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. **