Note: this cheat sheet is updated for the latest 2024 and 2025’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 🙂
Quantitative Methods is a key foundational topic for CFA Level 1, which forms a basis for Level 2 and Level 3 learnings.
2025’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.
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
\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
\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
Return \space on \space leveraged \space portfolio, R_L=R_P+\frac{V_B}{V_E}(R_P-r_D)
where RP= Return on the investment portfolio (unleveraged), rD= Cost of debt, VB = Debt/borrowed funds and VE = Equity of the portfolio
PV(Coupon Bond)=\frac{PMT_1}{(1+r)^1}+\frac{PMT_2}{(1+r)^2}+...+\frac{PMT_N+FV}{(1+r)^N}
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.
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
PV=\frac{D_1}{r-g}=\frac{D_0(1+g)}{r-g}, where \space r>g
Dn= Common dividend at time n, g = Constant dividend growth rate, r = Expected rate of return
\frac{PV}{E_1}=\frac{D_1/E_1}{r-g}
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 gS is the expected dividend growth rate in the first period and gL is the expected growth rate in the second period.
\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? 🙂
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 \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) 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}}
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 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)
E(X) = P(X1)X1 + P(X2)X2 + … + P(Xn)Xn
\sigma^2(X)=\displaystyle \sum_{i=1}^n P(X_i)[X_i-E(X)]^2
\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.
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)
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)
SF_{ratio}=\frac{E(R_P)-R_L}{\sigma_P} \\Shortfall \space risk = P(R_P < R_L) = N(−SF_{ratio})
The normal distribution is a continuous symmetric probability distribution that is completely described by two parameters: its mean, μ, and its variance σ2.
The properties of lognormal distributions are:
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 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.
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 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}}
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:
2) Via a table
H0 is true | H0 is false | |
---|---|---|
Reject H0 | Type 1 Error | Correct rejection |
Fail to Reject H0 | Correct decision | Type 2 Error |
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.
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}}
Type of Distribution | Known Variance? | Small sample, n<30 | Large sample, n>30 |
---|---|---|---|
Normal | Known ✔︎ | z-statistic | z-statistic |
Normal | Unknown ✘ | t-statistic | t or z-statistic, both are fine |
Non-Normal | Known ✔︎ | – | z-statistic |
Non-Normal | Unknown ✘ | – | t or z-statistic, both are fine |
Student’s t-distribution:
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 s2p 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.
Test statistic is calculated as:
t=\frac{\overline{d} - \mu_{d0}}{s_{\overline{d}}}, \space df= n-1
Test statistic is calculated as:
\chi^2=\frac{(n-1)s^2}{\sigma^2_0}
Chi-square distribution:
F-statistic is calculated as:
F=\frac{s^2_1}{s^2_2}
F-distribution:
t=\frac{r\sqrt{n-2}}{\sqrt{1-r^2}}\space,\space df=n-2
r_S=1-\frac{6\displaystyle \sum_{i=1}^n d_i^2}{n(n^2-1)}
\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}}}
Y = b0 + b1X + ɛ
where:
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 (Se) in simple linear regression:
S_e=\sqrt{MSE}=\sqrt{\frac{\sum^N_{i=1}(Y_i - \hat{Y_i})^2}{n-2}}
Introductory chapter about machine learning, AI and big data. Quite straightforward and know your facts here. No notes necessary 🙂
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.
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.
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.
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View Comments
Great Guide! However, some of the formulas in this cheat sheet are in an error saying Katex is not defined. Hopefully that can be fixed!
Oh no, what an unfortunate time to break. I double checked them and they seem to be working now. :)
Hi,
Thanks for the Cheat Sheet.
In the Introductory Table, if the usage/application of the topics is also mentioned, it will give a complete picture of the course.
If there is something like already available, i request you to kindly provide a link.
If there is something like a mind map on interconnections between topics that will be great.
Regards
Thanks for the suggestion Yadvend, definitely something we are looking to improve on in the new versions. Just lots to do :)
Thank you guys for those great materials!
I think there is a typo in bernoulli equation for probability of x successes, namely binomial coefficient should be nCx rather than nCr.
Hi Jimmy, thanks for spotting this typo, this is now corrected! I'm glad it is helpful for your CFA studies, and good to know that we have tons of readers out there checking my work :)
Hi, can I ask if we need to remember all the formula to calculate the test statistic, e.g. Chi-square test statistic formula, test statistic for correlation, independence, difference of mean, etc?
Yes I'm afraid so Kyle! Once you do some practice questions this should become easier and more intuitive. Anything in the curriculum is fair game for testing.
Hey Team 300 Hours
Is there a PDF version available for all the cheat sheets for offline studying?
Regards
Aditya Golani
Hi Aditya, unfortunately we don't make PDF versions available simply because we continuously update this page and improve it for better notes. PDF versions get outdated easily. For some of candidates, what they do is to just load the webpage on a sole tab (and disconnect the internet if easily distracted) for revisions.
I think the Type I and Type II error is wrapped for the Visual representation.
Hi, these cheat sheets have been really helpful, thank you. I think P(A or B) should be P(A) + P(B) - P(AB).
Thanks for spotting that Maisie! Good to know that you're finding it useful :) Good luck for your exams!
Hi,
I would like to thank you for all the amazing resources y'all provide.
Maise has correctly spotted a mistake. The LOS regarding Conditional and Joint Probabilities can confirm this.
You've acknowledged the error but haven't updated it on the page. I believe this might be misleading and it would be useful to update the page to avoid confusion.
Best,
Muhammad
Yes, that's corrected now Muhammad, thanks!
Hi! I believe the MAD formula should be using 1/n rather than 1/(n-1). Thank you for creating this resource!
So, what's about Linear regression?)
We're working on this - will be updated soon!
can we get them in a downloadable/printable format?
We haven't any plans to do that yet, but you could just print this page. Or, you know, support us and visit our website. And support our partners. Cause they support us. 😁