CFA (Chartered Financial Analyst) 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 CFA (Chartered Financial Analyst), not a summary of it.

1. 1

   Rates, Returns, and the Time Value of Money

   4 topics

   • Interest rates and rates of return
     • Nominal, real, and risk premiums
     • Holding period, arithmetic, and geometric returns
     • Money-weighted vs time-weighted returns
   • Time value of money for cash flows
   • Net present value, IRR, and yield measures
   • Annuities, perpetuities, and effective rates
2. 2

   Probability and Statistical Foundations

   4 topics

   • Descriptive statistics and data visualization
   • Measures of central tendency, dispersion, skewness, and kurtosis
   • Probability concepts
     • Conditional and joint probabilities, Bayes' formula
     • Expected value, variance, covariance, and correlation
   • Common probability distributions
     • Uniform, binomial, and normal distributions
     • Lognormal distribution and Monte Carlo simulation
3. 3

   Sampling and Hypothesis Testing

   4 topics

   • Sampling methods and the central limit theorem
   • Confidence intervals and resampling (bootstrap, jackknife)
   • Hypothesis testing
     • Null and alternative hypotheses, Type I and II errors
     • Tests of means, variances, and correlation
     • Parametric vs nonparametric tests
   • Tests of independence using contingency tables
4. 4

   Linear Regression and Machine Learning

   4 topics

   • Simple and multiple linear regression
     • Assumptions, ANOVA, and goodness of fit
     • Heteroskedasticity, serial correlation, and multicollinearity
   • Model misspecification and dummy variables
   • Time-series analysis and trend models
   • Big data and machine learning
     • Supervised and unsupervised learning
     • Fintech in investment management

Quantitative Methods flashcards for CFA (Chartered Financial Analyst)

19 of 67 cards from the Quantitative Methods deck — real questions with worked answers.

1. What is the difference between a nominal interest rate and a real interest rate?

   The nominal rate is the stated/quoted rate including compensation for inflation, while the real rate excludes inflation and reflects only the time value of money plus a real risk-free return. Approximately, $\text{nominal} \approx \text{real} + \text{inflation}$; exactly, $(1+r_{nominal}) = (1+r_{real})(1+\text{inflation})$.

2. State the components of a required interest rate (rate of return) on a security.

   $r = \text{real risk-free rate} + \text{inflation premium} + \text{default risk premium} + \text{liquidity premium} + \text{maturity premium}$. The nominal risk-free rate equals the real risk-free rate plus the inflation premium.

3. How is the holding period return (HPR) calculated for a single period?

   $$\text{HPR} = \frac{P_{1} - P_{0} + D_{1}}{P_{0}} = \frac{P_{1}+D_{1}}{P_{0}} - 1$$ where $P_{0}$ is the beginning price, $P_{1}$ the ending price, and $D_{1}$ any cash flow (dividend/interest) received.

4. How do you compute a multi-period holding period return from individual period returns?

   Chain-link (compound) the single-period returns: $$\text{HPR} = (1+R_{1})(1+R_{2})\cdots(1+R_{n}) - 1$$

5. What is the formula for the arithmetic mean return?

   $$\bar{R}_{A} = \frac{1}{n}\sum_{i=1}^{n} R_{i}$$ It is the simple average of periodic returns; it is the best estimate of a single period's expected return.

6. What is the formula for the geometric mean return and when is it used?

   $$\bar{R}_{G} = \left[\prod_{i=1}^{n}(1+R_{i})\right]^{1/n} - 1$$ It measures the compound annual growth rate of an investment over multiple periods and is always $\leq$ the arithmetic mean (equal only when all returns are identical).

7. Compare the money-weighted rate of return (MWRR) and the time-weighted rate of return (TWRR).

   MWRR is the IRR of all portfolio cash flows—it is sensitive to the timing and size of deposits/withdrawals. TWRR compounds the returns of each subperiod between cash flows, removing the effect of external cash flow timing, and is the preferred industry standard for evaluating manager performance.

8. What is the harmonic mean and a common finance 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 (averaging purchase prices). Order: harmonic $\leq$ geometric $\leq$ arithmetic mean.

9. What is the future value of a single cash flow under compound interest?

   $$FV = PV(1+r)^{N}$$ where $r$ is the periodic rate and $N$ is the number of compounding periods.

10. What is the present value of a single future cash flow?

    $$PV = \frac{FV}{(1+r)^{N}} = FV(1+r)^{-N}$$

11. What is the future value of an ordinary annuity?

    $$FV = A \times \frac{(1+r)^{N} - 1}{r}$$ where $A$ is the periodic payment made at the end of each period.

12. What is the present value of an ordinary annuity?

    $$PV = A \times \frac{1 - (1+r)^{-N}}{r}$$ where $A$ is the end-of-period payment.

13. How does an annuity due differ from an ordinary annuity in valuation?

    An annuity due has payments at the beginning of each period. Its value equals the ordinary annuity value multiplied by $(1+r)$: $$PV_{due} = PV_{ordinary}\times(1+r)$$

14. What is the present value of a perpetuity?

    $$PV = \frac{A}{r}$$ where $A$ is the constant periodic payment and $r$ is the discount rate. A growing perpetuity is $PV = \frac{A}{r-g}$.

15. What is the formula for the effective annual rate (EAR) given a stated annual rate?

    $$EAR = \left(1 + \frac{r_{s}}{m}\right)^{m} - 1$$ where $r_{s}$ is the stated annual rate and $m$ is the number of compounding periods per year. With continuous compounding, $EAR = e^{r_{s}} - 1$.

16. Define net present value (NPV) and give its decision rule.

    $$NPV = \sum_{t=0}^{N} \frac{CF_{t}}{(1+r)^{t}}$$ Accept a project if $NPV > 0$ (it adds value); reject if $NPV < 0$.

17. Define the internal rate of return (IRR) and its decision rule.

    IRR is the discount rate that makes NPV equal to zero: $$\sum_{t=0}^{N} \frac{CF_{t}}{(1+IRR)^{t}} = 0$$ Accept if $IRR > $ required rate of return (cost of capital).

18. When can the NPV and IRR rules give conflicting rankings for mutually exclusive projects?

    Conflicts arise from differences in project scale or in the timing/pattern of cash flows. When they conflict, NPV is preferred because it directly measures the dollar value added and assumes reinvestment at the discount rate rather than at the IRR.

19. What is the bank discount yield (BDY) for a money market instrument?

    $$r_{BD} = \frac{D}{F} \times \frac{360}{t}$$ where $D$ is the dollar discount, $F$ is the face value, and $t$ is days to maturity. It is based on face value and a 360-day year, understating the true yield.

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