# Quantitative Methods • Formulas CFA® Level 1 • 365 Financial Analyst (2023)

Need an all-in-one list with the Quantitative Methods formulas included in the CFA® Level 1 Exam? We have compiled them for you here. The relevant formulas have been organized and presented by chapter. In this section, we will cover the following topics — Time Value of Money, Statistical Concepts and Market Returns, Probability, Distribution, Sampling, Estimation, and Hypothesis Testing.

## 1. Time Value of Money

Effective Annual Rate (EAR)

Effective~annual~rate = \bigg(1 + \frac {Stated~annual~rate}{m}\bigg)^m- 1

Single Cash Flow (simplified formula)

FV{_N} = PV \times (1 + r){^N}
PV = \frac {FV{_N}} {(1 + r){^N}}

r = interest rate per period
PV = present value of the investment
FV{_N} = future value of the investment N periods from today

Investments paying interest more than once a year

FV{_N} = PV \times \bigg(1+\frac{r{_s}}{m}\bigg){^{mN}}
PV = \frac{FV{_N}}{\bigg(1+\frac{r{_s}}{m}\bigg){^{mN}}}

r{_s} = Stated annual interest rate
m = Number of compounding periods per year
N = Number of years

Future Value (FV) of an Investment with Continuous Compounding

FV{_N} = PVe{^r{_s}}{^N}

Ordinary Annuity

FV{_N} = A \times \bigg[ \frac {(1+r){^N-1}}{r} \bigg]
PV = A \times \Bigg[ \frac {1-\frac{1}{(1+r){^N}}}{r} \Bigg]

N = Number of time periods
A = Annuity amount
r = Interest rate per period

Annuity Due

FV~A{_{Due}} = FV~A{_{Ordinary}} \times (1+r) = A \times \bigg[ \frac {(1+r){^N}-1}{r}\bigg] \times (1+r)
PV~A{_{Due}} = PV~A{_{Ordinary}} \times (1+r) = A \times \Bigg[ \frac {1-\frac{1}{(1+r){^N}}}{r} \Bigg] \times (1+r)

A = Annuity amount
r = the interest rate per period corresponding to the frequency of annuity payments (for example, annual, quarterly, or monthly)
N = the number of annuity payments

Present Value (PV) of a Perpetuity

PV{_{Perpetuity}} = \frac {A}{r}

A = Annuity amount

Future value (FV) of a series of unequal cash flows

FV{_N} = Cash~flow{_1}(1 + r){^1} + Cash~flow{_2}(1 + r){^2} … Cash~flow{_N}(1 + r){^N}

Net Present Value (NPV)

NPV=\displaystyle\sum_{t=0}^N \frac{CF_{t}}{(1+r)^t}

CF{_t} = Expected net cash flow at time t
N = The investment’s projected life
r = The discount rate or opportunity cost of capital

Internal Rate of Return (IRR)

NPV= CF{_0} + \frac {CF{_1}}{(1+IRR){^1}} + \frac {CF{_2}}{(1+IRR){^2}} + … + \frac {CF{_N}}{(1+IRR){^N}} = 0

Holding Period Return (HPR)

No cash flows

HPR = \frac {Ending~value - Beginning~value}{Beginning~value}

Holding Period Return (HPR)

Cash flows occur at the end of the period

HPR = \frac {Ending~value - Beginning~value+ Cash~flows~received}{Beginning~value} = \frac {P{_1} - P{_0} + D{_1}}{Beginning~value}

P{_1} = Ending Value
P{_0} = Beginning Value
D = Cash flow/dividend received

Yield on a Bank Discount Basis (BDY)

r{_{BD}}= \frac {D}{F} \times \frac {360}{t}
(Video) CFA Level I | How to Study for Quantitative Methods

r{_{BD}} = Annualized yield on a bank discount basis
D = Dollar discount, which is equal to the difference between the face value of the bill (F) and its purchase price (P{_0})
F = Face value of the T-bill
t = Actual number of days remaining to maturity

Effective Annual Yield (EAY)

EAY = ( 1 + HPR) {^\frac {360}{t}}- 1

t = Time until maturity
HPR = Holding Period Return

Money Market Yield (CD Equivalent Yield)

Money~market~yield = HPR \times \bigg(\frac {360}{t}\bigg) = \frac {360 \times r{_{Bank~Discount}}}{360-(t \times r{_{Bank~Discount}})}

## 2. Statistical Concepts and Market Returns

Interval Width

Interval~Width = \frac {Range}{k}

Range = Largest observation number – Smallest Observation or number
k = Number of desired intervals

Relative Frequency

Relative~frequency = \frac {Interval~frequency}{Observations~in~data~set}

Population Mean

\mu = \frac {\displaystyle\sum_{i=1…n}^Nx{_i}}{N}= \frac {{x{_1}} + {x{_2}} + {x{_3}} + … +{x{_N}}} {N}

N = Number of observations in the entire population
x{_i} = the iᵗʰ observation

Sample Mean

\overline x= \frac {\displaystyle\sum_{i=1…n}^nx{_i}}{n} = \frac {{x{_1}} + {x{_2}} + {x{_3}} + … +{x{_n}}} {n}

Geometric Mean

G=\sqrt[n]{x{_1}{x{_2}{x{_3}}}…{x{_n}}}

n = Number of observations

Harmonic Mean

\overline x{_n}= \frac {n}{\displaystyle\sum_{i=1…n}^n \bigg(\frac{1}{x{_i}}\bigg)}

Median for odd numbers

Median= \Bigg\{ \frac {(n+1)}{2} \Bigg\}

Median for even numbers

Median= \Bigg\{ \frac {(n+2)}{2} \Bigg\}
Median= \frac {n}{2}

Weighted Mean

\overline x{_w} = \displaystyle\sum_{i=1…n}^n w{_i}x{_i}

w = Weights
x = Observations
Sum of all weights = 1

Portfolio Rate of Return

r{_p} = w{_a}r{_a} + w{_b}r{_b} + w{_c}r{_c} + … + w{_n}r{_n}

w = Weights
r = Returns

Position of the Observation at a Given Percentile y

L{_y} = \bigg\{ {(n+1)}\frac{y}{100} \bigg\}

y = The percentage point at which we are dividing the distribution
L{_y} = The location (L) of the percentile (Py) in the array sorted in ascending order

Range

Range= Maximum~value - Minimum~value

Mean Absolute Deviation

MAD =\frac {\displaystyle\sum_{i=1…n}^n |X{_i-\overline X}|}{n}

x = The sample mean
n = Number of observations in the sample

Population Variance

(Video) 1. CFA Level 1 Quantitative Methods Time Value of Money LO1 and LO2

\sigma{^2} = \frac {\displaystyle\sum_{i=1…n}^n (X{_i-\mu}){^2}}{N}

μ = Population mean
N = Size of the population

Population Standard Deviation

\sigma= \sqrt { \frac {\displaystyle\sum_{i=1…n}^N (X{_i-\mu}){^2}}{N}}

μ = Population mean
N = Size of the population

Sample Variance

S{^2} = \frac {\displaystyle\sum_{i=1}^n (X{_i-\overline X}){^2}}{n-1}

x = Sample mean
n = Number of observations in the sample

Sample Standard Deviation

s = \sqrt { \frac {\displaystyle\sum_{i=1}^n (X{_i-\overline X}){^2}}{n-1}}

x = Sample mean
n = Number of observations in the sample

Semi-Variance

Semi–variance = \frac {1}{n}\displaystyle\sum_{r{_t} < Mean}^n (Mean-r{_t}){^2}

n = Total number of observations below the mean
r{_t} = Observed value

Chebyshev Inequality

Percentage~of~observations~within~k~standard~deviations~of~the~arithmetic~mean > 1-\frac{1}{k{^2}}

k = Number of standard deviations from the mean

Coefficient of Variation

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

s = Sample standard deviation
\overline X = Sample mean

Sharpe Ratio

Sharpe~Ratio = \frac {R{_p}-R{_f}}{\sigma{_p}}

R{_p} = Mean return to the portfolio
R{_f} = Mean return to a risk-free asset
σ{_p} = Standard deviation of return on the portfolio

Skewness

s{_k}= \bigg[ \frac {n}{(n-1)(n-2)} \bigg] \times \frac {\displaystyle\sum_{i=1…n}^n (X{_i}-\overline X){^3}}{s{^3}}

n = Number of observations in the sample
s = Sample standard deviation

Kurtosis

K{_E}=\Bigg[ \frac {n (n+1)}{(n - 1)(n - 2)(n - 3)} \times \frac {\displaystyle\sum_{i=1…n}^n (X{_i}-\overline X){^4}}{s{^4}}\Bigg] \times \frac {3~(n-1){^2}}{(n - 2)(n - 3)}

n = Sample size
s = Sample standard deviation

## 3. Probability Concepts

Odds FOR E

Odds~FOR~E = \frac {P(E)}{1-P(E)}

E = Odds for event
P(E) = Probability of event

Conditional Probability

P(A|B) = \frac {P (A \cap B)}{P (B)}

where P(B) ≠ 0

Additive Law (The Addition Rule)

P(A \cup B) = P(A) + P(B) - P(A \cap B)

The Multiplication Rule (Joint Probability)

P(A \cap B) = P(A|B) \times P(B)

The Total Probability Rule

P(A) = P(A|S1) \times P(S{_1}) + P(A|S{_2}) \times P(S{_2}) + … + P(A|S{_n}) \times P(S{_n})
(Video) CFA Level I - Quantitative Methods - Revision Webinar

S{_1}, S{_2}, …, S{_n} are mutually exclusive and exhaustive scenarios or events

Expected Value

E(X) = P(A)X{_A} + P(B)X{_B} + ... + P(n)X{_n}

P(n) = Probability of an variable
X{_n} = Value of the variable

Covariance

COV {_{xy}}= \frac {(x-\overline x)(y-\overline y)}{n-1}

x = Value of x
\overline x = Mean of x values
y = Value of y
\overline y = Means of y
n = Total number of values

Correlation

\rho = \frac {cov{_{xy}}}{\sigma{_x}\sigma{_y}}

σ{_x} = Standard Deviation of x
σ{_y} = Standard Deviation of y
cov{_{xy}} = Covariance of x and y

Variance of a Random Variable

\sigma{^2} x= \displaystyle\sum_{i=1…n}^n \big(x - E(x)\big){^2} \times P(x)

The sum is taken over all values of x for which p(x) > 0

Portfolio Expected Return

E(R{_P}) = E(w{_1}r{_1} + w{_2}r{_2} + w{_3}r{_3} + … + w{_n}r{_n})

w = Constant
r = Random variable

Portfolio Variance

Var(R{_P}) = E\big[(R{_p} - E(R{_p}){^2} \big] = \big[w{_1}{^2} \sigma{_1}{^2} + w{_2}{^2}\sigma{_2}{^2} + w{_3}{^2}σ{_3}{^2} + 2w{_1}w{_2}Cov(R{_1}R{_2}) + 2w{_2}w{_3}Cov(R{_2}R{_3}) + 2w{_1}w{_3}Cov(R{_1}R{_3})\big]

R{_p} = Return on Portfolio

Bayes’ Formula

P(A|B) = \frac {P(B|A) \times P(A)}{P(B)}

The Combination Formula

nC{_r} = \binom{n}{c} = \frac {n!}{(n - r)! r!}

n = Total objects
r = Selected objects

The Permutation Formula

nP{_r} = \frac {n!}{(n - r)!}

## 4. Common Probability Distributions

The Binomial Probability Formula

P(x) = \frac {n!}{(n - x)! x!}p{^x} \times (1 - p){^{n - x}}

n = Number of trials
x = Up moves
p{^x} = Proability of up moves
(1 - p){^{n - x}} = Probability of down moves

Binomial Random Variable

E(X) = np
Variance = np(1 - p)

n = Number of trials
p = Probability

For a Random Normal Variable X

90% confidence interval for X is \overline x - 1.65s;~ \overline x + 1.65s
95% confidence interval for X is \overline x - 1.96s;~ \overline x + 1.96s
99% confidence interval for X is \overline x - 2.58s;~ \overline x + 2.58s

s = Standard error
1.65 = Reliability factor
x = Point estimate

Safety-First Ratio

SF{_{Ratio}}=\bigg[ \frac {E(R{_p}) - R{_L}}{\sigma{_p}} \bigg]

R{_p} = Portfolio Return
R{_L} = Threshhold level
σ{_p} = Standard Deviation

Continuously Compounded Rate of Return

FV = PV \times e{^{i \times t}}
(Video) Level 1 Chartered Financial Analyst (CFA ®): Conditional, unconditional and joint probabilities

i = Interest rate
t = Time
ln~e = 1
e = the exponential function, equal to 2.71828

## 5. Sampling and Estimation

Sampling Error of the Mean

Sample~Mean - Population~Mean

Standard Error of the Sample Mean (Known Population Variance)

SE = \frac {\sigma}{\sqrt n}

n = Number of samples
σ = Standard deviation

Standard Error of the Sample Mean (Unknown Population Variance)

SE = \frac {S}{\sqrt n}

S = Standard deviation in unknown population’s sample

Z-score

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

x = Observed value
σ = Standard deviation
μ = Population mean

Confidence Interval for Population Mean with z

\overline X - {Z{_{\frac {\alpha}{2}}}} \times \frac {\sigma}{\sqrt n}; \overline X + {Z{_{\frac {\alpha}{2}}}} \times \frac {\sigma}{\sqrt n}

Z{_{\frac {\alpha}{2}}} = Reliability factor
X = Mean of sample
σ = Standard deviation
n = Number of trials/size of the sample

Confidence Interval for Population Mean with t

\overline X - {t{_{\frac {\alpha}{2}}}} \times \frac {S}{\sqrt n}; \overline X + {t{_{\frac {\alpha}{2}}}} \times \frac {S}{\sqrt n}

t{_{\frac {\alpha}{2}}} = Reliability factor
n = Size of the sample
S = Standard deviation

Z or t-statistic?

Z \longrightarrow known population, standard deviation σ, no matter the sample size
t \longrightarrow unknown population, standard deviation s, and sample size below 30
Z \longrightarrow unknown population, standard deviation s, and sample size above 30

## 6. Hypothesis Testing

Test Statistics: Population Mean

z{_\alpha} = \frac {\overline X- \mu} {\frac {\sigma}{\sqrt n}}; t{_{n-1, \alpha}} = \frac {\overline X- \mu} {\frac {s}{\sqrt n}}

t{_{n-1}} = t-statistic with n–1 degrees of freedom (n is the sample size)
\overline X = Sample mean
μ = The hypothesized value of the population mean
s = Sample standard deviation

Test Statistics: Difference in Means – Sample Variances Assumed Equal (independent samples)

t–statistic = \frac {(\overline X{_1} - \overline X{_2}) - (μ{_1} - μ{_2})}{\Big( \frac {s{_p}{^2}}{n{_1}} + \frac {s{_p}{^2}}{n{_2}} \Big){^{\frac {1}{2}}}}
s{_p}{^2}= \frac {(n{_1}-1)s{_1}{^2}+(n{_2}-1)s{_2}{^2}}{n{_1}+n{_2}-2}

Number of degrees of freedom = n{_1} + n{_2} − 2

Test Statistics: Difference in Means – Sample Variances Assumed Unequal (independent samples)

t–statistic = \frac {(\overline x{_1} - \overline x{_2}) - (μ{_1} - μ{_2})}{\Big( \frac {s{_1}{^2}}{n{_1}} + \frac {s{_2}{^2}}{n{_2}} \Big){^{\frac {1}{2}}}}
degrees~of~freedom = \frac {\Big( \frac {s{_1}{^2}}{n{_1}} + \frac {s{_2}{^2}}{n{_2}} \Big){^2}} { \frac {\big(\frac {s{_1}{^2}}{n{_1}} \big){^2}}{n{_1}} + \frac {\big(\frac {s{_2}{^2}}{n{_2}} \big){^2}}{n{_2}}}

s = Standard deviation of respective sample
n = Total number of observations in the respective population

Test Statistics: Difference in Means – Paired Comparisons Test (dependent samples)

t = \frac {\overline d - \mu {_{dz}}}{S{_d}}, where \overline d = \frac {1}{n} \displaystyle\sum_{i=1…n}^n d{_i}

degrees of freedom: n–1
n = Number of paired observations
d = Sample mean difference
S{_d} = Standard error of d

Test Statistics: Variance Chi-square Test

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

degrees of freedom = n - 1
s{^2} = sample variance
\sigma{_0}^2 = hypothesized variance

Test Statistics: Variance F-Test

F = \frac {s{_1}^2}{s{_2}^2}, where {s{_1}^2} > {s{_2}^2}

degrees of freedom = n{_1} - 1 and n{_2} - 1
{s{_1}^2} = larger sample variance
{s{_2}^2} = smaller sample variance

Follow the links to find more formulas on Economics, Corporate Finance, Alternative Investments, Financial Reporting and Analysis, Portfolio Management, Equity Investments, Fixed-Income Investments, and Derivatives, included in the CFA® Level 1 Exam.

(Video) Probability Concepts (2023 CFA® Level I Exam – Quantitative Methods – Module 3)

## FAQs

### Is there a formula sheet for CFA Level 1? ›

Does the CFA exam provide a formula sheet? The CFA Institute does not provide a formula sheet for the CFA exam, and that's where Wiley's trusted formula sheets offer an advantage to CFA candidates.

How to study quants in CFA Level 1? ›

It is also important in the first two levels of the CFA exam, representing 12% in Level 1 and 5-10% in Level 2.
...
CFA Quant Study Tips
1. Manage your progress. ...
2. Focus on the logic, not memorizing the formula. ...
3. Watch Khan Academy channel. ...
4. Practice, practice and practice. ...
5. Allow time for quick review.

Which topic is the hardest for CFA Level 1? ›

CFA exam candidates who take their level 1 exam usually indicate FI, Derivatives, and FSA (or FRA)* to be the most difficult.

Which is the easiest topic in CFA Level 1? ›

Corporate Finance is one of the easiest topics of the Level one exam this is because this subject is very logical and intuitive and the formulas are easy. The scope is also quite limited and includes concepts related to NPV, IRR, corporate governance and capital management.

How to memorize formulas for CFA Level 1? ›

For those hard to remember key formulas, rewrite them every week. it will take you about 5min, and it will help you with the memorization process.
...
But when you reach that stage:
1. Create your own flashcards.
2. Stick them up near your study table.
3. Every 3 days, read through the entire formulae in your handwriting.
Aug 2, 2012

What is the hardest CFA exam? ›

Many CFA charterholders consider the Level 3 CFA Exam the most difficult because of the time and thought needed to answer the constructed responses successfully. While the typical Level 3 CFA Exam pass rates are the highest of the CFA Exams, only around 56% of CFA candidates pass the exam.

Can you complete CFA Level 1/3 months? ›

Let the one below be a sample 3-month schedule then... The FEB 2023 level 1 exam window lasts from 14 to 20 February 2023. So, you can have from 3 to 4 weeks of – what we call – Final Review. Basically that's when you should do as many mock exams as possible!

What is CFA Level 1 Expected salary? ›

A Chartered Financial Analyst (CFA) can expect an average starting salary of ₹3,92,500. The highest salaries can exceed ₹16,00,000.

Is Passing CFA Level 1 hard? ›

Overall, the CFA exams are very difficult, but candidates can increase their chances of passing by studying for over 300 hours, utilizing alternative prep materials, answering as many practice questions as possible, and creating a structured study plan.

Which is the easiest CFA level? ›

Corporate Finance is one of the easiest topics of the Level one exam this is because this subject is very logical and intuitive and the formulas are easy.

### How many times can you fail the CFA Level 1? ›

CFA Program exam results do not expire, and you are not required to enroll each year. There is no limit to the amount of time you have to complete the CFA Program. Beginning with the 2021 computer-based exams, each level exam can be taken twice each year, with a total of six maximum attempts per exam level.

How many hours a day should I study for CFA Level 1? ›

If you read two hours per day on weekdays and eight hours on the weekends, then it will take about 17 weeks (four months) to complete all of the readings. Weekends and nonworking days are the best time to study for the CFA.

Does CFA Level 1 look good on resume? ›

The Chartered Financial Analyst (CFA) qualification is a big asset for an investment professional and should be highlighted on a resume.

How many hours a week should you study for CFA Level 1? ›

CFA 6-Month Study Guide: Month 1

For the majority of candidates, we recommend a 6-month study plan. This would break down to about 12 hours per week or two hours per day for 6 days per week.

How can I pass CFA Level 1 in a month? ›

Here are 10 steps and some important tips to help you pass the CFA level 1 exam on the first try.
1. Step 1: Count the days you've on hand : ...
2. Step 2: Plan your Study : ...
3. Step 3: Reading the study notes : ...
4. Step 4: Do the end of chapter questions (EOCs) : ...
5. Step 5: Attempt more questions topic-wise :

What is the fastest way to memorize formulas? ›

8 Easy Ways to Memorize Math Formulas
1. Develop a Keen Interest in the Subject.
2. Employ Visual Memory.
3. Try Different Learning Strategies.
4. Get Rid of Distractions.
5. Practice as Much as You can.
6. Use Memory Techniques.
7. Sleep on it.
8. Understand the Formulas.
Jul 20, 2022

How can I study formulas quickly? ›

So, here are 8 ways to memorize maths formulas in an easy way.
1. Grow your interest in the concept in which you are studying. ...
2. Connect your concept with a visual memory. ...
3. Knowing the process behind maths formulas. ...
4. Always solve the problems with math formulas. ...
5. Write down maths formulas. ...
6. Stick your written formulas on your wall.
Dec 22, 2021

What is the hardest formula? ›

For decades, a math puzzle has stumped the smartest mathematicians in the world. x3+y3+z3=k, with k being all the numbers from one to 100, is a Diophantine equation that's sometimes known as "summing of three cubes."

Can you pass CFA in 2 weeks? ›

At the end of the day, yes, I did pass with just two weeks of study. But I think I was INCREDIBLY lucky to have managed. I'm glad I studied hard and tried to pass it instead of waiting for the next exam.

Why are CFA passing score so low? ›

The September results for the final test follow historically low pass rates across all levels of the CFA exam in recent months. The institute has cited testing disruptions stemming from the Covid-19 pandemic as a reason for the trend.

### Is CFA toughest course in the world? ›

The CFA course takes at least 2 years or more to be completed. CA is considered the toughest course throughout the world with an average passing percentage of 0.5%. The passing percentage of CFA is much higher as compared to CA. The average pass percentage seems to be 10%.

Is cheat sheet allowed in CFA? ›

No they don't provide any formula sheet and neither do they allow you to bring one. You need to learn all the formulas. I understand that there is a lot given in the curriculum. One thing you can do is that write all the formulas everyday till the exam.

Is there a lot of math in CFA Level 1? ›

The first two CFA exam levels are significantly quantitative - there is literally a topic area called Quantitative Methods. This topic area covers subjects like time value of money, discounted cash flow, and lots of statistical concepts, probability distributions and hypothesis testing.

Which calculator is best for CFA Level 1? ›

The CFA exam calculator policy allows only two calculator models for the exam:
• Texas Instruments (TI) BA II Plus (including BA II Plus Professional)
• Hewlett Packard (HP) 12C (including the HP 12C Platinum, 12C Platinum 25th anniversary edition, 12C 30th-anniversary edition, and HP 12C Prestige)
Jun 20, 2022

What is the best study material for CFA Level 1? ›

• Best seller. ...
• 2022 | CFA Level I - JuiceNotes [Revision material] | Set of 4 Booklets. ...
• 2023 CFA Wiley Study Guide Level 1 (Set of 5) ...
• 2023 CFA Level 1 Question Bank - 1000 Critical Question based on 2023 CFA curriculum. ...
• 2023 Certificate in ESG Investing Curriculum: ESG Investing Official Training Manual.

## Videos

1. CFA level 1 Video- Formula Revision - Quant - Part 1(of 2)
(FinTree)
2. Quants Formula Review: CFA Exam Level 1
(Kaplan Professional Middle East)
3. CFA Level 1 Practice Mock Exam Questions Quantitative Methods part 1
(Cassandra' s Tutoring)
4. 2. CFA Level 1 Quantitative Methods Time Value of Money LO3 to LO6 Part 1
(Mark Meldrum)
5. CFA Level 1 - Introduction to Linear Regression - Part 1
(Financial Training Solutions)
6. CFA Level 1 Revision | Quantitative Methods | The Time Value Of Money | Reading 1
(The Excel Hub)
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