CFA Quantitative Methods – Formula Cheat Sheet

CFA Level I

Quantitative Methods

Formula Reference Sheet

Time Value of Money

Present Value

PV = FV(1 + r)n

Future Value

FV = PV · (1 + r)n

Perpetuity — Present Value

PV = PMTr / n

Position at Percentile y

(n + 1) × y100

Rates & Compounding

EFF — Periodic Rate

EFF = (1 + r)n − 1

EFF — APR & Periods

EFF = (1 + APRn)n − 1

APR from Periodic Rate

APR = r × n

Mean Types

Geometric Mean

[(1+r1)(1+r2)···(1+rn)]1/n − 1

Harmonic Mean (used for cost averaging)

n 1x1 + 1x2 + ··· + 1xn

Dispersion & Deviation

Mean Absolute Deviation (MAD)

Σi=1N |xi − x̄| n

Population Variance (σ²)

Σi=1N |xi − μ|2 n

Sample Variance (s²)

Σi=1N |xi − x̄|2 n − 1

Population Stdev

√ Pop. Variance

Sample Stdev

√ Sample Variance

Normal Distributions

Standard Deviation Rules

68%	falls within ± 1σ
90%	falls within ± 1.65σ
95%	falls within ± 1.96σ
99%	falls within ± 2.58σ

Z-Score

Z = x − μσ

(observed − population mean) / standard deviation

Confidence Intervals

Common Z-Values

Confidence Level	Z_α/2	Significance (α)	Each Tail
90%	1.645	10%	5%
95%	1.960	5%	2.5%
99%	2.575	1%	0.5%

Known Population Variance

x̄ ± Zα/2 · σ√n

Unknown Population Variance

x̄ ± tα/2 · s√n

Note: The t-test for correlation = r√(n−2) / √(1−r²) — always 2-tailed.

Standard Error of the Sample Mean

Known Population Variance

σ / √n

Unknown Population Variance

s / √n

Roy’s Safety-First Ratio

SFR — Higher is better

SFR = E(Rp) − RLσ = Expected Return − Threshold LevelStandard Deviation

Continuously Compounded Rate Equalities

Price Relative = Holding Period Return = e^Rcc

Final PriceInitial Price = (1 + HPR) = eRcc

Equivalently: FV / PV = e^{Rcc · n}

Linear Regression

Variation Decomposition

SSTTotal Variation

SSRExplained

SSEUnexplained

F-Statistic

F = MSRMSE = SSRk SSEn−(k+1)

k = number of independent variables (usually 1 in simple regression)

R² — Coefficient of Determination

R2 = SSRSST

Covariance (Joint Probability)

Calculate E(R) of both first, then:
P(Rx − E(Rx)) · (Ry − E(Ry))

Probability & Correlation

Coefficient of Variation

CV = Standard DeviationMean

Correlation of x & y

ρxy = Cov(xy)(σx)(σy)

Conditional Probability — P(B|A)

P(B|A) = P(A∩B)P(A)

Conditional Probability — P(A|B)

P(A|B) = P(A∩B)P(B)

Odds Against Event

1 − P(E)P(E)

= Probability of event not happening / Probability of event happening

Probability Rules

Multiplication Rule (Joint Probability)

P(AB) = P(A | B) × P(B)

Independent events: P(AB) = P(A) · P(B)
If P(A|B) = P(A), then A & B are independent

Addition Rule

P(A or B) = P(A) + P(B) − P(AB)

Total Probability Rule

P(A) = P(A|B1) × P(B1) + P(A|B2) × P(B2) + ··· + P(A|Bn) × P(Bn)

Expected Value & Portfolio Variance

Expected Value

EV = (X1)P(X1) + (X2)P(X2) + ··· + (Xn)P(Xn)

Variance of a 2-Stock Portfolio

wA2σA2 + wB2σB2 + 2wAwB · CovAB

Binomial Random Variable

Probability of “x” Successes in “n” Trials

p(x) = n! (n − x)! · x! · px · (1−p)n−x

Remember: 0! = 1

Expected Value — Binomial

E(x) = (n)(p)

Variance — Binomial

Var(x) = (n)(p)(1−p)

Note: Geometric Mean ≈ Arithmetic Mean − Harmonic Mean (relationship between the three mean types)