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Free CFA Level I Formula Sheet PDF, No Signup (2026)

Quantitative Methods 14 items

Present Value (single sum)

P V = \frac{F V}{( 1 + r ) ^{n}}

FV = future value, r = periodic rate, n = number of periods

Future Value of ordinary annuity

F V = P M T \times \frac{( 1 + r ) ^{n} - 1}{r}

PMT = periodic payment, r = periodic rate, n = periods

Present Value of ordinary annuity

P V = P M T \times \frac{1 - ( 1 + r ) ^{- n}}{r}

PMT = periodic payment, r = periodic rate, n = periods

Present Value of perpetuity

P V = \frac{P M T}{r}

PMT = periodic payment, r = discount rate

Population variance

σ^{2} = \frac{\sum _{i = 1}^{N} ( X _{i} - μ ) ^{2}}{N}

μ

= population mean, N = population size

Sample variance

s^{2} = \frac{\sum _{i = 1}^{n} ( X _{i} - X ˉ ) ^{2}}{n - 1}

X ˉ

= sample mean, n = sample size (uses n−1 for unbiasedness)

Sharpe ratio

S_{p} = \frac{R _{p} - R _{f}}{σ _{p}}

R_p = portfolio return, R_f = risk-free rate,

σ_{p}

= portfolio std dev

Excess return per unit of total risk

Holding Period Return (HPR)

H P R = \frac{P _{1} - P _{0} + D _{1}}{P _{0}}

P_1 = ending price, P_0 = beginning price, D_1 = cash distributions received

Bayes' Theorem

P (A ∣ B) = \frac{P ( B ∣ A ) \cdot P ( A )}{P ( B )}

Updates prior probability P(A) given new information B

P (B) = P (B ∣ A) P (A) + P (B ∣ A^{c}) P (A^{c})

Correlation coefficient

ρ_{X Y} = \frac{Cov ( X , Y )}{σ _{X} σ _{Y}}

Ranges from −1 to +1; unitless measure of linear association

Geometric mean return

R ˉ_{G} = [\prod_{t = 1}^{n} (1 + R_{t})]^{1/ n} - 1

— R_t = period t return, n = number of periods

Money-weighted return (MWR)

\sum_{t = 0}^{N} \frac{C F _{t}}{( 1 + r ) ^{t}} = 0

— CF_t = cash flow at time t (outflows negative, inflows positive), r = money-weighted return, N = number of periods

Fisher relation (real vs nominal return)

(1 + R_{n o mina l}) = (1 + R_{r e a l}) (1 + i)

; R_nominal = nominal, R_real = real, i = inflation. Subtracting inflation is only an approximation; it breaks down when inflation is high.

Time-weighted return (TWR)

(1 + T W R) = \prod_{i = 1}^{n} (1 + r_{i})

— r_i = holding-period return for sub-period i, n = number of sub-periods between external cash flows

Economics 6 items

GDP expenditure approach

G D P = C + I + G + (X - M)

C = consumption, I = investment, G = government spending

X = exports, M = imports, (X−M) = net exports

Money multiplier

m = \frac{1}{reserve requirement}

Maximum deposit expansion from a given reserve base

Δ Money supply = m \times Δ Reserves

Fisher effect

(1 + r_{n o m}) = (1 + r_{r e a l}) (1 + π)

Approx:

r_{n o m} \approx r_{r e a l} + π

r_nom = nominal rate, r_real = real rate,

π

= expected inflation

Breakeven and shutdown points

Breakeven:

P = A T C

(price covers all costs).

Shutdown (short run):

P < A V C

. If

A V C < P < A T C

, keep operating short-run because contribution covers some fixed cost.

Price elasticity of demand

E_{d} = \frac{%Δ Q _{d}}{%Δ P}

∣ E_{d} ∣ > 1

: elastic (revenue rises when P falls).

∣ E_{d} ∣ < 1

: inelastic.

= 1

: unit-elastic (revenue maximized).

Cross-rate calculation

\frac{A}{C} = \frac{A}{B} \times \frac{B}{C}

Multiply or divide quoted rates to derive a cross-rate. Bid-ask: use bid for one leg and ask for the other to be conservative.

Corporate Issuers 6 items

WACC

W A C C = w_{d} r_{d} (1 - t) + w_{p} r_{p} + w_{e} r_{e}

w = weights (market value), r = required returns

d = debt, p = preferred, e = equity, t = tax rate

Free Cash Flow to the Firm (FCFF)

F C F F = N I + N C C + I n t (1 - t) - Δ W C - C a pE x

NI = net income, NCC = non-cash charges, Int = interest expense

Δ W C

= change in working capital, t = tax rate

Net Present Value (NPV)

N P V = \sum_{t = 0}^{n} \frac{C F _{t}}{( 1 + r ) ^{t}}

CF_t = expected after-tax cash flow at time t, r = required rate of return

Accept when NPV > 0: the investment adds value in currency terms

Degree of Financial Leverage (DFL)

D F L = \frac{E B I T}{E B I T - I}

I = interest expense

% change in EPS per 1% change in EBIT

Return on Invested Capital (ROIC)

R O I C = \frac{After-tax operating profit}{Average invested capital}

Invested capital = total debt + total equity

ROIC above the cost of capital means the company is creating value

Free Cash Flow to Equity (FCFE)

F C F E = F C F F - Int (1 - t) + Δ Debt

Cash available to equity holders after all obligations and reinvestment

Financial Statement Analysis 8 items

3-factor DuPont decomposition

R O E = \frac{N I}{Sales} \times \frac{Sales}{Assets} \times \frac{Assets}{Equity}

Net profit margin × Asset turnover × Financial leverage

Current ratio

Current ratio = \frac{Current assets}{Current liabilities}

Measures short-term liquidity; higher = more liquid

Inventory turnover

Inventory turnover = \frac{COGS}{Average inventory}

Days on hand (DOH):

D O H = \frac{365}{Inventory turnover}

Receivables turnover and DSO

Receivables turnover = \frac{Revenue}{Avg Accounts Receivable}

DSO = \frac{365}{Receivables turnover}

Days Sales Outstanding — average collection period

Return on Assets (ROA)

R O A = \frac{Net income}{Average total assets}

Alternative:

R O A = Net profit margin \times Asset turnover

2-factor DuPont decomposition

Return on Equity (ROE)

R O E = \frac{Net income}{Avg total equity}

DuPont: ROE = Net margin × Asset turnover × Leverage. Drives sustainable growth:

g = b \times R O E

.

Cash flow interest coverage ratio

Coverage = \frac{C F O + Int paid + Tax paid}{Int paid}

; CFO = cash from operations. Distinct from accounting version EBIT/Interest; the exam loves to swap them.

Cash return on assets

Cash ROA = \frac{C F O}{Average total assets}

— CFO = cash flow from operations; denominator uses average of beginning and ending total assets

Equity Investments 20 items

Gordon Growth Model (DDM)

V_{0} = \frac{D _{1}}{r - g} = \frac{D _{0} ( 1 + g )}{r - g}

D_1 = next dividend, r = required return, g = constant growth rate

Requires r > g

Justified P/E (leading)

\frac{P _{0}}{E _{1}} = \frac{1 - b}{r - g}

b = retention ratio (1−b = payout ratio), r = required return, g = ROE × b

Enterprise Value (EV)

E V = Market cap + Debt + Preferred + Minority interest - Cash

E V / E B I T D A

= enterprise value multiple

Capital-structure-neutral valuation metric

Price-to-Book ratio

P/B = \frac{Market price per share}{Book value per share}

Justified P/B:

\frac{R O E - g}{r - g}

P/B > 1 implies market values assets above book

P/E ratio (trailing & leading)

Trailing:

\frac{P _{0}}{E P S _{0}}

— uses last 12 months EPS.

Leading:

\frac{P _{0}}{E P S _{1}}

— uses next 12 months / forecast EPS. Forward-looking variant.

Equity value per share from enterprise value

P_{0} = \frac{E V - D e b t + C a s h}{S ha r es}

— EV = enterprise value, Debt = interest-bearing debt, Cash = cash and equivalents, Shares = diluted shares outstanding

Terminal value via Gordon growth applied to FCFF

T V_{n} = \frac{F C F F _{n + 1}}{W A C C - g}

— FCFF_{n+1} = next-period free cash flow to firm, WACC = weighted avg cost of capital, g = sustainable long-run growth

Residual income

R I_{t} = N I_{t} - (r \times B V_{t - 1})

— NI = net income, r = cost of equity, BV = book value of equity at start of period

Arbitrage pricing theory (APT) expected return

E (R_{i}) = R_{f} + \sum_{k = 1}^{K} β_{i, k} λ_{k}

— R_f = risk-free rate, β_{i,k} = sensitivity of asset i to factor k, λ_k = risk premium per unit exposure to factor k

Two-stage dividend discount model

V_{0} = \sum_{t = 1}^{n} \frac{D _{t}}{( 1 + r ) ^{t}} + \frac{D _{n + 1} / ( r - g _{s} )}{( 1 + r ) ^{n}}

— D_t = dividend at time t, r = cost of equity, g_s = stable growth, n = explicit horizon

Carhart four-factor model

E (R_{i}) - R_{f} = β_{i, M} [E (R_{m}) - R_{f}] + β_{i, S} SMB + β_{i, V} HML + β_{i, W} WML

— SMB = size, HML = value, WML = momentum premiums; β = loadings

Single-stage FCF perpetuity enterprise value

E V = \frac{F C F _{0} \times ( 1 + g )}{W A C C - g}

— FCF₀ = current free cash flow, g = terminal growth rate, WACC = weighted-average cost of capital

Total return on an equity security

R_{t o t a l} = \frac{P _{1} - P _{0} + D _{1}}{P _{0}} = R_{p r i ce} + \frac{D _{1}}{P _{0}}

— P_0 = beginning price, P_1 = ending price, D_1 = dividends received

Price return on an equity security

R_{p r i ce} = \frac{P _{1} - P _{0}}{P _{0}}

— P_0 = beginning price, P_1 = ending price

Average daily volume (ADV)

ADV = \frac{\sum _{i = 1}^{n} V _{i}}{n}

— V_i = shares traded on day i, n = number of trading days in the window

Free float shares

Float = Shares Outstanding - Restricted Shares

— Restricted = insider lock-ups, strategic stakes, treasury shares, and government holdings

Justified trailing P/E from Gordon growth

\frac{P _{0}}{E _{0}} = \frac{( 1 - b ) ( 1 + g )}{r - g}

— b = retention ratio, (1-b) = payout ratio, r = required return on equity, g = sustainable growth rate

Implied price via method of comparables

P_{t a r g e t} = M_{p eer} \times F_{t a r g e t}

— M_peer = peer-group median multiple, F_target = target's per-share fundamental (EPS, BVPS, etc.)

Cumulative voting total votes available

V = S \times N

— V = total votes a shareholder may cast, S = shares owned, N = number of director seats up for election

Voting power share in a dual-class structure

V P = \frac{S _{A} v _{A} + S _{B} v _{B}}{\sum _{i} S _{i} v _{i}}

— S = shares held in class, v = votes per share in class, denominator = total votes cast across all classes

Fixed Income 14 items

Bond price

P = \sum_{t = 1}^{n} \frac{C}{( 1 + r ) ^{t}} + \frac{F V}{( 1 + r ) ^{n}}

C = coupon payment, r = periodic YTM, n = periods, FV = face value

Current yield

Current yield = \frac{Annual coupon}{Price}

Simplest yield measure; ignores capital gains/losses and time value

Macaulay duration

D_{M a c} = \frac{\sum _{t = 1}^{n} t \cdot \frac{C F _{t}}{( 1 + r ) ^{t}}}{P}

Weighted average time to receive cash flows; measured in years

Modified duration

D_{M o d} = \frac{D _{M a c}}{1 + r}

%Δ P \approx - D_{M o d} \times Δ y

r = periodic YTM,

Δ y

= change in yield

Forward rate from spot rates

(1 + z_{2})^{2} = (1 + z_{1}) (1 +_{1} f_{1})

General:

(1 + z_{n})^{n} = (1 + z_{n - 1})^{n - 1} (1 +_{n - 1} f_{1})

z = spot rate, f = implied forward rate

Price value of a basis point (PVBP)

P V B P = ∣ P_{y + 0.01%} - P_{y} ∣

Alternative:

P V B P = D_{mod} \times P \times 0.0001

Dollar price change for a 1 bp yield move

Floating-rate note price using discount margin

P V = \sum_{t = 1}^{N} \frac{( M R R + QM ) F V / m}{( 1 + ( M R R + D M ) / m ) ^{t}} + \frac{F V}{( 1 + ( M R R + D M ) / m ) ^{N}}

— MRR = reference rate, QM = quoted margin, DM = discount margin, m = periods/yr, FV = face

Bond equivalent yield for money market instruments

B E Y = \frac{F V - P V}{P V} \cdot \frac{365}{d a y s}

— FV = face value, PV = price, days = days to maturity

Debt-to-EBITDA leverage ratio

Debt/EBITDA = \frac{Total Debt}{EBITDA}

— Total Debt = all interest-bearing debt; EBITDA = earnings before interest, taxes, depreciation, amortization. Lower is stronger.

EBITDA-to-interest coverage ratio

EBITDA/Interest = \frac{EBITDA}{Interest Expense}

— EBITDA = earnings before interest, taxes, depreciation, amortization; Interest Expense = period interest. Higher is stronger.

Effective convexity

EffCon = \frac{P _{-} + P _{+} - 2 P _{0}}{P _{0} \times ( Δ y ) ^{2}}

— P₋ = price if yields fall, P₊ = price if yields rise, P₀ = initial price, Δy = yield shock (decimal)

Effective duration

EffDur = \frac{P _{-} - P _{+}}{2 \times P _{0} \times Δ y}

— P₋ = price if yields fall, P₊ = price if yields rise, P₀ = initial price, Δy = yield shock (decimal)

Approximate convexity

ApproxCon = \frac{P _{-} + P _{+} - 2 P _{0}}{P _{0} \times ( Δ y ) ^{2}}

— P₋ = price after yield falls by Δy, P₊ = price after yield rises, P₀ = starting full price

Bond percentage price change with convexity adjustment

%Δ P \approx - ModDur \times Δ y + \frac{1}{2} \times Con \times (Δ y)^{2}

— ModDur = modified duration, Con = annual convexity, Δy = yield change (decimal)

Derivatives 9 items

Put-call parity

C + \frac{X}{( 1 + r ) ^{T}} = P + S_{0}

C = call price, P = put price, S_0 = spot price, X = exercise price

r = risk-free rate, T = time to expiration

Forward contract price

F_{0} = S_{0} (1 + r)^{T}

With continuous dividends:

F_{0} = S_{0} e^{(r - q) T}

S_0 = spot, r = risk-free rate, T = time, q = dividend yield

Forward price with discrete income or cost

F_{0} (T) = (S_{0} - P V (I) + P V (C)) (1 + r)^{T}

I = discrete income (dividends, coupons) over T; C = carrying cost (storage). PV at risk-free rate. Income reduces the forward; cost raises it.

Option payoff at expiration

Long call:

max (S_{T} - X, 0)

; Long put:

max (X - S_{T}, 0)

S_{T}

= price at expiry, X = strike. Short positions are the negative of long. Subtract premium paid for profit.

Intrinsic value and time value

Call intrinsic:

max (S - X, 0)

; Put intrinsic:

max (X - S, 0)

Time value = Option price − intrinsic. ATM/OTM intrinsic = 0; deep ITM time value → 0 near expiry.

Lower bound on European options (no dividends)

Call:

c \geq max (S_{0} - X (1 + r)^{- T}, 0)

Put:

p \geq max (X (1 + r)^{- T} - S_{0}, 0)

Enforces no-arbitrage. Below these, the option is mispriced relative to the synthetic.

Value of a long forward contract at time t

V_{t} = \frac{F _{t} - F _{0}}{( 1 + r ) ^{T - t}}

— F_t = current forward price, F_0 = original forward price, r = risk-free rate, T - t = time remaining to expiration

Swap fixed rate (price) at initiation

s^{*} = \frac{1 - D ( t _{n} )}{\sum _{i = 1}^{n} D ( t _{i} )}

— D(tᵢ) = discount factor at settlement i, n = number of settlements

Swap value to the fixed-receiver after initiation

V_{swap, fixed receiver} = P V (fixed leg) - P V (floating leg)

— PVs use current discount factors; floating leg = notional at any reset date

Alternative Investments 6 items

NAV per share

N A V = \frac{Total assets - Total liabilities}{Shares outstanding}

Used for mutual funds, ETFs, private equity fund valuation

Incentive fee with a hard vs. soft hurdle

Hard hurdle: fee

= p \times max (0, r - h)

per unit of capital, charged only on the return above the hurdle.

Soft hurdle: once

r > h

, fee

= p \times r

, charged on the entire gain. Below the hurdle, neither structure pays.

Hedge fund fee structure (2-and-20)

Mgmt fee

= m \times A U M

(e.g. 2%). Incentive fee

= p \times max (0, Profit above hurdle)

(e.g. 20%).

Net investor return = gross − both fees.

High-water mark (HWM) incentive fee

Incentive fee

= p \times max (0, NAV_{g r oss} - HWM)

HWM = highest NAV on which an incentive fee was previously paid. After a drawdown the fund must climb back above the HWM before incentive fees resume, so investors never pay twice for the same gains.

Loan-to-Value (LTV)

L T V = \frac{Loan amount}{Property value}

Higher LTV = more leverage and credit risk. Typical max ≈ 80% commercial; 95%+ residential with mortgage insurance.

Management fee base: hedge funds vs. private capital

Hedge funds: fee

= m \times AUM

(net asset value, marked to market).

Private capital funds: fee

= m \times Committed capital

during the investment period (often shifting to invested capital afterward), so fees accrue even before capital is deployed.

Portfolio Management 8 items

Capital Market Line (CML)

E (R_{p}) = R_{f} + \frac{E ( R _{m} ) - R _{f}}{σ _{m}} \cdot σ_{p}

Sharpe ratio of market is slope; uses total risk

σ_{p}

(not beta)

CAPM / Security Market Line (SML)

E (R_{i}) = R_{f} + β_{i} [E (R_{m}) - R_{f}]

β_{i} = \frac{Cov ( R _{i} , R _{m} )}{σ _{m}^{2}}

Uses systematic risk only; SML plots expected return vs beta

Two-asset portfolio variance

σ_{p}^{2} = w_{1}^{2} σ_{1}^{2} + w_{2}^{2} σ_{2}^{2} + 2 w_{1} w_{2} σ_{1} σ_{2} ρ_{12}

w = weights,

σ

= std devs,

ρ_{12}

= correlation coefficient

Information ratio

I R = \frac{R _{p} - R _{B}}{σ _{R_{p} - R_{B}}} = \frac{α}{Tracking error}

R_B = benchmark return,

α

= active return, TE = active risk

Treynor ratio

T_{p} = \frac{R _{p} - R _{f}}{β _{p}}

Excess return per unit of systematic risk (beta)

Compare with Sharpe (uses total risk

σ_{p}

)

Jensen's alpha

α_{p} = R_{p} - [R_{f} + β_{p} (R_{m} - R_{f})]

Actual return minus CAPM-expected return

α > 0

means manager added value beyond compensation for risk

M-squared (M²) performance measure

M^{2} = (R_{p} - R_{f}) \cdot \frac{σ _{m}}{σ _{p}} - (R_{m} - R_{f})

— Rp = portfolio return, Rf = risk-free rate, Rm = market return, σp = portfolio σ, σm = market σ

Beta from correlation and standard deviations

β_{i} = ρ_{i, m} \cdot \frac{σ _{i}}{σ _{m}} = \frac{Cov ( R _{i} , R _{m} )}{σ _{m}^{2}}

— ρ = correlation with market, σi = asset σ, σm = market σ, Cov = covariance

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