FRM (Financial Risk Manager) Quantitative Analysis (Part I) Syllabus

Quantitative Analysis (Part I) syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Quantitative Analysis (Part I) in FRM (Financial Risk Manager), not a summary of it.

1. 1

   Probability and Statistical Distributions

   4 topics

   • Fundamentals of Probability
     • Conditional, joint, and marginal probability
     • Bayes' theorem and updating
   • Random Variables and Moments
     • Expected value, variance, skewness, kurtosis
     • Covariance and correlation
   • Common Distributions
     • Bernoulli, binomial, and Poisson
     • Normal, lognormal, and Student's t
     • Chi-squared, F, and exponential distributions
   • Sampling and Estimation
     • Law of large numbers and central limit theorem
     • Point estimators and confidence intervals
2. 2

   Hypothesis Testing and Inference

   3 topics

   • Hypothesis Testing Framework
     • Null and alternative hypotheses
     • Type I and Type II errors, p-values
   • Test Statistics
     • t-tests and z-tests
     • Chi-squared and F-tests
   • Measuring Population Parameters
     • Sample mean and variance estimation
     • Bessel's correction
3. 3

   Linear Regression and Time Series

   3 topics

   • Simple and Multiple Linear Regression
     • OLS estimation and assumptions
     • R-squared and adjusted R-squared
     • Hypothesis tests on coefficients
   • Regression Diagnostics
     • Heteroskedasticity and multicollinearity
     • Omitted variable bias
   • Time Series Modeling
     • Stationarity and autocorrelation
     • AR, MA, and ARMA processes
     • Random walks and unit roots
4. 4

   Volatility, Correlation, and Simulation

   4 topics

   • Modeling Volatility
     • EWMA approach
     • GARCH(1,1) models
   • Estimating Correlations and Copulas
     • Correlation vs. dependence
     • Gaussian and t-copulas
   • Monte Carlo and Bootstrapping
     • Random number generation
     • Variance reduction techniques
     • Bootstrapping for resampling
   • Machine Learning Foundations
     • Supervised vs. unsupervised learning
     • Overfitting, training and test sets

Quantitative Analysis (Part I) flashcards for FRM (Financial Risk Manager)

25 of 50 cards from the Quantitative Analysis (Part I) deck — real questions with worked answers.

1. What are the three axioms of probability (Kolmogorov's axioms)?

   For any event $A$: (1) Non-negativity, $P(A) \geq 0$; (2) Normalization, $P(\Omega) = 1$ for the sample space $\Omega$; (3) Additivity, for mutually exclusive events $A$ and $B$, $P(A \cup B) = P(A) + P(B)$.

2. State the general addition rule for the probability of the union of two events $A$ and $B$.

   $$P(A \cup B) = P(A) + P(B) - P(A \cap B).$$ Subtracting $P(A \cap B)$ avoids double-counting the overlap. If $A$ and $B$ are mutually exclusive, $P(A \cap B) = 0$.

3. Define conditional probability $P(A \mid B)$ and give its formula.

   The probability of event $A$ occurring given that $B$ has occurred: $$P(A \mid B) = \frac{P(A \cap B)}{P(B)}, \quad P(B) > 0.$$

4. What is the multiplication rule for the joint probability of two events?

   $$P(A \cap B) = P(A \mid B)\, P(B) = P(B \mid A)\, P(A).$$ For independent events this simplifies to $P(A \cap B) = P(A)\,P(B)$.

5. What does it mean for two events $A$ and $B$ to be statistically independent?

   The occurrence of one does not affect the probability of the other: $P(A \mid B) = P(A)$, equivalently $$P(A \cap B) = P(A)\,P(B).$$

6. Define marginal probability and explain how it relates to joint probabilities.

   The marginal (unconditional) probability of an event is its probability irrespective of other variables. It is obtained by summing joint probabilities over the other variable: $$P(A) = \sum_{i} P(A \cap B_i).$$

7. State the total probability rule using a partition $B_1, \dots, B_n$ of the sample space.

   If $\{B_i\}$ are mutually exclusive and exhaustive, then $$P(A) = \sum_{i=1}^{n} P(A \mid B_i)\, P(B_i).$$

8. State Bayes' theorem.

   $$P(A \mid B) = \frac{P(B \mid A)\, P(A)}{P(B)},$$ where $P(B) = \sum_i P(B \mid A_i) P(A_i)$ by the total probability rule. It updates a prior $P(A)$ into a posterior $P(A \mid B)$ given evidence $B$.

9. In Bayesian updating, identify the prior, the likelihood, and the posterior in $P(\theta \mid D) = \dfrac{P(D \mid \theta) P(\theta)}{P(D)}$.

   Prior: $P(\theta)$ (belief before data). Likelihood: $P(D \mid \theta)$ (probability of data given the hypothesis). Posterior: $P(\theta \mid D)$ (updated belief after observing data $D$). $P(D)$ is the normalizing evidence.

10. Distinguish a discrete random variable from a continuous random variable.

    A discrete random variable takes countably many values (e.g., $0,1,2,\dots$) and is described by a probability mass function (PMF). A continuous random variable takes values over an interval and is described by a probability density function (PDF), where $P(X = x) = 0$ for any single point.

11. What is a cumulative distribution function (CDF) and what properties must it satisfy?

    $F(x) = P(X \leq x)$. It is non-decreasing, right-continuous, with $\lim_{x \to -\infty} F(x) = 0$ and $\lim_{x \to +\infty} F(x) = 1$. For continuous $X$, $f(x) = \dfrac{dF(x)}{dx}$.

12. Define the expected value of (a) a discrete and (b) a continuous random variable.

    Discrete: $$E[X] = \sum_{i} x_i\, p(x_i).$$ Continuous: $$E[X] = \int_{-\infty}^{\infty} x\, f(x)\, dx.$$

13. State the linearity property of expectations for constants $a, b$ and random variables $X, Y$.

    $$E[aX + bY + c] = a\,E[X] + b\,E[Y] + c.$$ Expectation is linear regardless of whether $X$ and $Y$ are independent.

14. Define the variance of a random variable and give its computational (shortcut) formula.

    $$\sigma^2 = \operatorname{Var}(X) = E\big[(X - \mu)^2\big] = E[X^2] - (E[X])^2.$$ The standard deviation is $\sigma = \sqrt{\operatorname{Var}(X)}$.

15. How does variance scale under a linear transformation $aX + b$?

    $$\operatorname{Var}(aX + b) = a^2 \operatorname{Var}(X).$$ Adding a constant $b$ does not change variance; multiplying by $a$ scales variance by $a^2$.

16. Define skewness and state what positive, zero, and negative skewness indicate.

    Skewness is the standardized third moment: $$S = E\!\left[\left(\frac{X - \mu}{\sigma}\right)^3\right].$$ $S > 0$: long right tail (mean > median). $S = 0$: symmetric. $S < 0$: long left tail (mean < median).

17. Define kurtosis and excess kurtosis, and explain leptokurtic vs. platykurtic.

    Kurtosis is the standardized fourth moment: $$K = E\!\left[\left(\frac{X - \mu}{\sigma}\right)^4\right].$$ Excess kurtosis $= K - 3$ (normal has $K = 3$). Leptokurtic: $K > 3$ (fat tails); platykurtic: $K < 3$ (thin tails).

18. Define the covariance between two random variables $X$ and $Y$.

    $$\operatorname{Cov}(X, Y) = E\big[(X - \mu_X)(Y - \mu_Y)\big] = E[XY] - E[X]E[Y].$$ Positive covariance means the variables tend to move together; negative means they move oppositely.

19. Define the correlation coefficient and state its range.

    $$\rho_{X,Y} = \frac{\operatorname{Cov}(X, Y)}{\sigma_X \sigma_Y}, \qquad -1 \leq \rho_{X,Y} \leq 1.$$ It is a standardized, unitless measure of the strength and direction of the linear relationship.

20. Give the formula for the variance of a sum of two random variables.

    $$\operatorname{Var}(X + Y) = \operatorname{Var}(X) + \operatorname{Var}(Y) + 2\operatorname{Cov}(X, Y).$$ If $X$ and $Y$ are independent, $\operatorname{Cov}(X,Y) = 0$, so variances simply add.

21. Does zero correlation imply independence? Explain.

    No. Independence implies zero correlation, but zero correlation does not imply independence. Correlation only measures linear dependence; two variables can be nonlinearly dependent yet have $\rho = 0$.

22. Describe the Bernoulli distribution: its parameter, mean, and variance.

    A single trial with outcomes $1$ (success, probability $p$) or $0$ (failure, probability $1-p$). $$E[X] = p, \qquad \operatorname{Var}(X) = p(1-p).$$

23. State the binomial PMF and its mean and variance.

    For $n$ independent Bernoulli trials with success probability $p$: $$P(X = k) = \binom{n}{k} p^{k} (1-p)^{n-k}.$$ $E[X] = np$, $\operatorname{Var}(X) = np(1-p)$.

24. State the Poisson PMF and its key mean/variance property.

    For events at average rate $\lambda$ per interval: $$P(X = k) = \frac{\lambda^{k} e^{-\lambda}}{k!}, \quad k = 0,1,2,\dots$$ Distinctive property: mean equals variance, $E[X] = \operatorname{Var}(X) = \lambda$.

25. When can the Poisson distribution approximate the binomial distribution?

    When $n$ is large and $p$ is small (rule of thumb $n \geq 20$, $p \leq 0.05$), the binomial is well approximated by a Poisson with $\lambda = np$.

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