CFA Level I subject

Quantitative Methods practice questions

Quantitative Methods questions emphasize time value of money, probability, sampling, hypothesis testing, regression, and return statistics. This section currently includes 111 public practice questions across 23 topic modules, with explanations, formulas, traps, and key takeaways.

Indicative public exam weight: 6-9%. Start with a topic guide when you need focused review, or use adaptive mode for mixed practice and due reviews.

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Introduction to Big Data Techniques

11 public questions with explanations, formulas, and exam traps.

Arithmetic and Geometric Means

1 public question with explanations, formulas, and exam traps.

Bayes Formula

1 public question with explanations, formulas, and exam traps.

Confidence Intervals

1 public question with explanations, formulas, and exam traps.

Estimation and Inference

9 public questions with explanations, formulas, and exam traps.

Hypothesis Testing

7 public questions with explanations, formulas, and exam traps.

Hypothesis Testing and Regression

1 public question with explanations, formulas, and exam traps.

Money-Weighted and Time-Weighted Returns

1 public question with explanations, formulas, and exam traps.

No-Arbitrage Forward Rates

1 public question with explanations, formulas, and exam traps.

Nominal, Effective, and Continuous Compounding

1 public question with explanations, formulas, and exam traps.

Parametric and Non-Parametric Tests of Independence

9 public questions with explanations, formulas, and exam traps.

Portfolio Mathematics

9 public questions with explanations, formulas, and exam traps.

Portfolio Variance

1 public question with explanations, formulas, and exam traps.

Probability Trees and Conditional Expectations

8 public questions with explanations, formulas, and exam traps.

Rates and Returns

10 public questions with explanations, formulas, and exam traps.

Roy Safety-First

1 public question with explanations, formulas, and exam traps.

Sample Standard Deviation

1 public question with explanations, formulas, and exam traps.

Simple Linear Regression

9 public questions with explanations, formulas, and exam traps.

Simulation Methods

9 public questions with explanations, formulas, and exam traps.

Skewness, Kurtosis, and CV

1 public question with explanations, formulas, and exam traps.

Statistical Measures of Asset Returns

9 public questions with explanations, formulas, and exam traps.

Time Value of Money

1 public question with explanations, formulas, and exam traps.

Time Value of Money in Finance

9 public questions with explanations, formulas, and exam traps.

Moderate

Quantitative Methods

Introduction to Big Data Techniques

A machine learning model performs extremely well on historical training data but poorly on new out-of-sample data. The concern is most accurately described as:

Moderate

Quantitative Methods

Introduction to Big Data Techniques

A model uses satellite images of parking lots to forecast retail sales. The vendor later reveals that images are missing on cloudy days, which occur more often in some regions. The primary concern is:

Moderate

Quantitative Methods

Arithmetic and Geometric Means

A fund has annual returns of -12%, 18%, and 10%. The arithmetic and geometric mean annual returns are closest to:

Moderate

Quantitative Methods

Bayes Formula

A risk model flags 12% of borrowers as high risk. Among borrowers who later default, 70% had been flagged. The unconditional default probability is 3%. Given a high-risk flag, the default probability is closest to:

Moderate

Quantitative Methods

Confidence Intervals

A sample of 64 funds has mean active return 1.6% and sample standard deviation 4.8%. Using z=1.96, the 95% confidence interval for the population mean is closest to:

Easy

Quantitative Methods

Estimation and Inference

The population mean return is best described as a:

Easy

Quantitative Methods

Estimation and Inference

The standard error of the sample mean is calculated as:

Easy

Quantitative Methods

Estimation and Inference

Dividing a population into industry groups and randomly sampling from each group is best described as:

Moderate

Quantitative Methods

Estimation and Inference

The central limit theorem most directly supports the conclusion that, for large samples, the sample mean:

Moderate

Quantitative Methods

Estimation and Inference

A sample of 36 funds has mean excess return 6% and sample standard deviation 12%. Using 1.96 as the reliability factor, the approximate 95% confidence interval for the population mean is:

Moderate

Quantitative Methods

Estimation and Inference

A survey of investors is posted only to a brokerage firm website and completed voluntarily. The sampling method is most likely:

Moderate

Quantitative Methods

Estimation and Inference

If the sample size is quadrupled while the sample standard deviation remains unchanged, the standard error of the sample mean:

Easy

Quantitative Methods

Estimation and Inference

A jackknife procedure differs from a bootstrap procedure because the jackknife most commonly:

Very Difficult

Quantitative Methods

Estimation and Inference

An analyst estimates the mean return from 25 monthly observations and reports the sample standard deviation as the uncertainty of the estimated mean. The most likely error is:

Easy

Quantitative Methods

Hypothesis Testing

At a 5% significance level, a test produces p-value of 0.03. The appropriate decision is to:

Moderate

Quantitative Methods

Hypothesis Testing

An analyst tests whether a portfolio mean return differs from zero in either direction. The most appropriate alternative hypothesis is:

Moderate

Quantitative Methods

Hypothesis Testing

A sample mean is 5.4, the hypothesized mean is 4.8, sample standard deviation is 1.8, and sample size is 36. The test statistic is closest to:

Moderate

Quantitative Methods

Hypothesis Testing

Holding the significance level fixed, a lower Type II error probability implies:

Very Difficult

Quantitative Methods

Hypothesis Testing

An investment strategy has mean excess return that is statistically significant at the 1% level, but the estimated excess return is 0.02% per year before transaction costs. The most accurate conclusion is that the result is:

Moderate

Quantitative Methods

Hypothesis Testing

A test of H0: mean return \<= 1.0% versus H1: mean return \> 1.0% uses sample mean 1.4%, sample standard deviation 1.8%, and n = 81. The test statistic is closest to:

Very Difficult

Quantitative Methods

Hypothesis Testing

For highly non-normal paired before-and-after performance rankings in a small sample, the most appropriate test type is likely:

Moderate

Quantitative Methods

Hypothesis Testing and Regression

A one-tailed test has H0: mu \<= 0 and Ha: mu \> 0. The p-value is 0.041 and alpha is 5%. The most accurate decision is to:

Easy

Quantitative Methods

Introduction to Big Data Techniques

Text from earnings call transcripts is best described as:

Easy

Quantitative Methods

Introduction to Big Data Techniques

A model trained on labeled bond default observations is most likely an example of:

Easy

Quantitative Methods

Introduction to Big Data Techniques

Overfitting is best described as a model that:

Moderate

Quantitative Methods

Introduction to Big Data Techniques

A data scientist uses one dataset to estimate model parameters, a second to tune model complexity, and a third for final out-of-sample performance assessment. The second dataset is best described as the:

Moderate

Quantitative Methods

Introduction to Big Data Techniques

The relationship among artificial intelligence, machine learning, and Big Data is most accurately described as:

Moderate

Quantitative Methods

Introduction to Big Data Techniques

Before using alternative data in an investment model, the most appropriate first step is usually to:

Moderate

Quantitative Methods

Introduction to Big Data Techniques

A portfolio model performs extremely well on historical backtests after hundreds of variables were tried, but it performs poorly in live trading. The most likely explanation is:

Easy

Quantitative Methods

Introduction to Big Data Techniques

A natural language processing model scores sentiment in central bank speeches and uses the score as an input to bond allocation. This application is best described as using:

Easy

Quantitative Methods

Introduction to Big Data Techniques

A machine learning credit model is trained on historical lending decisions that reflected biased approval practices. The most relevant risk is:

Moderate

Quantitative Methods

Money-Weighted and Time-Weighted Returns

A client starts with 100,000. After six months the account is 90,000 and the client contributes 40,000. At year-end the account is 156,000. The time-weighted return and money-weighted return are closest to:

Moderate

Quantitative Methods

No-Arbitrage Forward Rates

The one-year spot rate is 4.0% and the two-year spot rate is 5.2%, both annual effective rates. The one-year forward rate beginning one year from today is closest to:

Moderate

Quantitative Methods

Nominal, Effective, and Continuous Compounding

A bank quotes a stated annual rate of 8.40% compounded monthly. The equivalent continuously compounded annual rate is closest to:

Easy

Quantitative Methods

Parametric and Non-Parametric Tests of Independence

A hypothesis test of whether the population correlation coefficient equals zero has the null hypothesis:

Easy

Quantitative Methods

Parametric and Non-Parametric Tests of Independence

For a contingency table with 3 rows and 4 columns, the degrees of freedom for a chi-square independence test are:

Easy

Quantitative Methods

Parametric and Non-Parametric Tests of Independence

In a chi-square test of independence, expected frequency for a cell equals:

Moderate

Quantitative Methods

Parametric and Non-Parametric Tests of Independence

A sample correlation is 0.40 based on 25 paired observations. The test statistic for H0: rho = 0 is closest to:

Moderate

Quantitative Methods

Parametric and Non-Parametric Tests of Independence

A chi-square test of independence rejects the null hypothesis. The most accurate interpretation is that the two classifications are:

Moderate

Quantitative Methods

Parametric and Non-Parametric Tests of Independence

For ordinal credit ratings and ordinal analyst recommendation categories, a nonparametric independence test is most likely appropriate because the data are:

Very Difficult

Quantitative Methods

Parametric and Non-Parametric Tests of Independence

A correlation of 0.75 between factor exposure and fund return is statistically significant. The most accurate conclusion is that:

Very Difficult

Quantitative Methods

Parametric and Non-Parametric Tests of Independence

A 2 x 2 contingency table has observed counts [30, 20; 20, 30]. Row totals and column totals are each 50, and the grand total is 100. The chi-square statistic is closest to:

Very Difficult

Quantitative Methods

Parametric and Non-Parametric Tests of Independence

A parametric correlation test is least appropriate when:

Easy

Quantitative Methods

Portfolio Mathematics

A portfolio is 30% invested in Asset A with expected return 4% and 70% in Asset B with expected return 10%. The portfolio expected return is closest to:

Easy

Quantitative Methods

Portfolio Mathematics

A negative covariance between two assets most likely indicates that their returns:

Easy

Quantitative Methods

Portfolio Mathematics

Roy's safety-first ratio is best described as:

Moderate

Quantitative Methods

Portfolio Mathematics

A two-asset portfolio has weights of 40% and 60%, standard deviations of 10% and 15%, and correlation of 0.20. The portfolio standard deviation is closest to:

Moderate

Quantitative Methods

Portfolio Mathematics

The covariance between two asset returns is 0.0030. Their standard deviations are 12% and 20%. The correlation is closest to:

Moderate

Quantitative Methods

Portfolio Mathematics

For two risky assets with unchanged individual risks and weights, diversification benefit is greatest when the correlation is:

Very Difficult

Quantitative Methods

Portfolio Mathematics

A joint probability model has three states with probabilities 30%, 40%, and 30%. Asset A returns are 10%, 4%, and -2%; Asset B returns are 12%, 3%, and -6%. The covariance of returns is closest to:

Very Difficult

Quantitative Methods

Portfolio Mathematics

Three portfolios have the following expected returns and standard deviations. The minimum acceptable return is 2%: Portfolio A, 8% and 10%; Portfolio B, 9% and 14%; Portfolio C, 7% and 8%. Roy safety-first selects:

Very Difficult

Quantitative Methods

Portfolio Mathematics

Two assets each have a standard deviation of 10%. A 50/50 portfolio of the assets has a standard deviation of 0%. The correlation between the assets is:

Moderate

Quantitative Methods

Portfolio Variance

A portfolio has 40% in Asset A and 60% in Asset B. Standard deviations are 18% and 10%, and correlation is -0.25. The portfolio standard deviation is closest to:

Easy

Quantitative Methods

Probability Trees and Conditional Expectations

An asset has returns of -10%, 5%, and 15% with probabilities 20%, 50%, and 30%. The expected return is closest to:

Easy

Quantitative Methods

Probability Trees and Conditional Expectations

If P(A and B) = 0.18 and P(B) = 0.30, P(A | B) is closest to:

Moderate

Quantitative Methods

Probability Trees and Conditional Expectations

A model assigns a 60% probability to expansion. If expansion occurs, the probability of a positive earnings surprise is 80%; if recession occurs, it is 30%. The unconditional probability of a positive surprise is closest to:

Moderate

Quantitative Methods

Probability Trees and Conditional Expectations

If the economy expands, a stock is expected to return 12%; if it contracts, it is expected to return -6%. Given a 70% probability of expansion, the expected return is closest to:

Moderate

Quantitative Methods

Probability Trees and Conditional Expectations

For returns of -10%, 5%, and 15% with probabilities 20%, 50%, and 30%, the standard deviation is closest to:

Very Difficult

Quantitative Methods

Probability Trees and Conditional Expectations

Thirty percent of analysts are skilled. A favorable signal occurs 80% of the time for a skilled analyst and 25% of the time for an unskilled analyst. Given a favorable signal, the probability the analyst is skilled is closest to:

Moderate

Quantitative Methods

Probability Trees and Conditional Expectations

A strategy loses 4 if a signal is wrong and gains 6 if the signal is correct. The signal is correct with probability 65%. The expected payoff is closest to:

Very Difficult

Quantitative Methods

Probability Trees and Conditional Expectations

A risk report gives P(Default | Downgrade) = 18% and P(Downgrade | Default) = 60%. The analyst uses 60% as the probability of default after a downgrade. The analyst most likely confused:

Easy

Quantitative Methods

Rates and Returns

An interest rate used by an investor to discount an expected cash flow is best described as a:

Easy

Quantitative Methods

Rates and Returns

For a series of annual returns, the geometric mean return is most appropriate for measuring:

Easy

Quantitative Methods

Rates and Returns

A holding period return of 6.18% is closest to a continuously compounded return of:

Moderate

Quantitative Methods

Rates and Returns

A share is purchased for 40, pays a dividend of 1 during the year, and is sold for 43 at year-end. The holding period return is closest to:

Moderate

Quantitative Methods

Rates and Returns

A fund earns -5%, 12%, and 8% over three consecutive years. Its annualized compound return is closest to:

Moderate

Quantitative Methods

Rates and Returns

An investment requires an outflow of 1,000 today and pays 500 at the end of year 1 and 700 at the end of year 2. The money-weighted rate of return is closest to:

Very Difficult

Quantitative Methods

Rates and Returns

A portfolio starts with 100. At midyear, before an external contribution of 50, the portfolio value is 110. At year-end, the portfolio value is 192. The time-weighted rate of return is closest to:

Very Difficult

Quantitative Methods

Rates and Returns

A portfolio earns a nominal return of 8.50% when inflation is 3.20%. The real return is closest to:

Very Difficult

Quantitative Methods

Rates and Returns

A fund reports a 10.00% gross return before a 1.00% management fee charged on ending assets. The net return is closest to:

Moderate

Quantitative Methods

Rates and Returns

An investor buys a fund at 42.00, receives a 0.70 distribution, and sells 195 days later at 44.10. Using a 365-day year, the annualized effective holding period return is closest to:

Moderate

Quantitative Methods

Roy Safety-First

Minimum acceptable return is 2%. Portfolios A, B, and C have expected return/standard deviation of 8%/12%, 7%/8%, and 10%/18%, respectively. Roy's safety-first criterion selects:

Moderate

Quantitative Methods

Sample Standard Deviation

A sample of monthly excess returns is 2%, -4%, 5%, and 1%. The sample standard deviation is closest to:

Easy

Quantitative Methods

Simple Linear Regression

In the regression equation Return = 1.5 + 0.8 x MarketReturn, the slope coefficient is best interpreted as:

Easy

Quantitative Methods

Simple Linear Regression

If actual Y is 12 and predicted Y is 9, the residual is:

Easy

Quantitative Methods

Simple Linear Regression

An R-squared of 0.64 in simple linear regression indicates that:

Moderate

Quantitative Methods

Simple Linear Regression

A regression slope estimate is 0.65 with standard error 0.20. The t-statistic for testing whether the slope equals zero is closest to:

Moderate

Quantitative Methods

Simple Linear Regression

A simple regression has SSE = 180 and n = 12 observations. The standard error of estimate is closest to:

Moderate

Quantitative Methods

Simple Linear Regression

A simple regression has SSR = 75, SSE = 25, and n = 12. The F-statistic for overall fit is closest to:

Very Difficult

Quantitative Methods

Simple Linear Regression

A prediction interval for a new dependent variable observation is wider than a confidence interval for the mean dependent variable at the same X value primarily because the prediction interval includes:

Very Difficult

Quantitative Methods

Simple Linear Regression

A residual plot shows residuals with increasing spread as the independent variable rises. The regression assumption most likely violated is:

Very Difficult

Quantitative Methods

Simple Linear Regression

In a log-log regression ln(Sales) = 2.0 + 1.2 ln(Advertising), the slope coefficient is best interpreted as:

Easy

Quantitative Methods

Simulation Methods

Asset prices are often modeled as lognormal because a lognormal variable:

Easy

Quantitative Methods

Simulation Methods

Monte Carlo simulation is best described as a method that:

Easy

Quantitative Methods

Simulation Methods

Bootstrap resampling most likely involves:

Moderate

Quantitative Methods

Simulation Methods

An analyst assumes continuously compounded monthly returns are normally distributed. The corresponding one-month asset price distribution is most likely:

Moderate

Quantitative Methods

Simulation Methods

A simulation produces 10,000 one-year portfolio returns, of which 1,300 are below 0%. The simulated probability of a loss is closest to:

Moderate

Quantitative Methods

Simulation Methods

For a short data history with clear non-normal tail behavior, a bootstrap simulation is most likely preferred to a normal parametric simulation because it:

Very Difficult

Quantitative Methods

Simulation Methods

A Monte Carlo model estimates a 5% probability of a portfolio loss greater than 20%. The most accurate interpretation is that:

Moderate

Quantitative Methods

Simulation Methods

An analyst builds a simulation using expected return, volatility, and correlation assumptions that were estimated during an unusually calm period. The most relevant concern is:

Very Difficult

Quantitative Methods

Simulation Methods

If simple return R is -100%, the continuously compounded return ln(1 + R) is:

Moderate

Quantitative Methods

Skewness, Kurtosis, and CV

Two strategies have the same positive mean return. Strategy A has median above mean, high excess kurtosis, and lower coefficient of variation. The most accurate interpretation is that A has:

Easy

Quantitative Methods

Statistical Measures of Asset Returns

For the return series 2%, 4%, 4%, 9%, and 11%, the median return is:

Easy

Quantitative Methods

Statistical Measures of Asset Returns

The measure of central tendency most appropriate for averaging purchase prices when the same currency amount is invested each period is the:

Easy

Quantitative Methods

Statistical Measures of Asset Returns

A sample variance calculation uses n - 1 in the denominator primarily to:

Moderate

Quantitative Methods

Statistical Measures of Asset Returns

A portfolio consists of 40% in Asset X with expected return 6% and 60% in Asset Y with expected return 11%. The weighted mean return is closest to:

Moderate

Quantitative Methods

Statistical Measures of Asset Returns

For sample returns of 2%, 4%, 6%, and 8%, the sample standard deviation is closest to:

Moderate

Quantitative Methods

Statistical Measures of Asset Returns

An investment has expected return of 8% and standard deviation of 12%. Its coefficient of variation is closest to:

Easy

Quantitative Methods

Statistical Measures of Asset Returns

A return distribution has a long left tail and excess kurtosis greater than zero. The distribution is best described as:

Moderate

Quantitative Methods

Statistical Measures of Asset Returns

Using a target return of 0% and the sample target downside deviation formula with n - 1 in the denominator, the target downside deviation for returns of -6%, -2%, 3%, and 7% is closest to:

Easy

Quantitative Methods

Statistical Measures of Asset Returns

A covariance of returns is positive. The most accurate interpretation is that the two return series tend to:

Moderate

Quantitative Methods

Time Value of Money

A project requires five end-of-year payments of 25,000 beginning one year from today. At a 7.2% annual discount rate, the present value is closest to:

Easy

Quantitative Methods

Time Value of Money in Finance

At an annual discount rate of 5%, the present value of 1,000 received in three years is closest to:

Easy

Quantitative Methods

Time Value of Money in Finance

A stated annual rate of 12% compounded monthly has an effective annual rate closest to:

Easy

Quantitative Methods

Time Value of Money in Finance

For the same payment amount, term, and discount rate, the present value of an annuity due is:

Moderate

Quantitative Methods

Time Value of Money in Finance

A two-year bond has a 5% annual coupon, a par value of 1,000, and a required return of 6%. Its value is closest to:

Moderate

Quantitative Methods

Time Value of Money in Finance

A preferred share pays a constant annual dividend of 4.00 and investors require an 8.00% return. Its value is closest to:

Moderate

Quantitative Methods

Time Value of Money in Finance

A stock is valued at 60 using next year dividend of 3 and a required return of 9%. The implied constant growth rate is closest to:

Very Difficult

Quantitative Methods

Time Value of Money in Finance

A project has cash flows of -1,000 today, 300 in year 1, 400 in year 2, and 500 in year 3. At an 8% discount rate, the NPV is closest to:

Very Difficult

Quantitative Methods

Time Value of Money in Finance

The one-year spot rate is 3.00% and the two-year spot rate is 4.00%, both annual effective rates. The one-year forward rate one year from today is closest to:

Very Difficult

Quantitative Methods

Time Value of Money in Finance

An analyst values three cash flows separately and then adds their present values to value a package of the three claims. This process is best described as applying: