CFA Level 1 Quantitative Methods Cheat Sheet 2026

This comprehensive cheat sheet covers all 11 CFA Level 1 Quantitative Methods Learning Modules (6-9% exam weight): Time Value of Money, statistical measures, probability, portfolio mathematics, hypothesis testing, and regression.

Includes 50+ essential formulas, when to use z-stat vs t-stat, Type I/II error mnemonics, and calculator tips. Bookmark for quick review before practice sessions and exam day.

Quantitative Methods is the foundational topic for all CFA levels, covering the statistical and mathematical concepts you’ll apply throughout Financial Statement Analysis, Fixed Income, Equity, Portfolio Management, and Risk Management.

This cheat sheet condenses all 11 Learning Modules (LMs) into essential formulas, decision frameworks, and quick-reference tables for efficient review. ☕

What’s covered:

• Time Value of Money calculations, descriptive statistics (mean, variance, standard deviation), probability theory, portfolio mathematics, hypothesis testing (z-tests, t-tests, chi-square, F-tests), and simple linear regression.
• This represents 6-9% of the CFA Level 1 exam (11-16 questions out of 180), but the concepts appear in 60%+ of questions across other topics.

How to use this cheat sheet:

• Bookmark this page for quick formula lookup during practice sessions.
• Review each Learning Module section after completing that reading in your study materials.
• Use the decision tables (z-stat vs t-stat, which distribution to use) during mock exams.
• The formulas and concepts here form the foundation for Levels 2 and 3, so invest time to truly understand them, not just memorize.

What is Quantitative Methods and why does it matter for CFA Level 1?

Quantitative Methods teaches you the statistical and mathematical toolkit for financial analysis: how to calculate expected returns, measure risk, test hypotheses about investments, and make data-driven decisions with confidence intervals and probability distributions.

Why it’s foundational beyond its 6-9% weight:

• Time Value of Money (TVM): Used in Fixed Income bond pricing, Equity dividend discount models, Corporate Finance NPV/IRR, and Portfolio Management return calculations
• Statistics (mean, variance, standard deviation): Required for Portfolio Management risk metrics, Performance Measurement benchmarks, and Derivatives pricing
• Probability & Expected Values: Applied in Alternative Investments risk assessment, Equity scenario analysis, and Economics forecasting
• Hypothesis Testing: Used in Portfolio Management factor analysis, Equity fundamental analysis statistical significance, and Performance Measurement benchmark comparison
• Regression: Foundation for Level 2 Multiple Regression, Level 3 Factor Models, and Equity beta estimation

The deceptive weight: While Quant is only 6-9% as a standalone topic, candidates who struggle with these concepts typically fail at least 2-3 other high-weight topics that depend on quantitative literacy. Conversely, mastering Quant creates a multiplier effect across the entire curriculum.

Study approach: Start with Quant early in your CFA preparation (weeks 1-3). The concepts build sequentially – TVM enables bond valuation, statistics enables hypothesis testing, regression builds on both. Spending 30-40 hours mastering Quant saves 50+ hours struggling with quantitative concepts embedded in other topics.

CFA Level 1 Quantitative Methods: An Overview

Quantitative Methods is a key foundational topic for CFA Level 1, which forms a basis for Level 2 and Level 3 learnings.

2026’s CFA Level 1 Quantitative Methods’ topic weighting is 6-9%, which means 11-16 questions of the 180 questions of CFA Level 1 exam is centered around this topic.

It is covered in Topic 1 which contains 11 Learning Modules (LMs).

However, this 6-9% weighting figure is deceptively low because, as will be discussed below, the material covered in this topic area overlaps significantly with material covered in other areas of the curriculum.

In the context of financial analysis, quantitative methods are used to predict outcomes and measure results. Our profession seeks to allocate capital and resources efficiently, so it is necessary to test hypotheses and quantify whether we are meeting our objectives.

Here’s the summary of CFA Quantitative Methods’ readings in Level 1:

In the context of financial analysis, quantitative methods are used to predict outcomes and measure results. Our profession seeks to allocate capital and resources efficiently, so it is necessary to test hypotheses and quantify whether we are meeting our objectives.

In a nutshell, the CFA Level 1 Quantitative Methods readings teaches you:

– How to predict the most likely outcomes, or range of outcomes, for future events;
– How confident you can be in those predictions;
– How to calculate the impact of events once they have occurred.

LM1: Rates and Returns

Determinants of interest rates

Interest rate = Real risk-free rate + Inflation premium + Default risk premium + Liquidity premium + Maturity premium

(1 + Nominal risk-free rate) = (1 + Real risk-free rate) x (1 + Inflation premium)

Nominal risk-free rate ≈ Real risk-free rate + Inflation premium

Arithmetic, geometric and harmonic mean returns

\begin{align*}
Arithmetic \space mean \space(i.e. \space simple \space average)&=\frac{\sum R_i}{T}

\\Geometric \space mean&=[(1+R_1) \times (1+R_2) \times ... \space (1+R_n)]^{\frac{1}{n}}-1

\\Harmonic \space mean&=\frac{N}{\displaystyle \sum_{i=1}^{N} \Big (\frac{1}{X_i}\Big)}
\end{align*}

Remember that for the same dataset:

Arithmetic Mean > Geometric Mean > Harmonic Mean

(Geometric Mean)² = Arithmetic Mean x Harmonic Mean

Present Value (PV) and Future Value (FV) of cash flows

\begin{align*}
PV&=\frac{FV}{\Big(1+\frac{r}{n}\Big)^{nt}}

\\FV&=PV\Big(1+\frac{r}{n}\Big)^{nt}
\end{align*}

where r= discount rate, n= number of discounting period per year, t= number of years

Leveraged returns

Return \space on \space leveraged \space portfolio, R_L=R_P+\frac{V_B}{V_E}(R_P-r_D)

where R_P= Return on the investment portfolio (unleveraged), r_D= Cost of debt, V_B = Debt/borrowed funds and VE = Equity of the portfolio

LM2: The Time Value of Money in Finance

💡 Calculator Tip: TVM Efficiency Hacks

Master these BA II Plus calculator shortcuts to save time on exam day:

1. Always clear TVM registers first: [2nd] [CLR TVM] before every new problem
2. Cash flow sign convention: Outflows = negative (PV or PMT), Inflows = positive (FV)
3. BGN vs END mode: Annuity due (payments at period start) = BGN mode. Ordinary annuity (payments at period end) = END mode. Default is END – only change for annuity due problems.
4. Quick checks: If calculating PV and FV > PV, you entered something wrong (money grows over time at positive rates)
5. Solve for any variable: Enter any 4 of the 5 TVM variables (N, I/Y, PV, PMT, FV), solve for the 5th by pressing [CPT] + that variable

Practice until these become muscle memory – you’ll solve TVM problems in 30 seconds instead of 2 minutes.

Present value of coupon bond

PV(Coupon Bond)=\frac{PMT_1}{(1+r)^1}+\frac{PMT_2}{(1+r)^2}+...+\frac{PMT_N+FV}{(1+r)^N}

Present value of perpetual bond

If n goes to infinity, the formula simplifies to:

PV_{perpetuity} = \frac{PMT}{r}

where PMT is the amount of each payment, r is discount rate.

Present value of annuity instruments

PV=A\Bigg[\frac{1-(1+r)^{-t}}{r}\Bigg]

where A = Periodic cash flow, r = Market interest rate per period, t = Number of payment periods

Price of common share with constant dividend growth rate

PV=\frac{D_1}{r-g}=\frac{D_0(1+g)}{r-g}, where \space r>g

D_n= Common dividend at time n, g = Constant dividend growth rate, r = Expected rate of return

Forward P/E ratio

\frac{PV}{E_1}=\frac{D_1/E_1}{r-g}

Two stage dividend discount model

PV=\displaystyle\sum_{t=1}^n\frac{D_0(1+g_S)}{(1+r)^t}+\frac{D_0(1+g_S)^n(1+g_L)}{(1+r)^n(r-g_L)}

where g_S is the expected dividend growth rate in the first period and g_L is the expected growth rate in the second period.

LM3: Statistical Measures of Asset Return

Population vs sample variance’s formulae

\begin{align*}
Population \space variance&= \sigma^2 =\frac{\displaystyle \sum_{i=1}^{N}(x_i-\mu)^2}{N}

\\Sample \space variance&= s^2 =\frac{\displaystyle \sum_{i=1}^{N}(x_i-\bar{x})^2}{n-1}

\end{align*}

And the standard deviations for population and sample is simply just the square root of the corresponding variance.

Easy right? 🙂

Sample target semideviation

s_{target}=\sqrt{\frac{\displaystyle\sum_{X_i \leq B}^n(X_i-B)^2}{n-1}}

where B = target, n = total number of sample observations

Mean absolute deviation (MAD)

Mean \space absolute \space deviation (MAD)=\frac{\displaystyle \sum_{i=1}^n |x_i-\bar{x}|}{n}

MAD is a measure of the average of the absolute values of deviations from the mean in a data set. Since the sum of deviations from the mean in a dataset is always 0, we must use absolute values.

Coefficient of variation (CV)

Coefficient of variation (CV) is used to compare the relative dispersion between datasets, as it shows how much dispersion exists relative to a mean of a distribution.

CV is calculated by dividing the standard deviation of a distribution with its mean/expected value:

CV=\frac{s}{\overline {X}}

LM4: Probability Trees and Conditional Expectations

Conditional and joint probability

P(A | B) = P (AB) / P(B)

P(AB) = P(A | B) x P(B). But If A and B are independent, P(AB) = P(A) x P(B)

P(A or B) = P(A) + P(B) – P(AB)

Bayes’ formula

Bayes’ formula is basically a method of updating probabilities given new information:

P(Event \space | \space Information) = \frac{P(Information \space | \space Event)}{P(Information)}\times P(Event)

Expected value of a random variable X

E(X) = P(X₁)X₁ + P(X₂)X₂ + … + P(X_n)X_n

Probabilistic variance

\sigma^2(X)=\displaystyle \sum_{i=1}^n P(X_i)[X_i-E(X)]^2

LM5: Portfolio Mathematics

Correlation and covariance of returns

\rho, corr(R_A,R_B)=\frac{COV(R_A,R_B)}{\sigma (R_A) \sigma(R_B)}

Correlation equals covariance divided by the product of 2 standard deviations.

Expected return on a portfolio

E(R_p)=\displaystyle \sum_{i=1}^n w_iE(R_i)=w_1E(R_1)+w_2E(R_2)+ ... \space w_nE(R_n)

Variance of a 2 stock portfolio

Var(R_p)=w_A^2\sigma^2(R_A)+w_B^2\sigma^2(R_B)+2w_Aw_B\rho(R_A,R_B)\sigma(R_A)\sigma(R_B)
\\=w_A^2\sigma^2(R_A)+w_B^2\sigma^2(R_B)+2w_Aw_BCOV(R_A,R_B)

Roy’s safety-first ratio

SF_{ratio}=\frac{E(R_P)-R_L}{\sigma_P}

\\Shortfall \space risk = P(R_P < R_L) = N(−SF_{ratio})

LM6: Simulation Methods

Normal distribution

The normal distribution is a continuous symmetric probability distribution that is completely described by two parameters: its mean, μ, and its variance σ².

• 68% of observations lie in between μ +/- 1σ;
• 90% of observations lie in between μ +/- 1.65σ;
• 95% of observations lie in between μ +/- 1.96σ;
• 99% of observations lie in between μ +/- 2.58σ;

Lognormal distribution

The properties of lognormal distributions are:

• It cannot be negative;
• The upper end of its range extends to infinity;
• It is positively skewed.

For these reasons, lognormal distribution is suitable for model asset prices, not asset returns (since returns can be negative).

If X follows a lognormal distribution, then ln(X) follows a normal distribution.

Monte Carlo simulations

Monte Carlo simulations use computers to generate many random samples to produce a distribution of outcomes. It is used in financial planning, complex derivatives valuation and VAR estimates.

LM7: Estimation and Inference

Central Limit Theorem

The central limit theorem states that when we have simple random samples each of size n from a population with a mean μ and variance σ², the sample mean X approximately has a normal distribution with mean μ and variance σ²/n as n (sample size) becomes large, i.e. greater or equal to 30.

Standard Error

Standard error of sample mean = standard deviation of distribution of the sample means.

If population variance is known, standard error of sample mean is:

\sigma_{\overline{x}}=\frac{\sigma}{\sqrt{n}}

If population variance is unknown, standard error of sample mean is:

s_{\overline{x}}=\frac{s}{\sqrt{n}}

LM8: Hypothesis Testing

6 steps of hypothesis testing

1. State the hypothesis (null H₀ and alternative H_a)
2. Identify the appropriate test statistic
3. Specify significance level
4. State decision rule
5. Collect data and calculate test statistic
6. Make a decision

Type I vs. Type II error

An area most candidates often get confused on.

Here are 3 different ways to present this concept (taken from our article on ways to improve study memory) which I hope helps your understanding:

1. Visually

2) Via a table

3) Using letters, for rote memorization if desperate.

OK, you need to have watched Lord of the Rings for this to make sense, but here goes:

A Type I error is when you reject the null when you shouldn’t, just like Frodo rejecting the help of Sam, his loyal friend.

A Type II error is when you fail to reject the null when you should, just like how Frodo listened to Gollum even though he was a dangerous liar.

To distinguish between them, note that the two ll’s in Gollum look like the Roman numeral II, for a Type II error.

Confidence Intervals

For a given probability, confidence interval provides a range of values the mean value will be between.

With a known population variance, the confidence interval formula based on z-statistic is:

Confidence \space interval = \overline{X} \space ± \space z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}

For unknown population variance, the confidence interval formula based on t-statistic is:

Confidence \space interval = \overline{x} \space ± \space t_{\alpha/2} \times \frac{s}{\sqrt{n}}

Tests concerning a single mean: when should I use z-statistic or t-statistic?

Student’s t-distribution:

• t-distribution is defined by 1 parameter, i.e. degrees of freedom (df) = n-1.
• It is symmetrical, similar bell-shaped distribution to Normal distribution, but has fatter tails and a lower peak.
• As df increases, t-distribution approximates Normal distribution.

Tests concerning differences between means (independent samples)

Test statistic is calculated as:

t=\frac{(\overline{X}_1 - \overline{X}_2)-(\mu_1 - \mu_2)}{(\frac{s^2_p}{n_1}+\frac{s^2_p}{n_2})^{1/2}}

where s²_p is the pooled estimator of the variance, calculated with this formula

s^2_p=\frac{(n_1-1) s^2_1 + (n_2-1)s^2_2}{n_1+n_2-2}

where the number of degrees of freedom is n₁ + n₂ – 2.

Tests concerning differences between means (dependent samples)

Test statistic is calculated as:

t=\frac{\overline{d} - \mu_{d0}}{s_{\overline{d}}}, \space df= n-1

Test of a single variance (Chi-square test)

Test statistic is calculated as:

\chi^2=\frac{(n-1)s^2}{\sigma^2_0}

Chi-square distribution:

• A chi-square test is used to establish whether a hypothesized value of variance is equal to, less than, or greater than the true population variance.
• Chi-square distribution is the sum of squares of independent normal variables. Hence it cannot be negative.
• It is asymmetrical and is defined by one parameter, i.e. degrees of freedom (df) = n-1.
• The shape of curve is different for each df.
• As the df increase, the chi-square curve approximates Normal distribution.

Test of equality of two variances (F-test)

F-statistic is calculated as:

F=\frac{s^2_1}{s^2_2}

F-distribution:

• F-distribution is a ratio of 2 Chi-square distributions with 2 degrees of freedom, hence it cannot be negative either.
• As both the numerator and the denominator’s degrees of freedoms increase, the F-distribution approximates Normal distribution.

LM9: Parametric and Non-Parametric Tests of Independence

Test of a correlation

t=\frac{r\sqrt{n-2}}{\sqrt{1-r^2}}\space,\space df=n-2

Spearman Rank correlation coefficient

r_S=1-\frac{6\displaystyle \sum_{i=1}^n d_i^2}{n(n^2-1)}

Test of independence using category table data

\chi^2=\displaystyle\sum_{i=1}^m\frac{(O_{ij}-E_{ij})^2}{E_{ij}} \space, \space df=(r-1)(c-1)
\\Standardized \space residual=\frac{O_{ij}-E_{ij}}{\sqrt{E_{ij}}}

LM10: Simple Linear Regression

Basic model of simple linear regression

Y = b₀ + b₁X + ɛ

where:

• Y is the dependent (explained) variable;
• X is the independent (explanatory) variable;
• b₀ is the intercept;
• b₁ is the slope coefficient;
• ɛ is the error term.

Assumptions of simple linear regression

• Linear relationship between X and Y
• Homoscedasticity (constant variance of residuals for all observations)
• Independence between X and Y (residuals are uncorrelated for all observations)
• Normality of the residuals

Analysis of variance (ANOVA)

Sum of squares error (SSE) = the unexplained variation in Y

SSE=\sum^N_{i=1}(Y_i - \hat{Y_i})^2

Sum of squares regression (SSR) = the explained variation in Y

SSR=\sum^N_{i=1}(\hat{Y_i}-\overline{Y})^2

Sum of squares total (SST) = the total variation in Y = SSR + SSE

SST=\sum^N_{i=1}({Y_i}-\overline{Y})^2

Coefficient of determination (R²) measures the % of total variation in the dependent variable that is explained by the independent variable.

R^2=\frac{SSR}{SST}

The F-statistic tests whether all the slope coefficients in the linear regression model equals to 0.

F=\frac{Mean \space Square \space Regression}{Mean \space Square \space Error} = \frac{MSR}{MSE}= \frac{SSR/k}{(SSE/(n-(k+1))}

Standard error of estimate (S_e) in simple linear regression:

S_e=\sqrt{MSE}=\sqrt{\frac{\sum^N_{i=1}(Y_i - \hat{Y_i})^2}{n-2}}

LM11: Introduction to Big Data Techniques

Introductory chapter about machine learning, AI and big data. Quite straightforward and know your facts here. No notes necessary 🙂

Quick reference: When to use which test statistic

At-a-glance decision table

Detailed decision frameworks

One of the most confusing aspects of Quantitative Methods is knowing which statistical test to use. Here’s your decision framework:

Testing a Single Mean

Use z-statistic when:

• Population variance is KNOWN, OR
• Sample size is large (n ≥ 30) AND distribution is normal

Use t-statistic when:

• Population variance is UNKNOWN AND
• Sample size is small (n < 30) OR distribution is non-normal

Testing Difference Between Two Means

Independent samples (unrelated groups):

• Use pooled variance t-test
• Assumes equal variances between groups
• df = n₁ + n₂ – 2

Dependent samples (paired observations):

• Use paired t-test
• Same subjects measured twice (before/after)
• df = n – 1

Testing Variance

Single variance:

• Use chi-square test
• Tests if sample variance equals hypothesized population variance

Two variances (equality):

• Use F-test
• F = larger variance / smaller variance
• Always put larger variance in numerator

Testing Correlation or Independence

Correlation coefficient:

• Use t-test for correlation
• Tests if correlation is statistically different from zero

Independence (categorical data):

• Use chi-square test for independence
• Compares observed vs expected frequencies in contingency tables

Rank correlation (non-parametric):

• Use Spearman rank correlation
• When data is ordinal or contains outliers

Most frequently tested formulas: Quick reference

CFA Level 1 Quantitative Methods Tips

Start your studies (early) with Quantitative Methods

One simple approach to studying for any exam is to start on page 1 and read through to the end.

However, it is not uncommon for candidates to question whether to study the Ethical and Professional Standards readings first – because they don’t neatly fit in with the rest of the curriculum. As a result, many recommend saving the Ethics material for last.

According to our best CFA Level 1 topic study order, it’s a good idea to start with Quantitative Methods first, or at least early in your preparation.

These readings introduce essential topics that must be mastered in order to be successful on exam day because they are the absolute foundation of the Level 1 syllabus. Moreover, this material will show up repeatedly throughout the curriculum at every level as you progress towards your CFA charter.

Understand the concepts, don’t just memorize formulae

There is a natural tendency among candidates to view the Quantitative Methods material as a long list of equations to be memorized and worked through to produce a correct answer.

There are definitely a number of equations with which you are well-advised to become intimately familiar and you will likely give your calculator quite a workout when answering questions on this topic, but there is more to mastering this material than number crunching.

For example, the knowledge that a dataset’s harmonic mean is always less than its geometric mean, which is always less than its arithmetic mean, is just as important as memorizing the formulae used to calculate these measures.

Similarly, you don’t need to use your calculator to know that a security’s bank discount yield is less than its effective annual yield. 

You will see Quantitative Methods in other topic areas

As mentioned previously, Quantitative Methods topics are foundational knowledge with which you must be familiar with because it will show up repeatedly as you progress through the curriculum. 

For example, developing a solid understanding of the yield measures presented in these readings can only benefit you when you get to the readings on fixed-income and corporate finance.
​
In yet another example, the concept of Value-at-Risk, which is covered in the Study Session on portfolio management, is first introduced in the reading on probability distributions.

Indeed, it can be helpful to refer back to these readings if for no other reason than to remind yourself that you have covered these topics already and you probably understand them better than you give yourself credit for.

Frequently asked questions about CFA Level 1 Quantitative Methods

More Cheat Sheet articles will be updated and published continuously. Get ahead of other CFA candidates by signing up to our member’s list to get notified.

Meanwhile, here are other related articles that may be of interest:

• CFA Level 1 Cheat Sheets series: Economics | FSA | Corporate Issuers | Derivatives | Fixed Income | Equity Investments | Ethics | Alt Investments | Portfolio Management
• CFA Level 1: How to Prepare and Pass CFA in 18 Months
• CFA Level 1 Tips: Top 10 Advice from Previous Candidates
• 18 Actionable Ways to Improve Your Study Memory
• How to Study Effectively: Proven Methods that Work for CFA, FRM and CAIA
• The Ultimate Guide to CFA Practice Questions