CFA (Chartered Financial Analyst) Quantitative Methods Syllabus

What is the effective annual yield (EAY) from a holding period yield?

$$EAY = (1 + HPY)^{\frac{365}{t}} - 1$$ where $t$ is the number of days in the holding period.

Distinguish a population from a sample, and a parameter from a statistic.

A population is the complete set of all members of interest; a sample is a subset drawn from it. A parameter (e.g., $\mu$, $\sigma^{2}$) describes a population characteristic; a statistic (e.g., $\bar{X}$, $s^{2}$) describes a sample characteristic and estimates the parameter.

Name the four measurement scales in order of increasing information.

Nominal (categories, no order), Ordinal (ranked categories), Interval (equal differences, arbitrary zero, e.g. temperature), and Ratio (equal differences with a true zero, e.g. returns, money). Each higher scale supports the operations of the lower ones plus more.

What is a frequency distribution, and how is relative frequency computed?

A frequency distribution groups data into mutually exclusive intervals (bins) and tabulates the count (absolute frequency) in each. Relative frequency $= \frac{\text{absolute frequency of interval}}{\text{total observations}}$; cumulative relative frequency sums these up to a given interval.

Match common data-visualization tools to their purpose: histogram, frequency polygon, contingency table, scatter plot.

Histogram: distribution of one variable's frequencies. Frequency polygon: shape of a distribution via connected midpoints. Contingency (two-way) table: joint frequencies of two categorical variables. Scatter plot: joint relationship between two numerical variables.

Give the population and sample variance formulas.

Population variance: $$\sigma^{2} = \frac{1}{N}\sum_{i=1}^{N}(X_{i}-\mu)^{2}$$ Sample variance: $$s^{2} = \frac{1}{n-1}\sum_{i=1}^{n}(X_{i}-\bar{X})^{2}$$ The $n-1$ divisor makes $s^{2}$ an unbiased estimator.

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

$$CV = \frac{s}{\bar{X}}$$ It measures risk (dispersion) per unit of mean return—relative dispersion. A lower CV indicates less risk per unit of return, allowing comparison across data sets with different units or means.

Define skewness and interpret positive vs negative skew.

Skewness measures asymmetry of a distribution. Positive (right) skew has a long right tail with mean $>$ median $>$ mode. Negative (left) skew has a long left tail with mean $<$ median $<$ mode. A symmetric distribution has skewness of 0.

Define kurtosis and the three classifications relative to the normal distribution.

Kurtosis measures the peakedness and tail-weight of a distribution. Leptokurtic: more peaked, fatter tails (excess kurtosis $>0$); platykurtic: flatter, thinner tails (excess kurtosis $<0$); mesokurtic: same as normal (excess kurtosis $=0$). Excess kurtosis $=$ kurtosis $- 3$.

State Chebyshev's inequality.

For any distribution, the proportion of observations within $k$ standard deviations of the mean is at least $$1 - \frac{1}{k^{2}}, \quad k > 1$$ e.g., at least 75% lie within 2 SDs and at least 89% within 3 SDs.

Define the three approaches to estimating probability: empirical, a priori, and subjective.

Empirical probability is based on historical/observed frequency of an event. A priori probability is based on formal reasoning/logical analysis of the situation. Subjective probability is based on personal judgment or belief. Empirical and a priori are objective.

What is the multiplication rule for the joint probability of two events?

$$P(AB) = P(A\mid B)\,P(B) = P(B\mid A)\,P(A)$$ If $A$ and $B$ are independent, then $P(AB) = P(A)P(B)$.

State the addition rule for the probability of $A$ or $B$.

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

What is the total probability rule for an unconditional probability?

$$P(A) = \sum_{i=1}^{n} P(A\mid S_{i})\,P(S_{i})$$ where $S_{1},\dots,S_{n}$ are mutually exclusive and exhaustive scenarios. It weights conditional probabilities by the probabilities of each scenario.

State Bayes' formula and explain its purpose.

$$P(A\mid B) = \frac{P(B\mid A)\,P(A)}{P(B)}$$ It updates a prior probability $P(A)$ into a posterior probability given new information $B$, by combining the prior with the likelihood $P(B\mid A)$.

Distinguish independent events from mutually exclusive events.

Independent events: the occurrence of one does not affect the probability of the other, so $P(A\mid B)=P(A)$. Mutually exclusive events cannot occur together, so $P(AB)=0$. Two events that are mutually exclusive (with nonzero probabilities) cannot be independent.

Give the formulas for combinations and permutations.

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

Define expected value of a random variable.

$$E(X) = \sum_{i=1}^{n} P(x_{i})\,x_{i}$$ the probability-weighted average of all possible outcomes.

How is the variance of a random variable computed from its probability distribution?

$$\sigma^{2}(X) = \sum_{i=1}^{n} P(x_{i})\,[x_{i} - E(X)]^{2} = E(X^{2}) - [E(X)]^{2}$$

Define covariance and correlation between two random variables.

$$\text{Cov}(X,Y) = E\{[X-E(X)][Y-E(Y)]\}$$ Correlation standardizes covariance: $$\rho_{XY} = \frac{\text{Cov}(X,Y)}{\sigma_{X}\,\sigma_{Y}}, \quad -1 \leq \rho_{XY} \leq 1$$

What is the expected return and variance of a two-asset portfolio?

$$E(R_{p}) = w_{1}E(R_{1}) + w_{2}E(R_{2})$$ $$\sigma_{p}^{2} = w_{1}^{2}\sigma_{1}^{2} + w_{2}^{2}\sigma_{2}^{2} + 2w_{1}w_{2}\,\text{Cov}(R_{1},R_{2})$$

Distinguish a discrete from a continuous random variable, and a pdf from a cdf.

A discrete variable takes countable distinct values; a continuous variable takes any value in an interval. The probability function/pdf gives probabilities (or density) at outcomes; the cumulative distribution function (cdf) gives $F(x)=P(X\leq x)$, the probability of being at or below $x$.

Describe the discrete uniform and continuous uniform distributions.

Discrete uniform: each of $n$ outcomes is equally likely, $P(x)=1/n$. Continuous uniform over $[a,b]$: constant density $f(x)=\frac{1}{b-a}$, and $$P(x_{1}\leq X \leq x_{2}) = \frac{x_{2}-x_{1}}{b-a}$$

State the binomial probability function and its mean and variance.

$$P(X=x) = \binom{n}{x} p^{x}(1-p)^{n-x}$$ for $n$ independent Bernoulli trials with success probability $p$. Mean $=np$; variance $=np(1-p)$.

List the key properties of the normal distribution.

It is symmetric (skewness 0, kurtosis 3, mean = median = mode), fully described by $\mu$ and $\sigma^{2}$, has tails extending to $\pm\infty$, and a linear combination of normally distributed variables is also normal. About 68%, 95%, and 99% of observations lie within 1, 2, and 3 SDs.

What is the standard normal distribution and the z-score formula?

The standard normal is a normal distribution with $\mu=0$ and $\sigma=1$. Standardize any normal variable with $$Z = \frac{X - \mu}{\sigma}$$ giving the number of standard deviations $X$ is from the mean.

What confidence intervals correspond to roughly 90%, 95%, and 99% for a normal variable?

$$\mu \pm 1.645\sigma \;(90\%), \quad \mu \pm 1.96\sigma \;(95\%), \quad \mu \pm 2.58\sigma \;(99\%)$$ (z-values 1.645, 1.96, and 2.58 respectively.)

Define Roy's safety-first ratio and its relation to shortfall risk.

$$SFRatio = \frac{E(R_{p}) - R_{L}}{\sigma_{p}}$$ where $R_{L}$ is the threshold (minimum acceptable) return. The optimal portfolio maximizes the SFRatio, minimizing the probability that return falls below $R_{L}$ (shortfall risk). If returns are normal, $P(R_{p}<R_{L})=N(-SFRatio)$.

Describe the lognormal distribution and why it is used for asset prices.

A variable is lognormally distributed if its natural log is normally distributed. It is bounded below by 0 and skewed to the right, making it suitable for modeling asset prices (which cannot be negative). Continuously compounded returns are normally distributed when prices are lognormal.

What is the continuously compounded return for a holding period, and how does it relate to the HPR?

$$r_{cc} = \ln(1 + HPR) = \ln\!\left(\frac{P_{1}}{P_{0}}\right)$$ Continuously compounded returns are additive across periods, and the multi-period rate is $T$ times the single-period rate when i.i.d.

What is Monte Carlo simulation and what are its limitations?

Monte Carlo simulation generates many random outcomes from specified probability distributions to value complex instruments or analyze risk. Its limitations: it is only as good as the assumptions/distributions used, is computationally intensive, and provides statistical (not analytic) answers, offering no insight into cause-and-effect.

Compare simple random, stratified random, cluster, systematic, and convenience sampling.

Simple random: every item equally likely. Stratified: population divided into strata, random samples drawn proportionally from each. Cluster: population split into clusters, entire clusters sampled. Systematic: select every $k$th item. Convenience: easily accessible items (non-probability, prone to bias).

State the central limit theorem (CLT).

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

What is the standard error of the sample mean?

When the population standard deviation $\sigma$ is known: $$\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}}$$ When unknown, estimate it with the sample standard deviation: $s_{\bar{X}} = \frac{s}{\sqrt{n}}$. It measures the dispersion of the sampling distribution of the mean.

Distinguish a point estimate from a confidence interval, and list the desirable properties of an estimator.

A point estimate is a single value for a parameter; a confidence interval is a range expected to contain the parameter with a given probability. Good estimators are unbiased (expected value equals the parameter), efficient (smallest variance), and consistent (converges to the parameter as $n$ grows).

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

$$\bar{X} \pm (\text{reliability factor}) \times (\text{standard error})$$ Use $z$ when $\sigma$ is known (or large $n$): $\bar{X} \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$. Use $t$ when $\sigma$ is unknown and the population is approximately normal: $\bar{X} \pm t_{\alpha/2}\frac{s}{\sqrt{n}}$.

When should the t-distribution be used instead of the z-distribution, and how does it differ?

Use the t-distribution when the population variance is unknown (especially for small samples) and the population is approximately normal. It is symmetric and bell-shaped but has fatter tails than the normal; as degrees of freedom ($n-1$) increase, it approaches the standard normal.

Compare the bootstrap and jackknife resampling methods.

Both estimate the sampling distribution/standard error of a statistic. The bootstrap repeatedly draws samples of size $n$ with replacement from the original sample. The jackknife systematically leaves out one observation at a time (computing the statistic $n$ times). The bootstrap is more computationally intensive but more flexible.

Define data-snooping (data-mining) bias, sample selection bias, survivorship bias, look-ahead bias, and time-period bias.

Data-snooping: repeatedly testing data until a pattern appears. Sample selection: systematic exclusion of certain data. Survivorship: a form of selection bias where failed entities are excluded. Look-ahead: using information not available at the decision date. Time-period: results specific to a chosen, possibly unrepresentative, period.

Distinguish the null hypothesis from the alternative hypothesis.

The null hypothesis ($H_{0}$) is the statement being tested, generally containing equality (=, $\leq$, or $\geq$) and presumed true unless rejected. The alternative hypothesis ($H_{a}$) is what is concluded if $H_{0}$ is rejected and reflects the researcher's belief.

List the steps in hypothesis testing.

1) State $H_{0}$ and $H_{a}$; 2) identify the appropriate test statistic and its distribution; 3) specify the significance level $\alpha$; 4) state the decision rule (critical values); 5) compute the test statistic; 6) make the statistical decision (reject or fail to reject $H_{0}$); 7) make the economic/investment decision.

Define Type I and Type II errors and relate them to the significance level and power.

Type I error: rejecting a true $H_{0}$; its probability is the significance level $\alpha$. Type II error: failing to reject a false $H_{0}$; its probability is $\beta$. Power of a test $=1-\beta$ (probability of correctly rejecting a false null). Lowering $\alpha$ raises $\beta$ for a fixed sample size.

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

A two-tailed test examines whether a parameter differs from a value ($H_{a}: \theta \neq \theta_{0}$), splitting $\alpha$ into both tails. A one-tailed test examines a direction ($H_{a}: \theta > \theta_{0}$ or $\theta < \theta_{0}$), placing all of $\alpha$ in one tail.

Define the p-value and give its decision rule.

The p-value is the smallest significance level at which the null hypothesis can be rejected—the probability of obtaining a test statistic at least as extreme as observed if $H_{0}$ is true. Reject $H_{0}$ if p-value $< \alpha$.

What is the test statistic for testing a single population mean?

With unknown population variance (most common): $$t = \frac{\bar{X} - \mu_{0}}{s/\sqrt{n}}$$ with $n-1$ degrees of freedom. If $\sigma$ is known, use the z-statistic $z = \frac{\bar{X}-\mu_{0}}{\sigma/\sqrt{n}}$.

Distinguish statistical significance from economic significance.

Statistical significance means a result is unlikely due to chance (e.g., p-value $<\alpha$). Economic significance means the result is large enough to matter for decisions after accounting for transaction costs, taxes, and risk. A statistically significant result may lack economic significance.

What test statistics are used for the variance of a normal population and for comparing two variances?

For a single variance, use the chi-square statistic $$\chi^{2} = \frac{(n-1)s^{2}}{\sigma_{0}^{2}}$$ with $n-1$ degrees of freedom. To compare two population variances, use the F-statistic $$F = \frac{s_{1}^{2}}{s_{2}^{2}}$$ (larger variance in the numerator).