Financial Risk Manager (FRM) Quantitative Analysis Syllabus

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.

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.

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.

What is the standard error of the sample mean, and how does it scale with sample size?

$\text{SE}(\bar{X}) = \dfrac{\sigma}{\sqrt{n}}$ (estimated as $\dfrac{s}{\sqrt{n}}$). It decreases proportionally to $\dfrac{1}{\sqrt{n}}$, so quadrupling the sample size halves the standard error.

Define an unbiased estimator and give the bias formula.

An estimator $\hat{\theta}$ is unbiased if $E[\hat{\theta}] = \theta$. Bias is $E[\hat{\theta}] - \theta$; an unbiased estimator has zero bias. The sample mean is an unbiased estimator of the population mean.

List the desirable properties of an estimator: efficiency, consistency, and BLUE.

Efficiency: smallest variance among unbiased estimators. Consistency: $\hat{\theta} \xrightarrow{p} \theta$ as $n \to \infty$. Linearity. BLUE = Best Linear Unbiased Estimator, the minimum-variance unbiased estimator within the class of linear estimators (per Gauss-Markov).

Why is the sample variance divided by $n - 1$ rather than $n$?

Dividing by $n - 1$ (Bessel's correction) makes the sample variance an unbiased estimator of the population variance: $s^{2} = \dfrac{1}{n-1}\sum_{i=1}^{n}(X_i - \bar{X})^{2}$. Dividing by $n$ would systematically underestimate $\sigma^{2}$.

Outline the steps of the hypothesis testing framework.

State the null $H_0$ and alternative $H_1$; choose a significance level $\alpha$; compute a test statistic; determine the critical value(s) / rejection region or the p-value; reject $H_0$ if the statistic falls in the rejection region (or $p < \alpha$), otherwise fail to reject.

Distinguish Type I and Type II errors and define the power of a test.

Type I error: rejecting a true $H_0$, with probability $\alpha$ (the significance level). Type II error: failing to reject a false $H_0$, with probability $\beta$. Power $= 1 - \beta$, the probability of correctly rejecting a false null.

What is a p-value, and how is it used in a decision rule?

The p-value is the probability of obtaining a test statistic at least as extreme as the observed one, assuming $H_0$ is true. Reject $H_0$ if the p-value $< \alpha$; a smaller p-value indicates stronger evidence against the null.

Contrast a one-tailed and a two-tailed hypothesis test.

A two-tailed test ($H_1: \theta \neq \theta_0$) splits $\alpha$ between both tails and detects deviations in either direction. A one-tailed test ($H_1: \theta > \theta_0$ or $\theta < \theta_0$) places all of $\alpha$ in one tail and detects deviation in only the specified direction.

Describe maximum likelihood estimation (MLE).

MLE chooses parameter values that maximize the likelihood function $L(\theta) = \prod_i f(x_i; \theta)$ (or equivalently the log-likelihood $\ell(\theta) = \sum_i \ln f(x_i; \theta)$). The MLE is the parameter set that makes the observed data most probable; it is consistent and asymptotically efficient and normal.

Describe the method of moments estimation procedure.

The method of moments equates population (theoretical) moments to their corresponding sample moments and solves for the parameters. For example, set the population mean equal to the sample mean and the population variance equal to the sample variance, then solve.

Write the population simple linear regression model and identify each term.

$$Y_i = \beta_0 + \beta_1 X_i + \varepsilon_i,$$ where $\beta_0$ is the intercept, $\beta_1$ is the slope, $X_i$ is the regressor, $\varepsilon_i$ is the error term, and $Y_i$ is the dependent variable.

Give the OLS estimator of the slope $\hat{\beta}_1$ in simple regression.

$$\hat{\beta}_1 = \frac{\operatorname{Cov}(X, Y)}{\operatorname{Var}(X)} = \frac{\sum_{i}(X_i - \bar{X})(Y_i - \bar{Y})}{\sum_{i}(X_i - \bar{X})^{2}}.$$ The intercept is $\hat{\beta}_0 = \bar{Y} - \hat{\beta}_1 \bar{X}$.

What does OLS minimize, and what is the residual?

OLS minimizes the sum of squared residuals $\sum_i \hat{\varepsilon}_i^{2} = \sum_i (Y_i - \hat{Y}_i)^{2}$. The residual is $\hat{\varepsilon}_i = Y_i - \hat{Y}_i$, the difference between the observed and the fitted value.

Define $R^{2}$ and explain what it measures.

$R^{2} = \dfrac{\text{ESS}}{\text{TSS}} = 1 - \dfrac{\text{SSR}}{\text{TSS}}$, the fraction of total variation in $Y$ explained by the regression. It ranges from $0$ to $1$; higher values indicate a better in-sample fit.

Write the multiple regression model and explain how slope coefficients are interpreted.

$$Y_i = \beta_0 + \beta_1 X_{1i} + \beta_2 X_{2i} + \cdots + \beta_k X_{ki} + \varepsilon_i.$$ Each $\beta_j$ is the expected change in $Y$ for a one-unit increase in $X_j$, holding all other regressors constant (the partial/ceteris paribus effect).

List the key OLS (Gauss-Markov) assumptions for the classical linear regression model.

Linearity in parameters; the error has zero conditional mean $E[\varepsilon \mid X] = 0$; homoskedasticity (constant error variance); no autocorrelation of errors; no perfect multicollinearity among regressors. Under these, OLS is BLUE. Normality of errors is added for exact inference.

Why is adjusted $R^{2}$ preferred over $R^{2}$ in multiple regression?

$R^{2}$ never decreases when regressors are added, even useless ones. Adjusted $R^{2} = 1 - \dfrac{(1 - R^{2})(n - 1)}{n - k - 1}$ penalizes for the number of regressors $k$, so it can fall when an added variable does not improve fit enough to justify the lost degree of freedom.

Define multicollinearity and describe its consequences in OLS.

Multicollinearity is a high correlation among regressors. Perfect multicollinearity makes OLS estimates non-identifiable. Imperfect multicollinearity inflates coefficient standard errors, making individual coefficients statistically insignificant and unstable even when the overall regression fits well.

Define heteroskedasticity and explain its effect on OLS inference.

Heteroskedasticity is non-constant error variance, $\operatorname{Var}(\varepsilon_i \mid X) \neq \text{constant}$. OLS coefficients remain unbiased and consistent but are no longer efficient, and the usual standard errors are biased, invalidating t- and F-tests. Remedy: use robust (White) standard errors.

Explain omitted variable bias and the two conditions required for it.

Omitting a relevant variable biases the included coefficients if the omitted variable is both (1) a determinant of $Y$ and (2) correlated with an included regressor. This violates $E[\varepsilon \mid X] = 0$, making OLS biased and inconsistent.

How do you test the significance of a single regression coefficient?

Use a t-test: $t = \dfrac{\hat{\beta}_j - \beta_{j,0}}{\text{SE}(\hat{\beta}_j)}$, typically with $\beta_{j,0} = 0$. Compare $|t|$ to the critical value with $n - k - 1$ degrees of freedom (or check the p-value); reject if it exceeds the critical value.

What does the F-test assess in a multiple regression, and what is its null hypothesis?

The F-test assesses the joint significance of a group of (or all) slope coefficients. For overall significance the null is $H_0: \beta_1 = \beta_2 = \cdots = \beta_k = 0$. A large F-statistic relative to its critical value rejects the null that the regressors jointly have no explanatory power.

Define a stationary (covariance-stationary) time series.

A series is covariance-stationary if its mean, variance, and autocovariances are constant over time and the autocovariance depends only on the lag $h$, not on time $t$: $E[Y_t] = \mu$, $\operatorname{Var}(Y_t) = \sigma^{2}$, and $\operatorname{Cov}(Y_t, Y_{t-h}) = \gamma_h$.

Define white noise and autocorrelation.

White noise is a series of uncorrelated random variables with constant mean (usually zero) and constant variance, with no autocorrelation at any nonzero lag. Autocorrelation is the correlation of a series with its own lagged values: $\rho_h = \dfrac{\operatorname{Cov}(Y_t, Y_{t-h})}{\operatorname{Var}(Y_t)}$.

Write the AR(1) model and state its stationarity condition.

$$Y_t = c + \phi Y_{t-1} + \varepsilon_t,$$ where $\varepsilon_t$ is white noise. It is covariance-stationary if $|\phi| < 1$. Its unconditional mean is $\dfrac{c}{1 - \phi}$, and the autocorrelation function decays geometrically as $\phi^{h}$.

Write an MA(1) model and contrast MA and AR processes.

$$Y_t = \mu + \varepsilon_t + \theta\,\varepsilon_{t-1}.$$ An MA process expresses $Y_t$ as a finite linear combination of current and past white-noise shocks and always has finite memory (autocorrelations cut off after lag $q$). An AR process depends on past values of $Y$ and has infinite, geometrically decaying memory. An ARMA(p,q) combines both.

Compare the EWMA and GARCH(1,1) approaches to volatility estimation.

EWMA: $\sigma_t^{2} = \lambda\sigma_{t-1}^{2} + (1 - \lambda)r_{t-1}^{2}$, weighting recent returns more (no long-run mean). GARCH(1,1): $\sigma_t^{2} = \omega + \alpha r_{t-1}^{2} + \beta\sigma_{t-1}^{2}$, which adds a constant $\omega$ and mean-reverts to a long-run variance $\dfrac{\omega}{1 - \alpha - \beta}$ when $\alpha + \beta < 1$. EWMA is the special case with $\omega = 0$, $\alpha = 1-\lambda$, $\beta = \lambda$.

Define the Pearson correlation coefficient and contrast it with a copula's role in dependence modeling.

Pearson correlation $\rho_{X,Y} = \dfrac{\operatorname{Cov}(X, Y)}{\sigma_X \sigma_Y}$ measures only linear dependence and lies in $[-1, 1]$. A copula links marginal distributions into a joint distribution, capturing the full dependence structure (including nonlinear and tail dependence) separately from the marginals, per Sklar's theorem. This is important because correlation alone can miss joint extreme (tail) events.