The biggest mistake candidates make in Quantitative Analysis isn't a calculation error. It's misidentifying the question's core concept under time pressure, which leads to using the right formula for the wrong problem. The FRM exam doesn't just test if you can run the numbers; it tests if you know which numbers to run and why. This is a judgment test disguised as a math test.
These free FRM Quantitative Analysis practice questions test your ability to apply concepts like Bayes' theorem, hypothesis testing, and regression under exam pressure. They reveal if you are falling for common traps, like misinterpreting p-values or choosing the wrong statistical test, which is a primary reason candidates fail this section.
Key facts
- Exam Section
- FRM Part I, Book 2: Quantitative Analysis
- Blueprint Weighting
- 20% of the exam (20 questions)
- Key Topics
- Probability, statistics, regression, time series analysis, volatility models
- Official Body
- Global Association of Risk Professionals (GARP)
- Approved Calculators
- Texas Instruments BA II Plus (including Professional), HP 10BII+, or HP 12C
- Est. Section Study Hours
- 50-60 hours (as part of 200-250 total Part I hours)
The Global Association of Risk Professionals (GARP) reported a 44% pass rate for the May 2024 Part I FRM exam (GARP), underscoring the exam's difficulty.
What Do These Free Quantitative Analysis Questions Cover?
These questions are not random trivia. They are designed by FRM experts to mirror the style and difficulty of the actual FRM Part I exam, based on the 2026 GARP curriculum. They represent a sample from VoraPrep's complete course, which includes over 1,800 practice questions, detailed lessons, and our 24/7 personal tutor, Vory. Try VoraPrep's free FRM practice questions to experience our adaptive learning platform.
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This sample includes a mix of:
- Conceptual questions: Testing your understanding of the "why" behind the models.
- Computational questions: Requiring application of formulas using an approved financial calculator.
- Interpretive questions: Asking you to draw conclusions from data, such as a regression output.
They are all aligned with the official GARP blueprint, ensuring you are studying what will actually be on the exam.
Quantitative Analysis Practice Questions
Work through these questions one by one. Read the explanations carefully, especially for any you get wrong. Pay attention not just to what the right answer is, but why the other options are tempting but incorrect distractors.
Question 1: Probability
A portfolio manager estimates that there is a 60% probability that the economy will be in an expansionary state and a 40% probability it will be in a recessionary state. If the economy is expansionary, a particular stock has a 70% chance of increasing in value. If the economy is recessionary, the stock has a 30% chance of increasing in value. Given that the stock has increased in value, what is the probability that the economy is in an expansionary state?
D
Explanation: This is a classic application of Bayes' theorem, which is fundamentally about updating your beliefs based on new evidence.Let A = "Stock increases in value" Let B₁ = "Economy is expansionary" Let B₂ = "Economy is recessionary"
We are given: P(B₁) = 0.60 P(B₂) = 0.40 P(A | B₁) = 0.70 (Probability of stock increase given expansion) P(A | B₂) = 0.30 (Probability of stock increase given recession)
We need to find P(B₁ | A), the probability of an expansion given the stock increased.
Bayes' Theorem: P(B₁ | A) = [P(A | B₁) × P(B₁)] / P(A)
First, we calculate the total probability of the stock increasing, P(A), using the law of total probability: P(A) = P(A | B₁) × P(B₁) + P(A | B₂) × P(B₂) P(A) = (0.70 × 0.60) + (0.30 × 0.40) P(A) = 0.42 + 0.12 = 0.54
Now, plug this into Bayes' theorem: P(B₁ | A) = (0.70 × 0.60) / 0.54 P(B₁ | A) = 0.42 / 0.54 ≈ 0.7778 or 78%
Question 2: Hypothesis Testing
A risk analyst wants to test whether the average daily volatility of a currency pair has changed from its historical average of 15 basis points (bps). The analyst takes a sample of 36 daily volatilities and finds a sample mean of 16.5 bps and a sample standard deviation of 4 bps. Assuming the population is normally distributed, what is the appropriate test and conclusion at a 5% significance level? (Critical t-value with 35 df ≈ 2.03)
C
Explanation: The first judgment call is selecting the right tool. The population standard deviation is unknown; we only have the sample standard deviation.This is the only fact you need to make your first decision.
When the population variance is unknown, you must use a t-test. This eliminates options A and D immediately.
Now, we perform the t-test:
- Null Hypothesis (H₀): μ = 15 bps (The mean volatility has not changed)
- Alternative Hypothesis (H₁): μ ≠ 15 bps (The mean volatility has changed)
This is a two-tailed test.
Calculate the t-statistic: t = (Sample Mean − Population Mean) / (Sample Std Dev / √n) t = (16.5 − 15) / (4 / √36) t = 1.5 / (4 / 6) t = 1.5 / 0.6667 = 2.25
The calculated t-statistic is 2.25. The critical t-value for a two-tailed test at 5% significance with n-1 = 35 degrees of freedom is given as approximately 2.03.
Since our calculated t-statistic (2.25) is greater than the critical value (2.03), it falls in the rejection region. We reject the null hypothesis. The evidence suggests the average daily volatility has changed.
Question 3: Linear Regression Interpretation
An analyst runs a regression of monthly hedge fund returns (Y) against the monthly returns of a market index (X). The results are below:
| Variable | Coefficient | p-value |
|---|---|---|
| Intercept (α) | 0.003 | 0.04 |
| Market Index (β) | 0.85 | 0.001 |
The R-squared of the regression is 0.65. Which of the following is the most accurate interpretation?
D
Explanation: This question tests if you can read the "story" told by the regression output, not just identify the numbers.- Intercept (alpha): The coefficient is 0.003, or 0.3%. This is the fund's excess return after accounting for market movements. Its p-value is 0.04.
- Market Index (beta): The coefficient is 0.85, indicating the fund's sensitivity to the market. Its p-value is 0.001.
- R-squared: 0.65, meaning 65% of the variation in the hedge fund's returns (the Y variable) is explained by the market index's returns (the X variable).
Let's evaluate the options:
- A is incorrect. A beta less than 1 simply means the fund is less volatile than the market. This is a characteristic, not a flaw.
- B is incorrect. The p-value for alpha is 0.04. To be significant at the 1% level, the p-value would need to be less than 0.01.
- C is incorrect. This reverses the interpretation of R-squared. The independent variable (market) explains the dependent variable (fund), not the other way around. This is a very common trap for candidates who have memorized a definition without understanding the relationship.
- D is correct. The p-value for alpha (0.04) is less than the 5% significance level (0.05). This means we can reject the null hypothesis that alpha is zero. The fund's alpha is statistically significant.
Question 4: Time Series
Which of the following processes is most likely to be non-stationary?
C
Explanation: Stationarity is a critical concept in time series analysis. A stationary series has a constant mean, constant variance, and constant autocovariance structure over time. It tends to revert to a long-run value.- A) A white noise process is the definition of a simple stationary series. It has a mean of zero, constant variance, and zero autocorrelation.
- B) A GARCH process models changing variance (heteroskedasticity), but the underlying process can be covariance stationary if the sum of its parameters (α + β) is less than 1. The unconditional variance is constant.
- C) A random walk has a unit root. Its variance increases infinitely with time, and it does not revert to a long-run mean. It is the classic example of a non-stationary process.
- D) An AR(1) process is stationary as long as the absolute value of the autoregressive coefficient is less than 1. Since the coefficient is 0.7, this process is stationary and will revert to its mean.
Question 5: Kurtosis
A distribution with fatter tails than a normal distribution is referred to as:
C
Explanation: Kurtosis measures the "tailedness" of a distribution, which is a core concept for risk managers.- A normal distribution is mesokurtic and has a kurtosis of 3. Its excess kurtosis (kurtosis - 3) is 0.
- A leptokurtic distribution has fatter tails and is more peaked than a normal distribution. It has a kurtosis greater than 3 (positive excess kurtosis). This is typical of financial returns, which exhibit more extreme events than a normal distribution would predict.
- A platykurtic distribution has thinner tails and is flatter than a normal distribution. It has a kurtosis less than 3 (negative excess kurtosis).
- Skewness refers to the asymmetry of a distribution, not its tails.
The exam requires you to know this terminology cold. Leptokurtosis is fundamental to risk management because it acknowledges that extreme losses happen more often than a simple normal model would suggest.
Question 6: Standard Error
An analyst draws a sample of 100 stocks from an index and calculates the average return. If the analyst were to increase the sample size to 400 stocks, how would the standard error of the sample mean be expected to change?
B
Explanation: The formula for the standard error of the mean is: SE = σ / √n, where σ is the population standard deviation and n is the sample size.Let SE₁ be the standard error with n=100: SE₁ = σ / √100 = σ / 10 Let SE₂ be the standard error with n=400: SE₂ = σ / √400 = σ / 20
The new standard error (σ / 20) is exactly half of the original standard error (σ / 10).
This is an intuitive concept you should internalize: to double the precision of your estimate (halve the error), you must quadruple your sample size.
Question 7: GARCH Model
In a GARCH(1,1) model, σ²(t) = ω + αε²(t-1) + βσ²(t-1), the parameter 'α' represents the:
C
Explanation: Let's break down the GARCH(1,1) model components:- ω (omega): A constant term related to the long-run variance.
- α (alpha): The coefficient for the ARCH term, ε²(t-1). This is the squared residual (or "shock") from the previous period. It measures the reaction of volatility to past market shocks.
- β (beta): The coefficient for the GARCH term, σ²(t-1). This is the variance from the previous period. It measures the persistence of volatility.
Therefore, 'α' represents the weight assigned to the previous period's squared residual. A large 'α' means that volatility reacts sharply to market movements.
Question 8: OLS Assumptions
Which of the following is NOT a key assumption required for an Ordinary Least Squares (OLS) estimator to be the Best Linear Unbiased Estimator (BLUE)?
C
Explanation: This question tests a subtle but critical distinction. The properties of the OLS estimator are guaranteed by the Gauss-Markov assumptions.- A (Homoskedasticity), D (No perfect multicollinearity), and the assumption that the error term has a conditional mean of zero are all core Gauss-Markov assumptions.
- B (Non-stochastic independent variables) is a standard assumption in the classical linear regression model framework that helps establish the BLUE properties.
- C (Normality of the error term) is the outlier. The OLS estimators are BLUE even if the errors are not normally distributed. The normality assumption is an additional requirement of the Classical Linear Model (CLM) needed for hypothesis testing (like t-tests and F-tests) to be valid in small samples. It is not required for the estimator to be unbiased or efficient.
A portfolio manager is evaluating a new trading strategy. Over the past 64 days, the strategy has produced an average daily return of 0.05% with a standard deviation of 0.4%. The manager wants to test if the strategy's true mean return is greater than zero.
Question: Calculate the test statistic and determine if the manager can be 95% confident that the strategy's mean return is positive. (One-tailed critical t-value with 63 df ≈ 1.67)Step 1: Formulate the Hypotheses
This is a one-tailed test because the manager is specifically testing if the return is greater than zero.- Null Hypothesis (H₀): μ ≤ 0
- Alternative Hypothesis (H₁): μ > 0
Step 2: Choose the Correct Test
The population standard deviation is unknown. Even with a sample size over 30, the t-test is the correct choice.Step 3: Calculate the Test Statistic
t = (Sample Mean − Hypothesized Mean) / (Sample Std Dev / √n) t = (0.0005 − 0) / (0.004 / √64) t = 0.0005 / (0.004 / 8) t = 0.0005 / 0.0005 t = 1.0Step 4: Compare and Conclude
The calculated t-statistic is 1.0. The one-tailed critical t-value for 95% confidence (a 5% significance level) with n-1 = 63 degrees of freedom is approximately 1.67.Since our test statistic (1.0) is less than the critical value (1.67), we fail to reject the null hypothesis. The manager cannot be 95% confident that the strategy's true mean return is greater than zero. The observed positive return could simply be due to random chance.
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How Did You Score?
Count up your correct answers. This is a diagnostic tool, not a final grade.
- 8 or more correct: You have a strong command of these core concepts. Your focus should be on speed and spotting subtle traps. Your next step this week: take a 20-question timed quiz in VoraPrep on just Quantitative Analysis to simulate exam pressure.
- 5 to 7 correct: You have a decent foundation, but there are clear conceptual gaps. You likely understand the "what" but not always the "why." Go back to the questions you missed. Was it a specific topic like GARCH or a type of problem like hypothesis testing? Your next step this week: use VoraPrep's lessons to review one of your weak topics, then drill 25 adaptive questions on it.
- Less than 5 correct: This indicates a significant knowledge gap, which is valuable information. It means you need to return to first principles. Your next step this week: start with the "Fundamentals of Probability" chapter in the VoraPrep course and work through the interactive lessons. Build the foundation correctly now.
The goal is to internalize the logic used to solve these problems. This sample covers key areas, but a full review must also include topics like confidence intervals, other probability distributions, and Maximum Likelihood Estimation (MLE). For a deeper look at exam difficulty, our analysis of FRM pass rates and historical data provides more context.
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