Financial Risk Manager (FRM) Quantitative Analysis Syllabus

Quantitative Analysis 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 in Financial Risk Manager (FRM), not a summary of it.

1. 1

   Probability and Random Variables

   3 topics

   • Probability fundamentals
     • Conditional and unconditional probability and Bayes' rule
     • Independent versus mutually exclusive events
   • Random variables and probability functions
     • PMF, PDF, and cumulative distribution functions
     • Quantiles and the inverse CDF
   • Expectations and moments
     • Mean, variance, skewness, and kurtosis
     • Covariance, correlation, and coskewness
2. 2

   Common Probability Distributions

   4 topics

   • Discrete distributions: Bernoulli, binomial, and Poisson
   • Continuous distributions: uniform, normal, and lognormal
   • Fat-tailed and sampling distributions
     • Student's t, chi-squared, and F distributions
     • Mixture distributions and tail behavior
   • The Central Limit Theorem and the law of large numbers
3. 3

   Statistical Inference and Hypothesis Testing

   3 topics

   • Sampling, estimators, and their properties
     • Bias, efficiency, and consistency
     • Standard error and confidence intervals
   • Hypothesis testing framework
     • Null and alternative hypotheses, Type I and Type II errors
     • Test statistics, p-values, and significance levels
     • One-tailed versus two-tailed tests
   • Maximum likelihood estimation and method of moments
4. 4

   Regression Analysis

   4 topics

   • Linear regression with one regressor
     • OLS estimation and interpretation of coefficients
     • R-squared, standard error of regression, and goodness of fit
   • Multiple regression and the OLS assumptions
   • Regression diagnostics
     • Heteroskedasticity and serial correlation
     • Multicollinearity and omitted-variable bias
   • Hypothesis tests on coefficients and the F-test
5. 5

   Time Series Analysis and Volatility Modeling

   4 topics

   • Stationarity, autocorrelation, and white noise
   • AR, MA, and ARMA models
     • Model identification and forecasting
     • Unit roots and random walks
   • Volatility estimation
     • Exponentially weighted moving average (EWMA)
     • GARCH(1,1) and volatility clustering
   • Correlation, copulas, and dependence modeling
6. 6

   Simulation and Numerical Methods

   3 topics

   • Monte Carlo simulation
     • Generating random draws and inverse transform sampling
     • Estimating standard errors of simulated quantities
   • Variance reduction techniques
     • Antithetic variates and control variates
   • Bootstrapping and resampling methods

Quantitative Analysis flashcards for Financial Risk Manager (FRM)

23 of 51 cards from the Quantitative Analysis deck — real questions with worked answers.

1. State the three axioms of probability for events in a sample space.

   For any event $A$: $P(A) \geq 0$. The probability of the entire sample space is $P(\Omega) = 1$. For mutually exclusive events $A_1, A_2, \ldots$, $P\left(\bigcup_i A_i\right) = \sum_i P(A_i)$.

2. Write the definition of conditional probability and Bayes' theorem.

   Conditional probability: $P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}$ for $P(B) > 0$. Bayes' theorem: $P(A \mid B) = \dfrac{P(B \mid A)\,P(A)}{P(B)}$.

3. What does it mean for two events $A$ and $B$ to be independent, and how does it simplify the joint probability?

   Independence means the occurrence of one does not affect the other: $P(A \mid B) = P(A)$. Then the joint probability factors as $P(A \cap B) = P(A)\,P(B)$.

4. State the total probability rule for an event $A$ given a partition $\{B_i\}$.

   $$P(A) = \sum_{i} P(A \mid B_i)\,P(B_i),$$ where the events $B_i$ are mutually exclusive and exhaustive.

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

   A discrete random variable takes a countable set of values, described by a probability mass function (PMF) where $P(X = x) > 0$. A continuous random variable takes values on an interval, described by a probability density function (PDF) where $P(X = x) = 0$ and probabilities come from integrating the PDF.

6. Define the cumulative distribution function (CDF) and list its key properties.

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

7. Define the expected value of a discrete and a continuous random variable.

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

8. Give the two equivalent formulas for the variance of a random variable.

   $$\operatorname{Var}(X) = E\big[(X - E[X])^{2}\big] = E[X^{2}] - \big(E[X]\big)^{2}.$$

9. How do linear transformations affect the mean and variance? Give $E[aX+b]$ and $\operatorname{Var}(aX+b)$.

   $E[aX + b] = aE[X] + b$ and $\operatorname{Var}(aX + b) = a^{2}\operatorname{Var}(X)$. Adding a constant shifts the mean but not the variance.

10. Define skewness and kurtosis, and state the kurtosis of a normal distribution.

    Skewness is the standardized third moment $\dfrac{E[(X-\mu)^{3}]}{\sigma^{3}}$, measuring asymmetry. Kurtosis is the standardized fourth moment $\dfrac{E[(X-\mu)^{4}]}{\sigma^{4}}$, measuring tail heaviness. The normal distribution has kurtosis $= 3$; excess kurtosis $=$ kurtosis $- 3$.

11. For the covariance of two random variables, give the definition and the variance of a sum $X + Y$.

    $\operatorname{Cov}(X, Y) = E[(X - \mu_X)(Y - \mu_Y)] = E[XY] - E[X]E[Y]$. Then $\operatorname{Var}(X + Y) = \operatorname{Var}(X) + \operatorname{Var}(Y) + 2\operatorname{Cov}(X, Y)$.

12. Describe a Bernoulli random variable and give its mean and variance.

    A Bernoulli variable takes value $1$ (success) with probability $p$ and $0$ (failure) with probability $1-p$. Mean $E[X] = p$; variance $\operatorname{Var}(X) = p(1-p)$.

13. State the PMF, mean, and variance of a binomial random variable.

    $P(X = k) = \binom{n}{k} p^{k}(1-p)^{n-k}$ for $k = 0,1,\ldots,n$. Mean $= np$; variance $= np(1-p)$. It counts successes in $n$ independent Bernoulli trials.

14. State the PMF, mean, and variance of a Poisson random variable.

    $P(X = k) = \dfrac{\lambda^{k} e^{-\lambda}}{k!}$ for $k = 0,1,2,\ldots$. Both the mean and the variance equal $\lambda$. It models the number of events in a fixed interval given an average rate $\lambda$.

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

    When $n$ is large and $p$ is small, the binomial is well approximated by a Poisson with $\lambda = np$. The Poisson is the limiting case of the binomial as $n \to \infty$ and $p \to 0$ with $np$ held constant.

16. Give the PDF, mean, and variance of a continuous uniform distribution on $[a, b]$.

    $f(x) = \dfrac{1}{b - a}$ for $a \leq x \leq b$. Mean $= \dfrac{a + b}{2}$; variance $= \dfrac{(b - a)^{2}}{12}$.

17. Write the PDF of the normal distribution and state its key shape properties.

    $$f(x) = \frac{1}{\sigma\sqrt{2\pi}}\exp\!\left(-\frac{(x - \mu)^{2}}{2\sigma^{2}}\right).$$ It is symmetric and bell-shaped, fully described by $\mu$ and $\sigma^{2}$, with skewness $0$ and kurtosis $3$.

18. How do you standardize a normal random variable, and what fraction of observations lie within $\pm 1.96\sigma$?

    Standardize via $Z = \dfrac{X - \mu}{\sigma} \sim N(0,1)$. About $95\%$ of observations lie within $\mu \pm 1.96\sigma$ (and roughly $99\%$ within $\pm 2.58\sigma$).

19. Define the lognormal distribution and explain why it is used for asset prices.

    If $\ln(X) \sim N(\mu, \sigma^{2})$, then $X$ is lognormal. It is right-skewed and bounded below by zero, making it suitable for asset prices and other variables that cannot be negative.

20. What characterizes a fat-tailed (leptokurtic) distribution relative to the normal?

    A fat-tailed distribution has excess kurtosis $> 3$ (positive excess kurtosis), meaning more probability mass in the tails and a higher peak. Extreme outcomes are more likely than the normal would predict, which is important for risk modeling.

21. Describe the Student's t-distribution and how it relates to the normal as degrees of freedom increase.

    The Student's t is symmetric and bell-shaped but has fatter tails than the normal, controlled by degrees of freedom $\nu$. Its variance is $\dfrac{\nu}{\nu - 2}$ for $\nu > 2$. As $\nu \to \infty$, the t-distribution converges to the standard normal.

22. State the Central Limit Theorem.

    For i.i.d. random variables with mean $\mu$ and finite variance $\sigma^{2}$, the sample mean's distribution approaches normal as $n$ grows: $$\frac{\bar{X} - \mu}{\sigma / \sqrt{n}} \xrightarrow{d} N(0, 1).$$ This holds regardless of the underlying distribution's shape.

23. State the law of large numbers and contrast it with the CLT.

    The (weak) law of large numbers states the sample mean converges in probability to the population mean: $\bar{X} \xrightarrow{p} \mu$ as $n \to \infty$. The LLN concerns convergence of the mean to a constant, whereas the CLT describes the distributional shape (normality) of the standardized mean.

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