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CFA 一级公式表

CFA 一级中文公式表：2027 年考纲讲义中的全部公式，按科目和章节排列，可打印或免费下载 PDF。每一章都链接到讲解这些公式的讲义；如果还要看本节要点和答题技巧，请用速查表。

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各科目公式

本表收录讲义中的 501 个公式，例题里的计算过程不收录。公式最多的科目是数量方法（133 个）、财务报表分析（95 个）、权益投资（92 个）。道德与职业标准没有公式，本节要点见速查表。各科目考试权重是 CFA 协会公布的 2027 年一级考试权重范围：道德与职业标准 10–15%；数量方法、财务报表分析、权益投资、固定收益各 11–14%；投资组合管理 8–12%；经济学、公司金融、衍生品、另类投资各 6–9%。

数量方法（Quantitative Methods）考试权重 11–14% 133 个公式 · 11 章中 10 章有公式

第 1 章金融资产与金融工具的收益率（Returns of Financial Assets and Instruments）

单期收益率: $Price return = \frac{P _{1} - P _{0}}{P _{0}}$ $Capital distribution return = \frac{D}{P _{0}}$ $Total (holding period) return = \frac{P _{1} - P _{0} + D}{P _{0}} = price return + capital distribution return$
多期收益率的平均: $R ˉ_{arith} = \frac{R _{1} + R _{2} + \dots + R _{n}}{n}$ $R_{geo} = [(1 + R_{1}) (1 + R_{2}) \dots (1 + R_{n})]^{1/ n} - 1$
持有期收益率的年化: $Annualized return = (1 + H P R)^{365/ days} - 1$ $(1 + R_{period})^{c} - 1$
复利频率（compounding frequency）与连续复利（continuous compounding）: $Effective annual rate = (1 + \frac{r _{s}}{m})^{m} - 1$ $P V = \frac{F V}{( 1 + r _{s} / m ) ^{m}} (one year ahead)$ $r_{cc} = ln (1 + H P R)$ $H P R = e^{r_{cc}} - 1$
必要收益率的组成部分: $Required return = nominal risk-free rate real risk-free rate + expected inflation + risk premium(s)$
实际收益率（real return）: $1 + r_{real} = \frac{1 + r _{nominal}}{1 + π}$ $π = \frac{1 + r _{nominal}}{1 + r _{real}} - 1$
税前与税后名义收益率: $r_{after-tax} = price return \times (1 - t_{c g}) + distribution return \times (1 - t_{d})$
杠杆收益率（leveraged return）: $r_{L} = \frac{r _{P} ( V _{E} + V _{D} ) - r _{D} V _{D}}{V _{E}} = r_{P} + \frac{V _{D}}{V _{E}} (r_{P} - r_{D})$

第 2 章金融收益率的类型（Types of Financial Returns）

收益的两个来源: $Total return = capital appreciation return + capital distribution return$
利率与无风险利率: $H P R = \frac{Face value - Price}{Price}$ $r_{annual} = (1 + H P R)^{365/ t} - 1$
汇率: $1 + r_{domestic} = (1 + r_{local}) (1 + r_{F X})$
风险与收益的权衡: $short-term government bills < government bonds < corporate bonds < large-cap equities < small-cap equities$
投资组合预期收益率（portfolio expected return）: $E (R_{p}) = i \sum w_{i} E (R_{i})$ $total return = capital appreciation return + capital distribution return$

第 3 章收益率基准（Benchmarking Returns）

资金加权收益率（MWR）: $t = 0 \sum T \frac{C F _{t}}{( 1 + M W R ) ^{t}} = 0$
时间加权收益率（TWR）: $H P R_{i} = \frac{V _{e n d, i} - V _{b e g in, i} + D _{i}}{V _{b e g in, i}}$ $T W R_{ann u a l} = [(1 + H P R_{1}) (1 + H P R_{2}) \dots (1 + H P R_{n})]^{1/ years} - 1$
指数是什么，收益率怎样衡量: $R_{period} = (1 + R_{1}) (1 + R_{2}) \dots (1 + R_{n}) - 1$ $V_{n} = V_{0} (1 + R_{1}) (1 + R_{2}) \dots (1 + R_{n})$
价格加权指数: $Index = \frac{\sum P _{i}}{D}$ $w_{i} = \frac{P _{i}}{\sum P _{j}}$
市值加权（价值加权）指数: $w_{i} = \frac{P _{i} N _{i}}{\sum P _{j} N _{j}}$ $Index = \frac{\sum P _{i} N _{i}}{D}, D = \frac{base-period market cap}{base value}$

第 4 章金融中的货币时间价值（The Time Value of Money in Finance）

货币时间价值: $F V = P V (1 + r)^{t}$ $P V = \frac{F V}{( 1 + r ) ^{t}}$
固定票息债券: $P = t = 1 \sum N \frac{C}{( 1 + Y T M ) ^{t}} + \frac{F}{( 1 + Y T M ) ^{N}}$
永续债券: $P V_{perpetuity} = \frac{P M T}{r}$
摊销债券与年金: $P V_{annuity} = P M T \times \frac{1 - ( 1 + r ) ^{- N}}{r}$ $P M T = \frac{r \times P V}{1 - ( 1 + r ) ^{- N}}$
权益工具: $V_{preferred} = \frac{D _{p}}{r _{p}}$ $V_{0} = \frac{D _{1}}{r _{e} - g _{c}}$ $D_{1} = D_{0} (1 + g_{c})$
零息债券（zero-coupon bond）: $r = (\frac{F V}{P V})^{1/ N} - 1$
权益（equity）：隐含必要收益率: $r_{e} = \frac{D _{1}}{P _{0}} + g_{c} = dividend yield + constant growth rate$
权益：隐含增长率: $g_{c} = r_{e} - \frac{D _{1}}{P _{0}} = required return - dividend yield$
远期利率: $(1 + S_{2})^{2} = (1 + S_{1}) (1 + 1 y 1 y) \Rightarrow 1 y 1 y = \frac{( 1 + S _{2} ) ^{2}}{1 + S _{1}} - 1$ $(1 + S_{3})^{3} = (1 + S_{1}) (1 + 1 y 1 y) (1 + 2 y 1 y)$
远期汇率（forward exchange rates）: $\frac{Forward}{Spot} = \frac{( 1 + r _{price} ) ^{T}}{( 1 + r _{base} ) ^{T}}$ $\frac{Forward}{Spot} = \frac{1 + r _{price, period}}{1 + r _{base, period}}$
期权价值：单期二叉树模型: $h = \frac{c _{u} - c _{d}}{S _{u} - S _{d}}$

第 5 章资产收益率的统计特征（Statistical Characteristics of Asset Returns）

总体与样本: $Sample mean: X ˉ = \frac{\sum _{i = 1}^{n} X _{i}}{n}$ $Population mean: μ = \frac{\sum _{i = 1}^{N} X _{i}}{N}$
极差与平均绝对偏差: $range = maximum value - minimum value$ $MAD = \frac{\sum _{i = 1}^{n} ∣ X _{i} - X ˉ ∣}{n}$
偏度: $skewness = \frac{1}{n} i = 1 \sum n (\frac{X _{i} - X ˉ}{s})^{3}$
峰度: $kurtosis = \frac{1}{n} i = 1 \sum n (\frac{X _{i} - X ˉ}{s})^{4}$ $excess kurtosis = kurtosis - 3$
协方差: $Cov_{X Y} = \frac{\sum _{i = 1}^{n} ( X _{i} - X ˉ ) ( Y _{i} - Y ˉ )}{n - 1} (sample)$
相关性: $ρ_{X Y} = \frac{Cov _{X Y}}{s _{X} s _{Y}}$ $Cov_{X Y} = ρ_{X Y} s_{X} s_{Y}$
半方差与目标下行偏差: $σ_{semi}^{2} = \frac{\sum _{X_{i} < X ˉ} ( X _{i} - X ˉ ) ^{2}}{n}$ $s_{target} = \frac{\sum _{X_{i} < B} ( X _{i} - B ) ^{2}}{n - 1}$
变异系数: $C V = \frac{s}{X ˉ}$ $(or \frac{σ}{μ})$

第 6 章金融资产价格与收益率的统计分布（Statistical Distributions for Financial Asset Prices and Returns）

累积分布函数（CDF）: $F (x) = P (X \leq x)$ $P (a < X \leq b) = F (b) - F (a)$
无条件期望值（均值）: $E (X) = i = 1 \sum n P (x_{i}) x_{i}$
无条件方差与标准差: $σ^{2} (X) = i = 1 \sum n P (x_{i}) [x_{i} - E (X)]^{2}$ $σ (X) = σ^{2} (X)$
无条件协方差: $Cov (X, Y) = i = 1 \sum n P (x_{i}, y_{i}) [x_{i} - E (X)] [y_{i} - E (Y)]$
离散分布: $E (X) = n p$ $σ^{2} (X) = n p (1 - p)$ $P (k events in interval t) = \frac{( λ t ) ^{k} e ^{- λ t}}{k !}$ $mean = variance = λ (for one unit of time, t = 1)$
连续分布: $P (x_{1} \leq X \leq x_{2}) = \frac{x _{2} - x _{1}}{b - a}$ $mean = \frac{a + b}{2}$ $σ^{2} = \frac{( b - a ) ^{2}}{12}$
条件均值、条件方差和条件协方差: $E (X ∣ S) = i \sum P (x_{i} ∣ S) x_{i}$ $Var (X ∣ S) = i \sum P (x_{i} ∣ S) [x_{i} - E (X ∣ S)]^{2} = E (X^{2} ∣ S) - [E (X ∣ S)]^{2}$
用贝叶斯公式更新概率: $P (E ∣ I) = \frac{P ( I ∣ E )}{P ( I )} \times P (E) = \frac{P ( I ∣ E ) P ( E )}{P ( I ∣ E ) P ( E ) + P ( I ∣ E ^{c} ) P ( E ^{c} )}$

第 7 章估计与假设检验（Estimation and Hypothesis Testing）

中心极限定理（CLT）: $X ˉ \sim ˙ N (μ, \frac{σ ^{2}}{n}) for large n (rule of thumb: n \geq 30)$
标准误（standard error）与抽样误差: $σ_{X ˉ} = \frac{σ}{n} (σ known)$ $s_{X ˉ} = \frac{s}{n} (σ unknown, the usual case)$
置信区间: $point estimate \pm margin of error (reliability factor \times standard error)$
风险价值（VaR）：单尾区间: $VaR_{α} = z_{α} \times σ \times portfolio value$
均值的检验统计量: $test statistic = \frac{X ˉ - μ _{0}}{s / n} (use σ / n if σ is known)$
斯皮尔曼秩相关检验: $r_{s} = 1 - \frac{6 \sum d _{i}^{2}}{n ( n ^{2} - 1 )}$
用列联表做独立性检验: $E_{i, j} = \frac{( total for row i ) \times ( total for column j )}{total number of observations}$ $χ^{2} = i \sum j \sum \frac{( O _{i, j} - E _{i, j} ) ^{2}}{E _{i, j}}$ $df = (r - 1) (c - 1)$

第 8 章金融投资组合的收益与风险（The Return and Risk of a Financial Portfolio）

投资组合的预期收益率: $E (R_{P}) = i = 1 \sum N w_{i} E (R_{i})$ $i = 1 \sum N w_{i} = 1$ $R_{P, t} = i = 1 \sum N w_{i, t - 1} R_{i, t}$
投资组合方差（portfolio variance）: $σ_{P}^{2} = i = 1 \sum N j = 1 \sum N w_{i} w_{j} Cov (R_{i}, R_{j})$ $σ_{P}^{2} = w_{1}^{2} σ_{1}^{2} + w_{2}^{2} σ_{2}^{2} + 2 w_{1} w_{2} Cov_{1, 2}$ $σ_{P} = σ_{P}^{2}$
协方差与相关性: $Cov_{1, 2} = ρ_{1, 2} σ_{1} σ_{2} ⟹ σ_{P}^{2} = w_{1}^{2} σ_{1}^{2} + w_{2}^{2} σ_{2}^{2} + 2 w_{1} w_{2} ρ_{1, 2} σ_{1} σ_{2}$
更多资产：方差-协方差矩阵（variance-covariance matrix）: $number of unique covariances = \frac{N ^{2} - N}{2}$ $σ_{P}^{2} = i = 1 \sum N j = 1 \sum N w_{i} w_{j} ρ_{i, j} σ_{i} σ_{j}$ $ρ_{i, i} = 1$
大型投资组合：方差趋近于平均协方差: $σ_{P}^{2} = \frac{σ ˉ ^{2}}{N} + \frac{N - 1}{N} Cov$ $σ_{P}^{2} \approx ρ ˉ σ ˉ^{2}$
两项风险资产的最小方差投资组合: $w_{A} = \frac{σ _{B}^{2} - Cov _{A, B}}{σ _{A}^{2} + σ _{B}^{2} - 2 Cov _{A, B}}$ $w_{B} = 1 - w_{A}$
加入无风险资产: $Sharpe ratio = \frac{E ( R _{P} ) - R _{f}}{σ _{P}}$ $E (R_{C}) = w_{P} E (R_{P}) + (1 - w_{P}) R_{f}$ $σ_{C} = w_{P} σ_{P}$ $w_{P} = \frac{r _{target} - R _{f}}{E ( R _{P} ) - R _{f}}$
效用函数: $U = E (r) - \frac{1}{2} A σ^{2}$
资本配置线（capital allocation line, CAL）: $E (R_{C}) = R_{f} + [\frac{E ( R _{P} ) - R _{f}}{σ _{P}}] σ_{C}$
资本市场线（capital market line, CML）: $E (R_{P}) = R_{f} + [\frac{E ( R _{M} ) - R _{f}}{σ _{M}}] σ_{P}$
收益生成模型与 CAPM: $E (R_{i}) = R_{f} + β_{i} [E (R_{M}) - R_{f}]$
贝塔系数: $β_{i} = \frac{Cov ( R _{i} , R _{M} )}{σ _{M}^{2}} = \frac{ρ _{i, M} σ _{i} σ _{M}}{σ _{M}^{2}} = \frac{ρ _{i, M} σ _{i}}{σ _{M}}$

第 9 章金融资产价格与收益率的模拟（Simulation of Financial Asset Prices and Returns）

多变量模拟与 Cholesky 分解: $X_{1} = σ_{1} Z_{1}$ $X_{2} = σ_{2} (ρ Z_{1} + 1 - ρ^{2} Z_{2})$

第 10 章一元线性回归在金融中的应用（Applications of Simple Linear Regression in Finance）

模型与回归线: $Y_{i} = b_{0} + b_{1} X_{i} + ε_{i} (population model)$ $Y^_{i} = b^_{0} + b^_{1} X_{i} (estimated regression line, the line of best fit)$
最小二乘准则: $SSE = i = 1 \sum n (Y_{i} - Y^_{i})^{2}$ $b^_{1} = \frac{Cov ( X , Y )}{σ _{X}^{2}}$ $b^_{0} = Y ˉ - b^_{1} X ˉ$
方差分析（ANOVA）: $SST = SSR + SSE$
拟合优度指标: $SEE = MSE = \frac{SSE}{n - 2}$ $R^{2} = \frac{SSR}{SST} = 1 - \frac{SSE}{SST}$
F 检验: $F = \frac{MSR}{MSE}$ $df = 1 (numerator), n - 2 (denominator)$
回归系数的 t 检验: $t = \frac{b ^ _{1} - b _{1}}{s _{b^_{1}}}$ $df = n - 2$
预测值（predicted value）: $Y^= b^_{0} + b^_{1} X_{p}$
预测值的置信区间（预测区间（prediction interval））: $Y^\pm t_{c} s_{f}$ $df = n - 2$ $s_{f}^{2} = SEE^{2} [1 + \frac{1}{n} + \frac{( X _{p} - X ˉ ) ^{2}}{( n - 1 ) s _{X}^{2}}]$
作为单因子回归的 CAPM: $E (R_{i}) = R_{f} + β_{i} [E (R_{M}) - R_{f}]$ $Z_{i, t} = b_{0} + b_{1} Z_{M, t} + ε_{t}$ $Z_{i, t} = R_{i, t} - R_{f, t}, Z_{M, t} = R_{M, t} - R_{f, t}$

经济学（Economics）考试权重 6–9% 23 个公式 · 8 章中 6 章有公式

第 12 章厂商与市场结构（The Firm and Market Structures）

两种集中度指标: $N-firm concentration ratio = i = 1 \sum N s_{i}$ $HHI = i = 1 \sum N s_{i}^{2}$

第 14 章财政政策（Fiscal Policy）

财政政策工具：支出和收入，优点和缺点: $fiscal multiplier = \frac{1}{1 - M P C ( 1 - t )}$ $Δ A D_{tax} = - Δ T \times M P C \times fiscal multiplier$

第 15 章货币政策（Monetary Policy）

有效的中央银行、目标制度与政策的局限: $neutral interest rate = real trend rate of economic growth + inflation target$

第 17 章国际贸易（International Trade）

关税的福利计算公式（小国）: $Loss of consumer surplus = \frac{Q _{D, 0} + Q _{D, 1}}{2} Δ P$ $Gain in producer surplus = \frac{Q _{S, 0} + Q _{S, 1}}{2} Δ P$ $Tariff revenue = M_{1} Δ P = (Q_{D, 1} - Q_{S, 1}) Δ P$ $Deadweight loss = consumer surplus loss - producer surplus gain - tariff revenue = \frac{1}{2} (Q_{S, 1} - Q_{S, 0}) Δ P + \frac{1}{2} (Q_{D, 0} - Q_{D, 1}) Δ P$

第 18 章资本流动与外汇市场（Capital Flows and the FX Market）

名义汇率与实际汇率: $real P/B rate = nominal P/B rate \times \frac{C P I _{base currency}}{C P I _{price currency}}$
货币价值的百分比变化: $%Δ base = \frac{new P/B}{old P/B} - 1$ $%Δ price = \frac{1/ new P/B}{1/ old P/B} - 1 = \frac{old P/B}{new P/B} - 1$
汇率、贸易流动与资本流动: $(X - M) \equiv (S - I) + (T - G)$

第 19 章汇率计算（Exchange Rate Calculations）

货币交叉汇率: $\frac{A}{C} = \frac{A}{B} \times \frac{B}{C}$
无套利关系：利率平价（interest rate parity）: $(1 + r_{d}) = \frac{1}{S _{d / f}} (1 + r_{f}) F_{d / f}$ $\frac{F _{d / f}}{S _{d / f}} = \frac{1 + r _{d}}{1 + r _{f}}$ $F_{d / f} = S_{d / f} \frac{1 + r _{d}}{1 + r _{f}}$ $S_{d / f} = F_{d / f} \frac{1 + r _{f}}{1 + r _{d}}$ $\frac{F _{d / f}}{S _{d / f}} - 1 = \frac{r _{d} - r _{f}}{1 + r _{f}} \approx r_{d} - r_{f}$ $F = S \times \frac{1 + r _{price} ( days /360 )}{1 + r _{base} ( days /360 )}$
以点数或百分比表示的远期报价: $F = S + (points \times point size)$ $F = S \times (1 + percentage)$
远期升水或贴水: $forward premium (+) or discount (-) = \frac{F}{S} - 1$

公司金融（Corporate Finance）考试权重 6–9% 18 个公式 · 7 章中 4 章有公式

第 20 章组织形式、公司发行人的特征与所有权（Organizational Forms, Corporate Issuer Features, and Ownership）

公司发行人的主要特征: $effective tax rate on distributed profit = t_{c} + t_{d} (1 - t_{c})$
公众公司与私人公司: $free float (%) = \frac{shares outstanding - shares held by insiders, strategic investors and sponsors}{shares outstanding}$

第 23 章营运资本与流动性（Working Capital and Liquidity）

现金转换周期: $CCC = DOH + DSO - DPO$
商业信用及其成本: $EAR of supplier financing = (1 + \frac{a}{1 - a})^{365/ (c - b)} - 1$
比较 CCC: $total working capital = current assets - current liabilities$ $net working capital = (current assets - cash and marketable securities) - (current liabilities - short-term and current debt)$
流动性比率: $current ratio = \frac{current assets}{current liabilities}$ $quick ratio = \frac{cash and marketable securities + receivables}{current liabilities}$ $cash ratio = \frac{cash and marketable securities}{current liabilities}$

第 24 章资本投资与资本配置（Capital Investments and Capital Allocation）

净现值（NPV）: $N P V = C F_{0} + \frac{C F _{1}}{( 1 + r ) ^{1}} + \frac{C F _{2}}{( 1 + r ) ^{2}} + \dots + \frac{C F _{n}}{( 1 + r ) ^{n}} = t = 0 \sum n \frac{C F _{t}}{( 1 + r ) ^{t}}$
内部收益率（IRR）: $0 = t = 0 \sum n \frac{C F _{t}}{( 1 + I R R ) ^{t}}$
投入资本回报率（ROIC）: $R O I C = \frac{NOPAT}{average book value of invested capital} = operating margin after tax \frac{NOPAT}{Sales} \times capital turnover \frac{Sales}{average invested capital}$
实物期权: $Project value = N P V_{without options} + value of options - cost of options$

第 25 章资本结构（Capital Structure）

计算和解读 WACC: $W A C C = w_{d} r_{d} (1 - t) + w_{e} r_{e}$
MM 命题 II（无税）：权益资本成本与杠杆: $r_{e} = r_{0} + \frac{D}{E} (r_{0} - r_{d})$
有税条件下的 MM 定理: $V_{L} = V_{U} + t D$ $r_{e} = r_{0} + \frac{D}{E} (r_{0} - r_{d}) (1 - t)$
静态权衡理论: $V_{L} = V_{U} + t D - P V (costs of financial distress)$

财务报表分析（Financial Statement Analysis）考试权重 11–14% 95 个公式 · 12 章中 10 章有公式

第 28 章利润表分析（Analyzing Income Statements）

在一段时间内确认收入（长期合同）: $Cumulative revenue_{t} = \frac{Cumulative cost to date}{Total expected cost} \times Transaction price$
折旧方法: $Straight-line = \frac{Cost - Salvage}{Useful life}$
资本化利息: $Adjusted coverage = \frac{EBIT + Depreciation of capitalized interest}{Interest expense + Interest capitalized}$
基本每股收益（basic EPS）: $Basic EPS = \frac{Net income - Preferred dividends}{Weighted average number of common shares outstanding}$
稀释每股收益（diluted EPS）与假设转换法（if-converted method）: $Diluted EPS = \frac{NI - Pref. div. + Conv. pref. div. + Conv. debt interest \times ( 1 - t )}{Wtd. avg. shares + Shares from conv. pref. + Shares from conv. debt + Shares from options/warrants}$ $Incremental shares = N - \frac{N \times X}{P ˉ} = \frac{P ˉ - X}{P ˉ} \times N$
稀释性测试: $Conv. preferred: \frac{Preferred dividends}{Shares on conversion}$ $Conv. bonds: \frac{Interest \times ( 1 - t )}{Shares on conversion}$
纵向共同比利润表: $Common-size line item = \frac{Line item}{Revenue}$ $Effective tax rate = \frac{Income tax expense}{Pretax income}$
利润率比率: $Gross profit margin = \frac{Revenue - COGS}{Revenue}$ $Operating profit margin = \frac{Operating profit (EBIT)}{Revenue}$ $Pretax margin = \frac{Earnings before tax}{Revenue}$ $Net profit margin = \frac{Net income}{Revenue}$

第 29 章资产负债表分析（Analyzing Balance Sheets）

商誉: $Goodwill = purchase price - fair value of identifiable net assets acquired$
纵向共同比资产负债表: $Common-size item = \frac{balance sheet item}{total assets}$
流动性比率（短期债务）: $current ratio = \frac{current assets}{current liabilities}$ $quick ratio = \frac{cash + marketable securities + receivables}{current liabilities}$ $cash ratio = \frac{cash + marketable securities}{current liabilities}$
偿债能力比率（长期债务）: $long-term debt-to-equity = \frac{long-term debt}{total equity}$ $total debt-to-equity = \frac{total debt}{total equity}$ $debt ratio = \frac{total debt}{total assets}$ $financial leverage ratio = \frac{total assets}{total equity}$

第 30 章现金流量表分析 I（Analyzing Statements of Cash Flows I）

为什么要单独编现金流量表？: $CFO + CFI + CFF = Δ Cash; Beginning cash + Δ Cash = Ending cash$
时间差异体现在资产负债表上: $Beginning AR + Sales - Cash collected = Ending AR$
直接法: $Cash collected = Net sales - Δ AR + Δ Unearned revenue$ $Purchases = COGS + Ending inventory - Beginning inventory = COGS + Δ Inventory$ $Cash paid to suppliers = Purchases - Δ AP = COGS + Δ Inventory - Δ AP$ $Cash paid for wages = Wage expense - Δ Wages payable$ $Cash paid for interest = Interest expense - Δ Interest payable$ $Cash paid for insurance = Insurance expense + Δ Prepaid insurance$ $Cash taxes paid = Tax expense - Δ Taxes payable - Δ Deferred tax liability + Δ Deferred tax asset$
起点：净利润: $CFO = NI + NCC - WC_{INV}$
投资活动（CFI）: $Proceeds = Carrying value of asset sold + Gain (or - Loss)$ $Gross cost of disposals = Beginning gross PP&E + Purchases - Ending gross PP&E$ $Accumulated depreciation removed = Beginning AD + Depreciation expense - Ending AD$ $Carrying value disposed = Beginning net PP&E - Depreciation + Purchases - Ending net PP&E$
融资活动（CFF）: $CFF = (New borrowing - Principal repaid) + (Stock issued - Stock repurchased) - Cash dividends paid$ $Dividends declared = Beginning retained earnings + Net income - Ending retained earnings$ $Dividends paid = Dividends declared - Δ Dividends payable$

第 31 章现金流量表分析（二）（Analyzing Statements of Cash Flows II）

公司自由现金流（FCFF）: $F C F F = N I + N C C + [I n t \times (1 - t)] - F C_{I N V} - W C_{I N V}$ $F C F F = C F O + [I n t \times (1 - t)] - F C_{I N V}$
股权自由现金流（FCFE）: $F C F E = C F O - F C_{I N V} + net borrowing$ $F C F E = F C F F - [I n t \times (1 - t)] + net borrowing$

第 32 章存货分析（Analysis of Inventories）

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

第 33 章长期资产分析（Analysis of Long-Term Assets）

商誉: $Goodwill = Purchase price - Fair value of identifiable net assets acquired$
持有使用资产的减值：国际财务报告准则（International Financial Reporting Standards, IFRS）与美国公认会计原则（US Generally Accepted Accounting Principles, US GAAP）: $IFRS loss = C V - max (FV - costs to sell, VIU)$ $US GAAP loss = C V - FV (only if C V > undiscounted CF)$
用披露数据计算的比率: $Fixed asset turnover = \frac{Revenue}{Average fixed assets}$ $Average age \approx \frac{Accumulated depreciation}{Annual depreciation expense}$ $Total useful life \approx \frac{Gross (historical cost) PP&E}{Annual depreciation expense}$ $Remaining useful life \approx \frac{Net PP&E}{Annual depreciation expense} = Total useful life - Average age$

第 34 章长期负债与权益专题（Topics in Long-Term Liabilities and Equity）

承租人会计处理: $Lease liability_{0} = t = 1 \sum N \frac{P M T}{( 1 + r ) ^{t}} = P M T \times \frac{1 - ( 1 + r ) ^{- N}}{r}$
设定提存计划与设定受益计划: $Funded status = Fair value of plan assets - PV of the defined benefit obligation$

第 35 章所得税分析（Analysis of Income Taxes）

所得税费用等式: $income tax expense = taxes payable + Δ D T L - Δ D T A$

第 36 章财务报告质量（Financial Reporting Quality）

折旧、摊销和减值: $Annual depreciation (straight-line) = \frac{cost - salvage value}{useful life}$
把信号和被操纵的对象对应起来: $Days payables = \frac{accounts payable}{COGS (or purchases) /365}$
现金流与盈余: $CFO-to-net-income ratio = \frac{cash flow from operations}{net income}$

第 37 章财务分析技术（Financial Analysis Techniques）

营运能力比率: $receivables turnover = \frac{revenue}{average receivables}$ $days of sales outstanding (DSO) = \frac{365}{receivables turnover}$ $inventory turnover = \frac{cost of goods sold}{average inventory}$ $days of inventory on hand (DOH) = \frac{365}{inventory turnover}$ $payables turnover = \frac{cost of goods sold}{average trade payables}$ $number of days of payables (DPO) = \frac{365}{payables turnover}$ $purchases = ending inventory - beginning inventory + COGS$ $total asset turnover = \frac{revenue}{average total assets}$ $fixed asset turnover = \frac{revenue}{average net fixed assets}$ $working capital turnover = \frac{revenue}{average working capital}$
流动性比率: $current ratio = \frac{current assets}{current liabilities}$ $quick ratio = \frac{cash + marketable securities + receivables}{current liabilities}$ $cash ratio = \frac{cash + marketable securities}{current liabilities}$ $defensive interval = \frac{cash + marketable securities + receivables}{average daily expenditures}$ $average daily expenditures = \frac{COGS + SG&A + R&D - noncash charges (e.g., depreciation, amortization)}{365}$ $cash conversion cycle = DSO + DOH - number of days of payables$
偿债能力比率: $debt-to-equity = \frac{total debt}{total shareholders’ equity}$ $debt-to-capital = \frac{total debt}{total debt + total shareholders’ equity}$ $debt-to-assets = \frac{total debt}{total assets}$ $financial leverage ratio = \frac{average total assets}{average total equity}$ $interest coverage = \frac{EBIT}{interest payments}$ $debt-to-EBITDA = \frac{total debt}{EBITDA}$ $fixed charge coverage = \frac{EBIT + lease payments}{interest payments + lease payments}$
盈利能力比率: $gross profit margin = \frac{gross profit}{revenue}$ $operating profit margin = \frac{operating income}{revenue} or \frac{EBIT}{revenue}$ $pretax margin = \frac{EBT}{revenue}$ $net profit margin = \frac{net income}{revenue}$ $ROA = \frac{net income}{average total assets}$ $adjusted ROA = \frac{net income + interest expense ( 1 - tax rate )}{average total assets}$ $operating ROA = \frac{operating income (or EBIT)}{average total assets}$ $ROE = \frac{net income}{average total equity}$ $return on common equity = \frac{net income - preferred dividends}{average common equity}$
两因素分解: $ROE = \frac{net income}{average total equity} = ROA \frac{net income}{average total assets} \times financial leverage \frac{average total assets}{average total equity}$
原始（三因素）杜邦公式: $ROE = net profit margin \frac{net income}{revenue} \times total asset turnover \frac{revenue}{average total assets} \times financial leverage \frac{average total assets}{average total equity}$
扩展（五因素）杜邦公式: $ROE = tax burden \frac{net income}{EBT} \times interest burden \frac{EBT}{EBIT} \times EBIT margin \frac{EBIT}{revenue} \times total asset turnover \frac{revenue}{average total assets} \times financial leverage \frac{average total assets}{average total equity}$
经营风险与变异系数: $CV = \frac{standard deviation of the item}{mean (expected value) of the item}$

权益投资（Equities）考试权重 11–14% 92 个公式 · 12 章中 11 章有公式

第 39 章权益工具的特征（Equity Instrument Features）

优先股: $Annual preferred dividend = dividend rate \times par value$

第 41 章股票发行与交易（Equity Issuance and Trading）

报价与价差: $bid-ask spread = lowest ask - highest bid$ $bid-ask spread (%) = \frac{lowest ask - highest bid}{lowest ask}$ $mid-market price = \frac{lowest ask + highest bid}{2}$
VWAP: $VWAP = \frac{\sum _{i} ( price _{i} \times volume _{i} )}{\sum _{i} volume _{i}}$
自由流通量（market float）、成交量及其他流动性（liquidity）指标: $market float = shares outstanding - restricted shares or shares not available for trading$ $average daily volume = \frac{\sum _{i = 1}^{n} ( shares traded on day i )}{n}$ $turnover ratio = \frac{average daily volume}{market float}$

第 42 章权益收益的来源（Sources of Equity Returns）

权益（equity）的价格收益率与总收益率: $price return = \frac{P _{1} - P _{0}}{P _{0}}$ $total return = \frac{P _{1} - P _{0} + D}{P _{0}}$
股息再投资: $r_{E} = \frac{P _{1} ( 1 + \frac{D}{P _{P D}} ) - P _{0}}{P _{0}}$
权益价值等于现值: $P_{0} = \frac{E ( P _{1} ) + E ( D _{1} )}{1 + r _{E}}$

第 43 章权益估值导论（Introduction to Equity Valuation）

账面价值、市值与企业价值: $book value = share capital + additional paid-in capital + retained earnings + accumulated other comprehensive income (and other reserves) - treasury stock$ $book value per share (BVPS) = \frac{book value of equity}{shares outstanding}$
账面价值的局限: $P/B = \frac{market price per share}{BVPS} = \frac{market capitalization}{book value of equity}$ $EV = market value of common and preferred stock + market value of debt - cash and short-term investments$

第 44 章折现现金流与增长模型（Discounted Cash Flow (DCF) and Growth Models）

现值模型的现金流指标与估值步骤: $V_{0} = t = 1 \sum n \frac{E ( C F _{t} )}{( 1 + r ) ^{t}} + \frac{E ( T V _{n} )}{( 1 + r ) ^{n}}$
股权自由现金流: $F C F E = net income + depreciation - W C I n v - C a pE x + net borrowing$ $F C F E = C F O - F C I n v + net borrowing$ $net investment = C a pE x - depreciation + W C I n v$ $F C F E = net income - net investment + net borrowing$ $net borrowing = D R \times net investment$ $F C F E = net income - net investment + D R \times net investment = net income - (1 - D R) (net investment)$
公司自由现金流: $F C F F = C F O - C a pE x + interest (1 - t)$ $F C F F = F C F E + interest (1 - t) - net borrowing$
EBITDA: $F C F F = E B I T D A (1 - t) + (depreciation \times t) - F C I n v - W C I n v$
剩余收益: $R I_{t} = E P S_{t} - r \times B V P S_{t - 1}$
固定增长模型，即戈登增长模型（Gordon growth model）: $V_{0} = \frac{D _{1}}{r - g} = \frac{D _{0} ( 1 + g )}{r - g}$ $g = r - \frac{D _{1}}{P _{0}}$
多阶段（两阶段）模型（multistage (two-stage) models）: $V_{0} = t = 1 \sum n \frac{D _{t}}{( 1 + r ) ^{t}} + \frac{V _{n}}{( 1 + r ) ^{n}}$ $V_{n} = \frac{D _{n + 1}}{r - g _{L}}$
估计固定增长率: $g = (1 - dividend payout ratio) \times R O E = b \times R O E$ $S G R_{F C F E} = (1 - \frac{F C F E}{net income}) (\frac{net income}{beginning book value})$ $S G R_{F C F F} = (1 - \frac{F C F F}{N O P A T}) (\frac{N O P A T}{beginning book value of total capital})$
加权平均资本成本: $W A C C = \frac{M V _{d}}{M V _{d} + M V _{e}} r_{d} (1 - t) + \frac{M V _{e}}{M V _{d} + M V _{e}} r_{e}$
不可赎回、不可转换的优先股: $V_{0} = t = 1 \sum N \frac{D _{p}}{( 1 + r _{p} ) ^{t}} + \frac{F V}{( 1 + r _{p} ) ^{N}}$ $V_{0} = \frac{D _{p}}{r _{p}}$

第 45 章相对价值法股票估值（Relative Value Equity Valuation Approaches）

预测基本面法: $\frac{P _{0}}{E _{1}} = \frac{D _{1} / E _{1}}{r - g} = \frac{payout ratio}{r - g}$ $\frac{P _{0}}{B _{0}} = \frac{R O E - g}{r - g}$
两种方法的调和: $P E G = \frac{P / E}{g} (g in whole percentage points)$ $P / B_{i} = b_{0} + b_{1} R O E_{i} + b_{2} g_{i}$
价值 = 现有资产 + 增长机会: $P_{0} = \frac{E _{1}}{r} + P V GO$ $\frac{P V GO}{P _{0}} = 1 - \frac{E _{1} / P _{0}}{r}$
市销率（P/S）: $\frac{P _{0}}{S _{0}} = \frac{P _{0}}{E _{0}} \times \frac{E _{0}}{S _{0}} = \frac{payout \times ( 1 + g )}{r - g} \times profit margin$
企业价值乘数: $E V = market value of common and preferred stock + market value of debt - cash and short-term investments$ $E V = \frac{F C F F _{1}}{W A C C - g}$ $F C F F = E B I T D A (1 - t) + Dep \times t - CapEx - WCInv$
资产负债表类乘数: $\frac{E V}{B V of capital} = \frac{R O C \times ( 1 - reinvestment rate )}{W A C C - g}$
现金流乘数与收益率: $\frac{D _{0}}{P _{0}} = \frac{D _{0}}{E _{0}} \times \frac{E _{0}}{P _{0}} (payout \times earnings yield)$ $\frac{D _{0}}{P _{0}} = \frac{r - g}{1 + g}$
使用预测的未来值: $T V_{n} = justified leading P/E_{n} \times E_{n + 1}$ $justified leading P/E_{n} = \frac{payout}{r - g}$

第 46 章权益估值中的财务报表预测（Financial Statement Forecasting in Equity Valuation）

收入建模: $g_{r e v e n u e} = g_{ma r k e t} + (\frac{new share}{old share} - 1)$
费用（利润）建模: $Δ Expense = b_{0} + b_{1} \times Δ Revenue$ $COGS = forecast revenue \times (1 - gross margin)$
融资建模: $financing needs = Δ operating assets - Δ non-debt liabilities - Δ retained earnings$
初创期：风险投资的计算: $ROI multiple = (1 + r)^{n} ⟹ r = (ROI multiple)^{1/ n} - 1$ $post-money value = \frac{expected revenue \times ( P / S )}{ROI multiple}$ $pre-money value = post-money value - VC investment$ $f = \frac{VC investment}{post-money value}$
成长期: $T V_{n} = N I_{n} \times P / E$ $V_{0} = \frac{T V _{n}}{( 1 + r ) ^{n}} or V_{0} = \frac{T V _{n}}{ROI multiple}$
衰退期: $value = P (going concern) \times going-concern value + P (distress) \times distress sale value$
企业价值与 EV 倍数: $EV = market value of common and preferred stock + market value of debt - cash and short-term investments$
根据预测模型的输出给权益估值: $S G R_{F C F F} = (1 - \frac{F C F F}{N O P A T}) (\frac{N O P A T}{beginning book value of total capital})$

第 47 章行业与竞争分析（Industry and Competitive Analysis）

盈利能力: $operating profit margin = \frac{operating income}{total revenue}$ $ROIC = \frac{NOPAT _{t}}{average invested capital}$
市场份额与集中度: $market share = \frac{company sales}{industry size}$ $H H I = i = 1 \sum N (market share_{i})^{2}$

第 48 章公司分析：过去、现在与未来（Company Analysis: Past, Present, and Future）

初创期和成长期公司的市场潜力: $Serviceable market = TAM \times technical fit (%) \times geographic reach (%)$
经营盈利能力: $EBITDA = revenue - COGS - operating expenses (excluding D&A)$ $EBIT = EBITDA - depreciation and amortization$ $Net income = EBIT - interest expense - taxes$
经营杠杆: $Operating profit = Q \times (P - V C) - F C$
股权自由现金流（free cash flow to equity, FCFE）: $FCFE = NI + D&A - capital expenditures - Δ working capital + net borrowing$
营运资本管理: $CCC = DOH + DSO - DPO$
资本投资: $Net capital expenditure = total capital expenditure - D&A$ $ROIC = \frac{NOPAT}{average invested capital}$ $invested capital \approx total long-term liabilities + total equity$
资本结构: $DFL = \frac{%Δ net income}{%Δ operating income}$

第 49 章权益分析师研究报告（Equity Analyst Research Reports）

两阶段 FCFE 估值: $V_{0} = \frac{F C F E _{1}}{1 + r} + \frac{F C F E _{2} / ( r - g )}{1 + r}$

第 50 章资本资产定价模型、市场模型和其他基于因子的股票模型（The Capital Asset Pricing Model, Market Model, and Other Factor-Based Equity Models）

CAPM 与证券市场线（security market line, SML）: $E (R_{i}) = R_{f} + β_{i} [E (R_{m}) - R_{f}]$
调整后贝塔：均值回归（mean reversion）: $adjusted beta = ω \times raw beta + (1 - ω) \times 1.0$ $ω commonly 2/3$
用可比公司估计贝塔：纯业务法（pure-play approach）: $β_{U} = \frac{β _{L, peers}}{1 + ( 1 - t _{peers} ) \frac{D}{E} _{peers}}$ $β_{L, subject} = β_{U} [1 + (1 - t_{subject}) \frac{D}{E}_{subject}]$
估计市场风险溢价: $E (R_{m}) = \frac{D _{1}}{P _{0}} + g$ $M R P = \frac{D _{1}}{P _{0}} + g - R_{f}$
新兴市场：国家风险溢价: $C R P = sovereign bond yield spread \times \frac{σ _{equity index}}{σ _{sovereign debt}}$ $E (R_{i}) = R_{f} + β_{i} (M R P + C R P)$
套利定价理论（APT）: $E (R_{i}) = R_{f} + β_{1 i} f_{1} + β_{2 i} f_{2} + \dots + β_{k i} f_{k}$
四因子模型（four-factor model）: $E (R_{i}) = R_{f} + β_{mk t} [E (R_{m}) - R_{f}] + β_{S M B} E (S M B) + β_{H M L} E (H M L) + β_{U M D} E (U M D)$

固定收益（Fixed Income）考试权重 11–14% 62 个公式 · 19 章中 11 章有公式

第 54 章公司发行人的固定收益市场（Fixed-Income Markets for Corporate Issuers）

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

第 56 章固定收益债券估值：价格与收益率（Fixed-Income Bond Valuation: Prices and Yields）

价格 = 承诺现金流的现值: $P = \frac{C}{( 1 + Y T M )} + \frac{C}{( 1 + Y T M ) ^{2}} + \dots + \frac{C + F V}{( 1 + Y T M ) ^{N}}$ $P = t = 1 \sum 2 N \frac{C /2}{( 1 + Y T M /2 ) ^{t}} + \frac{F V}{( 1 + Y T M /2 ) ^{2 N}}$
应计利息、净价和全价: $A I = \frac{t}{T} \times P M T$
矩阵定价法: $Y T M_{t a r g e t} = Y T M_{s h or t} + \frac{T _{t a r g e t} - T _{s h or t}}{T _{l o n g} - T _{s h or t}} (Y T M_{l o n g} - Y T M_{s h or t})$

第 57 章固定利率债券的收益率与收益率利差指标（Yield and Yield Spread Measures for Fixed-Rate Bonds）

年付息次数与有效年收益率: $E A Y = (1 + \frac{stated annual yield}{n})^{n} - 1$ $stated yield_{m} = m [(1 + E A Y)^{1/ m} - 1]$
当期收益率与简单收益率: $current yield = \frac{annual cash coupon}{flat price}$ $simple yield = \frac{annual coupon \pm straight-line amortization}{flat price}$
期权调整价格与期权调整收益率: $callable bond value = straight bond value - call option value$
Z-spread: $price = t = 1 \sum N \frac{C F _{t}}{( 1 + S _{t} + Z ) ^{t}}$
期权调整利差（OAS）: $OAS = Z-spread - option value (bp)$ $option value = Z-spread - OAS$

第 58 章浮动利率工具的收益率与收益率利差指标（Yield and Yield Spread Measures for Floating-Rate Instruments）

重置日的简化估值: $V = t = 1 \sum N \frac{( MRR + QM ) \times F V / m}{( 1 + \frac{MRR + DM}{m} ) ^{t}} + \frac{F V}{( 1 + \frac{MRR + DM}{m} ) ^{N}}$
关键公式：货币市场收益率: $Add-on yield (365-day) = H P Y \times \frac{365}{days}$ $Discount yield = \frac{F V - P}{F V} \times \frac{360}{days}$ $P = F V \times (1 - discount yield \times \frac{days}{year})$ $F V = P V \times (1 + add-on yield \times \frac{days}{year})$
与半年付息债券比较: $E A Y = (1 + H P Y)^{365/ days} - 1$ $semiannual bond basis = 2 [(1 + E A Y)^{1/2} - 1]$

第 59 章利率期限结构：即期利率曲线、平价收益率曲线与远期利率曲线（The Term Structure of Interest Rates: Spot, Par, and Forward Curves）

用即期利率定价: $P = \frac{P M T}{( 1 + S _{1} )} + \frac{P M T}{( 1 + S _{2} ) ^{2}} + \dots + \frac{P M T + F V}{( 1 + S _{N} ) ^{N}}$
平价利率: $100 = \frac{c}{1 + S _{1}} + \frac{c}{( 1 + S _{2} ) ^{2}} + \dots + \frac{c + 100}{( 1 + S _{N} ) ^{N}} \Rightarrow c = \frac{100 [ 1 - \frac{1}{( 1 + S _{N} ) ^{N}} ]}{\sum _{t = 1}^{N} \frac{1}{( 1 + S _{t} ) ^{t}}}$
远期利率及其表示法: $(1 + S_{3})^{3} = (1 + S_{1}) (1 +_{1 y 1 y}) (1 +_{2 y 1 y}) = (1 + S_{2})^{2} (1 +_{2 y 1 y})$ $S_{N} = [(1 + S_{1}) (1 +_{1 y 1 y}) \dots (1 +_{(N - 1) y 1 y})]^{1/ N} - 1$ $_{A y B y} = [\frac{( 1 + S _{A + B} ) ^{A + B}}{( 1 + S _{A} ) ^{A}}]^{1/ B} - 1$
用远期利率为债券估值: $P = \frac{P M T}{1 + S _{1}} + \frac{P M T}{( 1 + S _{1} ) ( 1 + _{1 y 1 y} )} + \frac{P M T + F V}{( 1 + S _{1} ) ( 1 + _{1 y 1 y} ) ( 1 + _{2 y 1 y} )}$

第 60 章利率风险与收益（Interest Rate Risk and Return）

投资期收益率: $Horizon yield = (\frac{sale price (or principal) + coupons + reinvestment income}{purchase price})^{1/ H} - 1$
持有期收益率（holding period return）、麦考利久期与投资期限: $Duration gap = Macaulay duration - investment horizon$
计算和解释麦考利久期: $MacDur = t = 1 \sum N t \times w_{t}$ $w_{t} = \frac{C F _{t} / ( 1 + r ) ^{t}}{full price}$ $t \sum w_{t} = 1$

第 61 章基于收益率的债券久期指标及其性质（Yield-Based Bond Duration Measures and Properties）

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

第 62 章基于收益率的债券凸性与组合特性（Yield-Based Bond Convexity and Portfolio Properties）

由现金流计算凸性: $convexity_{t} = \frac{t ( t + 1 )}{( 1 + r ) ^{2}}$
近似凸性: $approximate convexity = \frac{V _{-} + V _{+} - 2 V _{0}}{( Δ Y T M ) ^{2} V _{0}}$
用久期和凸性估计价格变化: $%Δ P V^{F u l l} \approx - AnnModDur \times Δ Y T M + \frac{1}{2} \times AnnConvexity \times (Δ Y T M)^{2}$
金额凸性: $MoneyCon = AnnConvexity \times P V^{F u l l} of the position$ $Δ P V^{F u l l} \approx - (MoneyDur \times Δ Y T M) + \frac{1}{2} \times MoneyCon \times (Δ Y T M)^{2}$

第 63 章基于曲线的固定收益风险度量与经验度量（Curve-Based and Empirical Fixed-Income Risk Measures）

含嵌入式期权（embedded options）的债券为什么用有效久期和有效凸性: $effective duration (EffDur) = \frac{V _{-} - V _{+}}{2 V _{0} Δ Curve}$ $effective convexity (EffCon) = \frac{V _{-} + V _{+} - 2 V _{0}}{( Δ Curve ) ^{2} V _{0}}$
基准收益率变动时的价格百分比变动: $%Δ P V^{F u l l} \approx - EffDur \times Δ Curve + \frac{1}{2} \times EffCon \times (Δ Curve)^{2}$
关键利率久期与形状风险: $%Δ V \approx - k \sum KeyRateDur_{k} \times Δ r_{k}$

第 64 章信用风险（Credit Risk）

信用风险的衡量: $expected loss = probability of default \times loss given default$ $LGD (in money) = expected exposure \times (1 - recovery rate)$ $LGD% (rate) = loss severity = 1 - recovery rate$ $credit spread \approx PD \times LGD% = PD \times (1 - recovery rate)$
市场因素：流动性: $%Δ P_{full} \approx - ModDur \times Δ Spread + \frac{1}{2} Convexity \times (Δ Spread)^{2}$

第 69 章抵押贷款支持证券（MBS）的工具与市场特征（Mortgage-Backed Security (MBS) Instrument and Market Features）

抵押贷款转手证券与 CMO: $WAC = i \sum w_{i} \times mortgage rate_{i}$ $WAM = i \sum w_{i} \times remaining term_{i}$ $pass-through rate = WAC - fees$
商业抵押贷款支持证券: $DSCR = \frac{net operating income}{debt service}$ $LTV = \frac{loan amount}{appraised value}$

衍生品（Derivatives）考试权重 6–9% 41 个公式 · 10 章中 10 章有公式

第 70 章衍生工具与衍生品市场的特征（Derivative Instrument and Derivative Market Features）

多头敞口与空头敞口: $Payoff to long at settlement = S_{T} - F$ $Payoff to short = F - S_{T}$

第 71 章远期承诺与或有索取权的特征及工具（Forward Commitment and Contingent Claim Features and Instruments）

期权多头、空头头寸到期时的价值和利润: $c_{T} = max (0, S_{T} - X)$ $p_{T} = max (0, X - S_{T})$ $Profit to buyer = value at expiration - premium paid$ $Profit to writer = premium received - value at expiration$

第 72 章衍生品的好处、风险以及发行人和投资者的运用（Derivative Benefits, Risks, and Issuer and Investor Uses）

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

第 73 章衍生品定价中的套利、复制与持有成本（Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives）

复制远期（标的资产没有持有成本和持有收益）: $F_{0} (T) = S_{0} (1 + R_{f})^{T}$
即期价格与预期未来价格，以及持有成本（cost of carry）: $F_{0} (T) = [S_{0} + P V_{0} (cost) - P V_{0} (benefit)] (1 + R_{f})^{T}$ $F_{0} (T) = S_{0} e^{r T}$ $F_{0} (T) = S_{0} e^{(r + c) T}$ $F_{0} (T) = S_{0} e^{(r + c - b) T}$
货币远期: $F = S_{0} \times (\frac{1 + R _{price}}{1 + R _{base}})^{T}$ $or, continuously compounded, F = S_{0} e^{(r_{price} - r_{base}) T}$

第 74 章远期合约的定价与估值及不同期限标的资产的远期定价（Pricing and Valuation of Forward Contracts and for an Underlying With Varying Maturities）

远期的价值与价格：签订时、存续期间、到期时: $V_{t} (T) = [S_{t} + P V_{t} (cost) - P V_{t} (benefit)] - F_{0} (T) (1 + R_{f})^{- (T - t)}$
隐含（无套利）远期利率: $(1 + Z_{A + B})^{A + B} = (1 + Z_{A})^{A} (1 + F_{A, B})^{B} so F_{A, B} = [\frac{( 1 + Z _{A + B} ) ^{A + B}}{( 1 + Z _{A} ) ^{A}}]^{1/ B} - 1$ $1 + Z_{long} \frac{d _{long}}{360} = (1 + Z_{short} \frac{d _{short}}{360}) (1 + F \frac{d _{long} - d _{short}}{360})$
远期利率协议（FRA）: $Payment to long = \frac{Notional \times ( MRR - FRA rate ) \times \frac{days}{360}}{1 + MRR \times \frac{days}{360}}$

第 75 章期货合约的定价与估值（Pricing and Valuation of Futures Contracts）

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

第 76 章利率互换及其他互换的定价与估值（Pricing and Valuation of Interest Rates and Other Swaps）

求平价互换利率: $n = 1 \sum N \frac{MRR _{n}}{( 1 + S _{n} ) ^{n}} = n = 1 \sum N \frac{F}{( 1 + S _{n} ) ^{n}}$ $F = \frac{1 - \frac{1}{( 1 + S _{N} ) ^{N}}}{n = 1 \sum N \frac{1}{( 1 + S _{n} ) ^{n}}}$
存续期间的价值: $Value to fixed-rate payer = P V (expected floating payments) - P V (remaining fixed payments)$

第 77 章期权的定价与估值（Pricing and Valuation of Options）

价值状态、行权价值和时间价值: $Call exercise value = max (0, S - X)$ $Put exercise value = max (0, X - S)$ $Option premium = exercise value + time value, so time value = premium - exercise value$

第 78 章用看跌—看涨期权平价复制期权（Option Replication Using Put-Call Parity）

看跌—看涨期权平价: $c + X (1 + R_{f})^{- T} = S + p$
看跌—看涨期权远期平价及期权在公司金融中的应用: $F_{0} (T) (1 + R_{f})^{- T} + p_{0} = c_{0} + X (1 + R_{f})^{- T}$ $p_{0} - c_{0} = [X - F_{0} (T)] (1 + R_{f})^{- T}$
用期权看公司（权益和债务）: $V_{0} = c_{0} + P V (D) - p_{0}$ $Equity_{T} = max (0, V_{T} - D)$ $Debt_{T} = min (V_{T}, D) = D - max (0, D - V_{T})$

第 79 章用单期二叉树模型为衍生品估值（Valuing a Derivative Using a One-Period Binomial Model）

看涨期权：买入 $h$ 股，卖出一份看涨期权: $h S_{1}^{u} - c_{1}^{u} = h S_{1}^{d} - c_{1}^{d} \Rightarrow h = \frac{c _{1}^{u} - c _{1}^{d}}{S _{1}^{u} - S _{1}^{d}}$ $c_{0} = h S_{0} - V_{0}$
看跌期权：买入 $h$ 股，买入一份看跌期权: $h = \frac{p _{1}^{d} - p _{1}^{u}}{S _{1}^{u} - S _{1}^{d}}$ $V_{0} = h S_{0} + p_{0} \Rightarrow p_{0} = V_{0} - h S_{0}$
风险中性: $π_{U} = \frac{1 + R _{f} - R ^{d}}{R ^{u} - R ^{d}}$ $π_{D} = 1 - π_{U}$ $c_{0} = \frac{π _{U} c _{1}^{u} + π _{D} c _{1}^{d}}{1 + R _{f}}$ $p_{0} = \frac{π _{U} p _{1}^{u} + π _{D} p _{1}^{d}}{1 + R _{f}}$

另类投资（Alternative Investments）考试权重 6–9% 11 个公式 · 7 章中 4 章有公式

第 80 章另类投资的特征、投资方式与结构（Alternative Investment Features, Methods, and Structures）

业绩报酬的计算: $Fee_{soft hurdle} = p \times R$ $Fee_{hard hurdle} = p \times (R - h)$ $full catch-up reached at R = \frac{h}{1 - p}$

第 81 章另类投资的业绩与收益率（Alternative Investment Performance and Returns）

生命周期与 J 曲线: $MOIC = \frac{total capital returned + value of remaining assets}{total capital paid in}$
杠杆: $leveraged return = \frac{r ( V _{0} + V _{B} ) - r _{B} V _{B}}{V _{0}}$ $leveraged return = r + \frac{V _{B}}{V _{0}} (r - r_{B})$
扣费后收益率的计算: $total fees = m V_{1} + max [0, p (V_{1} - V_{0})]$ $r_{after fees} = \frac{V _{1} - V _{0} - total fees}{V _{0}}$

第 84 章自然资源（Natural Resources）

大宗商品估值：持有成本（cost of carry）: $futures price \approx spot price \times (1 + r_{f}) + storage costs - convenience yield$

第 85 章对冲基金（Hedge Funds）

收益来源、指数偏差和分散化: $R_{fund} = broad market market beta + sector/strategy exposure strategy beta + manager skill α$ $C V = \frac{σ}{R ˉ}$

投资组合管理（Portfolio Construction）考试权重 8–12% 26 个公式 · 6 章中 3 章有公式

第 87 章投资组合的风险与收益（第一部分）（Portfolio Risk and Return: Part I）

效用函数: $U = E (r) - \frac{1}{2} A σ^{2}$
加入无风险资产：资本配置线: $E (R_{p}) = w E (R_{r i s k y}) + (1 - w) R_{f}$ $σ_{p} = w σ_{r i s k y}$
单一资产: $σ^{2} = \frac{\sum _{t = 1}^{T} ( R _{t} - μ ) ^{2}}{T} (population)$ $s^{2} = \frac{\sum _{t = 1}^{T} ( R _{t} - R ˉ ) ^{2}}{T - 1} (sample)$
两项资产：协方差（covariance）: $Cov_{1, 2} = \frac{\sum _{t = 1}^{T} ( R _{1, t} - R ˉ _{1} ) ( R _{2, t} - R ˉ _{2} )}{T - 1}$
相关性（correlation）: $ρ_{1, 2} = \frac{Cov _{1, 2}}{σ _{1} σ _{2}}$ $Cov_{1, 2} = ρ_{1, 2} σ_{1} σ_{2}$
投资组合标准差（portfolio standard deviation）: $E (R_{p}) = w_{1} E (R_{1}) + w_{2} E (R_{2})$ $σ_{p}^{2} = w_{1}^{2} σ_{1}^{2} + w_{2}^{2} σ_{2}^{2} + 2 w_{1} w_{2} Cov_{1, 2} = w_{1}^{2} σ_{1}^{2} + w_{2}^{2} σ_{2}^{2} + 2 w_{1} w_{2} ρ_{1, 2} σ_{1} σ_{2}$ $σ_{p} = σ_{p}^{2}$
投资于不完全相关的资产: $σ_{p}^{2} = (w_{1} σ_{1} + w_{2} σ_{2})^{2} \Rightarrow σ_{p} = w_{1} σ_{1} + w_{2} σ_{2}$ $w_{1} = \frac{σ _{2}}{σ _{1} + σ _{2}}$

第 88 章投资组合的风险与收益（第二部分）（Portfolio Risk and Return: Part II）

在风险投资组合中加入无风险资产: $E (R_{C}) = w E (R_{P}) + (1 - w) R_{f}$ $σ_{C} = w σ_{P}$
资本配置线（CAL）与资本市场线（CML）: $E (R_{P}) = R_{f} + [\frac{E ( R _{M} ) - R _{f}}{σ _{M}}] σ_{P}$
系统性风险与非系统性风险: $total risk = systematic risk + unsystematic risk$
收益生成模型与市场模型: $E (R_{i}) - R_{f} = β_{i, 1} E (F_{1}) + β_{i, 2} E (F_{2}) + \dots + β_{i, k} E (F_{k})$ $R_{i} = α_{i} + β_{i} R_{m} + e_{i}$
贝塔系数的计算与解释: $β_{i} = \frac{C o v ( R _{i} , R _{m} )}{σ _{m}^{2}} = \frac{ρ _{i, m} σ _{i} σ _{m}}{σ _{m}^{2}} = ρ_{i, m} \frac{σ _{i}}{σ _{m}}$
资本资产定价模型（capital asset pricing model, CAPM）、CAPM 的假设与证券市场线（security market line, SML）: $E (R_{i}) = R_{f} + β_{i} [E (R_{m}) - R_{f}]$ $β_{i} = \frac{C o v ( R _{i} , R _{m} )}{σ _{m}^{2}}$
CAPM 与 SML 的应用：找出定价错误的证券: $forecast return = \frac{P _{1} - P _{0} + D _{1}}{P _{0}}$ $required return = R_{f} + β [E (R_{m}) - R_{f}]$

第 89 章投资组合管理概述（Portfolio Management: An Overview）

衡量分散化收益（diversification benefit）：分散化比率: $Diversification ratio = \frac{σ of an equally weighted portfolio of the n securities}{average σ of the n individual securities}$
共同基金和其他集合投资工具: $N A V = \frac{fund assets - fund liabilities}{shares outstanding}$

道德与职业标准（Ethical and Professional Standards）考试权重 10–15% 无公式

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