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CFA Level I formula sheet

Every core formula for CFA® Level I, grouped by topic, with a note on when each one applies.

Quantitative Methods

Returns, time value of money, statistics and simple regression.

Holding period return

HPR = (P₁ − P₀ + D₁) / P₀

Price change plus income, over the starting price.

Geometric mean return

R_G = [(1 + R₁)(1 + R₂)…(1 + R_n)]^1/n − 1

The compound annual rate. Always at or below the arithmetic mean.

Harmonic mean

X_H = n / Σ(1 / X_i)

Average cost per share under cost averaging.

Annualising a return

R_annual = (1 + R_period)^c − 1

c = number of periods in a year.

Continuously compounded return

r_cc = ln(1 + HPR) = ln(P₁ / P₀)

Continuously compounded returns add across periods.

Real and nominal returns

(1 + r_nominal) = (1 + r_real)(1 + π)

π = inflation. The approximation r_nominal ≈ r_real + π is only an approximation.

Future value

FV = PV × (1 + r/m)^mN

m = compounding periods per year, N = years. Continuous: FV = PV × e^rN.

Effective annual rate

EAR = (1 + r/m)^m − 1

Continuous compounding: EAR = e^r − 1.

Present value of an ordinary annuity

PV = A × [1 − (1 + r)^−N] / r

Annuity due: multiply by (1 + r).

Present value of a perpetuity

PV = A / r

Growing perpetuity: PV = A₁ / (r − g), with r > g.

Sample variance and standard deviation

s² = Σ(X_i − X̄)² / (n − 1) s = √s²

Divide by n − 1 for a sample, n for a population.

Coefficient of variation

CV = s / X̄

Risk per unit of return.

Covariance and correlation

ρ_XY = Cov(X, Y) / (σ_X σ_Y)

Correlation is bounded between −1 and +1.

Two-asset portfolio variance

σ_p² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρ₁₂σ₁σ₂

Diversification comes from ρ below +1.

Safety-first ratio

SFRatio = [E(R_p) − R_L] / σ_p

Choose the portfolio with the highest ratio. R_L = threshold return.

Bayes' formula

P(A | B) = P(B | A) × P(A) / P(B)

Updates a prior probability with new information.

Standard error of the mean

s_X̄ = s / √n

Shrinks with the square root of the sample size.

Confidence interval and test statistic

X̄ ± z_α/2 × s / √n t = (X̄ − μ₀) / (s / √n)

Use t with n − 1 degrees of freedom when σ is unknown.

Simple linear regression

b̂₁ = Cov(X, Y) / Var(X) b̂₀ = Ȳ − b̂₁X̄

R² = SSR / SST. SEE = √[SSE / (n − 2)]. F = MSR / MSE.

Work it through: TVM keystrokes · EAR keystrokes · Standard deviation keystrokes.

Economics

Elasticity, national income, money and exchange rates.

Price elasticity of demand

E_P = %ΔQ_d / %ΔP

|E| > 1 is elastic. Income and cross-price elasticities use the same form.

GDP, expenditure approach

GDP = C + I + G + (X − M)

Consumption, investment, government spending, net exports.

Fiscal multiplier

1 / [1 − MPC × (1 − t)]

t = tax rate, MPC = marginal propensity to consume.

Money multiplier

1 / reserve requirement

Simple deposit multiplier.

Quantity theory of money

M × V = P × Y

Money supply × velocity = price level × real output.

Fisher effect

R_nominal ≈ R_real + π^e

Expected inflation is priced into nominal rates.

Forward exchange rate

F_f/d = S_f/d × (1 + i_f × Actual/360) / (1 + i_d × Actual/360)

Covered interest rate parity. The higher-rate currency trades at a forward discount.

Financial Statement Analysis

Earnings per share, ratios, cash flow and inventory.

Basic EPS

(Net income − Preferred dividends) / Weighted average shares

Diluted EPS adds back the effects of convertibles, options and warrants if dilutive.

DuPont analysis

ROE = (NI / Sales) × (Sales / Assets) × (Assets / Equity)

Margin × turnover × leverage.

Liquidity ratios

Current = CA / CL Quick = (Cash + ST securities + Receivables) / CL

Cash ratio uses cash and marketable securities only.

Activity ratios

Inventory turnover = COGS / Avg inventory DOH = 365 / Inventory turnover

DSO = 365 / Receivables turnover. DPO = 365 / Payables turnover.

Cash conversion cycle

CCC = DOH + DSO − DPO

Days of financing the operating cycle needs.

Solvency ratios

Debt-to-equity = Total debt / Total equity Interest coverage = EBIT / Interest

Financial leverage = Avg assets / Avg equity.

Free cash flow to the firm

FCFF = CFO + Interest × (1 − t) − FCInv

Under IFRS, check where interest paid is classified.

Free cash flow to equity

FCFE = CFO − FCInv + Net borrowing

Cash available to common shareholders.

Inventory identity

COGS = Beginning inventory + Purchases − Ending inventory

FIFO inventory = LIFO inventory + LIFO reserve.

Depreciation

Straight-line = (Cost − Salvage) / Life DDB = (2 / Life) × Beginning book value

DDB ignores salvage until book value reaches it.

Corporate Issuers

Cost of capital, leverage and capital structure.

Weighted average cost of capital

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

Use target or market-value weights.

Cost of equity, CAPM

r_e = R_f + β × [E(R_m) − R_f]

Bond yield plus risk premium is the common alternative.

Net present value

NPV = Σ CF_t / (1 + r)^t − CF₀

IRR is the rate that sets NPV to zero. Prefer NPV when they conflict.

Degree of operating leverage

DOL = Q(P − V) / [Q(P − V) − F]

DFL = EBIT / (EBIT − Interest). DTL = DOL × DFL.

Breakeven quantity

Q_BE = (F + C) / (P − V)

F = fixed operating costs, C = fixed financing costs. Operating breakeven drops C.

Modigliani–Miller with taxes

V_L = V_U + t × D

The debt tax shield adds value until distress costs bite.

Work it through: NPV and IRR keystrokes.

Equity Investments

Dividend discount models, multiples and margin.

Gordon growth model

V₀ = D₁ / (r − g)

Requires r > g and constant growth.

Justified forward P/E

P₀ / E₁ = (D₁ / E₁) / (r − g)

Payout ratio over (r − g).

Sustainable growth rate

g = b × ROE

b = retention rate = 1 − payout ratio.

Preferred stock value

V₀ = D_p / r_p

A perpetuity for non-callable, non-convertible preferred.

Leverage ratio and margin call price

Leverage = 1 / IM P_call = P₀ × (1 − IM) / (1 − MM)

IM = initial margin, MM = maintenance margin.

Enterprise value

EV = Market value of equity + Debt + Preferred − Cash

EV / EBITDA compares firms with different leverage.

Fixed Income

Bond pricing, yield measures, duration, convexity and credit.

Bond price

P = Σ C / (1 + r)^t + FV / (1 + r)^N

Per-period coupon and yield for semi-annual bonds.

Full price and accrued interest

Full price = Flat price + AI AI = (t / T) × PMT

t = days since last coupon, T = days in the coupon period.

Current yield

Annual coupon / Flat price

Ignores pull to par.

Modified duration

ModDur = MacDur / (1 + r)

r = yield per period. %ΔP ≈ −ModDur × Δy.

Approximate duration and convexity

ApproxModDur = (PV₋ − PV₊) / (2 × Δy × PV₀) ApproxCon = (PV₋ + PV₊ − 2PV₀) / (Δy² × PV₀)

Effective duration uses the same form with curve shifts, for bonds with options.

Price change with convexity

%ΔP ≈ −ModDur × Δy + ½ × Convexity × (Δy)²

Δy in decimals: 100 bp = 0.01.

Money duration and PVBP

MoneyDur = ModDur × Full price PVBP = (PV₋ − PV₊) / 2

PVBP uses a 1 bp shift.

Forward rates from spot rates

(1 + S₂)² = (1 + S₁) × (1 + f_1,1)

No-arbitrage link between spot and forward rates.

Expected credit loss

Expected loss = PD × LGD LGD = Exposure × (1 − Recovery rate)

Yield spread ≈ PD × LGD, as a first approximation.

Work it through: Duration and convexity calculator · Bond price keystrokes.

Derivatives

Forwards, option payoffs, put–call parity and the binomial model.

Forward price

F₀(T) = S₀ × (1 + r)^T

With income and costs: F₀(T) = [S₀ − PV(I) + PV(C)] × (1 + r)^T.

Value of a long forward during its life

V_t = S_t − F₀(T) / (1 + r)^{T − t}

Zero at initiation.

Option payoffs at expiry

Call = max(0, S_T − X) Put = max(0, X − S_T)

Profit subtracts the premium paid.

Put–call parity

c₀ + X / (1 + r)^T = p₀ + S₀

Put–call forward parity replaces S₀ with F₀(T) / (1 + r)^T.

One-period binomial model

q = (1 + r − d) / (u − d) c₀ = [q × c_u + (1 − q) × c_d] / (1 + r)

q is the risk-neutral probability, not a real-world one.

Swap value

Value to payer = PV(floating leg) − PV(fixed leg)

Par swap rate = (1 − DF_N) / Σ DF_t.

Work it through: Swap valuation calculator.

Alternative Investments

Fees and real estate valuation.

Hedge fund fees

Fee = Management % × AUM + Incentive % × (Return above hurdle, net of high-water mark)

Check whether the incentive fee is calculated net of the management fee.

Capitalisation rate

Cap rate = NOI / Property value

Value = NOI₁ / Cap rate.

Multiple of invested capital

MOIC = (Realised + Unrealised value) / Invested capital

Ignores timing, unlike IRR.

Portfolio Management and Construction

CAPM, the capital market line and performance measures.

Portfolio expected return

E(R_p) = Σ w_i E(R_i)

Weights sum to one.

Utility

U = E(R) − ½ × A × σ²

A = risk aversion coefficient.

Capital market line

E(R_p) = R_f + {[E(R_m) − R_f] / σ_m} × σ_p

Efficient portfolios of the risk-free asset and the market.

Beta

β_i = Cov(R_i, R_m) / σ_m² = ρ_i,m σ_i / σ_m

Systematic risk only.

Security market line (CAPM)

E(R_i) = R_f + β_i × [E(R_m) − R_f]

Above the SML = undervalued.

Sharpe and Treynor ratios

Sharpe = (R_p − R_f) / σ_p Treynor = (R_p − R_f) / β_p

Sharpe for total risk, Treynor for systematic risk.

M-squared and Jensen's alpha

M² = (R_p − R_f) × σ_m / σ_p − (R_m − R_f) α_p = R_p − [R_f + β_p(R_m − R_f)]

Both measure risk-adjusted outperformance.

Work it through: VaR calculator.

Practise with them

CFA® Level I practice questionsExam-style questions with explanations.
CFA® Level I topic weightsWhere each formula sits in the exam.
Finance calculatorsVaR, duration and convexity, swap valuation.
BA II Plus keystrokesThe exam calculator, key by key.

A study aid, not the official formula list: the exam does not provide formulas, and the curriculum is the authority on notation. CFA® and FRM® are registered trademarks of their respective owners; GroundLearn is not affiliated with CFA Institute or GARP.

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