CFA Level 1 Quantitative Methods: Our Cheat Sheet

Note: this cheat sheet is updated for the latest 2023’s curriculum.

CFA exams are tough, we get it. We have gone through them ourselves. Quantitative Methods is the foundation you need to get right for the rest of the topics.

That’s why we created our Cheat Sheets to help your review sessions and refresh your memory on important CFA Level 1 concepts.☕

Each of our Cheat Sheet article focuses on one specific topic area for one specific CFA level.

More Cheat Sheets will be published in the coming weeks, sign up to our member’s list to be notified first.

By referring to the CFA Learning Outcome Statements (LOS), we prioritize and highlight the absolute key concepts and formulas you need to know for each topic. With some tips at the end too!

Use the Cheat Sheets during your practice sessions to get you to a flying start.

Let’s dive in – this is a long article, bookmark and come back to it often 🙂

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.

2023’s CFA Level 1 Quantitative Methods’ topic weighting is 8%-12%, which means 14-21 questions of the 180 questions of CFA Level 1 exam is centered around this topic.

It is covered in Topic 1 which contains 7 Learning Modules (LMs).

However, this 8%-12% 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: The Time Value of Money

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

Annuity due vs ordinary annuity

Specifically:

• Ordinary annuity = cash flow at the end-of-time period.
• Annuity due = cash flow at the beginning-of-time period.

\begin{align*}
PV_{annuity \space due}&=PV_{ordinary \space annuity} \times (1+r)

\\FV_{annuity \space due}&=FV_{ordinary \space annuity} \times (1+r)
\end{align*}

Perpetuity is just an annuity with infinite life

And the formula simplifies to:

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

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

LM2: Organizing, Visualizing and Describing Data

Arithmetic, Geometric and Harmonic Mean

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

\\Geometric \space mean&=(X_1 \times X_2 \times ... \space X_n)^{\frac{1}{n}}

\\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

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? 🙂

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}}

LM3: Probability Concepts

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)

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

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)

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)

LM4: Common Probability Distributions

Uniform distribution

Discrete uniform distribution:

P(x)=\frac{1}{n},\space where\space x=x_1,x_2,...,x_n

Continuous uniform distribution:

F(x)=\frac{x-a}{b-a};\space a\le x \le b

Binomial distribution

The binomial distribution is a sequence of n Bernoulli trials where the outcome of every trial can be a success (p) or a failure (1-p).

With the probability of success (p) the same for each trial, the probability of x successes in n trials is:

P(X=x)=\space^nC_xp^x(1-p)^n\space^-\space^x

The expected value of a binomial random variable is simply:

E(X)=n \times p

Variance of a binomial random variable

\sigma^2=n \times p \times (1-p)

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;

Compute Z-score

Z-score is used to standardize an observation from normal distribution. It shows you the number of standard deviation a given observation is from population mean.

z=\frac {x-\mu}{\sigma}

Safety-first ratio (SF ratio)

SF ratio is used to measure shortfall risk:

SF \space ratio = \frac{E(R_p)-R_{target}}{\sigma_P}

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).

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.

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.

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.

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.

LM5: Sampling and Estimations

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}}

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}}

LM6: 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.

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

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}

Test of equality of two variances (F-test)

F-statistic is calculated as:

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

LM7: Introduction to 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}}

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.

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 | FRA | 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