Chartered Financial Analyst (CFA) Quantitative Methods Syllabus

Write the formula for the variance of a two-asset portfolio.

$$\sigma_p^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1 w_2 \rho_{1,2}\sigma_1\sigma_2$$ Equivalently the covariance term is $2 w_1 w_2 Cov(R_1,R_2)$. Diversification benefit increases as $\rho_{1,2}$ decreases.

Distinguish counting principles: combinations versus permutations.

Combinations (order does not matter): $$\binom{n}{r} = \frac{n!}{(n-r)!\,r!}$$ Permutations (order matters): $$nPr = \frac{n!}{(n-r)!}$$ Use the multiplication rule of counting for sequential independent choices; labeling formula $\frac{n!}{n_1!n_2!\cdots n_k!}$ for assigning to groups.

Describe the binomial distribution: its parameters, mean, and variance.

Models the number of successes $X$ in $n$ independent Bernoulli trials with constant success probability $p$. $$P(X=x) = \binom{n}{x}p^x(1-p)^{n-x}$$ Mean $= np$; Variance $= np(1-p)$.

What are the defining characteristics of the normal distribution?

Symmetric, bell-shaped; fully described by mean $\mu$ and variance $\sigma^2$; skewness $=0$, excess kurtosis $=0$. A linear combination of normally distributed variables is also normal. Approximately 68% of observations fall within $\pm1\sigma$, 95% within $\pm2\sigma$ (1.96 exactly), and 99% within $\pm2.58\sigma$.

How do you standardize a normal random variable, and what is the z-score?

$$Z = \frac{X - \mu}{\sigma}$$ The standard normal $Z$ has mean 0 and standard deviation 1. The z-score expresses how many standard deviations an observation lies from the mean, allowing probability lookups from the standard normal table.

Define Roy's safety-first ratio and explain how it is used.

$$SFRatio = \frac{E(R_P) - R_L}{\sigma_P}$$ where $R_L$ is the minimum (threshold) return. The optimal portfolio maximizes the SFRatio—minimizing the probability that return falls below $R_L$. It mirrors the Sharpe ratio when $R_L$ equals the risk-free rate.

What is the lognormal distribution and why is it used to model asset prices?

A variable is lognormally distributed if its natural log is normally distributed. It is bounded below by 0 and positively skewed, making it suitable for asset prices (which can't be negative). If continuously compounded returns are normal, prices are lognormal.

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

Symmetric, bell-shaped, defined by degrees of freedom (df). It has fatter tails than the normal distribution, giving more conservative (wider) confidence intervals. As df increases, it approaches the standard normal. Used when the population variance is unknown and/or the sample is small.

State the Central Limit Theorem and its key implication for the sampling distribution of the mean.

For a population with mean $\mu$ and finite variance $\sigma^2$, the sampling distribution of the sample mean $\bar{X}$ approaches a normal distribution as sample size $n$ grows large (typically $n \geq 30$), regardless of the population's distribution, with mean $\mu$ and variance $\frac{\sigma^2}{n}$.

What is the standard error of the sample mean, with known and unknown population variance?

Known population standard deviation: $$\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}}$$ Unknown (estimated from sample): $$s_{\bar{X}} = \frac{s}{\sqrt{n}}$$ The standard error decreases as $n$ increases.

Distinguish probability sampling methods: simple random, stratified, cluster, and systematic sampling.

Simple random: every item equally likely. Stratified: population divided into strata, random samples drawn from each proportionally (reduces variability). Cluster: population split into clusters, entire clusters sampled. Systematic: select every $k$th item. Non-probability methods (convenience, judgmental) risk bias.

Construct a confidence interval for the population mean and identify the reliability factors.

$$\bar{X} \pm (\text{reliability factor}) \times \frac{s}{\sqrt{n}}$$ For 90% CI use $z = 1.645$, 95% use $z = 1.96$, 99% use $z = 2.58$ (normal, known variance). Use the $t$-distribution reliability factor when variance is unknown (especially small samples).

Compare data snooping bias, sample selection bias, look-ahead bias, and time-period bias.

Data snooping: repeatedly mining the same data until a pattern appears by chance. Sample selection (incl. survivorship) bias: systematically excluding part of the population. Look-ahead bias: using information not available at the time of analysis. Time-period bias: results specific to a particular time period that don't generalize.

List the steps in hypothesis testing.

1) State the null ($H_0$) and alternative ($H_a$) hypotheses; 2) Identify the appropriate test statistic and its distribution; 3) Specify the significance level $\alpha$; 4) Collect data and compute the test statistic; 5) Make the statistical decision (reject or fail to reject $H_0$ by comparing to critical value / p-value); 6) State the economic/investment conclusion.

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

Type I error: rejecting a true null hypothesis (probability $= \alpha$, the significance level). Type II error: failing to reject a false null (probability $= \beta$). Power of a test $= 1 - \beta$, the probability of correctly rejecting a false null. There is a trade-off: lowering $\alpha$ raises $\beta$.

What is a p-value, and how is the decision rule stated using it?

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

Which test statistic is used to test a single population mean with unknown variance, and what is its formula?

A $t$-test with $n-1$ degrees of freedom: $$t = \frac{\bar{X} - \mu_0}{s / \sqrt{n}}$$ where $\mu_0$ is the hypothesized mean. If the population variance is known and the sample is large, a $z$-test may be used instead.

What test statistics are used for hypotheses about a single variance and the equality of two variances?

Single variance: chi-square test, $\chi^2 = \frac{(n-1)s^2}{\sigma_0^2}$ with $n-1$ df (asymmetric, non-negative). Equality of two variances: F-test, $F = \frac{s_1^2}{s_2^2}$ (larger variance in the numerator) with $n_1-1$ and $n_2-1$ df.

When do you use a paired comparisons (paired t) test versus a difference-in-means test?

Use a paired comparisons test when samples are dependent (e.g., before/after on the same subjects, or correlated series)—it tests the mean of the differences. Use a difference-in-means (two-sample) test when the two samples are independent. Using the wrong one ignores or wrongly assumes correlation.

What is a nonparametric test and when is it appropriate?

A nonparametric test makes minimal assumptions about the population distribution (or tests something other than a parameter). Appropriate when data don't meet distributional assumptions, when ranks/ordinal data are involved, or when testing for randomness (e.g., Spearman rank correlation, runs test).

Write the simple linear regression model and identify the dependent and independent variables.

$$Y_i = b_0 + b_1 X_i + \varepsilon_i$$ $Y$ is the dependent (explained) variable, $X$ is the independent (explanatory) variable, $b_0$ is the intercept, $b_1$ is the slope, and $\varepsilon_i$ is the random error term. Ordinary least squares estimates $b_0, b_1$ by minimizing $\sum \hat{\varepsilon}_i^2$.

List the key assumptions of the classical (simple) linear regression model.

1) Linearity: the relationship between $Y$ and $X$ is linear in coefficients; 2) Homoskedasticity: constant error variance; 3) Independence: errors are uncorrelated across observations; 4) Normality: errors are normally distributed. (X is not random / not perfectly correlated with the error.)

How is the slope coefficient in simple linear regression computed?

$$\hat{b}_1 = \frac{Cov(X,Y)}{Var(X)} = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sum (X_i - \bar{X})^2}$$ and the intercept $\hat{b}_0 = \bar{Y} - \hat{b}_1 \bar{X}$, so the regression line passes through $(\bar{X}, \bar{Y})$.

Define the coefficient of determination ($R^2$) and the decomposition of total variation.

$$R^2 = \frac{SSR}{SST} = 1 - \frac{SSE}{SST}$$ where $SST = SSR + SSE$ (Total = Regression/explained + Error/unexplained). $R^2$ is the fraction of total variation in $Y$ explained by the regression. In simple regression, $R^2 = r^2$ (the squared correlation).

What is the standard error of the estimate (SEE) and how does it relate to the F-statistic in regression?

$$SEE = \sqrt{\frac{SSE}{n-2}} = \sqrt{MSE}$$ a measure of the typical regression residual (lower is better fit). The F-statistic tests overall significance: $F = \frac{MSR}{MSE} = \frac{SSR/k}{SSE/(n-k-1)}$; in simple regression $k=1$.

How do you test the statistical significance of a regression slope coefficient?

$$t = \frac{\hat{b}_1 - b_1}{s_{\hat{b}_1}}$$ with $n-2$ degrees of freedom (testing $b_1 = 0$). Reject if $|t|$ exceeds the critical value or p-value $< \alpha$. A confidence interval is $\hat{b}_1 \pm t_c \, s_{\hat{b}_1}$.

List common functional forms used to model nonlinear relationships in regression.

Log-lin model: $\ln Y = b_0 + b_1 X$ (constant relative change in Y). Lin-log model: $Y = b_0 + b_1 \ln X$. Log-log (double log): $\ln Y = b_0 + b_1 \ln X$ (slope is an elasticity). Choice is guided by the pattern of residuals and goodness of fit.

In multiple regression, how is an estimated slope coefficient interpreted, and what is adjusted $R^2$?

Each slope $b_j$ is the expected change in $Y$ for a one-unit change in $X_j$, holding all other independent variables constant. Adjusted $R^2$ penalizes added variables: $$\bar{R}^2 = 1 - \left(\frac{n-1}{n-k-1}\right)(1 - R^2)$$ It can decrease when an irrelevant variable is added.

Compare the three regression assumption violations—heteroskedasticity, serial correlation, and multicollinearity—including a detection method for each.

Heteroskedasticity: non-constant error variance; detected by Breusch–Pagan test; biases standard errors. Serial correlation: correlated errors over time; detected by Durbin–Watson or Breusch–Godfrey; biases standard errors (and coefficients if lagged dependent variable). Multicollinearity: high correlation among independents; detected by high $R^2$/significant F but insignificant t-stats, or high VIF ($>5$ or $10$); inflates coefficient standard errors.

What conditions must a time series satisfy to be covariance stationary, and what is a unit root?

Covariance stationarity requires constant mean, constant variance, and constant covariance with lagged values over time. A unit root means the lag coefficient equals 1 (e.g., a random walk), making the series nonstationary; tested with the Dickey–Fuller test. Nonstationary series are often first-differenced to achieve stationarity before modeling.