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}

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 *i*ᵗʰ 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**

\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})

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}}

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. **

## 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? ›**

...

**CFA Quant Study Tips**

- Manage your progress. ...
- Focus on the logic, not memorizing the formula. ...
- Watch Khan Academy channel. ...
- Practice, practice and practice. ...
- 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? ›**

...

**But when you reach that stage:**

- Create your own flashcards.
- Stick them up near your study table.
- Every 3 days, read through the entire formulae in your handwriting.

**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.**

- Step 1: Count the days you've on hand : ...
- Step 2: Plan your Study : ...
- Step 3: Reading the study notes : ...
- Step 4: Do the end of chapter questions (EOCs) : ...
- Step 5: Attempt more questions topic-wise :

**What is the fastest way to memorize formulas? ›**

**8 Easy Ways to Memorize Math Formulas**

- Develop a Keen Interest in the Subject.
- Employ Visual Memory.
- Try Different Learning Strategies.
- Get Rid of Distractions.
- Practice as Much as You can.
- Use Memory Techniques.
- Sleep on it.
- Understand the Formulas.

**How can I study formulas quickly? ›**

**So, here are 8 ways to memorize maths formulas in an easy way.**

- Grow your interest in the concept in which you are studying. ...
- Connect your concept with a visual memory. ...
- Knowing the process behind maths formulas. ...
- Always solve the problems with math formulas. ...
- Write down maths formulas. ...
- Stick your written formulas on your wall.

**What is the hardest formula? ›**

For decades, a math puzzle has stumped the smartest mathematicians in the world. **x ^{3}+y^{3}+z^{3}=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)

**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.