative Methods, or at least learn it early in your preparation. These readings introduce fundamental concepts that make up the Level 1 syllabus and that you must grasp to do well on exam day. Additionally, as you go toward your CFA charter, this content will recur throughout the program at every level.

Prioritize the rationale over memorization of the formula
You would do well to become very familiar with a number of the equations; you'll probably use your calculator a lot when responding to questions on this subject. However, mastering this content involves more than just crunching numbers. While it is important to apply the concepts, it’ll be easier to recall how to do that if you understand the context as well. In some cases, if you really understand the concepts, you won’t have to rely on your calculator at all.

Level 2 is more conceptual than Level 1, but you’ll still have to do the math
Level 2 Quantitative Methods is more: more complex models, more variables, more formulae, but also more theory and understanding of why the math works. But you will still have to crunch the numbers.

Put the time in understanding and practicing your itemset/vignette exam technique
Quant is a great place to misread and misuse numbers, so be sure you have a plan for reading through the vignette and then doubling back to find the necessary data—and to ignore the unnecessary data—as you answer specific questions. Always be aware of what the vignette does and doesn’t give you.

Be sure to master calculator functions
During the exam, you don’t want to be wishing that you had paid more attention to learning the calculator functions. They can be a huge time saver, keeping you from fumbling and wasting time. And anyways, these are skills that will always come in handy.
For more information, visit our CFA Level 1 Study Guide and Level 2 Study Guide