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 🙂

**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.

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 Number | Sub-topic | Description |
---|---|---|

6 | The Time Value of Money | The valuation of assets and securities in different points in time |

7 | Statistical Concepts and Market Returns | Measuring central tendencies and dispersion |

8 | Probability Concepts | Using probabilities to predict outcomes when faced with uncertainty |

9 | Common Probability Distributions | A discussion of normal and non-normal distributions |

10 | Sampling and Estimation | Using a sample to estimate characteristics of a population |

11 | Hypothesis Testing | Determining 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

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

$PV=\frac{FV}{{\left(1+{\displaystyle \frac{r}{n}}\right)}^{n*t}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}FV=PV{\left(1+\frac{r}{n}\right)}^{n*t}$

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.

$P{V}_{annuitydue}=P{V}_{ordinaryannuity}\times (1+r)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}F{V}_{annuitydue}=F{V}_{ordinaryannuity}\times (1+r)$

### Perpetuity is just an annuity with infinite life

And the formula simplifies to:

$P{V}_{perpetuity}=\frac{PMT}{r}\phantom{\rule{0ex}{0ex}}$

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

## Reading 7: Statistical Concepts and Market Returns

### Arithmetic, Geometric and Harmonic Mean

$Arithmeticmean(i.e.simpleaverage)=\frac{\sum _{}{X}_{i}}{N}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Geometricmean={({X}_{1}*{X}_{2}*...{X}_{n})}^{\frac{1}{n}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Harmonicmean=\frac{N}{{\displaystyle \underset{i=1}{\overset{N}{\sum \left(\frac{1}{{X}_{i}}\right)}}}}\phantom{\rule{0ex}{0ex}}$

Remember that for the

samedataset:

Arithmetic Mean > Geometric Mean > Harmonic Mean

### Population vs sample variance’s formulae

$Populationvariance={\sigma}^{2}=\frac{{\displaystyle \sum _{i=1}^{N}}{\left({x}_{i}\u2013\mu \right)}^{2}}{N}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Samplevariance={s}^{2}=\frac{{\displaystyle \sum _{i=1}^{n}}{\left({x}_{i}\u2013\overline{x}\right)}^{2}}{n\u20131}$

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

### Mean absolute deviation (MAD)

$Meanabsolutedeviation\left(MAD\right)\hspace{0.17em}=\frac{{\displaystyle \sum _{i=1}^{n}}\left|{X}_{i}\u2013\overline{X}\right|}{n\u20131}$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 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:

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

### 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.

$Sharperatio=\frac{{r}_{p}\u2013{r}_{f}}{{\sigma}_{p}}$

where r_{p} = portfolio return, r_{f}= risk free rate, σ_{p}= standard deviation of portfolio returns

## Reading 8: Probability Concepts

### Expected value of a random variable X

#### $E\left(X\right)=P\left({X}_{1}\right){X}_{1}+P\left({X}_{2}\right){X}_{2}+...P\left({X}_{n}\right){X}_{n}$

### Probabilistic variance

${\sigma}^{2}(X)=\sum _{i=1}^{n}P\left({X}_{i}\right){\left[{X}_{i}\u2013E\left(X\right)\right]}^{2}$

### Correlation and covariance of returns

$\rho ,corr\left({R}_{A},{R}_{B}\right)=\frac{COV\left({R}_{A},{R}_{B}\right)}{\sigma \left({R}_{A}\right)\sigma \left({R}_{B}\right)}$

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

### Expected return on a portfolio

$E\left({R}_{p}\right)=\sum _{i=1}^{n}{w}_{i}E\left({R}_{i}\right)={w}_{1}E\left({R}_{1}\right)+{w}_{2}E\left({R}_{2}\right)+...{w}_{n}E\left({R}_{n}\right)$

### Variance of a 2 stock portfolio

$Var\left({R}_{P}\right)={w}_{A}^{2}{\sigma}^{2}\left({R}_{A}\right)+{w}_{B}^{2}{\sigma}^{2}\left({R}_{B}\right)+2{w}_{A}{w}_{B}\rho ({R}_{A},{R}_{B})\sigma \left({R}_{A}\right)\sigma \left({R}_{B}\right)\phantom{\rule{0ex}{0ex}}={w}_{A}^{2}{\sigma}^{2}\left({R}_{A}\right)+{w}_{B}^{2}{\sigma}^{2}\left({R}_{B}\right)+2{w}_{A}{w}_{B}COV({R}_{A},{R}_{B})$

## Reading 9: Common Probability Distributions

### 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:

$P(X=x)=C_{x}^{n}{p}^{x}{\left(1\u2013p\right)}^{n\u2013x}$

The expected value of a binomial random variable is simply:

$E\left(X\right)=n\times p$

Variance of a binomial random variable

${\sigma}^{2}=n\times p\times (1\u2013p)$

### Normal distribution

The normal distribution is a continuous symmetric probability distribution that is completely described by two parameters: its mean, μ, and its variance σ^{2}.

- 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.

$z=\frac{x\u2013\mu}{\sigma}$

### Roy’s safety-first ratio (SF ratio)

$SF\hspace{0.17em}ratio=\frac{E\left({R}_{P}\right)\u2013{R}_{Target}}{{\sigma}_{P}}$

## Reading 10: 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 σ^{2}, the sample mean X approximately has a normal distribution with mean μ and variance σ^{2}/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:

$Confidenceinterval=\overline{x}\pm {z}_{\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{\sigma}{\sqrt{n}}$

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

$Confidenceinterval=\overline{x}\pm {t}_{\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{s}{\sqrt{n}}$

## 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:

**Visually**

**2) Via a table**

H_{0} is true | H_{0} is false | |
---|---|---|

Reject H_{0} | Type 1 Error | Correct rejection |

Fail to Reject H_{0} | 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 TypeIIerror 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 Go

llum look like the Roman numeral II, for a TypeIIerror.

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

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 |

### 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.

${\chi}^{2}=\frac{(n\u20131){s}^{2}}{{\sigma}_{0}^{2}}$

### F-test for equality of variances of 2 populations

$F=\frac{{s}_{1}^{2}}{{s}_{2}^{2}}$

**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.

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