CFA Program — Quantitative Methods

Question 1

In a multiple regression model, a variance inflation factor (VIF) greater than 10 for an independent variable most likely indicates:

• A) Severe heteroskedasticity in the residuals
• B) Significant multicollinearity involving that independent variable
• C) The model has no explanatory power
• D) The dependent variable is misspecified

Question 2

In time-series analysis, a series that has a constant mean, constant variance, and constant autocovariance structure over time is described as:

• A) Non-stationary
• B) Covariance stationary
• C) Heteroskedastic
• D) Autoregressive of infinite order

Question 3

When residuals in a regression exhibit heteroskedasticity, which of the following is most directly affected?

• A) The regression coefficients become biased
• B) The standard errors of the coefficients become unreliable, affecting hypothesis tests, even though the coefficients themselves may remain unbiased
• C) The dependent variable becomes non-stationary
• D) R-squared becomes undefined

Question 4

A Monte Carlo simulation is best described as a technique that:

• A) Uses randomly generated inputs based on specified probability distributions to model a large number of possible outcomes
• B) Guarantees the exact future value of an investment
• C) Only applies to fixed income securities
• D) Replaces the need for any historical data analysis

Question 5

In an event study measuring abnormal stock returns around an earnings announcement, the abnormal return is generally calculated as:

• A) The stock's actual return minus its expected (normal) return based on a benchmark model
• B) The stock's dividend yield minus the risk-free rate
• C) The stock's total return divided by trading volume
• D) The market index return alone, ignoring the individual stock

Question 6

A machine learning model performs very well on training data but performs poorly on new, out-of-sample data. This is most likely an example of:

• A) Underfitting
• B) Overfitting
• C) Perfect generalization
• D) Homoskedasticity

Question 7

An analyst builds a multiple regression model to explain stock returns using several independent variables and finds a high R-squared but several individually insignificant t-statistics on the coefficients. This pattern is most likely indicative of:

• A) A guarantee that the model is free of all statistical problems.
• B) Multicollinearity among the independent variables, which can inflate standard errors and reduce individual coefficient significance even when the overall model fits well.
• C) A model with no independent variables included.
• D) Proof that none of the variables have any true relationship with returns.

Question 8

In time-series analysis, a series that exhibits a consistent upward trend over time and whose statistical properties (mean, variance) change over time is best described as:

• A) Non-stationary
• B) Stationary
• C) Homoskedastic with constant mean
• D) A pure white noise process

Question 9

Which of the following best describes "heteroskedasticity" in a regression model?

• A) A condition where all independent variables are perfectly uncorrelated.
• B) A condition that only occurs in time-series models, never cross-sectional models.
• C) A guarantee that the regression coefficients are unbiased.
• D) A condition where the variance of the regression's error terms is not constant across all observations.

Question 10

An analyst estimating a company's cost of equity using a multi-factor model, rather than the single-factor CAPM, does so primarily to:

• A) Guarantee a more accurate result in every single case.
• B) Avoid the use of any historical data.
• C) Capture additional systematic risk factors beyond just market risk (such as size or value factors) that may help explain the cross-section of expected returns.
• D) Eliminate the need to consider any risk factors whatsoever.

Question 11

Which of the following best describes a Type II error in hypothesis testing?

• A) Correctly failing to reject a true null hypothesis.
• B) Failing to reject a null hypothesis that is actually false.
• C) Rejecting a null hypothesis that is actually true.
• D) Correctly rejecting a false null hypothesis.

Question 12

An analyst runs a regression and observes a Durbin-Watson statistic significantly different from 2. This is most likely evidence of:

• A) Serial correlation (autocorrelation) among the regression's residuals.
• B) Perfect multicollinearity among independent variables.
• C) A guarantee that the model has no explanatory power.
• D) Evidence that all coefficients are statistically significant.

Question 13

An analyst calculates a 90% confidence interval and a 99% confidence interval for the same sample mean, using the same underlying data. Compared to the 90% interval, the 99% confidence interval will be:

• A) Wider, since achieving greater confidence that the true parameter falls within the interval requires a broader range.
• B) Narrower, since higher confidence always corresponds to a tighter range.
• C) Exactly the same width regardless of the confidence level chosen.
• D) Impossible to calculate using the same underlying data.

Question 14

Which of the following best describes "look-ahead bias" in backtesting an investment strategy?

• A) A bias that occurs only when using exclusively real-time, point-in-time data.
• B) Using information in a backtest that would not actually have been available to investors at the time the simulated investment decision was made, which can artificially inflate the backtested strategy's apparent performance.
• C) A bias that only affects forward-looking, not historical, analysis.
• D) A bias that has no effect on backtested investment strategy performance.

Question 15

Which of the following best describes a key advantage of using a Monte Carlo simulation approach to analyze a complex financial planning or valuation problem, compared to a single deterministic scenario analysis?

• A) Monte Carlo simulation requires no assumptions about the underlying variables.
• B) Monte Carlo simulation can only be used for equity valuation, never for other applications.
• C) Monte Carlo simulation generates a full distribution of possible outcomes by running many iterations with randomly varying inputs, providing insight into the range and likelihood of different results rather than a single point estimate.
• D) Monte Carlo simulation always produces a single, definitively correct answer.

Question 16

An analyst uses a logistic regression model to estimate the probability that a bond issuer will default within one year. This choice of model, rather than standard linear regression, is most appropriate because:

• A) Logistic regression eliminates the need for any independent variables.
• B) The dependent variable (default or no default) is binary, and logistic regression constrains predicted probabilities to fall between 0 and 1, unlike linear regression, which could produce nonsensical predicted probabilities outside this range.
• C) Logistic regression can only be used when all independent variables are also binary.
• D) Linear regression always produces more accurate binary outcome predictions than logistic regression.

Question 17

An analyst notes that a regression model's residuals are not normally distributed, exhibiting significant skewness. What is a potential consequence of this violation of the normality assumption?

• A) The regression coefficients themselves become mathematically impossible to calculate.
• B) This violation has no effect whatsoever on any aspect of the regression analysis.
• C) Non-normal residuals always indicate the regression model has zero explanatory power.
• D) Standard hypothesis tests on the regression coefficients, which often rely on an assumption of normally distributed errors (particularly in smaller samples), may become less reliable.

Question 18

Which of the following best describes the purpose of "out-of-sample" testing when evaluating a quantitative investment model?

• A) Assessing how well the model performs on data not used in its original estimation, providing a more realistic gauge of the model's likely future (out-of-sample) predictive power and helping detect overfitting.
• B) Out-of-sample testing guarantees the model will perform identically in the future.
• C) Out-of-sample testing is only relevant for models with no independent variables.
• D) Out-of-sample testing eliminates the need for any in-sample model fitting.

Question 19

In a multiple regression, adding an additional independent variable to the model will cause the adjusted R² to:

• A) Remain unchanged, since adjusted R² is calculated independently of the number of independent variables
• B) Increase only if the new variable's addition improves the model enough to offset the penalty for reduced degrees of freedom, and can decrease if the variable adds little explanatory power
• C) Always increase, just like the unadjusted R²
• D) Always decrease, because adjusted R² penalizes every additional variable regardless of its explanatory power

Question 20

In a multiple regression of a stock's monthly return on two factors, the estimated slope coefficient on the value factor is 0.85 with a standard error of 0.21. Using a t-test with a critical value of approximately 2.02, the analyst should conclude that the coefficient is:

• A) Statistically significant, because the calculated t-statistic of about 4.05 exceeds the critical value
• B) Not statistically significant, because the calculated t-statistic of about 4.05 is less than the critical value
• C) Statistically significant only at the 1% level but not at conventional levels used in practice
• D) Impossible to test without first knowing the sample's mean return

Question 21

A three-factor regression uses a sample of 40 monthly observations and produces an R² of 0.42. To test whether the regression is significant overall (all slope coefficients jointly equal zero) using an F-test, the calculated F-statistic is closest to:

• A) 0.42
• B) 4.20
• C) 14.50
• D) 8.69

Question 22

An analyst estimates a regression coefficient of 1.20 with a standard error of 0.40 and 97 degrees of freedom. Using a two-tailed critical t-value of 1.98, the 95% confidence interval for the coefficient is closest to:

• A) -0.79 to 3.19
• B) 1.16 to 1.24
• C) 0.41 to 1.99
• D) 0.80 to 1.60

Question 23

A regression of employee bonus (in $000s) on years of experience includes a dummy variable equal to 1 for employees in the trading department and 0 otherwise. The estimated coefficient on the dummy variable is 15.2, holding years of experience constant. The most appropriate interpretation is:

• A) Trading department employees are estimated to earn, on average, $15,200 more in bonus than otherwise similar non-trading employees with the same years of experience
• B) Each additional year of experience is associated with a $15,200 higher bonus for trading employees only
• C) The trading department accounts for 15.2% of total bonus variation explained by the model
• D) Trading department employees earn bonuses that are 15.2 times higher than non-trading employees

Question 24

A regression model for stock returns includes an interaction term formed by multiplying a firm-size dummy variable (1 = large-cap, 0 = small-cap) by market beta. The purpose of including this interaction term is most likely to test whether:

• A) Large-cap and small-cap firms have different average returns regardless of beta
• B) Beta and firm size are perfectly collinear and should be combined into a single variable
• C) The regression residuals are heteroskedastic across firm-size groups
• D) The effect (slope) of beta on returns differs between large-cap and small-cap firms

Question 25

An analyst runs a multiple regression of a stock's return on five macroeconomic factors. The overall F-statistic is highly significant and the R-squared is 0.89, yet none of the five individual slope coefficients has a t-statistic significant at the 5% level. This pattern is most likely evidence of:

• A) Positive serial correlation of the residuals, which by itself would not produce this specific combination of a highly significant F-statistic and uniformly insignificant individual t-statistics.
• B) Conditional heteroskedasticity, which primarily affects the reliability of standard errors but does not typically produce this specific joint-versus-individual significance pattern as its hallmark symptom.
• C) A correctly specified model with no statistical concerns, since a high R-squared alone always confirms individual coefficients are reliably estimated.
• D) Multicollinearity among the independent variables, since a strong joint explanatory relationship combined with individually insignificant coefficients is a classic symptom of high correlation among the independent variables inflating individual coefficient standard errors.

Question 26

An analyst estimates that the unconditional (prior) probability of a recession occurring in the next year is 20%. Historically, a yield curve inversion has preceded 80% of actual recessions, but has also occurred (as a false signal) ahead of 10% of years in which no recession subsequently occurred. Given that a yield curve inversion has just occurred, what is the updated (posterior) probability of a recession in the next year, using Bayes' formula?

• A) Approximately 66.7%: P(recession | inversion) = [P(inversion | recession) x P(recession)] / [P(inversion|recession)P(recession) + P(inversion|no recession)P(no recession)] = (0.80 x 0.20) / [(0.80 x 0.20) + (0.10 x 0.80)] = 0.16 / 0.24 = 0.667.
• B) Approximately 80.0%, which is simply the conditional probability of inversion given recession, restated without applying Bayes' formula to update on the new information.
• C) Approximately 20.0%, which is simply the original unconditional prior probability, unchanged and not updated to reflect the new inversion evidence.
• D) Approximately 88.9%, which results from incorrectly using only the recession-scenario probabilities in both the numerator and denominator of Bayes' formula.

Question 27

A data science team builds a machine learning model to predict corporate bond default and achieves 98% classification accuracy on the training data, but only 61% accuracy on a separate, previously unseen test data set. This large gap is most likely evidence of:

• A) Underfitting, since a 98% training accuracy figure by itself is generally considered too low to be a useful predictive model.
• B) A perfectly well-specified, robust model, since achieving very high training accuracy is always the single most important criterion in evaluating a machine learning model's quality.
• C) Conditional heteroskedasticity in the underlying feature data, a concept that applies specifically to linear time-series regression rather than to classification model generalization.
• D) Overfitting, where the model has learned patterns, including noise, specific to the training data that do not generalize well to new data; techniques such as k-fold cross-validation, regularization, or pruning (for tree-based models) are commonly used to reduce this problem.

Question 28

A regression of a company's annual revenue growth (Y, in %) on its annual advertising spending growth (X, in %) produces the equation Y = 3.2 + 0.85X, based on a sample of 42 years, with the slope coefficient's standard error equal to 0.30. Using a 5% two-tailed significance level (critical t approximately 2.02 for the relevant degrees of freedom), which conclusion is most accurate regarding the slope coefficient?

• A) The calculated t-statistic is 0.30 / 0.85 = 0.35, which is below the critical value, so the null hypothesis cannot be rejected and the relationship is not statistically significant.
• B) The calculated t-statistic is 2.83, but since it does not exceed 3.00, the coefficient must be considered statistically insignificant at any conventional significance level.
• C) The calculated t-statistic is 0.85 / 0.30 = 2.83, which exceeds the critical value of approximately 2.02, so the null hypothesis that the slope coefficient equals zero is rejected -- advertising spending growth has a statistically significant relationship with revenue growth at the 5% level.
• D) Hypothesis testing on an individual slope coefficient is not possible without also knowing the full value of R-squared for the regression.

Question 29

An analyst tests a monthly interest rate time series for a unit root using an Augmented Dickey-Fuller test and fails to reject the null hypothesis of a unit root. What does this result most directly suggest, and what is the generally appropriate next step?

• A) The series is confirmed stationary, and no further transformation of the data is needed before use in a regression model.
• B) The series is most likely nonstationary (a random walk, potentially with drift), meaning it lacks a finite, stable mean and variance and would produce spurious regression results if used in its current form; the analyst should generally first-difference the series and retest for stationarity before use in the regression.
• C) The test result indicates the series exhibits strong seasonality that must be removed using a seasonal dummy variable approach instead of differencing.
• D) Failing to reject the null in a Dickey-Fuller test has no implications for whether the series may be safely used in a linear regression.

Question 30

An analyst regresses monthly hedge fund returns on several risk factors and suspects the error terms exhibit conditional heteroskedasticity, where the variance of residuals is related to the level of one of the independent variables. If left uncorrected, what is the most likely consequence for hypothesis tests on the regression coefficients?

• A) Standard errors will typically be incorrectly estimated (often understated), causing t-statistics to be overstated and increasing the risk of incorrectly rejecting a true null hypothesis (a Type I error), even though the coefficient point estimates themselves generally remain unbiased.
• B) The coefficient point estimates themselves become biased and inconsistent, invalidating the entire regression regardless of any correction applied.
• C) Heteroskedasticity has no effect whatsoever on any aspect of regression inference and may always be safely ignored.
• D) Standard errors become systematically overstated in every case, making hypothesis tests overly conservative rather than overly liberal.