Level I guide · 2026 curriculum

CFA® Level I formula sheet

The 76 formulas you are most likely to need on the Level I exam, grouped by topic. You don't get a formula sheet in the exam, so the aim is to know each one well enough to use it without looking it up.

Each formula comes with a short note on when to use it or the mistake candidates most often make. Ethics has no formulas and is left out. For what each topic tests and how to study it, follow the link to its full guide.

Quantitative Methods

Holding-period return: $\displaystyle R = \frac{P_1 - P_0 + D_1}{P_0}$ Include income received during the period. Divide by the starting price, not the ending price.
Geometric mean return: $\displaystyle R_G = \left[(1 + R_1)(1 + R_2)\cdots(1 + R_T)\right]^{1/T} - 1$ The compound rate per period. It is never above the arithmetic mean, and the gap grows with volatility.
Effective annual rate: $\displaystyle \text{EAR} = \left(1 + \frac{r_s}{m}\right)^{m} - 1, \quad \text{continuous: } e^{r_s} - 1$ r_s is the stated annual rate and m the number of compounding periods per year. More frequent compounding gives a higher EAR.
Bayes' formula: $\displaystyle P(E \mid I) = \frac{P(I \mid E)}{P(I)} \times P(E)$ P(E) is the prior. Find the unconditional P(I) first with the total probability rule.
Two-asset portfolio variance: $\displaystyle \sigma_p^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\rho_{12}\sigma_1\sigma_2$ The last term can also be written 2w_1w_2Cov(R_1, R_2). Take the square root for the standard deviation.
Safety-first ratio: $\displaystyle \text{SFRatio} = \frac{E(R_p) - R_L}{\sigma_p}$ R_L is the threshold return. Choose the portfolio with the highest ratio; with normal returns, the shortfall probability is N(−SFRatio).
Test statistic for a single mean: $\displaystyle t = \frac{\bar{X} - \mu_0}{s / \sqrt{n}}$ The denominator is the standard error of the mean. Use n − 1 degrees of freedom; use z if the population variance is known.
Test of a correlation: $\displaystyle t = \frac{r\sqrt{n - 2}}{\sqrt{1 - r^2}}$ n − 2 degrees of freedom. The same statistic is used for a Spearman rank correlation in large samples.
Regression fit: $\displaystyle R^2 = \frac{\text{SSR}}{\text{SST}}, \quad s_e = \sqrt{\frac{\text{SSE}}{n - 2}}, \quad F = \frac{\text{SSR}/1}{\text{SSE}/(n - 2)}$ SST = SSR + SSE. In simple regression, R² is the squared correlation and F is the square of the slope's t-statistic.

Full Quantitative Methods study guide

Economics

Breakeven and shutdown (short run): $\displaystyle P = \text{ATC}_{\min} \text{ (breakeven)}, \quad P < \text{AVC}_{\min} \text{ (shut down)}$ Between the two, keep producing in the short run: revenue still covers variable cost and part of fixed cost.
Herfindahl–Hirschman index: $\displaystyle \text{HHI} = \sum_{i=1}^{N} s_i^2$ s is each firm's market share. Use the same units throughout, decimals or percentages.
Fiscal multiplier: $\displaystyle \frac{1}{1 - c(1 - t)}$ c is the marginal propensity to consume and t the tax rate.
Cross rate: $\displaystyle S_{A/C} = S_{A/B} \times S_{B/C}$ The common currency B cancels. If it doesn't, invert one quote first.
Forward rate: $\displaystyle F_{P/B} = S_{P/B} \times \frac{1 + i_P \times \frac{\text{days}}{360}}{1 + i_B \times \frac{\text{days}}{360}}$ P is the price currency, B the base currency. The base currency trades at a forward premium when its interest rate is the lower one.
Forward points: $\displaystyle (F - S) \times 10{,}000$ Most currency pairs are quoted to four decimals. Yen quotes, which use two decimals, use 100.
Real exchange rate: $\displaystyle \text{real } S_{d/f} = S_{d/f} \times \frac{P_f}{P_d}$ d is the domestic currency, f the foreign. A rise means domestic goods have become cheaper in real terms.
Change in a currency's value: $\displaystyle \frac{S_1}{S_0} - 1 \text{ for the base}, \quad \frac{S_0}{S_1} - 1 \text{ for the price currency}$ The two percentages are not equal and opposite.

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Financial Statement Analysis

Basic EPS: $\displaystyle \text{EPS} = \frac{\text{Net income} - \text{Preferred dividends}}{\text{Weighted average shares outstanding}}$ Weight shares by the fraction of the year they were outstanding. Stock splits and stock dividends are applied retroactively.
Diluted EPS: $\displaystyle \frac{\text{NI} - \text{Pref div} + \text{Conv. pref div} + \text{Conv. debt interest}\,(1 - t)}{\text{WASO} + \text{Shares from conversion} + \text{Incremental option shares}}$ Treasury stock method: incremental shares = shares issued − (exercise proceeds ÷ average market price). Leave out any security that would raise EPS.
DuPont (three-step): $\displaystyle \text{ROE} = \frac{\text{Net income}}{\text{Revenue}} \times \frac{\text{Revenue}}{\text{Average total assets}} \times \frac{\text{Average total assets}}{\text{Average equity}}$ Net profit margin × total asset turnover × financial leverage.
DuPont (five-step): $\displaystyle \text{ROE} = \frac{\text{NI}}{\text{EBT}} \times \frac{\text{EBT}}{\text{EBIT}} \times \frac{\text{EBIT}}{\text{Revenue}} \times \frac{\text{Revenue}}{\text{Average assets}} \times \frac{\text{Average assets}}{\text{Average equity}}$ Tax burden × interest burden × EBIT margin × asset turnover × leverage. A higher tax burden ratio means a lower tax rate.
Cash conversion cycle: $\displaystyle \text{CCC} = \text{DOH} + \text{DSO} - \text{DPO}$ Days = 365 ÷ turnover. Inventory and payables turnover use cost of sales (or purchases for payables); receivables turnover uses revenue.
Free cash flow: $\displaystyle \text{FCFF} = \text{CFO} + \text{Int}(1 - t) - \text{FCInv}, \quad \text{FCFE} = \text{CFO} - \text{FCInv} + \text{Net borrowing}$ FCInv is capital spending net of proceeds from asset sales. Add back after-tax interest only if interest paid was deducted in CFO.
LIFO to FIFO: $\displaystyle \text{Inv}_{\text{FIFO}} = \text{Inv}_{\text{LIFO}} + \text{LR}, \quad \text{COGS}_{\text{FIFO}} = \text{COGS}_{\text{LIFO}} - (\text{LR}_{\text{end}} - \text{LR}_{\text{beg}})$ LR is the LIFO reserve. Only the change in the reserve affects cost of sales for the year.
Double-declining balance depreciation: $\displaystyle D_t = \frac{2}{\text{Useful life}} \times \text{Opening carrying amount}$ Start from gross cost, not cost minus residual value, and stop once the carrying amount reaches residual value.
Income tax expense: $\displaystyle \text{Tax expense} = \text{Taxes payable} + \Delta\text{DTL} - \Delta\text{DTA}$ Temporary differences create deferred taxes; permanent differences make the effective rate differ from the statutory rate.

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Corporate Issuers

Cash conversion cycle: $\displaystyle \text{CCC} = \text{DOH} + \text{DSO} - \text{DPO}$ DOH and DPO use cost of goods sold; DSO uses sales. A shorter cycle means less cash tied up in operations.
Cost of trade credit (effective annual rate): $\displaystyle \left(1 + \frac{\text{Discount}}{1 - \text{Discount}}\right)^{\frac{365}{\text{Payment period} - \text{Discount period}}} - 1$ The cost of forgoing the discount. Compare it with the bank rate: if trade credit costs more, borrow and take the discount.
Net present value: $\displaystyle \text{NPV} = \sum_{t=0}^{n} \frac{\text{CF}_t}{(1 + r)^t}$ CF0 is usually the negative initial outlay. Accept independent projects with a positive NPV.
Internal rate of return: $\displaystyle \sum_{t=0}^{n} \frac{\text{CF}_t}{(1 + \text{IRR})^t} = 0$ Solve with the calculator's cash flow function. Accept when the IRR exceeds the required return.
Return on invested capital: $\displaystyle \text{ROIC} = \frac{(1 - t) \times \text{Operating profit}}{\text{Average total long-term liabilities and equity}}$ Value is created when ROIC exceeds the cost of capital.
Weighted average cost of capital: $\displaystyle \text{WACC} = w_d r_d (1 - t) + w_p r_p + w_e r_e$ Use market-value (or target) weights. Only the cost of debt is adjusted for tax.
MM Proposition I with taxes: $\displaystyle V_L = V_U + tD$ Without taxes, V_L = V_U: capital structure does not affect value.
MM Proposition II with taxes: $\displaystyle r_e = r_0 + (r_0 - r_d)(1 - t)\frac{D}{E}$ Without taxes, drop the (1 − t) term. r0 is the cost of capital of the all-equity firm.

Full Corporate Issuers study guide

Equity Investments

Leverage ratio (maximum): $\displaystyle \text{Leverage ratio} = \frac{1}{\text{Initial margin}}$ Ignoring interest and commissions, the return on equity equals the leverage ratio times the return on the shares.
Margin call price (long position): $\displaystyle P = P_0 \times \frac{1 - \text{Initial margin}}{1 - \text{Maintenance margin}}$ For a short position it becomes P0 × (1 + initial margin) / (1 + maintenance margin), and the call comes when the price rises above it.
Index price and total return: $\displaystyle \text{PR} = \frac{V_1 - V_0}{V_0}, \quad \text{TR} = \frac{V_1 - V_0 + \text{Inc}}{V_0}$ Inc is the income (dividends) received on the constituents over the period. Multi-period index values link returns geometrically.
Price-weighted index: $\displaystyle V_{PW} = \frac{\sum_{i=1}^{N} P_i}{D}$ After a split or a change of constituent, choose the new divisor D so that the index value is unchanged.
Market-cap weight: $\displaystyle w_i = \frac{Q_i P_i}{\sum_{j=1}^{N} Q_j P_j}$ Q is shares outstanding. For a float-adjusted index, multiply each Q by the fraction of shares that trade freely.
Gordon growth model: $\displaystyle V_0 = \frac{D_1}{r - g} = \frac{D_0(1 + g)}{r - g}$ Requires r > g. In a multistage model the terminal value Vn = Dn+1 / (r − g) is dated at time n, so discount it n years.
Sustainable growth rate: $\displaystyle g = b \times \text{ROE}$ b is the retention rate, 1 − dividend payout ratio.
Justified forward P/E: $\displaystyle \frac{P_0}{E_1} = \frac{D_1 / E_1}{r - g}$ The numerator is the payout ratio. Multiplying the result by E1 gives the same value as the Gordon growth model.
Enterprise value: $\displaystyle \text{EV} = \text{Market cap} + \text{Preferred} + \text{Debt} - \text{Cash and short-term investments}$ EV multiples such as EV/EBITDA suit comparisons between companies with different capital structures.

Full Equity Investments study guide

Fixed Income

Bond price from yield-to-maturity: $\displaystyle PV = \sum_{t=1}^{N} \frac{PMT}{(1 + r)^{t}} + \frac{FV}{(1 + r)^{N}}$ r is the yield per period and N the number of periods. With spot rates, discount each cash flow at its own rate, (1 + z_t)^t, instead.
Full price, accrued interest and flat price: $\displaystyle PV^{Full} = PV \times (1 + r)^{t/T}, \quad AI = \frac{t}{T} \times PMT, \quad PV^{Flat} = PV^{Full} - AI$ PV is the price on the last coupon date, t the days since that date and T the days in the coupon period, both under the bond's day count convention.
Implied forward rate from spot rates: $\displaystyle (1 + z_A)^{A} \times (1 + IFR_{A,B-A})^{B-A} = (1 + z_B)^{B}$ IFR_{A,B-A} is the rate for B − A years starting in year A, so IFR_{2,1} is the 2y1y rate.
Modified duration from Macaulay duration: $\displaystyle \text{ModDur} = \frac{\text{MacDur}}{1 + r}$ r is the yield per period. If MacDur is measured in periods, divide the result by the number of periods per year to annualise it.
Approximate modified duration: $\displaystyle \text{ApproxModDur} = \frac{PV_{-} - PV_{+}}{2 \times \Delta\text{Yield} \times PV_{0}}$ PV_− and PV_+ are the prices after the yield falls and rises by ΔYield. Enter ΔYield as a decimal.
Price change with duration and convexity: $\displaystyle \%\Delta PV^{Full} \approx -\text{AnnModDur} \times \Delta\text{Yield} + \tfrac{1}{2} \times \text{AnnConvexity} \times (\Delta\text{Yield})^{2}$ The convexity term is positive for an option-free bond, so it adds to gains and reduces losses.
Money duration and PVBP: $\displaystyle \text{MoneyDur} = \text{AnnModDur} \times PV^{Full}, \quad PVBP = \frac{PV_{-} - PV_{+}}{2}$ PV_− and PV_+ here use a 1 bp change in yield. PVBP is roughly money duration × 0.0001.
Effective duration: $\displaystyle \text{EffDur} = \frac{PV_{-} - PV_{+}}{2 \times \Delta\text{Curve} \times PV_{0}}$ The prices come from a model after the benchmark curve shifts down and up by ΔCurve. Use it for bonds with embedded options.
Expected loss: $\displaystyle \text{Expected loss} = \text{POD} \times \text{LGD}, \quad \text{LGD} = \text{Exposure} \times (1 - RR)$ RR is the recovery rate. As a rough rule, the credit spread is about POD × LGD when LGD is stated as a percentage.

Full Fixed Income study guide

Derivatives

Forward price with cost of carry (discrete): $\displaystyle F_0(T) = \left[S_0 - PV_0(I) + PV_0(C)\right](1 + r)^T$ I is income on the underlying (dividends, coupons) and C the carrying costs, such as storage. With neither, F = S × (1 + r)^T.
Forward price (continuous compounding): $\displaystyle F_0(T) = S_0 e^{(r + c - y)T}$ c is the storage cost rate and y the income or convenience yield. For an index, y is the dividend yield; for a currency, r and y are the price-currency and base-currency interest rates.
Value of a long forward during its life: $\displaystyle V_t(T) = \left[S_t - PV_t(I) + PV_t(C)\right] - \frac{F_0(T)}{(1 + r)^{T - t}}$ Only income and costs still to come count. At expiry this becomes S_T − F_0(T); the short's value is the negative.
Option exercise value: $\displaystyle c_T = \max(0, S_T - X), \quad p_T = \max(0, X - S_T)$ Time value is the option price minus exercise value. Profit to the buyer subtracts the premium paid.
Lower bounds for European options: $\displaystyle c_0 \ge \max\left[0, S_0 - \frac{X}{(1 + r)^T}\right], \quad p_0 \ge \max\left[0, \frac{X}{(1 + r)^T} - S_0\right]$ A price below the bound would allow arbitrage. For an out-of-the-money option the bound is zero.
Put–call parity: $\displaystyle S_0 + p_0 = c_0 + \frac{X}{(1 + r)^T}$ A protective put equals a fiduciary call. Both options must be European, with the same underlying, exercise price and expiry.
Put–call–forward parity: $\displaystyle \frac{F_0(T)}{(1 + r)^T} + p_0 = c_0 + \frac{X}{(1 + r)^T}$ The present value of the forward price replaces the spot price. So c_0 − p_0 equals the present value of F_0(T) − X.
One-period binomial model: $\displaystyle \pi = \frac{1 + r - d}{u - d}, \quad c_0 = \frac{\pi c_u + (1 - \pi) c_d}{1 + r}$ u and d are the up and down factors and r the risk-free rate per period. The same formula values a put with p_u and p_d.
Swap rate from discount factors: $\displaystyle s_N = \frac{1 - DF_N}{\sum_{i=1}^{N} DF_i}$ An existing swap is worth (s_new − s_old) × ΣDF × notional to the fixed-rate payer, summing over the remaining periods.

Full Derivatives study guide

Alternative Investments

Multiple of invested capital: $\displaystyle \text{MOIC} = \frac{\text{Realised value} + \text{Unrealised value}}{\text{Invested capital}}$ Invested capital is paid-in capital less management fees and fund expenses. MOIC ignores the timing of cash flows; IRR does not.
GP return with a hard hurdle: $\displaystyle r_{GP} = \max[0,\ p(r - r_h)]$ p is the performance fee, r the fund's return for the period and r_h the hurdle rate. Management fees are ignored.
GP return with a catch-up: $\displaystyle r_{GP} = \max[0,\ r_{cu} + p(r - r_h - r_{cu})]$ With a full catch-up, r_cu = r_h × p/(1 − p). If the return clears the hurdle plus the catch-up, the GP ends up with p of the whole return.
Hedge fund fees with a high-water mark: $\displaystyle \text{MF} = m \times V_1, \quad \text{IF} = p \times \max[0,\ (V_1 - \text{MF}) - \text{HWM}]$ V_1 is year-end value before fees; HWM is the highest earlier value after all fees, or the entry value for a new investor. If the fees are calculated independently, apply p to the gain before the management fee.
Leveraged return: $\displaystyle r_L = \frac{r(V_c + V_b) - V_b r_b}{V_c} = r + \frac{V_b}{V_c}(r - r_b)$ V_c is investors' own capital and V_b the amount borrowed at rate r_b. Leverage helps only when the portfolio return exceeds the borrowing rate.
Commodity forward price: $\displaystyle F_0(T) = S_0 e^{(r + c - i)T}$ r is the risk-free rate, c the cost of carry and i the convenience yield. When i exceeds r + c, forward prices sit below spot (backwardation); otherwise the curve is in contango.

Full Alternative Investments study guide

Portfolio Management

Two-asset portfolio variance: $\displaystyle \sigma_P^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\rho_{12}\sigma_1\sigma_2$ The last term is the covariance term. The lower the correlation, the lower the portfolio risk.
Utility of an investment: $\displaystyle U = E(r) - \frac{1}{2}A\sigma^2$ A is the risk-aversion coefficient: positive for a risk-averse investor, zero for risk-neutral. Use decimals, not percentages.
Capital allocation line / capital market line: $\displaystyle E(R_P) = R_f + \frac{E(R_M) - R_f}{\sigma_M}\,\sigma_P$ The slope is the Sharpe ratio of the risky portfolio. With the market portfolio as the risky portfolio, the line is the CML.
Beta: $\displaystyle \beta_i = \frac{\text{Cov}(R_i, R_m)}{\sigma_m^2} = \rho_{i,m}\frac{\sigma_i}{\sigma_m}$ Beta measures systematic risk only. The market has a beta of 1 and the risk-free asset a beta of 0.
CAPM / security market line: $\displaystyle E(R_i) = R_f + \beta_i\,[E(R_m) - R_f]$ A security whose forecast return lies above the SML is undervalued; below it, overvalued.
Sharpe ratio: $\displaystyle \frac{R_p - R_f}{\sigma_p}$ Excess return per unit of total risk.
Treynor ratio: $\displaystyle \frac{R_p - R_f}{\beta_p}$ Excess return per unit of systematic risk. Suitable only for well-diversified portfolios.
M² (risk-adjusted performance): $\displaystyle M^2 = (R_p - R_f)\frac{\sigma_m}{\sigma_p} + R_f$ The portfolio's return if it were scaled to the market's total risk. M² alpha is M² minus the market return; the ranking always matches the Sharpe ratio.
Jensen's alpha: $\displaystyle \alpha_p = R_p - [R_f + \beta_p(R_m - R_f)]$ Actual return minus the return the CAPM predicts for the portfolio's beta.

Full Portfolio Management study guide

How to learn the formulas

Learn what each input means, not just the letters. Most wrong answers use the right formula with one wrong input.
Practise each formula on your calculator until the keystrokes are automatic; speed matters as much as accuracy.
Watch the units: decimals versus percentages, periodic versus annual rates, and days versus years.
Test yourself with timed questions. Recognising which formula a question needs is half the work.

Put the formulas to work

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