Reference

CFA® Level I Formula Sheet

91 formulas across 9 topics. Search or filter to find what you need.

Quantitative Methods

Present value (single sum)

PV = \dfrac{FV}{(1+r)^{n}}

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

Future value of ordinary annuity

FV = PMT \times \dfrac{(1+r)^{n} - 1}{r}

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

Present value of ordinary annuity

PV = PMT \times \dfrac{1 - (1+r)^{-n}}{r}

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

Present value of perpetuity

PV = \dfrac{PMT}{r}

PMT = periodic payment, r = discount rate

Population variance

\sigma^{2} = \dfrac{\sum_{i=1}^{N} (X_i - \mu)^{2}}{N}

μ = population mean, N = population size

Sample variance

s^{2} = \dfrac{\sum_{i=1}^{n} (X_i - \bar{X})^{2}}{n - 1}

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

Sharpe ratio

S = \dfrac{R_p - R_f}{\sigma_p}

Rp = portfolio return, Rf = risk-free rate, σp = portfolio std dev. Excess return per unit of total risk

Holding period return (HPR)

HPR = \dfrac{P_1 - P_0 + D_1}{P_0}

P1 = ending price, P0 = beginning price, D1 = cash distributions received

Bayes' theorem

P(A \mid B) = \dfrac{P(B \mid A)\,P(A)}{P(B)}

P(B) = P(B \mid A)P(A) + P(B \mid A^{c})P(A^{c})

Updates prior probability P(A) given new information B

Correlation coefficient

\rho_{XY} = \dfrac{\mathrm{Cov}(X, Y)}{\sigma_X\,\sigma_Y}

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

Geometric mean return

R_G = \left[\prod_{t=1}^{n} (1 + R_t)\right]^{1/n} - 1

Rt = period t return, n = number of periods

Money-weighted return (MWR)

\sum_{t=0}^{N} \dfrac{CF_t}{(1+r)^{t}} = 0

CFt = cash flow at time t (outflows −, inflows +), r = money-weighted return

Fisher relation (real vs nominal)

(1 + R_{nominal}) = (1 + R_{real})(1 + i)

i = inflation. Subtracting inflation is only an approximation; it breaks down when inflation is high

Time-weighted return (TWR)

(1 + TWR) = \prod_{i=1}^{n} (1 + r_i)

ri = holding-period return for sub-period i; n = sub-periods between external cash flows

Economics

GDP — expenditure approach

GDP = C + I + G + (X - M)

C = consumption, I = investment, G = government spending, X = exports, M = imports, (X−M) = net exports

Money multiplier

m = \dfrac{1}{\text{reserve requirement}}

\text{Money supply} = m \times \text{Reserves}

Maximum deposit expansion from a given reserve base

Fisher effect

(1 + r_{nom}) = (1 + r_{real})(1 + \pi)

r_{nom} \approx r_{real} + \pi

rnom = nominal rate, rreal = real rate, π = expected inflation

Breakeven and shutdown points

\text{Breakeven:}\ \ P = ATC

\text{Shutdown (short run):}\ \ P < AVC

If AVC < P < ATC, keep operating short-run — contribution covers some fixed cost

Price elasticity of demand

E_d = \dfrac{\%\,\Delta Q_d}{\%\,\Delta P}

Ed > 1: elastic. Ed < 1: inelastic. = 1: unit-elastic (revenue maximized)

Cross-rate calculation

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

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

Corporate Issuers

WACC

WACC = w_d r_d (1 - t) + w_p r_p + w_e r_e

w = market-value weights, r = required returns; d = debt, p = preferred, e = equity, t = tax rate

FCFF

FCFF = NI + NCC + Int(1 - t) - \Delta WC - CapEx

NI = net income, NCC = non-cash charges, Int = interest expense, ΔWC = change in working capital, t = tax rate

Degree of operating leverage (DOL)

DOL = \dfrac{Q(P - V)}{Q(P - V) - F} = \dfrac{\text{Contribution}}{EBIT}

Q = units, P = price, V = variable cost/unit, F = fixed costs

Degree of financial leverage (DFL)

DFL = \dfrac{EBIT}{EBIT - I}

I = interest expense. % change in EPS per 1% change in EBIT

Degree of total leverage (DTL)

DTL = DOL \times DFL = \dfrac{Q(P - V)}{Q(P - V) - F - I}

% change in EPS per 1% change in sales

Free cash flow to equity (FCFE)

FCFE = FCFF - Int(1 - t) + \text{Net borrowing}

Cash available to equity holders after all obligations and reinvestment

Financial Statement Analysis

3-factor DuPont decomposition

ROE = \dfrac{NI}{Sales} \times \dfrac{Sales}{Assets} \times \dfrac{Assets}{Equity}

Net profit margin × Asset turnover × Financial leverage

Current ratio

\text{Current ratio} = \dfrac{\text{Current assets}}{\text{Current liabilities}}

Measures short-term liquidity; higher = more liquid

Inventory turnover

\text{Inv. turnover} = \dfrac{COGS}{\text{Avg inventory}}

DOH = \dfrac{365}{\text{Inv. turnover}}

Days on hand (DOH) = 365 / Inventory turnover

Receivables turnover and DSO

\text{Rec. turnover} = \dfrac{Revenue}{\text{Avg A/R}}

DSO = \dfrac{365}{\text{Rec. turnover}}

DSO = average collection period

Return on assets (ROA)

ROA = \dfrac{\text{Net income}}{\text{Avg total assets}}

Alternative: ROA = Net profit margin × Asset turnover

Return on equity (ROE)

ROE = \dfrac{\text{Net income}}{\text{Avg total equity}}

DuPont: ROE = Net margin × Asset turnover × Leverage. Sustainable growth g = b × ROE

Cash flow interest coverage

\text{Coverage} = \dfrac{CFO + \text{Int paid} + \text{Tax paid}}{\text{Int paid}}

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

Cash return on assets

\text{Cash ROA} = \dfrac{CFO}{\text{Avg total assets}}

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

Equity Investments

Gordon growth model (DDM)

V_0 = \dfrac{D_1}{r - g} = \dfrac{D_0(1 + g)}{r - g}

D1 = next dividend, r = required return, g = constant growth rate. Requires r > g

Justified leading P/E

\dfrac{P_0}{E_1} = \dfrac{1 - b}{r - g}

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

Enterprise value (EV)

EV = \text{Mkt cap} + \text{Debt} + \text{Pref} + \text{Minority} - \text{Cash}

Capital-structure-neutral metric. EV/EBITDA = enterprise value multiple

Price-to-book ratio

P/B = \dfrac{\text{Market price / share}}{\text{Book value / share}}

\text{Justified } P/B = \dfrac{ROE - g}{r - g}

P/B > 1 implies the market values assets above book

P/E ratio (trailing & leading)

\text{Trailing} = \dfrac{P_0}{EPS_0}

\text{Leading} = \dfrac{P_0}{EPS_1}

Leading P/E uses next-12-months forecast EPS — the forward-looking variant

Equity value per share from EV

P_0 = \dfrac{EV - \text{Debt} + \text{Cash}}{\text{Shares}}

Debt = interest-bearing debt, Cash = cash & equivalents, Shares = diluted shares outstanding

Terminal value via Gordon growth (FCFF)

TV_n = \dfrac{FCFF_{n+1}}{WACC - g}

FCFF(n+1) = next-period FCFF, WACC = weighted-avg cost of capital, g = sustainable long-run growth

Residual income

RI_t = NI_t - (r \times BV_{t-1})

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

Arbitrage pricing theory (APT)

E(R_i) = R_f + \sum_{k=1}^{K} \beta_{i,k}\,\lambda_k

Rf = risk-free rate, βi,k = sensitivity of asset i to factor k, λk = risk premium per unit of factor k

Two-stage dividend discount model

V_0 = \sum_{t=1}^{n} \dfrac{D_t}{(1+r)^{t}} + \dfrac{D_{n+1} / (r - g_s)}{(1+r)^{n}}

Dt = dividend at time t, r = cost of equity, gs = stable growth, n = explicit horizon

Carhart four-factor model

E(R_i) - R_f = \beta_M[E(R_m) - R_f] + \beta_S\,SMB + \beta_V\,HML + \beta_W\,WML

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

Single-stage FCF perpetuity EV

EV = \dfrac{FCF_0(1 + g)}{WACC - g}

FCF = current free cash flow, g = terminal growth rate, WACC = weighted-avg cost of capital

Total return on an equity security

R = \dfrac{P_1 - P_0 + D_1}{P_0} = R_{price} + \dfrac{D_1}{P_0}

P0 = beginning price, P1 = ending price, D1 = dividends received

Price return on an equity security

R_{price} = \dfrac{P_1 - P_0}{P_0}

P1 = ending price, P0 = beginning price

Average daily volume (ADV)

ADV = \dfrac{\sum_{i=1}^{n} V_i}{n}

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

Free float shares

\text{Float} = \text{Shares outstanding} - \text{Restricted shares}

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

Justified trailing P/E (Gordon growth)

\dfrac{P_0}{E_0} = \dfrac{(1 - b)(1 + g)}{r - g}

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

Implied price via comparables

P_{target} = M_{peer} \times F_{target}

Mpeer = peer-group median multiple, Ftarget = target's per-share fundamental (EPS, BVPS, etc.)

Cumulative voting — total votes

V = S \times N

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

Voting power (dual-class structure)

VP = \dfrac{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

Bond price

P = \sum_{t=1}^{n} \dfrac{C}{(1+r)^{t}} + \dfrac{FV}{(1+r)^{n}}

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

Current yield

\text{Current yield} = \dfrac{\text{Annual coupon}}{\text{Price}}

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

Macaulay duration

D_{Mac} = \dfrac{\sum_{t=1}^{n} t \cdot \dfrac{CF_t}{(1+r)^{t}}}{P}

Weighted average time to receive cash flows; measured in years

Modified duration

D_{Mod} = \dfrac{D_{Mac}}{1 + r}

\%\,\Delta P \approx -D_{Mod} \times \Delta y

r = periodic YTM, Δy = change in yield

Forward rate from spot rates

(1 + z_2)^{2} = (1 + z_1)(1 + f)

(1 + z_n)^{n} = (1 + z_{n-1})^{n-1}(1 + f)

z = spot rate, f = implied forward rate

Price value of a basis point (PVBP)

PVBP = D_{Mod} \times P \times 0.0001

Dollar price change for a 1 bp yield move

FRN price (discount margin)

PV = \sum_{t=1}^{N} \dfrac{(MRR+QM)\,FV/m}{\left(1+\frac{MRR+DM}{m}\right)^{t}} + \dfrac{FV}{\left(1+\frac{MRR+DM}{m}\right)^{N}}

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

Bond-equivalent yield (money market)

BEY = \dfrac{FV - PV}{PV} \times \dfrac{365}{\text{days}}

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

Debt-to-EBITDA leverage

\dfrac{\text{Debt}}{\text{EBITDA}} = \dfrac{\text{Total debt}}{\text{EBITDA}}

Total debt = all interest-bearing debt. Lower is stronger

EBITDA-to-interest coverage

\dfrac{\text{EBITDA}}{\text{Interest}} = \dfrac{\text{EBITDA}}{\text{Interest expense}}

Higher is stronger

Effective duration

\text{EffDur} = \dfrac{P_- - P_+}{2 \times P_0 \times \Delta y}

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

Effective convexity

\text{EffCon} = \dfrac{P_- + P_+ - 2P_0}{P_0 \times (\Delta y)^{2}}

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

Approximate convexity

\text{ApproxCon} = \dfrac{P_- + P_+ - 2P_0}{P_0 \times (\Delta y)^{2}}

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

Price change with convexity adjustment

\%\,\Delta P \approx -ModDur \times \Delta y + \tfrac{1}{2}\,Con \times (\Delta y)^{2}

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

Derivatives

Put-call parity

C + \dfrac{X}{(1+r)^{T}} = P + S_0

C = call price, P = put price, S0 = spot price, X = exercise price, r = risk-free rate, T = time to expiration

Forward contract price

F_0 = S_0(1 + r)^{T}

F_0 = S_0\,e^{(r - q)T}

With continuous dividends: F0 = S0·e^((r−q)T). S0 = spot, r = risk-free, T = time, q = dividend yield

Forward price with income or cost

F_0(T) = \big(S_0 - PV(I) + PV(C)\big)(1 + r)^{T}

I = discrete income (dividends, coupons), C = carrying cost (storage). Income reduces the forward; cost raises it

Option payoff at expiration

\text{Long call: } \max(S_T - X,\,0)

\text{Long put: } \max(X - S_T,\,0)

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

Intrinsic value and time value

\text{Call: } \max(S - X,\,0)\quad \text{Put: } \max(X - S,\,0)

\text{Time value} = \text{Option price} - \text{intrinsic}

ATM/OTM intrinsic = 0; deep-ITM time value ≈ 0 near expiry

Lower bound on European options

c \geq \max\!\big(S_0 - X(1+r)^{-T},\,0\big)

p \geq \max\!\big(X(1+r)^{-T} - S_0,\,0\big)

No-dividend case. Enforces no-arbitrage; below these the option is mispriced vs the synthetic

Value of a long forward at time t

V_t = \dfrac{F_t - F_0}{(1+r)^{T-t}}

Ft = current forward price, F0 = original forward price, r = risk-free rate, T−t = time remaining

Swap fixed rate (price) at initiation

s = \dfrac{1 - D(t_n)}{\sum_{i=1}^{n} D(t_i)}

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

Swap value to fixed-receiver

V_{swap} = PV(\text{fixed leg}) - PV(\text{floating leg})

After initiation. PVs use current discount factors; floating leg = notional at any reset date

Alternative Investments

NAV per share

NAV = \dfrac{\text{Total assets} - \text{Total liabilities}}{\text{Shares outstanding}}

Used for mutual funds, ETFs, and private-equity fund valuation

Capitalization rate (cap rate)

V = \dfrac{NOI}{r_{cap}}

\text{Cap rate} = \dfrac{NOI}{\text{Value}}

NOI = net operating income (stabilized), rcap = cap rate

Hedge fund fees (2-and-20)

\text{Mgmt fee} = m \times AUM

\text{Incentive} = p \times \max(0,\ \text{profit above hurdle})

Net investor return = gross − both fees

Net operating income (NOI)

NOI = \text{Effective gross income} - \text{Operating expenses}

Excludes financing (interest), income tax, depreciation, and amortization. Foundation of cap-rate valuation

Loan-to-value (LTV)

LTV = \dfrac{\text{Loan amount}}{\text{Property value}}

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

Debt service coverage ratio (DSCR)

DSCR = \dfrac{NOI}{\text{Debt service}}

Debt service = annual principal + interest. DSCR > 1 means cash flow covers debt; CRE lenders typically require ≥ 1.20–1.30

Portfolio Management

Capital market line (CML)

E(R_p) = R_f + \dfrac{E(R_m) - R_f}{\sigma_m} \times \sigma_p

Sharpe ratio of the market is the slope; uses total risk σp (not beta)

CAPM / Security market line (SML)

E(R_i) = R_f + \beta_i[E(R_m) - R_f]

\beta_i = \dfrac{\mathrm{Cov}(R_i, R_m)}{\sigma_m^{2}}

Uses systematic risk only; SML plots expected return vs beta

Two-asset portfolio variance

\sigma_p^{2} = w_1^{2}\sigma_1^{2} + w_2^{2}\sigma_2^{2} + 2 w_1 w_2 \sigma_1 \sigma_2 \rho_{12}

w = weights, σ = std devs, ρ12 = correlation coefficient

Information ratio

IR = \dfrac{R_p - R_B}{\text{Tracking error}}

RB = benchmark return, (Rp − RB) = active return, tracking error = active risk

Treynor ratio

T = \dfrac{R_p - R_f}{\beta_p}

Excess return per unit of systematic risk (beta). Compare with Sharpe, which uses total risk σp

Jensen's alpha

\alpha_p = R_p - \big[R_f + \beta_p(R_m - R_f)\big]

Actual return minus CAPM-expected return. α > 0 means the manager added value beyond compensation for risk

M-squared (M²)

M^{2} = (R_p - R_f)\dfrac{\sigma_m}{\sigma_p} - (R_m - R_f)

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

Beta from correlation & std deviations

\beta_i = \rho_{i,m}\dfrac{\sigma_i}{\sigma_m} = \dfrac{\mathrm{Cov}(R_i, R_m)}{\sigma_m^{2}}

ρi,m = correlation with market, σi = asset σ, σm = market σ