Chartered Financial Analyst (CFA) Quantitative Methods Syllabus

Quantitative Methods syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Quantitative Methods in Chartered Financial Analyst (CFA), not a summary of it.

1. 1

   Time Value of Money and Rates of Return

   3 topics

   • Interest Rates and the Time Value of Money
     • Nominal vs. real interest rates and risk premiums
     • Future value and present value of single cash flows and annuities
     • Effective annual rate and compounding frequency
   • Measures of Investment Return
     • Holding period, money-weighted, and time-weighted returns
     • Annualized returns and gross vs. net returns
     • Arithmetic vs. geometric mean returns
   • The Time Value of Money in Finance
     • Bond and equity valuation as discounted cash flows
     • Implied returns and yields
2. 2

   Statistical Concepts and Distributions

   4 topics

   • Organizing and Describing Data
     • Frequency distributions and visualization
     • Measures of central tendency and dispersion
     • Skewness, kurtosis, and the coefficient of variation
   • Probability Concepts
     • Conditional and joint probabilities, Bayes' formula
     • Expected value, variance, covariance, and correlation
     • Counting principles: combinations and permutations
   • Common Probability Distributions
     • Uniform, binomial, and normal distributions
     • Lognormal distribution and continuous compounding
     • Student's t, chi-square, and F distributions
   • Sampling and Estimation
     • Sampling methods and the central limit theorem
     • Confidence intervals and standard error
     • Resampling: bootstrap and jackknife
3. 3

   Hypothesis Testing and Linear Regression

   3 topics

   • Hypothesis Testing
     • Null and alternative hypotheses, Type I and II errors
     • Tests of a single mean, variance, and correlation
     • Parametric vs. nonparametric tests and p-values
   • Simple Linear Regression
     • Ordinary least squares assumptions and estimation
     • ANOVA, coefficient of determination, and standard error of estimate
     • Hypothesis tests on slope and prediction intervals
   • Multiple Regression and Model Issues
     • Multiple regression interpretation and adjusted R-squared
     • Heteroskedasticity, serial correlation, and multicollinearity
     • Model misspecification and dummy variables
4. 4

   Time Series and Machine Learning

   3 topics

   • Time-Series Analysis
     • Trend models and autoregressive (AR) models
     • Unit roots, random walks, and cointegration
     • Seasonality and ARCH models
   • Machine Learning
     • Supervised vs. unsupervised learning
     • Overfitting, regularization (LASSO), and penalized regression
     • Classification and regression trees, neural networks
   • Big Data Projects and Simulation
     • Data preparation, wrangling, and feature engineering
     • Model training and performance evaluation metrics
     • Monte Carlo simulation in finance

Quantitative Methods flashcards for Chartered Financial Analyst (CFA)

19 of 50 cards from the Quantitative Methods deck — real questions with worked answers.

1. What are the three components that make up a nominal (quoted) interest rate, and what is the formula?

   Nominal rate = real risk-free rate + inflation premium + risk premiums (default, liquidity, maturity). Approximately: $r_{nominal} \approx r_{real} + \text{inflation premium} + \text{risk premiums}$. The risk-free rate itself = real risk-free rate + expected inflation.

2. Write the formula for the future value of a single cash flow compounded at a stated annual rate $r$ with $m$ compounding periods per year over $N$ years.

   $$FV = PV\left(1 + \frac{r_s}{m}\right)^{mN}$$ where $r_s$ is the stated annual rate and $m$ is the number of compounding periods per year.

3. What is the formula for the effective annual rate (EAR) given a stated annual rate $r_s$ compounded $m$ times per year, and for continuous compounding?

   Periodic compounding: $$EAR = \left(1 + \frac{r_s}{m}\right)^{m} - 1$$ Continuous compounding: $$EAR = e^{r_s} - 1$$

4. Distinguish between the holding period return (HPR), arithmetic mean return, and geometric mean return.

   HPR over one period: $\frac{P_1 - P_0 + D_1}{P_0}$. Arithmetic mean = simple average of returns, $\frac{1}{n}\sum R_i$. Geometric mean = compound rate, $\left[\prod_{i=1}^{n}(1+R_i)\right]^{1/n} - 1$. The geometric mean is always $\leq$ arithmetic mean and better reflects compound growth over multiple periods.

5. What is the money-weighted rate of return and how does it differ from the time-weighted rate of return?

   The money-weighted return is the IRR of a portfolio's cash flows—it sets the NPV of all inflows/outflows to zero and is sensitive to the timing and size of contributions/withdrawals. The time-weighted return measures the compound growth of one unit of money, removing the effect of cash flow timing, and is the industry standard for evaluating manager performance.

6. What is the harmonic mean and a typical financial application?

   $$\bar{X}_H = \frac{n}{\sum_{i=1}^{n} \frac{1}{X_i}}$$ It is used to compute the average cost per share under dollar-cost averaging (e.g., averaging purchase prices). Relationship: harmonic mean $\leq$ geometric mean $\leq$ arithmetic mean (unless all values equal).

7. Define a money market yield (CD-equivalent yield) versus a bank discount yield for a T-bill.

   Bank discount yield: $r_{BD} = \frac{D}{F} \times \frac{360}{t}$ (based on face value $F$, discount $D$, 360 days). Money market yield: $r_{MM} = \frac{360 \times r_{BD}}{360 - (t \times r_{BD})}$, which is based on purchase price and is comparable to other money market instruments.

8. What is the formula relating present value and future value for an ordinary annuity?

   PV of an ordinary annuity: $$PV = A \times \frac{1 - (1+r)^{-N}}{r}$$ FV of an ordinary annuity: $$FV = A \times \frac{(1+r)^{N} - 1}{r}$$ where $A$ is the periodic payment, paid at the end of each period.

9. How do you value a perpetuity, and how is it adapted for a constantly growing perpetuity (Gordon growth)?

   Level perpetuity: $$PV = \frac{A}{r}$$ Growing perpetuity (Gordon growth model): $$PV = \frac{A_1}{r - g}$$ where $g$ is the constant growth rate and requires $r > g$.

10. What is the no-arbitrage relationship between spot rates and forward rates (one-period forward)?

    $$(1+z_2)^2 = (1+z_1)(1 + f_{1,1})$$ where $z_1, z_2$ are spot rates for 1 and 2 years and $f_{1,1}$ is the one-year forward rate one year from now. Generally: $(1+z_k)^k = (1+z_j)^j (1+f_{j,k-j})^{k-j}$.

11. Compare a population parameter with a sample statistic, and give the symbols for population vs sample mean and variance.

    A parameter describes a population (fixed, usually unknown); a statistic describes a sample (computed, varies). Population mean $\mu$ vs sample mean $\bar{X}$. Population variance $\sigma^2 = \frac{\sum(X_i - \mu)^2}{N}$ vs sample variance $s^2 = \frac{\sum(X_i - \bar{X})^2}{n-1}$ (uses $n-1$ for unbiasedness).

12. Define the four measurement scales (levels of measurement) used to classify data.

    Nominal (categories, no order), Ordinal (ranked but unequal intervals), Interval (equal intervals, arbitrary zero, e.g., temperature in Celsius), and Ratio (equal intervals and a true zero, e.g., returns, prices). Ratio is the most informative.

13. What is a frequency distribution and how are relative frequency and cumulative relative frequency defined?

    A frequency distribution tabulates observations into mutually exclusive intervals (bins). Relative frequency = (absolute frequency of a bin) / (total observations). Cumulative relative frequency = sum of relative frequencies up to and including a given bin.

14. Define skewness and interpret positive vs negative skew for a return distribution.

    Skewness measures asymmetry. Positive (right) skew: long right tail, mean > median > mode, more frequent small losses with occasional large gains. Negative (left) skew: long left tail, mean < median < mode, more frequent small gains with occasional large losses. Sample skewness $\approx \frac{1}{n}\frac{\sum(X_i-\bar{X})^3}{s^3}$.

15. Define kurtosis and excess kurtosis, and classify leptokurtic, platykurtic, and mesokurtic distributions.

    Kurtosis measures tail weight/peakedness. Excess kurtosis = kurtosis $- 3$. Leptokurtic: excess kurtosis $> 0$ (fat tails, more outlier risk). Platykurtic: excess kurtosis $< 0$ (thin tails). Mesokurtic: excess kurtosis $= 0$ (like the normal distribution).

16. What is the coefficient of variation (CV) and why is it useful?

    $$CV = \frac{s}{\bar{X}}$$ It measures risk (dispersion) per unit of mean return and is a unitless, scale-free measure that allows comparison of relative dispersion across data sets with different means or units. Lower CV indicates less risk per unit of return.

17. Define covariance and the correlation coefficient between two random variables.

    Covariance: $Cov(X,Y) = E[(X-\mu_X)(Y-\mu_Y)]$. Correlation: $$\rho_{X,Y} = \frac{Cov(X,Y)}{\sigma_X \sigma_Y}$$ Correlation is bounded $-1 \leq \rho \leq +1$, is unitless, and measures the strength of the linear relationship.

18. State the multiplication and addition rules of probability.

    Multiplication rule (joint probability): $P(AB) = P(A \mid B)P(B)$. Addition rule: $P(A \text{ or } B) = P(A) + P(B) - P(AB)$. For mutually exclusive events $P(AB)=0$; for independent events $P(AB)=P(A)P(B)$.

19. State the total probability rule and Bayes' formula.

    Total probability: $P(A) = \sum_{i} P(A \mid S_i)P(S_i)$ over mutually exclusive, exhaustive scenarios $S_i$. Bayes' formula: $$P(S_i \mid A) = \frac{P(A \mid S_i)P(S_i)}{P(A)}$$ updating a prior probability with new information.

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