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

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 $\rho = 0$.

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

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

State the binomial PMF and its mean and variance.

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

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, $E[X] = \operatorname{Var}(X) = \lambda$.

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 $\lambda = np$.

List the key properties of the normal distribution $N(\mu, \sigma^2)$.

Symmetric and bell-shaped; fully described by mean $\mu$ and variance $\sigma^2$; skewness $= 0$, kurtosis $= 3$; sums of independent normals are normal. Empirical rule: about $68\%$, $95\%$, $99.7\%$ of data lie within $\pm 1, \pm 2, \pm 3$ standard deviations.

How do you standardize a normal random variable, and what is the resulting distribution?

$$Z = \frac{X - \mu}{\sigma} \sim N(0, 1).$$ The standard normal has mean $0$ and variance $1$; $Z$ counts how many standard deviations $X$ is from its mean.

Define the lognormal distribution and a key application in finance.

$X$ is lognormal if $\ln(X) \sim N(\mu, \sigma^2)$. It is positively skewed and bounded below by $0$, making it suitable for modeling asset prices (which cannot be negative) when continuously compounded returns are normal.

Describe the Student's t-distribution and how it compares to the normal.

Symmetric and bell-shaped but with fatter tails than the normal, governed by degrees of freedom $df$. It has excess kurtosis ($>0$) for small $df$. As $df \to \infty$, the t-distribution converges to the standard normal.

When is the Student's t-distribution used instead of the normal (z) distribution?

When estimating the mean of a normally distributed population with an unknown population variance and a small sample size. The fatter tails account for the extra uncertainty from estimating $\sigma$ with the sample $s$.

Describe the chi-squared distribution and one common use.

The sum of squares of $k$ independent standard normal variables: $\chi^2_k = \sum_{i=1}^{k} Z_i^2$. It is asymmetric, non-negative, right-skewed (skew decreasing as $df$ rises). Used to test hypotheses about a population variance and for goodness-of-fit tests.

Describe the F-distribution and how it relates to the chi-squared distribution.

The ratio of two independent chi-squared variables each divided by their degrees of freedom: $$F = \frac{\chi^2_1 / d_1}{\chi^2_2 / d_2}.$$ Non-negative and right-skewed; used to test the equality of two population variances and in regression (ANOVA).

Describe the exponential distribution: its PDF, mean, and a key property.

Models waiting time between Poisson events: $f(x) = \lambda e^{-\lambda x}$ for $x \geq 0$. Mean $= \dfrac{1}{\lambda}$, variance $= \dfrac{1}{\lambda^2}$. It is memoryless: $P(X > s + t \mid X > s) = P(X > t)$.

State the Law of Large Numbers (LLN).

As the sample size $n$ increases, the sample mean $\bar{X}$ converges to the population mean $\mu$: $$\bar{X}_n \xrightarrow{n \to \infty} \mu.$$ Larger samples yield estimates that get arbitrarily close to the true mean.

State the Central Limit Theorem (CLT).

For a sample of size $n$ drawn from a population with mean $\mu$ and finite variance $\sigma^2$, the sampling distribution of the sample mean approaches a normal distribution as $n$ grows: $$\bar{X} \xrightarrow{\text{approx}} N\!\left(\mu, \frac{\sigma^2}{n}\right),$$ regardless of the population's shape (commonly $n \geq 30$).

What is the standard error of the sample mean and how does sample size affect it?

$$SE = \frac{\sigma}{\sqrt{n}} \quad (\text{or } \frac{s}{\sqrt{n}} \text{ if } \sigma \text{ unknown}).$$ It measures the dispersion of the sampling distribution of $\bar{X}$ and decreases as $n$ increases (proportional to $1/\sqrt{n}$).

What three desirable properties should a good point estimator have?

Unbiasedness: $E[\hat{\theta}] = \theta$. Efficiency: smallest variance among unbiased estimators. Consistency: $\hat{\theta} \to \theta$ as $n \to \infty$. (Some texts add linearity, giving BLUE.)

What is the difference between a point estimate and an interval estimate?

A point estimate is a single value (e.g., $\bar{X}$) approximating the parameter. An interval estimate (confidence interval) is a range of values, with an associated confidence level, that is likely to contain the true parameter.

Give the general form of a confidence interval for the population mean.

$$\bar{X} \pm (\text{reliability factor}) \times \frac{s}{\sqrt{n}}.$$ The reliability factor is a $z$-value (known $\sigma$ or large $n$) or a $t$-value (unknown $\sigma$, small $n$).

What reliability factors ($z$) correspond to 90%, 95%, and 99% confidence intervals for a two-sided normal?

$90\% \to z = 1.645$; $95\% \to z = 1.96$; $99\% \to z = 2.58$ (more precisely $2.576$). Higher confidence requires a wider interval.

Interpret what a 95% confidence interval actually means.

If the sampling procedure were repeated many times, about $95\%$ of the constructed intervals would contain the true population parameter. It is a statement about the long-run procedure, not a $95\%$ probability that one specific interval contains the parameter.

Define the null hypothesis $H_0$ and alternative hypothesis $H_a$.

$H_0$ is the statement being tested, typically a position of 'no effect' or equality, and always contains the equality (e.g., $=, \leq, \geq$). $H_a$ is the conclusion accepted if $H_0$ is rejected; it reflects what the researcher seeks to support.

Distinguish a one-tailed test from a two-tailed test.

Two-tailed tests an inequality, $H_0: \mu = \mu_0$ vs. $H_a: \mu \neq \mu_0$, splitting $\alpha$ into both tails. One-tailed tests direction, e.g. $H_0: \mu \leq \mu_0$ vs. $H_a: \mu > \mu_0$, placing all of $\alpha$ in one tail.

Define Type I and Type II errors in hypothesis testing.

Type I error: rejecting a true $H_0$ (false positive); its probability is the significance level $\alpha$. Type II error: failing to reject a false $H_0$ (false negative); its probability is $\beta$.

What is the power of a statistical test?

Power $= 1 - \beta$, the probability of correctly rejecting a false null hypothesis. Power increases with larger sample size, larger effect size, lower variance, and a higher $\alpha$.

Explain the trade-off between Type I and Type II errors.

For a fixed sample size, decreasing $\alpha$ (less chance of a Type I error) increases $\beta$ (more chance of a Type II error), and vice versa. Only increasing the sample size can reduce both simultaneously.

Define the p-value and state the decision rule using it.

The p-value is the smallest significance level at which $H_0$ can be rejected — the probability of obtaining a test statistic at least as extreme as observed, assuming $H_0$ is true. Decision rule: reject $H_0$ if p-value $< \alpha$.

What is a test statistic, and what is the general form for testing a population mean?

A standardized value computed from sample data and compared to a critical value. General form: $$\text{Test statistic} = \frac{\hat{\theta} - \theta_0}{\text{standard error}} = \frac{\bar{X} - \mu_0}{s / \sqrt{n}}.$$

When do you use a z-test versus a t-test for a population mean?

Use a z-test when the population variance is known (or $n$ is large by the CLT). Use a t-test (with $n-1$ degrees of freedom) when the population variance is unknown and the sample is small from an approximately normal population.

State the decision rule for a two-tailed test using the test statistic and critical value.

Reject $H_0$ if the absolute value of the test statistic exceeds the critical value, i.e. $|\text{test stat}| > z_{\alpha/2}$ (or $t_{\alpha/2, \, n-1}$). Otherwise, fail to reject $H_0$. The rejection region lies in both tails, each containing $\alpha/2$.