CFA Level 1 Quantitative Methods: Our Cheat Sheet

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CFA exams are tough, we get it. We have gone through them ourselves.

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

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

By referring to the CFA Learning Outcome Statements (LOS), we prioritize and highlight the absolute key concepts and formulae 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 🙂

Table Of Contents


CFA Level 1 Quantitative Methods: An Overview

cfa level 1 quantitative methods

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

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 Study Sessions 2 and 3, which includes Reading 6-11.

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 Quantitative Methods’ readings:

Reading NumberSub-topicDescription
6The Time Value of MoneyThe valuation of assets and securities in different points in time
7Statistical Concepts and Market ReturnsMeasuring central tendencies and dispersion
8Probability ConceptsUsing probabilities to predict outcomes when faced with uncertainty
9Common Probability DistributionsA discussion of normal and non-normal distributions
10Sampling and EstimationUsing a sample to estimate characteristics of a population
11Hypothesis TestingDetermining the level of confidence you can have in your conclusions

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.


Reading 6: The Time Value of Money

time management

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

PV = FV1+rnn*tFV = PV 1+rnn*t

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.

PVannuity due=PVordinary annuity× (1+r)FVannuity due=FVordinary annuity× (1+r)


Perpetuity is just an annuity with infinite life

And the formula simplifies to:

PVperpetuity=PMTr

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


Reading 7: Statistical Concepts and Market Returns

CFA Level 1 Quantitative Methods: Our Cheat Sheet 1

Arithmetic, Geometric and Harmonic Mean

Arithmetic mean (i.e. simple average) = XiNGeometric mean = (X1* X2*... Xn)1nHarmonic mean = N1Xii=1N

Remember that for the same dataset:

Arithmetic Mean > Geometric Mean > Harmonic Mean


Population vs sample variance’s formulae

Population variance = σ2=i=1Nxiμ2NSample variance = s2=i=1nxix¯2n1

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 absolute deviation (MAD)= i=1nXiX¯n1MAD 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 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=sX¯


Sharpe ratio

Sharpe ratio measures the risk-adjusted returns of a portfolio. The higher this ratio is, the more return you get per amount of risk, i.e. higher is better.

Sharpe ratio =rprfσp

where rp = portfolio return, rf= risk free rate, σp= standard deviation of portfolio returns


Reading 8: Probability Concepts

CFA Level 1 Quantitative Methods: Our Cheat Sheet 2

Expected value of a random variable X

E(X)=P(X1)X1+P(X2)X2+... P(Xn)Xn


Probabilistic variance

σ2(X)=i=1nPXiXiEX2


Correlation and covariance of returns

ρ, corrRA,RB=COVRA,RBσRAσRB

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


Expected return on a portfolio

E(Rp)=i=1nwiE(Ri)=w1E(R1)+w2E(R2)+...wnE(Rn)


Variance of a 2 stock portfolio

Var(RP)=wA2σ2(RA)+wB2σ2(RB)+2wAwB ρ(RA,RB)σ(RA)σ(RB)=wA2σ2(RA)+wB2σ2(RB)+2wAwB COV(RA,RB)


Reading 9: Common Probability Distributions

CFA Level 1 Quantitative Methods: Our Cheat Sheet 3

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)=Cxn px1pnx

The expected value of a binomial random variable is simply:

E(X)=n×p

Variance of a binomial random variable

σ2=n×p×(1p)


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 +/- 1.645σ;
  • 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=xμσ

Roy’s safety-first ratio (SF ratio)

SFratio=E(RP)RTargetσP


Reading 10: Sampling and Estimations

CFA Level 1 Quantitative Methods: Our Cheat Sheet 4

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:

σx¯=σn

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

sx¯=sn


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 interval =x¯±zα2σn

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

Confidence interval =x¯±tα2sn


Reading 11: Hypothesis Testing

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
type 1 vs type 2 error

2) Via a table

H0 is trueH0 is false
Reject H0Type 1 ErrorCorrect rejection
Fail to Reject H0Correct decisionType 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.


When should I use z-statistic or t-statistic?

Type of DistributionKnown Variance?Small sample, n<30Large sample, n>30
NormalKnown ✔︎z-statisticz-statistic
NormalUnknown ✘t-statistict or z-statistic, both are fine
Non-NormalKnown ✔︎z-statistic
Non-NormalUnknown ✘t or z-statistic, both are fine

Chi-square test of a single population variance

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.

χ2=(n1)s2σ02


F-test for equality of variances of 2 populations

F=s12s22


CFA Level 1 Quantitative Methods Tips

CFA Level 1 Quantitative Methods: Our Cheat Sheet 5

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

Picture

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

cfa study material shortlist

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 published over the coming weeks. 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:

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