CFA Quantitative Methods Syllabus

What is a key mathematical property of the arithmetic mean regarding deviations?

The sum of deviations from the arithmetic mean is always zero: $$\sum_{i=1}^{N}(X_i - \bar{X}) = 0$$

Define the geometric mean and state when it is used.

$$G = \sqrt[N]{X_1 \cdot X_2 \cdots X_N}$$ (requires $X_i \geq 0$). It is used to compute compound growth rates over time, e.g., average returns. For returns: $1+G = \sqrt[N]{\prod(1+R_i)}$.

Define the harmonic mean and a typical application.

$$\bar{X}_H = \frac{N}{\sum_{i=1}^{N} \frac{1}{X_i}}$$ It is used for averaging ratios such as cost per share in dollar-cost averaging.

State the relationship (inequality) among the arithmetic, geometric, and harmonic means for non-equal positive values.

$$\bar{X}_H \leq G \leq \bar{X}$$ They are equal only when all observations are identical.

How is the median determined, and what is its main advantage over the mean?

The median is the middle value of data ordered ascending (average of the two middle values if $N$ is even). Its advantage: it is not affected by extreme values (outliers).

What is the mode, and can a data set have more than one?

The mode is the most frequently occurring value. A data set can be unimodal, bimodal, or multimodal, or have no mode if no value repeats.

What is a weighted mean and its formula?

$$\bar{X}_w = \sum_{i=1}^{N} w_i X_i \quad \text{where} \quad \sum_{i=1}^{N} w_i = 1$$ Each observation is weighted by its proportional importance (e.g., portfolio weights).

How is the position of a percentile located in ordered data?

$$L_y = (N+1)\frac{y}{100}$$ where $L_y$ is the location of the $y$th percentile and $N$ is the number of observations; interpolate between adjacent values if $L_y$ is not an integer.

What does the range measure and what is its formula?

The range is the simplest measure of dispersion: $$\text{Range} = \text{Maximum value} - \text{Minimum value}$$ It only uses two data points and ignores the distribution between them.

What is the Mean Absolute Deviation (MAD)?

$$MAD = \frac{1}{N}\sum_{i=1}^{N}\left|X_i - \bar{X}\right|$$ It is the average of the absolute deviations from the mean.

Write the formula for population variance.

$$\sigma^2 = \frac{1}{N}\sum_{i=1}^{N}(X_i - \mu)^2$$ where $\mu$ is the population mean and $N$ the population size.

Write the formula for sample variance and explain why it divides by $n-1$.

$$s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(X_i - \bar{X})^2$$ Dividing by $n-1$ (degrees of freedom) corrects downward bias, giving an unbiased estimator of the population variance.

How is standard deviation related to variance?

Standard deviation is the positive square root of the variance: population $\sigma = \sqrt{\sigma^2}$ and sample $s = \sqrt{s^2}$. It is expressed in the same units as the data.

What is the coefficient of variation (CV) and what does it measure?

$$CV = \frac{s}{\bar{X}}$$ It measures the amount of risk (dispersion) per unit of return/mean — a unit-free relative measure useful for comparing data sets with different units or means. Lower CV means less risk per unit of mean.

State Chebyshev's inequality and give the proportion for $k=2$ and $k=3$.

For any distribution, the proportion of observations within $k$ standard deviations of the mean is at least $1 - \frac{1}{k^2}$ (for $k>1$). For $k=2$: at least $75\%$; for $k=3$: at least $\approx 88.9\%$.

What does the Sharpe ratio measure, and what is its formula?

It measures excess return per unit of total risk (standard deviation): $$\text{Sharpe} = \frac{\bar{R}_p - R_f}{s_p}$$ where $\bar{R}_p$ is portfolio return, $R_f$ the risk-free rate, and $s_p$ the standard deviation of returns. Higher is better.

Distinguish between the empirical (objective) and a priori probability.

A priori probability is based on formal reasoning/deduction (e.g., a fair die). Empirical probability is estimated from historical data/frequency of occurrence. Both are objective; a subjective probability is based on personal judgment.

State the multiplication rule for the joint probability of two events.

$$P(AB) = P(A \mid B) \cdot P(B)$$ The joint probability equals the conditional probability of $A$ given $B$ times the unconditional probability of $B$.

State the addition rule for the probability that A or B occurs.

$$P(A \text{ or } B) = P(A) + P(B) - P(AB)$$ For mutually exclusive events $P(AB)=0$, so it reduces to $P(A)+P(B)$.

What condition defines two events as independent in probability terms?

Events $A$ and $B$ are independent if $P(A \mid B) = P(A)$ (and equivalently $P(B\mid A)=P(B)$). Then their joint probability is $P(AB) = P(A)P(B)$.

State the total probability rule for an event A given mutually exclusive, exhaustive scenarios $S_i$.

$$P(A) = \sum_{i=1}^{n} P(A \mid S_i) P(S_i)$$ It computes an unconditional probability as a weighted average of conditional probabilities.

State Bayes' formula for updating probabilities.

$$P(A \mid B) = \frac{P(B \mid A) \cdot P(A)}{P(B)}$$ It updates a prior probability $P(A)$ into a posterior given new information $B$.

Define covariance and correlation between two random variables.

Covariance: $Cov(R_i,R_j) = E\{[R_i - E(R_i)][R_j - E(R_j)]\}$. Correlation standardizes it: $$\rho_{ij} = \frac{Cov(R_i,R_j)}{\sigma_i \sigma_j}, \quad -1 \leq \rho_{ij} \leq 1$$

Give the formulas for the number of permutations and combinations of $n$ items taken $r$ at a time.

Permutations (order matters): $$_nP_r = \frac{n!}{(n-r)!}$$ Combinations (order does not matter): $$_nC_r = \binom{n}{r} = \frac{n!}{(n-r)!\,r!}$$

What is the difference between the null hypothesis and the alternative hypothesis?

The null hypothesis ($H_0$) is the statement to be tested, always containing the equality (=, $\leq$, or $\geq$) and presumed true unless rejected. The alternative ($H_a$) is what the researcher wants to support and is accepted only if $H_0$ is rejected.

When do you use a two-tailed test versus a one-tailed test, with example hypotheses?

Two-tailed when testing whether a parameter differs from a value: $H_0: \theta = \theta_0$ vs $H_a: \theta \neq \theta_0$. One-tailed when testing direction: $H_0: \theta \leq \theta_0$ vs $H_a: \theta > \theta_0$ (or the reverse).

Define a Type I error and relate it to the significance level.

A Type I error is rejecting a true null hypothesis (a false positive). Its probability equals the significance level $\alpha$, which is chosen by the researcher (e.g., 0.05).

Define a Type II error and the power of a test.

A Type II error is failing to reject a false null hypothesis (a false negative); its probability is $\beta$. The power of a test is $1 - \beta$, the probability of correctly rejecting a false null.

Explain the trade-off between Type I and Type II error probabilities.

For a fixed sample size, decreasing $\alpha$ (Type I) increases $\beta$ (Type II), and vice versa. The only way to reduce both simultaneously is to increase the sample size $n$, which also raises power $(1-\beta)$.

What is a test statistic, and what is its general standardized form?

A test statistic standardizes the difference between a sample statistic and the hypothesized parameter: $$\text{Test statistic} = \frac{\text{sample statistic} - \text{hypothesized value}}{\text{standard error}}$$ It is compared to a critical value to decide on $H_0$.

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

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