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CFA Level 1 Formula Sheet

A free CFA Level 1 formula sheet: every formula in the notes for the 2027 curriculum, by topic and reading, free to print or save as a PDF. Each reading links to the notes that explain its formulas; for the key takeaways and exam shortcuts as well, use the cheat sheet.

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Formulas by topic

The sheet holds 501 formulas from the notes; worked numbers from the examples are left out. The topics with the most are Quantitative Methods (133), Financial Statement Analysis (95) and Equities (92). Ethical and Professional Standards has no formulas; the key takeaways are on the cheat sheet. Topic weights are the ranges CFA Institute publishes for the 2027 Level I exam: Ethical and Professional Standards 10–15%; Quantitative Methods, Financial Statement Analysis, Equities and Fixed Income 11–14% each; Portfolio Construction 8–12%; Economics, Corporate Finance, Derivatives and Alternative Investments 6–9% each.

Quantitative Methods11–14% of exam 133 formulas · 10 of 11 readings

Reading 1: Returns of Financial Assets and Instruments

Return over a single period: $Price return = \frac{P _{1} - P _{0}}{P _{0}}$ $Capital distribution return = \frac{D}{P _{0}}$ $Total (holding period) return = \frac{P _{1} - P _{0} + D}{P _{0}} = price return + capital distribution return$
Averaging returns over several periods: $R ˉ_{arith} = \frac{R _{1} + R _{2} + \dots + R _{n}}{n}$ $R_{geo} = [(1 + R_{1}) (1 + R_{2}) \dots (1 + R_{n})]^{1/ n} - 1$
Annualizing a holding period return: $Annualized return = (1 + H P R)^{365/ days} - 1$ $(1 + R_{period})^{c} - 1$
Compounding frequency and continuous compounding: $Effective annual rate = (1 + \frac{r _{s}}{m})^{m} - 1$ $P V = \frac{F V}{( 1 + r _{s} / m ) ^{m}} (one year ahead)$ $r_{cc} = ln (1 + H P R)$ $H P R = e^{r_{cc}} - 1$
Building blocks of a required return: $Required return = nominal risk-free rate real risk-free rate + expected inflation + risk premium(s)$
Real return: $1 + r_{real} = \frac{1 + r _{nominal}}{1 + π}$ $π = \frac{1 + r _{nominal}}{1 + r _{real}} - 1$
Pretax vs. after-tax nominal return: $r_{after-tax} = price return \times (1 - t_{c g}) + distribution return \times (1 - t_{d})$
Leveraged return: $r_{L} = \frac{r _{P} ( V _{E} + V _{D} ) - r _{D} V _{D}}{V _{E}} = r_{P} + \frac{V _{D}}{V _{E}} (r_{P} - r_{D})$

Reading 2: Types of Financial Returns

Two sources of return: $Total return = capital appreciation return + capital distribution return$
Interest rates and the risk-free rate: $H P R = \frac{Face value - Price}{Price}$ $r_{annual} = (1 + H P R)^{365/ t} - 1$
Foreign exchange rates: $1 + r_{domestic} = (1 + r_{local}) (1 + r_{F X})$
The risk-return tradeoff: $short-term government bills < government bonds < corporate bonds < large-cap equities < small-cap equities$
Portfolio expected return: $E (R_{p}) = i \sum w_{i} E (R_{i})$ $total return = capital appreciation return + capital distribution return$

Reading 3: Benchmarking Returns

Money-weighted rate of return (MWR): $t = 0 \sum T \frac{C F _{t}}{( 1 + M W R ) ^{t}} = 0$
Time-weighted rate of return (TWR): $H P R_{i} = \frac{V _{e n d, i} - V _{b e g in, i} + D _{i}}{V _{b e g in, i}}$ $T W R_{ann u a l} = [(1 + H P R_{1}) (1 + H P R_{2}) \dots (1 + H P R_{n})]^{1/ years} - 1$
What an index is and how its return is measured: $R_{period} = (1 + R_{1}) (1 + R_{2}) \dots (1 + R_{n}) - 1$ $V_{n} = V_{0} (1 + R_{1}) (1 + R_{2}) \dots (1 + R_{n})$
Price-weighted index: $Index = \frac{\sum P _{i}}{D}$ $w_{i} = \frac{P _{i}}{\sum P _{j}}$
Market-capitalization-weighted (value-weighted) index: $w_{i} = \frac{P _{i} N _{i}}{\sum P _{j} N _{j}}$ $Index = \frac{\sum P _{i} N _{i}}{D}, D = \frac{base-period market cap}{base value}$

Reading 4: The Time Value of Money in Finance

The time value of money: $F V = P V (1 + r)^{t}$ $P V = \frac{F V}{( 1 + r ) ^{t}}$
Fixed-coupon bonds: $P = t = 1 \sum N \frac{C}{( 1 + Y T M ) ^{t}} + \frac{F}{( 1 + Y T M ) ^{N}}$
Perpetual bonds: $P V_{perpetuity} = \frac{P M T}{r}$
Amortizing bonds and annuities: $P V_{annuity} = P M T \times \frac{1 - ( 1 + r ) ^{- N}}{r}$ $P M T = \frac{r \times P V}{1 - ( 1 + r ) ^{- N}}$
Equity instruments: $V_{preferred} = \frac{D _{p}}{r _{p}}$ $V_{0} = \frac{D _{1}}{r _{e} - g _{c}}$ $D_{1} = D_{0} (1 + g_{c})$
Zero-coupon bond: $r = (\frac{F V}{P V})^{1/ N} - 1$
Equity: implied required return: $r_{e} = \frac{D _{1}}{P _{0}} + g_{c} = dividend yield + constant growth rate$
Equity: implied growth rate: $g_{c} = r_{e} - \frac{D _{1}}{P _{0}} = required return - dividend yield$
Forward interest rates: $(1 + S_{2})^{2} = (1 + S_{1}) (1 + 1 y 1 y) \Rightarrow 1 y 1 y = \frac{( 1 + S _{2} ) ^{2}}{1 + S _{1}} - 1$ $(1 + S_{3})^{3} = (1 + S_{1}) (1 + 1 y 1 y) (1 + 2 y 1 y)$
Forward exchange rates: $\frac{Forward}{Spot} = \frac{( 1 + r _{price} ) ^{T}}{( 1 + r _{base} ) ^{T}}$ $\frac{Forward}{Spot} = \frac{1 + r _{price, period}}{1 + r _{base, period}}$
Option values: one-period binomial model: $h = \frac{c _{u} - c _{d}}{S _{u} - S _{d}}$

Reading 5: Statistical Characteristics of Asset Returns

Population vs. sample: $Sample mean: X ˉ = \frac{\sum _{i = 1}^{n} X _{i}}{n}$ $Population mean: μ = \frac{\sum _{i = 1}^{N} X _{i}}{N}$
Range and mean absolute deviation: $range = maximum value - minimum value$ $MAD = \frac{\sum _{i = 1}^{n} ∣ X _{i} - X ˉ ∣}{n}$
Skewness: $skewness = \frac{1}{n} i = 1 \sum n (\frac{X _{i} - X ˉ}{s})^{3}$
Kurtosis: $kurtosis = \frac{1}{n} i = 1 \sum n (\frac{X _{i} - X ˉ}{s})^{4}$ $excess kurtosis = kurtosis - 3$
Covariance: $Cov_{X Y} = \frac{\sum _{i = 1}^{n} ( X _{i} - X ˉ ) ( Y _{i} - Y ˉ )}{n - 1} (sample)$
Correlation: $ρ_{X Y} = \frac{Cov _{X Y}}{s _{X} s _{Y}}$ $Cov_{X Y} = ρ_{X Y} s_{X} s_{Y}$
Semivariance and target downside deviation: $σ_{semi}^{2} = \frac{\sum _{X_{i} < X ˉ} ( X _{i} - X ˉ ) ^{2}}{n}$ $s_{target} = \frac{\sum _{X_{i} < B} ( X _{i} - B ) ^{2}}{n - 1}$
Coefficient of variation: $C V = \frac{s}{X ˉ}$ $(or \frac{σ}{μ})$

Reading 6: Statistical Distributions for Financial Asset Prices and Returns

Cumulative distribution function (CDF): $F (x) = P (X \leq x)$ $P (a < X \leq b) = F (b) - F (a)$
Unconditional expected value (mean): $E (X) = i = 1 \sum n P (x_{i}) x_{i}$
Unconditional variance and standard deviation: $σ^{2} (X) = i = 1 \sum n P (x_{i}) [x_{i} - E (X)]^{2}$ $σ (X) = σ^{2} (X)$
Unconditional covariance: $Cov (X, Y) = i = 1 \sum n P (x_{i}, y_{i}) [x_{i} - E (X)] [y_{i} - E (Y)]$
Discrete distributions: $E (X) = n p$ $σ^{2} (X) = n p (1 - p)$ $P (k events in interval t) = \frac{( λ t ) ^{k} e ^{- λ t}}{k !}$ $mean = variance = λ (for one unit of time, t = 1)$
Continuous distributions: $P (x_{1} \leq X \leq x_{2}) = \frac{x _{2} - x _{1}}{b - a}$ $mean = \frac{a + b}{2}$ $σ^{2} = \frac{( b - a ) ^{2}}{12}$
Conditional mean, variance and covariance: $E (X ∣ S) = i \sum P (x_{i} ∣ S) x_{i}$ $Var (X ∣ S) = i \sum P (x_{i} ∣ S) [x_{i} - E (X ∣ S)]^{2} = E (X^{2} ∣ S) - [E (X ∣ S)]^{2}$
Updating probabilities with Bayes' formula: $P (E ∣ I) = \frac{P ( I ∣ E )}{P ( I )} \times P (E) = \frac{P ( I ∣ E ) P ( E )}{P ( I ∣ E ) P ( E ) + P ( I ∣ E ^{c} ) P ( E ^{c} )}$

Reading 7: Estimation and Hypothesis Testing

The central limit theorem (CLT): $X ˉ \sim ˙ N (μ, \frac{σ ^{2}}{n}) for large n (rule of thumb: n \geq 30)$
Standard error and sampling error: $σ_{X ˉ} = \frac{σ}{n} (σ known)$ $s_{X ˉ} = \frac{s}{n} (σ unknown, the usual case)$
Confidence intervals: $point estimate \pm margin of error (reliability factor \times standard error)$
Value at risk (VaR) — a one-tailed interval: $VaR_{α} = z_{α} \times σ \times portfolio value$
Test statistic for a mean: $test statistic = \frac{X ˉ - μ _{0}}{s / n} (use σ / n if σ is known)$
Spearman rank correlation test: $r_{s} = 1 - \frac{6 \sum d _{i}^{2}}{n ( n ^{2} - 1 )}$
Test of independence using a contingency table: $E_{i, j} = \frac{( total for row i ) \times ( total for column j )}{total number of observations}$ $χ^{2} = i \sum j \sum \frac{( O _{i, j} - E _{i, j} ) ^{2}}{E _{i, j}}$ $df = (r - 1) (c - 1)$

Reading 8: The Return and Risk of a Financial Portfolio

Expected return of a portfolio: $E (R_{P}) = i = 1 \sum N w_{i} E (R_{i})$ $i = 1 \sum N w_{i} = 1$ $R_{P, t} = i = 1 \sum N w_{i, t - 1} R_{i, t}$
Portfolio variance: $σ_{P}^{2} = i = 1 \sum N j = 1 \sum N w_{i} w_{j} Cov (R_{i}, R_{j})$ $σ_{P}^{2} = w_{1}^{2} σ_{1}^{2} + w_{2}^{2} σ_{2}^{2} + 2 w_{1} w_{2} Cov_{1, 2}$ $σ_{P} = σ_{P}^{2}$
Covariance and correlation: $Cov_{1, 2} = ρ_{1, 2} σ_{1} σ_{2} ⟹ σ_{P}^{2} = w_{1}^{2} σ_{1}^{2} + w_{2}^{2} σ_{2}^{2} + 2 w_{1} w_{2} ρ_{1, 2} σ_{1} σ_{2}$
More assets: the variance–covariance matrix: $number of unique covariances = \frac{N ^{2} - N}{2}$ $σ_{P}^{2} = i = 1 \sum N j = 1 \sum N w_{i} w_{j} ρ_{i, j} σ_{i} σ_{j}$ $ρ_{i, i} = 1$
Large portfolios: variance approaches average covariance: $σ_{P}^{2} = \frac{σ ˉ ^{2}}{N} + \frac{N - 1}{N} Cov$ $σ_{P}^{2} \approx ρ ˉ σ ˉ^{2}$
Minimum-variance portfolio of two risky assets: $w_{A} = \frac{σ _{B}^{2} - Cov _{A, B}}{σ _{A}^{2} + σ _{B}^{2} - 2 Cov _{A, B}}$ $w_{B} = 1 - w_{A}$
Adding a risk-free asset: $Sharpe ratio = \frac{E ( R _{P} ) - R _{f}}{σ _{P}}$ $E (R_{C}) = w_{P} E (R_{P}) + (1 - w_{P}) R_{f}$ $σ_{C} = w_{P} σ_{P}$ $w_{P} = \frac{r _{target} - R _{f}}{E ( R _{P} ) - R _{f}}$
Utility function: $U = E (r) - \frac{1}{2} A σ^{2}$
The capital allocation line (CAL): $E (R_{C}) = R_{f} + [\frac{E ( R _{P} ) - R _{f}}{σ _{P}}] σ_{C}$
The capital market line (CML): $E (R_{P}) = R_{f} + [\frac{E ( R _{M} ) - R _{f}}{σ _{M}}] σ_{P}$
Return-generating models and the CAPM: $E (R_{i}) = R_{f} + β_{i} [E (R_{M}) - R_{f}]$
Beta: $β_{i} = \frac{Cov ( R _{i} , R _{M} )}{σ _{M}^{2}} = \frac{ρ _{i, M} σ _{i} σ _{M}}{σ _{M}^{2}} = \frac{ρ _{i, M} σ _{i}}{σ _{M}}$

Reading 9: Simulation of Financial Asset Prices and Returns

Multivariate simulation and the Cholesky decomposition: $X_{1} = σ_{1} Z_{1}$ $X_{2} = σ_{2} (ρ Z_{1} + 1 - ρ^{2} Z_{2})$

Reading 10: Applications of Simple Linear Regression in Finance

The model and the regression line: $Y_{i} = b_{0} + b_{1} X_{i} + ε_{i} (population model)$ $Y^_{i} = b^_{0} + b^_{1} X_{i} (estimated regression line, the line of best fit)$
Least squares criterion: $SSE = i = 1 \sum n (Y_{i} - Y^_{i})^{2}$ $b^_{1} = \frac{Cov ( X , Y )}{σ _{X}^{2}}$ $b^_{0} = Y ˉ - b^_{1} X ˉ$
Analysis of variance (ANOVA): $SST = SSR + SSE$
Goodness-of-fit measures: $SEE = MSE = \frac{SSE}{n - 2}$ $R^{2} = \frac{SSR}{SST} = 1 - \frac{SSE}{SST}$
F-test: $F = \frac{MSR}{MSE}$ $df = 1 (numerator), n - 2 (denominator)$
t-test of a regression coefficient: $t = \frac{b ^ _{1} - b _{1}}{s _{b^_{1}}}$ $df = n - 2$
Predicted values: $Y^= b^_{0} + b^_{1} X_{p}$
Confidence (prediction) interval for a predicted value: $Y^\pm t_{c} s_{f}$ $df = n - 2$ $s_{f}^{2} = SEE^{2} [1 + \frac{1}{n} + \frac{( X _{p} - X ˉ ) ^{2}}{( n - 1 ) s _{X}^{2}}]$
The CAPM as a one-factor regression: $E (R_{i}) = R_{f} + β_{i} [E (R_{M}) - R_{f}]$ $Z_{i, t} = b_{0} + b_{1} Z_{M, t} + ε_{t}$ $Z_{i, t} = R_{i, t} - R_{f, t}, Z_{M, t} = R_{M, t} - R_{f, t}$

Economics6–9% of exam 23 formulas · 6 of 8 readings

Reading 12: The Firm and Market Structures

The two concentration measures: $N-firm concentration ratio = i = 1 \sum N s_{i}$ $HHI = i = 1 \sum N s_{i}^{2}$

Reading 14: Fiscal Policy

Fiscal policy tools: spending and revenue, advantages and disadvantages: $fiscal multiplier = \frac{1}{1 - M P C ( 1 - t )}$ $Δ A D_{tax} = - Δ T \times M P C \times fiscal multiplier$

Reading 15: Monetary Policy

Effective central banks, targeting regimes and limits of policy: $neutral interest rate = real trend rate of economic growth + inflation target$

Reading 17: International Trade

Tariff welfare formulas (small country): $Loss of consumer surplus = \frac{Q _{D, 0} + Q _{D, 1}}{2} Δ P$ $Gain in producer surplus = \frac{Q _{S, 0} + Q _{S, 1}}{2} Δ P$ $Tariff revenue = M_{1} Δ P = (Q_{D, 1} - Q_{S, 1}) Δ P$ $Deadweight loss = consumer surplus loss - producer surplus gain - tariff revenue = \frac{1}{2} (Q_{S, 1} - Q_{S, 0}) Δ P + \frac{1}{2} (Q_{D, 0} - Q_{D, 1}) Δ P$

Reading 18: Capital Flows and the FX Market

Nominal and real exchange rates: $real P/B rate = nominal P/B rate \times \frac{C P I _{base currency}}{C P I _{price currency}}$
Percentage change in a currency's value: $%Δ base = \frac{new P/B}{old P/B} - 1$ $%Δ price = \frac{1/ new P/B}{1/ old P/B} - 1 = \frac{old P/B}{new P/B} - 1$
Exchange rates, trade and capital flows: $(X - M) \equiv (S - I) + (T - G)$

Reading 19: Exchange Rate Calculations

Currency cross-rates: $\frac{A}{C} = \frac{A}{B} \times \frac{B}{C}$
The no-arbitrage (interest rate parity) relation: $(1 + r_{d}) = \frac{1}{S _{d / f}} (1 + r_{f}) F_{d / f}$ $\frac{F _{d / f}}{S _{d / f}} = \frac{1 + r _{d}}{1 + r _{f}}$ $F_{d / f} = S_{d / f} \frac{1 + r _{d}}{1 + r _{f}}$ $S_{d / f} = F_{d / f} \frac{1 + r _{f}}{1 + r _{d}}$ $\frac{F _{d / f}}{S _{d / f}} - 1 = \frac{r _{d} - r _{f}}{1 + r _{f}} \approx r_{d} - r_{f}$ $F = S \times \frac{1 + r _{price} ( days /360 )}{1 + r _{base} ( days /360 )}$
Forward quotes in points or percentages: $F = S + (points \times point size)$ $F = S \times (1 + percentage)$
Forward premium or discount: $forward premium (+) or discount (-) = \frac{F}{S} - 1$

Corporate Finance6–9% of exam 18 formulas · 4 of 7 readings

Reading 20: Organizational Forms, Corporate Issuer Features, and Ownership

Key features of corporate issuers: $effective tax rate on distributed profit = t_{c} + t_{d} (1 - t_{c})$
Public versus private companies: $free float (%) = \frac{shares outstanding - shares held by insiders, strategic investors and sponsors}{shares outstanding}$

Reading 23: Working Capital and Liquidity

The cash conversion cycle: $CCC = DOH + DSO - DPO$
Trade credit and its cost: $EAR of supplier financing = (1 + \frac{a}{1 - a})^{365/ (c - b)} - 1$
Comparing CCCs: $total working capital = current assets - current liabilities$ $net working capital = (current assets - cash and marketable securities) - (current liabilities - short-term and current debt)$
Liquidity ratios: $current ratio = \frac{current assets}{current liabilities}$ $quick ratio = \frac{cash and marketable securities + receivables}{current liabilities}$ $cash ratio = \frac{cash and marketable securities}{current liabilities}$

Reading 24: Capital Investments and Capital Allocation

Net present value (NPV): $N P V = C F_{0} + \frac{C F _{1}}{( 1 + r ) ^{1}} + \frac{C F _{2}}{( 1 + r ) ^{2}} + \dots + \frac{C F _{n}}{( 1 + r ) ^{n}} = t = 0 \sum n \frac{C F _{t}}{( 1 + r ) ^{t}}$
Internal rate of return (IRR): $0 = t = 0 \sum n \frac{C F _{t}}{( 1 + I R R ) ^{t}}$
Return on invested capital (ROIC): $R O I C = \frac{NOPAT}{average book value of invested capital} = operating margin after tax \frac{NOPAT}{Sales} \times capital turnover \frac{Sales}{average invested capital}$
Real options: $Project value = N P V_{without options} + value of options - cost of options$

Reading 25: Capital Structure

Calculating and interpreting the WACC: $W A C C = w_{d} r_{d} (1 - t) + w_{e} r_{e}$
MM Proposition II (no taxes): cost of equity and leverage: $r_{e} = r_{0} + \frac{D}{E} (r_{0} - r_{d})$
MM with taxes: $V_{L} = V_{U} + t D$ $r_{e} = r_{0} + \frac{D}{E} (r_{0} - r_{d}) (1 - t)$
Static trade-off theory: $V_{L} = V_{U} + t D - P V (costs of financial distress)$

Financial Statement Analysis11–14% of exam 95 formulas · 10 of 12 readings

Reading 28: Analyzing Income Statements

Revenue over time (long-term contracts): $Cumulative revenue_{t} = \frac{Cumulative cost to date}{Total expected cost} \times Transaction price$
Depreciation methods: $Straight-line = \frac{Cost - Salvage}{Useful life}$
Capitalized interest: $Adjusted coverage = \frac{EBIT + Depreciation of capitalized interest}{Interest expense + Interest capitalized}$
Basic EPS: $Basic EPS = \frac{Net income - Preferred dividends}{Weighted average number of common shares outstanding}$
Diluted EPS and the if-converted method: $Diluted EPS = \frac{NI - Pref. div. + Conv. pref. div. + Conv. debt interest \times ( 1 - t )}{Wtd. avg. shares + Shares from conv. pref. + Shares from conv. debt + Shares from options/warrants}$ $Incremental shares = N - \frac{N \times X}{P ˉ} = \frac{P ˉ - X}{P ˉ} \times N$
Testing for dilution: $Conv. preferred: \frac{Preferred dividends}{Shares on conversion}$ $Conv. bonds: \frac{Interest \times ( 1 - t )}{Shares on conversion}$
Vertical common-size income statement: $Common-size line item = \frac{Line item}{Revenue}$ $Effective tax rate = \frac{Income tax expense}{Pretax income}$
Margin ratios: $Gross profit margin = \frac{Revenue - COGS}{Revenue}$ $Operating profit margin = \frac{Operating profit (EBIT)}{Revenue}$ $Pretax margin = \frac{Earnings before tax}{Revenue}$ $Net profit margin = \frac{Net income}{Revenue}$

Reading 29: Analyzing Balance Sheets

Goodwill: $Goodwill = purchase price - fair value of identifiable net assets acquired$
Vertical common-size balance sheet: $Common-size item = \frac{balance sheet item}{total assets}$
Liquidity ratios (short-term obligations): $current ratio = \frac{current assets}{current liabilities}$ $quick ratio = \frac{cash + marketable securities + receivables}{current liabilities}$ $cash ratio = \frac{cash + marketable securities}{current liabilities}$
Solvency ratios (long-term obligations): $long-term debt-to-equity = \frac{long-term debt}{total equity}$ $total debt-to-equity = \frac{total debt}{total equity}$ $debt ratio = \frac{total debt}{total assets}$ $financial leverage ratio = \frac{total assets}{total equity}$

Reading 30: Analyzing Statements of Cash Flows I

Why a separate cash flow statement?: $CFO + CFI + CFF = Δ Cash; Beginning cash + Δ Cash = Ending cash$
Timing differences show up on the balance sheet: $Beginning AR + Sales - Cash collected = Ending AR$
The direct method: $Cash collected = Net sales - Δ AR + Δ Unearned revenue$ $Purchases = COGS + Ending inventory - Beginning inventory = COGS + Δ Inventory$ $Cash paid to suppliers = Purchases - Δ AP = COGS + Δ Inventory - Δ AP$ $Cash paid for wages = Wage expense - Δ Wages payable$ $Cash paid for interest = Interest expense - Δ Interest payable$ $Cash paid for insurance = Insurance expense + Δ Prepaid insurance$ $Cash taxes paid = Tax expense - Δ Taxes payable - Δ Deferred tax liability + Δ Deferred tax asset$
Starting point: net income: $CFO = NI + NCC - WC_{INV}$
Investing activities (CFI): $Proceeds = Carrying value of asset sold + Gain (or - Loss)$ $Gross cost of disposals = Beginning gross PP&E + Purchases - Ending gross PP&E$ $Accumulated depreciation removed = Beginning AD + Depreciation expense - Ending AD$ $Carrying value disposed = Beginning net PP&E - Depreciation + Purchases - Ending net PP&E$
Financing activities (CFF): $CFF = (New borrowing - Principal repaid) + (Stock issued - Stock repurchased) - Cash dividends paid$ $Dividends declared = Beginning retained earnings + Net income - Ending retained earnings$ $Dividends paid = Dividends declared - Δ Dividends payable$

Reading 31: Analyzing Statements of Cash Flows II

Free cash flow to the firm (FCFF): $F C F F = N I + N C C + [I n t \times (1 - t)] - F C_{I N V} - W C_{I N V}$ $F C F F = C F O + [I n t \times (1 - t)] - F C_{I N V}$
Free cash flow to equity (FCFE): $F C F E = C F O - F C_{I N V} + net borrowing$ $F C F E = F C F F - [I n t \times (1 - t)] + net borrowing$

Reading 32: Analysis of Inventories

Inventory ratios: $Inventory turnover = \frac{COGS}{average inventory}$ $Days of inventory on hand (DOH) = \frac{365}{inventory turnover}$

Reading 33: Analysis of Long-Term Assets

Goodwill: $Goodwill = Purchase price - Fair value of identifiable net assets acquired$
Impairment of assets held for use: IFRS and US GAAP: $IFRS loss = C V - max (FV - costs to sell, VIU)$ $US GAAP loss = C V - FV (only if C V > undiscounted CF)$
Ratios built from the disclosures: $Fixed asset turnover = \frac{Revenue}{Average fixed assets}$ $Average age \approx \frac{Accumulated depreciation}{Annual depreciation expense}$ $Total useful life \approx \frac{Gross (historical cost) PP&E}{Annual depreciation expense}$ $Remaining useful life \approx \frac{Net PP&E}{Annual depreciation expense} = Total useful life - Average age$

Reading 34: Topics in Long-Term Liabilities and Equity

Lessee accounting: $Lease liability_{0} = t = 1 \sum N \frac{P M T}{( 1 + r ) ^{t}} = P M T \times \frac{1 - ( 1 + r ) ^{- N}}{r}$
Defined contribution vs. defined benefit plans: $Funded status = Fair value of plan assets - PV of the defined benefit obligation$

Reading 35: Analysis of Income Taxes

The income tax expense equation: $income tax expense = taxes payable + Δ D T L - Δ D T A$

Reading 36: Financial Reporting Quality

Depreciation, amortization and impairment: $Annual depreciation (straight-line) = \frac{cost - salvage value}{useful life}$
Linking the signs to what is being manipulated: $Days payables = \frac{accounts payable}{COGS (or purchases) /365}$
Cash flow versus earnings: $CFO-to-net-income ratio = \frac{cash flow from operations}{net income}$

Reading 37: Financial Analysis Techniques

Activity ratios: $receivables turnover = \frac{revenue}{average receivables}$ $days of sales outstanding (DSO) = \frac{365}{receivables turnover}$ $inventory turnover = \frac{cost of goods sold}{average inventory}$ $days of inventory on hand (DOH) = \frac{365}{inventory turnover}$ $payables turnover = \frac{cost of goods sold}{average trade payables}$ $number of days of payables (DPO) = \frac{365}{payables turnover}$ $purchases = ending inventory - beginning inventory + COGS$ $total asset turnover = \frac{revenue}{average total assets}$ $fixed asset turnover = \frac{revenue}{average net fixed assets}$ $working capital turnover = \frac{revenue}{average working capital}$
Liquidity ratios: $current ratio = \frac{current assets}{current liabilities}$ $quick ratio = \frac{cash + marketable securities + receivables}{current liabilities}$ $cash ratio = \frac{cash + marketable securities}{current liabilities}$ $defensive interval = \frac{cash + marketable securities + receivables}{average daily expenditures}$ $average daily expenditures = \frac{COGS + SG&A + R&D - noncash charges (e.g., depreciation, amortization)}{365}$ $cash conversion cycle = DSO + DOH - number of days of payables$
Solvency ratios: $debt-to-equity = \frac{total debt}{total shareholders’ equity}$ $debt-to-capital = \frac{total debt}{total debt + total shareholders’ equity}$ $debt-to-assets = \frac{total debt}{total assets}$ $financial leverage ratio = \frac{average total assets}{average total equity}$ $interest coverage = \frac{EBIT}{interest payments}$ $debt-to-EBITDA = \frac{total debt}{EBITDA}$ $fixed charge coverage = \frac{EBIT + lease payments}{interest payments + lease payments}$
Profitability ratios: $gross profit margin = \frac{gross profit}{revenue}$ $operating profit margin = \frac{operating income}{revenue} or \frac{EBIT}{revenue}$ $pretax margin = \frac{EBT}{revenue}$ $net profit margin = \frac{net income}{revenue}$ $ROA = \frac{net income}{average total assets}$ $adjusted ROA = \frac{net income + interest expense ( 1 - tax rate )}{average total assets}$ $operating ROA = \frac{operating income (or EBIT)}{average total assets}$ $ROE = \frac{net income}{average total equity}$ $return on common equity = \frac{net income - preferred dividends}{average common equity}$
Two-part decomposition: $ROE = \frac{net income}{average total equity} = ROA \frac{net income}{average total assets} \times financial leverage \frac{average total assets}{average total equity}$
Original (three-part) DuPont equation: $ROE = net profit margin \frac{net income}{revenue} \times total asset turnover \frac{revenue}{average total assets} \times financial leverage \frac{average total assets}{average total equity}$
Extended (five-part) DuPont equation: $ROE = tax burden \frac{net income}{EBT} \times interest burden \frac{EBT}{EBIT} \times EBIT margin \frac{EBIT}{revenue} \times total asset turnover \frac{revenue}{average total assets} \times financial leverage \frac{average total assets}{average total equity}$
Business risk and the coefficient of variation: $CV = \frac{standard deviation of the item}{mean (expected value) of the item}$

Equities11–14% of exam 92 formulas · 11 of 12 readings

Reading 39: Equity Instrument Features

Preference shares (preferred stock): $Annual preferred dividend = dividend rate \times par value$

Reading 41: Equity Issuance and Trading

Quotes and the spread: $bid-ask spread = lowest ask - highest bid$ $bid-ask spread (%) = \frac{lowest ask - highest bid}{lowest ask}$ $mid-market price = \frac{lowest ask + highest bid}{2}$
VWAP: $VWAP = \frac{\sum _{i} ( price _{i} \times volume _{i} )}{\sum _{i} volume _{i}}$
Float, trading volume and other liquidity measures: $market float = shares outstanding - restricted shares or shares not available for trading$ $average daily volume = \frac{\sum _{i = 1}^{n} ( shares traded on day i )}{n}$ $turnover ratio = \frac{average daily volume}{market float}$

Reading 42: Sources of Equity Returns

Price return and total return on equity: $price return = \frac{P _{1} - P _{0}}{P _{0}}$ $total return = \frac{P _{1} - P _{0} + D}{P _{0}}$
Reinvested dividends: $r_{E} = \frac{P _{1} ( 1 + \frac{D}{P _{P D}} ) - P _{0}}{P _{0}}$
Equity value as a present value: $P_{0} = \frac{E ( P _{1} ) + E ( D _{1} )}{1 + r _{E}}$

Reading 43: Introduction to Equity Valuation

Book value, market capitalization and enterprise value: $book value = share capital + additional paid-in capital + retained earnings + accumulated other comprehensive income (and other reserves) - treasury stock$ $book value per share (BVPS) = \frac{book value of equity}{shares outstanding}$
Limits of book value: $P/B = \frac{market price per share}{BVPS} = \frac{market capitalization}{book value of equity}$ $EV = market value of common and preferred stock + market value of debt - cash and short-term investments$

Reading 44: Discounted Cash Flow (DCF) and Growth Models

Cash flow measures for present value models and the valuation process: $V_{0} = t = 1 \sum n \frac{E ( C F _{t} )}{( 1 + r ) ^{t}} + \frac{E ( T V _{n} )}{( 1 + r ) ^{n}}$
Free cash flow to equity: $F C F E = net income + depreciation - W C I n v - C a pE x + net borrowing$ $F C F E = C F O - F C I n v + net borrowing$ $net investment = C a pE x - depreciation + W C I n v$ $F C F E = net income - net investment + net borrowing$ $net borrowing = D R \times net investment$ $F C F E = net income - net investment + D R \times net investment = net income - (1 - D R) (net investment)$
Free cash flow to the firm: $F C F F = C F O - C a pE x + interest (1 - t)$ $F C F F = F C F E + interest (1 - t) - net borrowing$
EBITDA: $F C F F = E B I T D A (1 - t) + (depreciation \times t) - F C I n v - W C I n v$
Residual income: $R I_{t} = E P S_{t} - r \times B V P S_{t - 1}$
Constant growth (Gordon growth) model: $V_{0} = \frac{D _{1}}{r - g} = \frac{D _{0} ( 1 + g )}{r - g}$ $g = r - \frac{D _{1}}{P _{0}}$
Multistage (two-stage) models: $V_{0} = t = 1 \sum n \frac{D _{t}}{( 1 + r ) ^{t}} + \frac{V _{n}}{( 1 + r ) ^{n}}$ $V_{n} = \frac{D _{n + 1}}{r - g _{L}}$
Estimating the constant growth rate: $g = (1 - dividend payout ratio) \times R O E = b \times R O E$ $S G R_{F C F E} = (1 - \frac{F C F E}{net income}) (\frac{net income}{beginning book value})$ $S G R_{F C F F} = (1 - \frac{F C F F}{N O P A T}) (\frac{N O P A T}{beginning book value of total capital})$
Weighted average cost of capital: $W A C C = \frac{M V _{d}}{M V _{d} + M V _{e}} r_{d} (1 - t) + \frac{M V _{e}}{M V _{d} + M V _{e}} r_{e}$
Non-callable, non-convertible preferred: $V_{0} = t = 1 \sum N \frac{D _{p}}{( 1 + r _{p} ) ^{t}} + \frac{F V}{( 1 + r _{p} ) ^{N}}$ $V_{0} = \frac{D _{p}}{r _{p}}$

Reading 45: Relative Value Equity Valuation Approaches

Method of forecasted fundamentals: $\frac{P _{0}}{E _{1}} = \frac{D _{1} / E _{1}}{r - g} = \frac{payout ratio}{r - g}$ $\frac{P _{0}}{B _{0}} = \frac{R O E - g}{r - g}$
Reconciling the two methods: $P E G = \frac{P / E}{g} (g in whole percentage points)$ $P / B_{i} = b_{0} + b_{1} R O E_{i} + b_{2} g_{i}$
Value = assets in place + growth opportunities: $P_{0} = \frac{E _{1}}{r} + P V GO$ $\frac{P V GO}{P _{0}} = 1 - \frac{E _{1} / P _{0}}{r}$
Price-to-sales (P/S): $\frac{P _{0}}{S _{0}} = \frac{P _{0}}{E _{0}} \times \frac{E _{0}}{S _{0}} = \frac{payout \times ( 1 + g )}{r - g} \times profit margin$
Enterprise value multiples: $E V = market value of common and preferred stock + market value of debt - cash and short-term investments$ $E V = \frac{F C F F _{1}}{W A C C - g}$ $F C F F = E B I T D A (1 - t) + Dep \times t - CapEx - WCInv$
Balance-sheet multiples: $\frac{E V}{B V of capital} = \frac{R O C \times ( 1 - reinvestment rate )}{W A C C - g}$
Cash-flow multiples and yields: $\frac{D _{0}}{P _{0}} = \frac{D _{0}}{E _{0}} \times \frac{E _{0}}{P _{0}} (payout \times earnings yield)$ $\frac{D _{0}}{P _{0}} = \frac{r - g}{1 + g}$
Using projected future values: $T V_{n} = justified leading P/E_{n} \times E_{n + 1}$ $justified leading P/E_{n} = \frac{payout}{r - g}$

Reading 46: Financial Statement Forecasting in Equity Valuation

Revenue modeling: $g_{r e v e n u e} = g_{ma r k e t} + (\frac{new share}{old share} - 1)$
Expense (profit) modeling: $Δ Expense = b_{0} + b_{1} \times Δ Revenue$ $COGS = forecast revenue \times (1 - gross margin)$
Financing modeling: $financing needs = Δ operating assets - Δ non-debt liabilities - Δ retained earnings$
Start-up stage: venture capital arithmetic: $ROI multiple = (1 + r)^{n} ⟹ r = (ROI multiple)^{1/ n} - 1$ $post-money value = \frac{expected revenue \times ( P / S )}{ROI multiple}$ $pre-money value = post-money value - VC investment$ $f = \frac{VC investment}{post-money value}$
Growth stage: $T V_{n} = N I_{n} \times P / E$ $V_{0} = \frac{T V _{n}}{( 1 + r ) ^{n}} or V_{0} = \frac{T V _{n}}{ROI multiple}$
Decline stage: $value = P (going concern) \times going-concern value + P (distress) \times distress sale value$
Enterprise value and EV multiples: $EV = market value of common and preferred stock + market value of debt - cash and short-term investments$
Valuing equity from forecast model outputs: $S G R_{F C F F} = (1 - \frac{F C F F}{N O P A T}) (\frac{N O P A T}{beginning book value of total capital})$

Reading 47: Industry and Competitive Analysis

Profitability: $operating profit margin = \frac{operating income}{total revenue}$ $ROIC = \frac{NOPAT _{t}}{average invested capital}$
Market share and concentration: $market share = \frac{company sales}{industry size}$ $H H I = i = 1 \sum N (market share_{i})^{2}$

Reading 48: Company Analysis: Past, Present, and Future

Market potential for start-up and growth companies: $Serviceable market = TAM \times technical fit (%) \times geographic reach (%)$
Operating profitability: $EBITDA = revenue - COGS - operating expenses (excluding D&A)$ $EBIT = EBITDA - depreciation and amortization$ $Net income = EBIT - interest expense - taxes$
Operating leverage: $Operating profit = Q \times (P - V C) - F C$
Free cash flow to equity: $FCFE = NI + D&A - capital expenditures - Δ working capital + net borrowing$
Working capital management: $CCC = DOH + DSO - DPO$
Capital investments: $Net capital expenditure = total capital expenditure - D&A$ $ROIC = \frac{NOPAT}{average invested capital}$ $invested capital \approx total long-term liabilities + total equity$
Capital structure: $DFL = \frac{%Δ net income}{%Δ operating income}$

Reading 49: Equity Analyst Research Reports

Two-stage FCFE valuation: $V_{0} = \frac{F C F E _{1}}{1 + r} + \frac{F C F E _{2} / ( r - g )}{1 + r}$

Reading 50: The Capital Asset Pricing Model, Market Model, and Other Factor-Based Equity Models

CAPM and the security market line: $E (R_{i}) = R_{f} + β_{i} [E (R_{m}) - R_{f}]$
Adjusted beta (mean reversion): $adjusted beta = ω \times raw beta + (1 - ω) \times 1.0$ $ω commonly 2/3$
Beta from comparables (pure-play approach): $β_{U} = \frac{β _{L, peers}}{1 + ( 1 - t _{peers} ) \frac{D}{E} _{peers}}$ $β_{L, subject} = β_{U} [1 + (1 - t_{subject}) \frac{D}{E}_{subject}]$
Estimating the market risk premium: $E (R_{m}) = \frac{D _{1}}{P _{0}} + g$ $M R P = \frac{D _{1}}{P _{0}} + g - R_{f}$
Emerging markets: country risk premium: $C R P = sovereign bond yield spread \times \frac{σ _{equity index}}{σ _{sovereign debt}}$ $E (R_{i}) = R_{f} + β_{i} (M R P + C R P)$
Arbitrage pricing theory (APT): $E (R_{i}) = R_{f} + β_{1 i} f_{1} + β_{2 i} f_{2} + \dots + β_{k i} f_{k}$
The four-factor model: $E (R_{i}) = R_{f} + β_{mk t} [E (R_{m}) - R_{f}] + β_{S M B} E (S M B) + β_{H M L} E (H M L) + β_{U M D} E (U M D)$

Fixed Income11–14% of exam 62 formulas · 11 of 19 readings

Reading 54: Fixed-Income Markets for Corporate Issuers

Repurchase agreements: $purchase price = \frac{market value of collateral}{initial margin}$ $repurchase price = purchase price \times (1 + repo rate \times \frac{days}{360})$ $haircut = \frac{market value - purchase price}{market value} = 1 - \frac{1}{initial margin}$ $variation margin = initial margin \times adjusted purchase price - market value of collateral$

Reading 56: Fixed-Income Bond Valuation: Prices and Yields

Price = present value of the promised cash flows: $P = \frac{C}{( 1 + Y T M )} + \frac{C}{( 1 + Y T M ) ^{2}} + \dots + \frac{C + F V}{( 1 + Y T M ) ^{N}}$ $P = t = 1 \sum 2 N \frac{C /2}{( 1 + Y T M /2 ) ^{t}} + \frac{F V}{( 1 + Y T M /2 ) ^{2 N}}$
Accrued interest, flat price and full price: $A I = \frac{t}{T} \times P M T$
Matrix pricing: $Y T M_{t a r g e t} = Y T M_{s h or t} + \frac{T _{t a r g e t} - T _{s h or t}}{T _{l o n g} - T _{s h or t}} (Y T M_{l o n g} - Y T M_{s h or t})$

Reading 57: Yield and Yield Spread Measures for Fixed-Rate Bonds

Periodicity and the effective annual yield: $E A Y = (1 + \frac{stated annual yield}{n})^{n} - 1$ $stated yield_{m} = m [(1 + E A Y)^{1/ m} - 1]$
Current yield and simple yield: $current yield = \frac{annual cash coupon}{flat price}$ $simple yield = \frac{annual coupon \pm straight-line amortization}{flat price}$
Option-adjusted price and option-adjusted yield: $callable bond value = straight bond value - call option value$
Z-spread: $price = t = 1 \sum N \frac{C F _{t}}{( 1 + S _{t} + Z ) ^{t}}$
Option-adjusted spread (OAS): $OAS = Z-spread - option value (bp)$ $option value = Z-spread - OAS$

Reading 58: Yield and Yield Spread Measures for Floating-Rate Instruments

Simplified valuation on a reset date: $V = t = 1 \sum N \frac{( MRR + QM ) \times F V / m}{( 1 + \frac{MRR + DM}{m} ) ^{t}} + \frac{F V}{( 1 + \frac{MRR + DM}{m} ) ^{N}}$
Key formulas: money market yields: $Add-on yield (365-day) = H P Y \times \frac{365}{days}$ $Discount yield = \frac{F V - P}{F V} \times \frac{360}{days}$ $P = F V \times (1 - discount yield \times \frac{days}{year})$ $F V = P V \times (1 + add-on yield \times \frac{days}{year})$
Comparing with a semiannual-pay bond: $E A Y = (1 + H P Y)^{365/ days} - 1$ $semiannual bond basis = 2 [(1 + E A Y)^{1/2} - 1]$

Reading 59: The Term Structure of Interest Rates: Spot, Par, and Forward Curves

Pricing with spot rates: $P = \frac{P M T}{( 1 + S _{1} )} + \frac{P M T}{( 1 + S _{2} ) ^{2}} + \dots + \frac{P M T + F V}{( 1 + S _{N} ) ^{N}}$
Par rates: $100 = \frac{c}{1 + S _{1}} + \frac{c}{( 1 + S _{2} ) ^{2}} + \dots + \frac{c + 100}{( 1 + S _{N} ) ^{N}} \Rightarrow c = \frac{100 [ 1 - \frac{1}{( 1 + S _{N} ) ^{N}} ]}{\sum _{t = 1}^{N} \frac{1}{( 1 + S _{t} ) ^{t}}}$
Forward rates and notation: $(1 + S_{3})^{3} = (1 + S_{1}) (1 +_{1 y 1 y}) (1 +_{2 y 1 y}) = (1 + S_{2})^{2} (1 +_{2 y 1 y})$ $S_{N} = [(1 + S_{1}) (1 +_{1 y 1 y}) \dots (1 +_{(N - 1) y 1 y})]^{1/ N} - 1$ $_{A y B y} = [\frac{( 1 + S _{A + B} ) ^{A + B}}{( 1 + S _{A} ) ^{A}}]^{1/ B} - 1$
Valuing a bond with forward rates: $P = \frac{P M T}{1 + S _{1}} + \frac{P M T}{( 1 + S _{1} ) ( 1 + _{1 y 1 y} )} + \frac{P M T + F V}{( 1 + S _{1} ) ( 1 + _{1 y 1 y} ) ( 1 + _{2 y 1 y} )}$

Reading 60: Interest Rate Risk and Return

Horizon yield: $Horizon yield = (\frac{sale price (or principal) + coupons + reinvestment income}{purchase price})^{1/ H} - 1$
Holding period return, Macaulay duration and the investment horizon: $Duration gap = Macaulay duration - investment horizon$
Calculating and interpreting Macaulay duration: $MacDur = t = 1 \sum N t \times w_{t}$ $w_{t} = \frac{C F _{t} / ( 1 + r ) ^{t}}{full price}$ $t \sum w_{t} = 1$

Reading 61: Yield-Based Bond Duration Measures and Properties

Modified duration: $ModDur = \frac{MacDur}{1 + r} (annual-pay: r = YTM)$ $ModDur = \frac{MacDur}{1 + YTM /2} (semiannual-pay, YTM on a semiannual bond basis, durations in years)$ $%Δ P \approx - ModDur \times Δ YTM$
Approximate modified duration: $ApproxModDur = \frac{V _{-} - V _{+}}{2 \times V _{0} \times Δ YTM}$
Money duration and PVBP: $Money duration = ModDur \times full price of the position$ $Money duration per 100 of par = ModDur \times full price per 100$ $Δ V \approx - Money duration \times Δ YTM$ $PVBP = \frac{V _{-} - V _{+}}{2}$ $PVBP \approx Money duration \times 0.0001$

Reading 62: Yield-Based Bond Convexity and Portfolio Properties

Convexity from the cash flows: $convexity_{t} = \frac{t ( t + 1 )}{( 1 + r ) ^{2}}$
Approximate convexity: $approximate convexity = \frac{V _{-} + V _{+} - 2 V _{0}}{( Δ Y T M ) ^{2} V _{0}}$
Price change using duration and convexity: $%Δ P V^{F u l l} \approx - AnnModDur \times Δ Y T M + \frac{1}{2} \times AnnConvexity \times (Δ Y T M)^{2}$
Money convexity: $MoneyCon = AnnConvexity \times P V^{F u l l} of the position$ $Δ P V^{F u l l} \approx - (MoneyDur \times Δ Y T M) + \frac{1}{2} \times MoneyCon \times (Δ Y T M)^{2}$

Reading 63: Curve-Based and Empirical Fixed-Income Risk Measures

Why effective duration and effective convexity for bonds with embedded options: $effective duration (EffDur) = \frac{V _{-} - V _{+}}{2 V _{0} Δ Curve}$ $effective convexity (EffCon) = \frac{V _{-} + V _{+} - 2 V _{0}}{( Δ Curve ) ^{2} V _{0}}$
Percentage price change for a change in the benchmark yield: $%Δ P V^{F u l l} \approx - EffDur \times Δ Curve + \frac{1}{2} \times EffCon \times (Δ Curve)^{2}$
Key rate duration and shaping risk: $%Δ V \approx - k \sum KeyRateDur_{k} \times Δ r_{k}$

Reading 64: Credit Risk

Measuring credit risk: $expected loss = probability of default \times loss given default$ $LGD (in money) = expected exposure \times (1 - recovery rate)$ $LGD% (rate) = loss severity = 1 - recovery rate$ $credit spread \approx PD \times LGD% = PD \times (1 - recovery rate)$
Market factors: liquidity: $%Δ P_{full} \approx - ModDur \times Δ Spread + \frac{1}{2} Convexity \times (Δ Spread)^{2}$

Reading 69: Mortgage-Backed Security (MBS) Instrument and Market Features

Mortgage pass-throughs and CMOs: $WAC = i \sum w_{i} \times mortgage rate_{i}$ $WAM = i \sum w_{i} \times remaining term_{i}$ $pass-through rate = WAC - fees$
Commercial mortgage-backed securities: $DSCR = \frac{net operating income}{debt service}$ $LTV = \frac{loan amount}{appraised value}$

Derivatives6–9% of exam 41 formulas · 10 of 10 readings

Reading 70: Derivative Instrument and Derivative Market Features

Long and short exposure: $Payoff to long at settlement = S_{T} - F$ $Payoff to short = F - S_{T}$

Reading 71: Forward Commitment and Contingent Claim Features and Instruments

Value at expiration and profit for long and short option positions: $c_{T} = max (0, S_{T} - X)$ $p_{T} = max (0, X - S_{T})$ $Profit to buyer = value at expiration - premium paid$ $Profit to writer = premium received - value at expiration$

Reading 72: Derivative Benefits, Risks, and Issuer and Investor Uses

Risks: $%Δ margin = \frac{Δ P %}{m}$ $leverage = \frac{1}{m}$

Reading 73: Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives

Replicating forwards (underlying with no costs or benefits of holding): $F_{0} (T) = S_{0} (1 + R_{f})^{T}$
Spot versus expected future price, and the cost of carry: $F_{0} (T) = [S_{0} + P V_{0} (cost) - P V_{0} (benefit)] (1 + R_{f})^{T}$ $F_{0} (T) = S_{0} e^{r T}$ $F_{0} (T) = S_{0} e^{(r + c) T}$ $F_{0} (T) = S_{0} e^{(r + c - b) T}$
Currency forwards: $F = S_{0} \times (\frac{1 + R _{price}}{1 + R _{base}})^{T}$ $or, continuously compounded, F = S_{0} e^{(r_{price} - r_{base}) T}$

Reading 74: Pricing and Valuation of Forward Contracts and for an Underlying With Varying Maturities

Value and price of a forward: at initiation, during its life, at expiration: $V_{t} (T) = [S_{t} + P V_{t} (cost) - P V_{t} (benefit)] - F_{0} (T) (1 + R_{f})^{- (T - t)}$
Implied (no-arbitrage) forward rates: $(1 + Z_{A + B})^{A + B} = (1 + Z_{A})^{A} (1 + F_{A, B})^{B} so F_{A, B} = [\frac{( 1 + Z _{A + B} ) ^{A + B}}{( 1 + Z _{A} ) ^{A}}]^{1/ B} - 1$ $1 + Z_{long} \frac{d _{long}}{360} = (1 + Z_{short} \frac{d _{short}}{360}) (1 + F \frac{d _{long} - d _{short}}{360})$
Forward rate agreements (FRAs): $Payment to long = \frac{Notional \times ( MRR - FRA rate ) \times \frac{days}{360}}{1 + MRR \times \frac{days}{360}}$

Reading 75: Pricing and Valuation of Futures Contracts

Interest rate futures: $futures price = 100 - (100 \times MRR_{A, B - A})$ $BPV = notional principal \times period (in years) \times 0.0001$

Reading 76: Pricing and Valuation of Interest Rates and Other Swaps

Solving for the par swap rate: $n = 1 \sum N \frac{MRR _{n}}{( 1 + S _{n} ) ^{n}} = n = 1 \sum N \frac{F}{( 1 + S _{n} ) ^{n}}$ $F = \frac{1 - \frac{1}{( 1 + S _{N} ) ^{N}}}{n = 1 \sum N \frac{1}{( 1 + S _{n} ) ^{n}}}$
Value during the life: $Value to fixed-rate payer = P V (expected floating payments) - P V (remaining fixed payments)$

Reading 77: Pricing and Valuation of Options

Moneyness, exercise value and time value: $Call exercise value = max (0, S - X)$ $Put exercise value = max (0, X - S)$ $Option premium = exercise value + time value, so time value = premium - exercise value$

Reading 78: Option Replication Using Put-Call Parity

Put–call parity: $c + X (1 + R_{f})^{- T} = S + p$
Put–call forward parity and options in corporate finance: $F_{0} (T) (1 + R_{f})^{- T} + p_{0} = c_{0} + X (1 + R_{f})^{- T}$ $p_{0} - c_{0} = [X - F_{0} (T)] (1 + R_{f})^{- T}$
Option view of a firm (equity and debt): $V_{0} = c_{0} + P V (D) - p_{0}$ $Equity_{T} = max (0, V_{T} - D)$ $Debt_{T} = min (V_{T}, D) = D - max (0, D - V_{T})$

Reading 79: Valuing a Derivative Using a One-Period Binomial Model

Call: long $h$ shares, short one call: $h S_{1}^{u} - c_{1}^{u} = h S_{1}^{d} - c_{1}^{d} \Rightarrow h = \frac{c _{1}^{u} - c _{1}^{d}}{S _{1}^{u} - S _{1}^{d}}$ $c_{0} = h S_{0} - V_{0}$
Put: long $h$ shares, long one put: $h = \frac{p _{1}^{d} - p _{1}^{u}}{S _{1}^{u} - S _{1}^{d}}$ $V_{0} = h S_{0} + p_{0} \Rightarrow p_{0} = V_{0} - h S_{0}$
Risk neutrality: $π_{U} = \frac{1 + R _{f} - R ^{d}}{R ^{u} - R ^{d}}$ $π_{D} = 1 - π_{U}$ $c_{0} = \frac{π _{U} c _{1}^{u} + π _{D} c _{1}^{d}}{1 + R _{f}}$ $p_{0} = \frac{π _{U} p _{1}^{u} + π _{D} p _{1}^{d}}{1 + R _{f}}$

Alternative Investments6–9% of exam 11 formulas · 4 of 7 readings

Reading 80: Alternative Investment Features, Methods, and Structures

Calculating the performance fee: $Fee_{soft hurdle} = p \times R$ $Fee_{hard hurdle} = p \times (R - h)$ $full catch-up reached at R = \frac{h}{1 - p}$

Reading 81: Alternative Investment Performance and Returns

Life cycle and the J-curve: $MOIC = \frac{total capital returned + value of remaining assets}{total capital paid in}$
Leverage: $leveraged return = \frac{r ( V _{0} + V _{B} ) - r _{B} V _{B}}{V _{0}}$ $leveraged return = r + \frac{V _{B}}{V _{0}} (r - r_{B})$
After-fee return mechanics: $total fees = m V_{1} + max [0, p (V_{1} - V_{0})]$ $r_{after fees} = \frac{V _{1} - V _{0} - total fees}{V _{0}}$

Reading 84: Natural Resources

Commodity valuation: cost of carry: $futures price \approx spot price \times (1 + r_{f}) + storage costs - convenience yield$

Reading 85: Hedge Funds

Where returns come from, index biases and diversification: $R_{fund} = broad market market beta + sector/strategy exposure strategy beta + manager skill α$ $C V = \frac{σ}{R ˉ}$

Portfolio Construction8–12% of exam 26 formulas · 3 of 6 readings

Reading 87: Portfolio Risk and Return: Part I

Utility function: $U = E (r) - \frac{1}{2} A σ^{2}$
Adding the risk-free asset: the capital allocation line: $E (R_{p}) = w E (R_{r i s k y}) + (1 - w) R_{f}$ $σ_{p} = w σ_{r i s k y}$
One asset: $σ^{2} = \frac{\sum _{t = 1}^{T} ( R _{t} - μ ) ^{2}}{T} (population)$ $s^{2} = \frac{\sum _{t = 1}^{T} ( R _{t} - R ˉ ) ^{2}}{T - 1} (sample)$
Two assets: covariance: $Cov_{1, 2} = \frac{\sum _{t = 1}^{T} ( R _{1, t} - R ˉ _{1} ) ( R _{2, t} - R ˉ _{2} )}{T - 1}$
Correlation: $ρ_{1, 2} = \frac{Cov _{1, 2}}{σ _{1} σ _{2}}$ $Cov_{1, 2} = ρ_{1, 2} σ_{1} σ_{2}$
Portfolio standard deviation: $E (R_{p}) = w_{1} E (R_{1}) + w_{2} E (R_{2})$ $σ_{p}^{2} = w_{1}^{2} σ_{1}^{2} + w_{2}^{2} σ_{2}^{2} + 2 w_{1} w_{2} Cov_{1, 2} = w_{1}^{2} σ_{1}^{2} + w_{2}^{2} σ_{2}^{2} + 2 w_{1} w_{2} ρ_{1, 2} σ_{1} σ_{2}$ $σ_{p} = σ_{p}^{2}$
Investing in assets that are less than perfectly correlated: $σ_{p}^{2} = (w_{1} σ_{1} + w_{2} σ_{2})^{2} \Rightarrow σ_{p} = w_{1} σ_{1} + w_{2} σ_{2}$ $w_{1} = \frac{σ _{2}}{σ _{1} + σ _{2}}$

Reading 88: Portfolio Risk and Return: Part II

Adding a risk-free asset to a risky portfolio: $E (R_{C}) = w E (R_{P}) + (1 - w) R_{f}$ $σ_{C} = w σ_{P}$
The capital allocation line (CAL) and the capital market line (CML): $E (R_{P}) = R_{f} + [\frac{E ( R _{M} ) - R _{f}}{σ _{M}}] σ_{P}$
Systematic and unsystematic risk: $total risk = systematic risk + unsystematic risk$
Return generating models and the market model: $E (R_{i}) - R_{f} = β_{i, 1} E (F_{1}) + β_{i, 2} E (F_{2}) + \dots + β_{i, k} E (F_{k})$ $R_{i} = α_{i} + β_{i} R_{m} + e_{i}$
Calculating and interpreting beta: $β_{i} = \frac{C o v ( R _{i} , R _{m} )}{σ _{m}^{2}} = \frac{ρ _{i, m} σ _{i} σ _{m}}{σ _{m}^{2}} = ρ_{i, m} \frac{σ _{i}}{σ _{m}}$
The CAPM, its assumptions, and the security market line (SML): $E (R_{i}) = R_{f} + β_{i} [E (R_{m}) - R_{f}]$ $β_{i} = \frac{C o v ( R _{i} , R _{m} )}{σ _{m}^{2}}$
Applying the CAPM and the SML: finding mispriced securities: $forecast return = \frac{P _{1} - P _{0} + D _{1}}{P _{0}}$ $required return = R_{f} + β [E (R_{m}) - R_{f}]$

Reading 89: Portfolio Management: An Overview

Measuring the benefit: the diversification ratio: $Diversification ratio = \frac{σ of an equally weighted portfolio of the n securities}{average σ of the n individual securities}$
Mutual funds and other pooled investments: $N A V = \frac{fund assets - fund liabilities}{shares outstanding}$

Ethical and Professional Standards10–15% of exam No formulas

This topic has no formulas. Its key takeaways and exam shortcuts are on the cheat sheet.

CFA Level 1 Formula Sheet: common questions

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