CFA® Level I Quantitative Methods

Sitting the exam in 2027? This guide follows the 2026 curriculum. For 2027 the topic is rebuilt and its weight rises to 11–14%; Roy's safety-first criterion, the harmonic mean and jackknife resampling leave the curriculum. See what changed in 2027.

The 11 learning modules

1. 01 Rates and Returns

   How an interest rate is built from a real risk-free rate plus premiums for inflation, default, liquidity and maturity, and the different ways to measure a return. You calculate holding-period returns and arithmetic, geometric and harmonic means, and compare money-weighted returns (driven by the timing of the investor's cash flows) with time-weighted returns (which are not). The module ends with annualising returns, continuous compounding, and gross, net, after-tax, real and leveraged returns.

   10 lessons · 3 videos in the Academy →

2. 02 Time Value of Money in Finance

   Present and future values of single sums, annuities, perpetuities and uneven cash flows, applied to bonds and shares. You price zero-coupon, coupon and amortising instruments, value shares with constant or changing dividend growth, and work backwards to the implied return or growth rate in a market price. Cash flow additivity links it all together and is used to derive implied forward rates, forward exchange rates and the price of an option in a one-period binomial setting.

   8 lessons · 2 videos in the Academy →

3. 03 Statistical Measures of Asset Returns

   Describing a set of returns: the mean, median and mode, trimmed and winsorised means for dealing with outliers, and quantiles, including box and whisker plots. Dispersion covers the range, mean absolute deviation, sample variance and standard deviation, target downside deviation and the coefficient of variation. You also interpret skewness and excess kurtosis, and calculate covariance and correlation while knowing their limits: outliers, spurious correlation and non-linear relationships.

   8 lessons · 2 videos in the Academy →

4. 04 Probability Trees and Conditional Expectations

   Expected value and variance of a random variable from a set of outcomes and probabilities. Probability trees let you work out conditional expected values and combine them with the total probability rule. Bayes' formula then updates a prior probability when new information arrives, and it is one of the most reliably tested calculations in the topic.

   6 lessons · 3 videos in the Academy →

5. 05 Portfolio Mathematics

   The expected return and variance of a portfolio, built from weights, variances and covariances. You calculate covariance from a joint probability table, convert between covariance and correlation, and see how lower correlation reduces portfolio risk. The module also applies the normal distribution to risk: confidence intervals for returns, shortfall risk and Roy's safety-first ratio.

   7 lessons · 1 video in the Academy →

6. 06 Simulation Methods

   The lognormal distribution and why it suits asset prices, and continuously compounded returns, which add up over time and whose variance scales with the number of periods. You then compare two ways of generating outcomes: Monte Carlo simulation, which draws from assumed distributions, and bootstrapping, which resamples the observed data with replacement. Expect questions on the steps of a simulation and on the strengths and limits of each method.

   6 lessons · 1 video in the Academy →

7. 07 Estimation and Inference

   How to draw a sample: simple random, systematic, stratified and cluster sampling, plus non-probability methods such as convenience and judgmental sampling. The central limit theorem explains why the sample mean is approximately normal in large samples, and the standard error measures how precise it is. You also meet resampling methods, the bootstrap and the jackknife, as ways to estimate standard errors without a formula.

   8 lessons · 1 video in the Academy →

8. 08 Hypothesis Testing

   The steps of a hypothesis test, how to set up the null and alternative, and how to read significance levels, p-values, Type I and Type II errors and power. You test a single mean, the difference between two means (with a pooled variance for independent samples, or a paired comparisons test for dependent ones), a single variance with the chi-square test and two variances with the F-test. The module closes with when a non-parametric test is the better choice.

   7 lessons · 4 videos in the Academy →

9. 09 Parametric and Non-Parametric Tests of Independence

   Whether two variables are related. You test a Pearson correlation with a t-test, and use the Spearman rank correlation when the data are ranks, contain outliers or are not normal. For categorical data you build a contingency table, calculate expected frequencies and run a chi-square test of independence.

   6 lessons · 2 videos in the Academy →

10. 10 Simple Linear Regression

    Fitting a line by least squares, interpreting the slope and intercept, and the four assumptions: linearity, homoskedasticity, independence and normality of the residuals. You read an ANOVA table to get R², the F-statistic and the standard error of the estimate, test individual coefficients with t-tests, and build a prediction interval for a forecast. The module also covers indicator variables and the log-lin, lin-log and log-log functional forms.

    9 lessons · 4 videos in the Academy →

11. 11 Introduction to Big Data Techniques

    A descriptive module on how data science is used in investment analysis. It covers the characteristics of big data (volume, velocity, variety and veracity), sources of alternative data, and the difference between supervised, unsupervised and deep learning, along with overfitting and underfitting. The last part covers the data-science process, from capture and curation to visualisation, and the use of natural language processing on text.

    7 lessons in the Academy →

What the questions cover

• Holding-period, arithmetic, geometric and harmonic mean returns, and money- vs time-weighted returns
• Compounding, present and future values of annuities and perpetuities, and implied returns and growth rates
• Measures of location, dispersion, skewness, kurtosis and correlation
• Expected value and variance, probability trees, the total probability rule and Bayes' formula
• Portfolio variance with covariance and correlation, and the safety-first ratio
• Sampling, the central limit theorem, standard error, and hypothesis tests of means, variances and correlations
• Simple linear regression output, and the vocabulary of big data and machine learning

Key formulas

Holding-period return

$\displaystyle R = \frac{P_1 - P_0 + D_1}{P_0}$Include income received during the period. Divide by the starting price, not the ending price.Learn it: Holding period return →

Geometric mean return

$\displaystyle R_G = \left[(1 + R_1)(1 + R_2)\cdots(1 + R_T)\right]^{1/T} - 1$The compound rate per period. It is never above the arithmetic mean, and the gap grows with volatility.

Effective annual rate

$\displaystyle \text{EAR} = \left(1 + \frac{r_s}{m}\right)^{m} - 1, \quad \text{continuous: } e^{r_s} - 1$r_s is the stated annual rate and m the number of compounding periods per year. More frequent compounding gives a higher EAR.Learn it: Present value and discount (zero-coupon) bonds →

Bayes' formula

$\displaystyle P(E \mid I) = \frac{P(I \mid E)}{P(I)} \times P(E)$P(E) is the prior. Find the unconditional P(I) first with the total probability rule.Learn it: Bayes' formula: updating a probability with new information →

Two-asset portfolio variance

$\displaystyle \sigma_p^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\rho_{12}\sigma_1\sigma_2$The last term can also be written 2w_1w_2Cov(R_1, R_2). Take the square root for the standard deviation.Learn it: Correlation and diversification →

Safety-first ratio

$\displaystyle \text{SFRatio} = \frac{E(R_p) - R_L}{\sigma_p}$R_L is the threshold return. Choose the portfolio with the highest ratio; with normal returns, the shortfall probability is N(−SFRatio).Learn it: Shortfall risk and Roy's safety-first criterion →

Test statistic for a single mean

$\displaystyle t = \frac{\bar{X} - \mu_0}{s / \sqrt{n}}$The denominator is the standard error of the mean. Use n − 1 degrees of freedom; use z if the population variance is known.

Test of a correlation

$\displaystyle t = \frac{r\sqrt{n - 2}}{\sqrt{1 - r^2}}$n − 2 degrees of freedom. The same statistic is used for a Spearman rank correlation in large samples.

Regression fit

$\displaystyle R^2 = \frac{\text{SSR}}{\text{SST}}, \quad s_e = \sqrt{\frac{\text{SSE}}{n - 2}}, \quad F = \frac{\text{SSR}/1}{\text{SSE}/(n - 2)}$SST = SSR + SSE. In simple regression, R² is the squared correlation and F is the square of the slope's t-statistic.

The Quantitative Methods formula sheet: the key formulas explained

Common mistakes

• Confusing money-weighted and time-weighted returns. The money-weighted return depends on when the investor added or withdrew money, so it is the wrong measure for judging a manager who does not control those flows.
• Getting annuity timing wrong. The ordinary annuity and perpetuity formulas give a value one period before the first payment, so a perpetuity starting in year 5 must be discounted back four years, and an annuity due is worth the ordinary annuity times (1 + r).
• Dividing by the wrong number: n instead of n − 1 for a sample variance, or n instead of n − 2 for the standard error of the estimate in a regression.
• Using the standard deviation where the standard error is needed. Confidence intervals and tests of a mean use s/√n, not s.
• Stopping halfway through Bayes' formula, for example reporting P(information | event) or the joint probability instead of the updated probability of the event.
• Putting the claim to be tested in the null hypothesis. The claim goes in the alternative, and failing to reject the null is not the same as proving it true.
• Forgetting to convert back in log models. A log-lin regression forecasts ln Y, so you must take e to that power to get the forecast of Y itself.

How to study it

• Get fast with your calculator's time value, cash-flow and statistics functions; they save minutes per question.
• Write down what the question asks for (the standard error, not the standard deviation; the forecast of Y, not of ln Y) before calculating.
• Many distractors are the right formula with one wrong input, such as n instead of n − 1, or the wrong number of degrees of freedom.
• For every hypothesis test, write the null and alternative first. The alternative tells you whether the test is one- or two-sided and which tail to use.
• Don't skip the descriptive modules. Simulation, sampling methods and big data give quick marks for candidates who know the terms.

Try 3 Quantitative Methods questions

Practise and prove Quantitative Methods

The question bank and the mock exams are the practice half of CheapMocks Academy. Try the free sample questions, see the topic weights for the whole exam and the latest Level I pass rates.

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