6. Linear Factor Models
本章把 SDF 限定为因子的线性函数 \(m=a+\mathbf b'\tilde{\mathbf f}\)，它等价于收益的 Beta 定价表示 \(\mathbb E[R^i]=R^f+\boldsymbol\beta_i'\boldsymbol\lambda\)。三条主线：(i) 套利定价理论 (APT, Ross 1976)——纯粹由无套利（加因子结构）推出近似线性定价，是经验性表示而非经济模型；在精确因子定价下 SDF 严格线性，并给出风险溢价向量 \(\boldsymbol\lambda=-R^f\mathbf p(\tilde{\mathbf f})\)。(ii) CAPM 作为单因子模型——二次效用、CARA+正态、对数效用三种偏好都把问题化为均值方差，从而市场组合是唯一定价因子。(iii) 跨期 CAPM (ICAPM, Merton 1973)——连续时间动态最优化中，最优组合 = 均值方差（短视）需求 + 对冲需求 (hedging demand)；当投资机会集随状态变量变动时，除市场 beta 外还出现对冲组合 beta，得到多因子的 ICAPM 定价式。
This chapter restricts the SDF to a linear function of factors \(m=a+\mathbf b'\tilde{\mathbf f}\), equivalent to the Beta pricing representation of returns \(\mathbb E[R^i]=R^f+\boldsymbol\beta_i'\boldsymbol\lambda\). Three threads: (i) Arbitrage Pricing Theory (APT, Ross 1976) — approximate linear pricing derived from no-arbitrage plus a factor structure; an empirical representation, not an economic model. Under exact factor pricing the SDF is exactly linear, with risk-premium vector \(\boldsymbol\lambda=-R^f\mathbf p(\tilde{\mathbf f})\). (ii) CAPM as a single-factor model — quadratic utility, CARA with normal growth, and log utility each reduce the problem to mean-variance, so the market portfolio is the unique pricing factor. (iii) Intertemporal CAPM (ICAPM, Merton 1973) — in continuous-time dynamic optimization the optimal portfolio = mean-variance (myopic) demand + hedging demand; when the investment opportunity set moves with a state variable, a hedging-portfolio beta joins the market beta, giving a multi-factor ICAPM pricing equation.
若 SDF 是因子向量 \(\tilde{\mathbf f}\) 的线性函数 \(m=a+\mathbf b'\tilde{\mathbf f}\)（\(a,b\in\mathbb R\)），我们就称之为线性因子模型。它与下式等价：
If the SDF is a linear function of the factor vector \(\tilde{\mathbf f}\), \(m=a+\mathbf b'\tilde{\mathbf f}\) (with \(a,b\in\mathbb R\)), we call it a linear factor model. It is equivalent to
$$m=a+\mathbf b'\tilde{\mathbf f}\quad\Leftrightarrow\quad R^i=\mathbb E[R^i]+\boldsymbol\beta_i'\tilde{\mathbf f}.$$
6.1 Arbitrage Pricing Theory
套利定价理论 (APT) 由 Ross (1976) 提出。实证上，把股票按规模、价值等特征 (characteristics) 分组，组内股票往往同涨同跌；但单只股票仍有自身的特质 (idiosyncratic) 成分。APT 正是由这些经验事实出发。
Arbitrage Pricing Theory (APT) is due to Ross (1976). Empirically, stocks sorted by characteristics such as size and value tend to move together within a group, yet each stock keeps its own idiosyncratic component. APT is motivated by these empirical facts.
6.1.1 The Theorem
定理 6.1（APT）。 设 \(\mathcal X_R\) 是一组资产总收益，\(\tilde{\mathbf f}\) 是 \(N\) 个去均值因子 (de-meaned factors) 的集合，\(\boldsymbol\beta_i\) 是收益 \(R^i\) 的因子载荷向量。若每个 \(R^i\in\mathcal X_R\) 都由线性因子模型生成，
Theorem 6.1 (APT). Let \(\mathcal X_R\) be a collection of asset gross returns, \(\tilde{\mathbf f}\) a collection of \(N\) de-meaned factors, and \(\boldsymbol\beta_i\) the vector of factor loadings for return \(R^i\). If every \(R^i\in\mathcal X_R\) is generated by a linear factor model,
$$R^i=\mathbb E[R^i]+\boldsymbol\beta_i'\tilde{\mathbf f}+\varepsilon^i,\qquad \mathbb E[\varepsilon^i]=\mathbb E[\varepsilon^i\tilde f_j]=0\ \ \forall i,j,$$
则在一定假设下，存在一个线性于因子的贴现因子 \(m=a+\mathbf b'\tilde{\mathbf f}\) 为所有 \(R^i\in\mathcal X_R\) 定价。
then under certain assumptions there is a discount factor \(m=a+\mathbf b'\tilde{\mathbf f}\), linear in the factors, that prices all \(R^i\in\mathcal X_R\).
6.1.2 Exact Factor Pricing
先看无特质风险的理想世界，即每个总收益没有误差项：
Start with the ideal world of no idiosyncratic risk, i.e. no error term in each gross return:
$$R^i=\mathbb E[R^i]+\boldsymbol\beta_i'\tilde{\mathbf f}.\tag{6.1}$$
由 (6.1) 直接解出载荷 \(\boldsymbol\beta_i\)——正是把 \(R^i\) 对因子做 OLS 回归的系数：
From (6.1) the loading \(\boldsymbol\beta_i\) solves directly — it is exactly the OLS regression coefficient of \(R^i\) on the factors:
$$\boldsymbol\beta_i=\Big(\mathrm{Var}\big(\tilde{\mathbf f}\big)\Big)^{-1}\mathrm{Cov}\big(R^i,\tilde{\mathbf f}\big).\tag{6.2}$$
把常数并入因子，记增广因子 \(\bar{\mathbf f}=(1,\tilde{\mathbf f})'\)，价格向量 \(\mathbf p(\tilde{\mathbf f})\)。
命题 6.1。 设 \(f^*\) 为支付空间 \(\mathcal X\) 内的 SDF，则 \(f^*\) 线性于 \(\bar{\mathbf f}\)，给定 (6.3)。
Fold the constant into the factors: let the augmented factor be \(\bar{\mathbf f}=(1,\tilde{\mathbf f})'\) with price vector \(\mathbf p(\tilde{\mathbf f})\).
Proposition 6.1. Let \(f^*\) be the SDF in \(\mathcal X\). Then \(f^*\) is linear in \(\bar{\mathbf f}\), given by (6.3).
证明 / Proof：\(f^*=\mathbb E[\hat{\mathbf f}\hat{\mathbf f}']^{-1}\mathbf p(\hat{\mathbf f})'\hat{\mathbf f}=a+\mathbf b'\tilde{\mathbf f}\)
要 \(f^*=\mathbf c^{*\prime}\hat{\mathbf f}\) 为因子 \(\hat{\mathbf f}\) 定价：\(\mathbf p(\hat{\mathbf f})=\mathbb E[f^*\hat{\mathbf f}]=\mathbb E[\hat{\mathbf f}\hat{\mathbf f}']\mathbf c^*\)，于是
Require \(f^*=\mathbf c^{*\prime}\hat{\mathbf f}\) to price the factors \(\hat{\mathbf f}\): \(\mathbf p(\hat{\mathbf f})=\mathbb E[f^*\hat{\mathbf f}]=\mathbb E[\hat{\mathbf f}\hat{\mathbf f}']\mathbf c^*\), hence
$$\mathbf c^*=\mathbb E\big[\hat{\mathbf f}\hat{\mathbf f}'\big]^{-1}\mathbf p\big(\hat{\mathbf f}\big),\qquad f^*=\mathbb E\big[\hat{\mathbf f}\hat{\mathbf f}'\big]^{-1}\mathbf p\big(\hat{\mathbf f}\big)'\hat{\mathbf f}.\tag{6.3}$$
这显式给出 \(f^*\) 的存在，且它线性于 \(\tilde{\mathbf f}\)：
This gives the existence of \(f^*\) explicitly, and it is linear in \(\tilde{\mathbf f}\):
$$f^*=\underbrace{\mathbb E\big[\hat{\mathbf f}\hat{\mathbf f}'\big]^{-1}\mathbf p\big(\hat{\mathbf f}\big)'\hat{\mathbf f}}_{\equiv(a,\mathbf b')}=a+\mathbf b'\tilde{\mathbf f}.\quad\blacksquare\tag{6.4}$$
接着由一价定律为任意 \(R^i\in\mathcal X_R\subset\mathcal X\) 定价。从 (6.1) 出发 (6.5)–(6.9)：
Then price any \(R^i\in\mathcal X_R\subset\mathcal X\) by the Law of One Price. Starting from (6.1), (6.5)–(6.9):
$$p(R^i)=\mathbb E[R^i]\,\underbrace{p(1)}_{=1/R^f}+\boldsymbol\beta_i'\mathbf p(\tilde{\mathbf f})\ \Rightarrow\ 1=\mathbb E[R^i]\frac{1}{R^f}+\boldsymbol\beta_i'\mathbf p(\tilde{\mathbf f}).\tag{6.5–6.6}$$
$$\mathbb E[R^i]=R^f+\boldsymbol\beta_i'\underbrace{\Big(-R^f\,\mathbf p(\tilde{\mathbf f})\Big)}_{\equiv\boldsymbol\lambda}=R^f+\boldsymbol\beta_i'\boldsymbol\lambda.\tag{6.7–6.9}$$
风险溢价向量 \(\boldsymbol\lambda=-R^f\mathbf p(\tilde{\mathbf f})\) (6.10)。由于 \(\tilde{\mathbf f}\) 去均值，\(p(\tilde f_j)\) 只反映方差风险（若因子均值非零，\(\mathbb E[\tilde f_j]=0\)）。
命题 6.2。 (6.4)、(6.1)、(6.9) 等价，即
The risk-premium vector is \(\boldsymbol\lambda=-R^f\mathbf p(\tilde{\mathbf f})\) (6.10). Since \(\tilde{\mathbf f}\) is de-meaned, \(p(\tilde f_j)\) only reflects variance risk (the factors have mean zero, \(\mathbb E[\tilde f_j]=0\)).
Proposition 6.2. (6.4), (6.1) and (6.9) are equivalent, i.e.
$$f^*=a+\mathbf b'\tilde{\mathbf f}\quad\Leftrightarrow\quad R^i=\mathbb E[R^i]+\boldsymbol\beta_i'\tilde{\mathbf f}\quad\Leftrightarrow\quad \mathbb E[R^i]=R^f+\boldsymbol\beta_i'\boldsymbol\lambda.\tag{6.11}$$
证明 / Proof：\(f^*=a+\mathbf b'\tilde{\mathbf f}\ \Rightarrow\ R^i=\mathbb E[R^i]+\boldsymbol\beta_i'\tilde{\mathbf f}+\varepsilon^i\)
一个方向由命题 6.1 立得。另一方向：\(f^*\in\mathcal X\)，由定理 5.2 三分解 \(R^i=R^*+\omega^i R^{e*}+\eta^i\)，其中 \(R^*=f^*/\mathbb E[(f^*)^2]\) (5.32)。代入整理 (6.12)，并验证残差 \(\mathbb E[\varepsilon^i]=0\)、\(\mathbb E[\eta^i]=0\) (6.13)：
One direction is Proposition 6.1. For the other: \(f^*\in\mathcal X\), so by the Theorem 5.2 decomposition \(R^i=R^*+\omega^i R^{e*}+\eta^i\) with \(R^*=f^*/\mathbb E[(f^*)^2]\) (5.32). Substituting and simplifying (6.12), and checking \(\mathbb E[\varepsilon^i]=0\), \(\mathbb E[\eta^i]=0\) (6.13):
$$R^i=\mathbb E[R^i]+\mathbf k_i'\tilde{\mathbf f}+\varepsilon^i,\qquad \mathbf k_i=\frac{1}{\mathbb E[(f^*)^2]}\mathbf b.\tag{6.13}$$
但 \(\mathbf k_i'\tilde{\mathbf f}\) 中的因子与残差 \(\varepsilon^i\) 可能相关 \(\mathrm{Cov}(\tilde f_k,\varepsilon^i)=\mathbb E[\tilde f_k(R^i-R^*)]\neq0\)。调整残差使其与 \(\tilde{\mathbf f}\) 不相关：
But the factors in \(\mathbf k_i'\tilde{\mathbf f}\) may correlate with the residual \(\varepsilon^i\), \(\mathrm{Cov}(\tilde f_k,\varepsilon^i)=\mathbb E[\tilde f_k(R^i-R^*)]\neq0\). Adjust the residual to be uncorrelated with \(\tilde{\mathbf f}\):
$$\varepsilon^i=\epsilon^i-\big(\mathrm{Cov}(\epsilon^i,\tilde{\mathbf f})\big)'\mathbb E[\tilde{\mathbf f}\tilde{\mathbf f}']^{-1}\tilde{\mathbf f},$$
则 \(\mathbb E[\varepsilon^i]=0\) 且 \(\mathrm{Cov}(\varepsilon^i,\tilde{\mathbf f})=\mathbf 0\)，把调整项并入载荷得 \(\boldsymbol\beta_i\)，于是 \(R^i=\mathbb E[R^i]+\boldsymbol\beta_i'\tilde{\mathbf f}+\varepsilon^i\)，\(\varepsilon^i\) 为特质波动。精确因子定价下 \(\varepsilon^i=0\)。\(\blacksquare\)
Then \(\mathbb E[\varepsilon^i]=0\) and \(\mathrm{Cov}(\varepsilon^i,\tilde{\mathbf f})=\mathbf 0\); folding the adjustment into the loading gives \(\boldsymbol\beta_i\), so \(R^i=\mathbb E[R^i]+\boldsymbol\beta_i'\tilde{\mathbf f}+\varepsilon^i\) with \(\varepsilon^i\) the idiosyncratic variation. Under exact factor pricing \(\varepsilon^i=0\). \(\blacksquare\)
关于命题 6.2 的几点观察：
- (6.1) 与 (6.9) 给出 \(R^i=R^f+\boldsymbol\beta_i'(\boldsymbol\lambda+\tilde{\mathbf f})\) (6.14)。
- (6.14) 说明 \(\boldsymbol\lambda+\tilde{\mathbf f}\) 的每个分量都落在 (5.33) 定义的超额收益空间 \(\mathcal Z\) 中：\(p(\lambda_j+\tilde f_j)=0\)。
- 估计上分两步：先时序回归用风险 \(\boldsymbol\beta_i\)，再截面回归 \(\mathbb E[R^i]\) 对 \(\boldsymbol\beta_i\) 估计 \(\boldsymbol\lambda\)。
A few observations on Proposition 6.2:
- (6.1) and (6.9) give \(R^i=R^f+\boldsymbol\beta_i'(\boldsymbol\lambda+\tilde{\mathbf f})\) (6.14).
- (6.14) implies each component of \(\boldsymbol\lambda+\tilde{\mathbf f}\) lies in the excess-return space \(\mathcal Z\) of (5.33): \(p(\lambda_j+\tilde f_j)=0\).
- Estimation is two-step: first a time-series regression for the risk loadings \(\boldsymbol\beta_i\), then a cross-sectional regression of \(\mathbb E[R^i]\) on \(\boldsymbol\beta_i\) to estimate \(\boldsymbol\lambda\).
6.1.3 Approximate APT
现实中存在特质风险。APT 的力量在于：充分分散 (well-diversified) 的组合会把特质风险消掉。
命题 6.3（充分分散）。 设 \(R^p\) 是 \(N\) 个资产的等权组合，特质风险互不相关 \(\mathbb E[\varepsilon^i\varepsilon^j]=0\ (i\neq j)\)，则当 \(N\to\infty\) 时 \(\varepsilon^p\to0\)。
In reality idiosyncratic risk is present. The power of APT: a well-diversified portfolio diversifies idiosyncratic risk away.
Proposition 6.3 (well-diversification). Let \(R^p\) be the equally weighted portfolio of \(N\) assets with uncorrelated idiosyncratic risks \(\mathbb E[\varepsilon^i\varepsilon^j]=0\ (i\neq j)\). Then \(\varepsilon^p\to0\) as \(N\to\infty\).
证明 / Proof：\(\lim_{N\to\infty}\mathrm{Var}(\varepsilon^p)=0\)
设各特质方差有界 \(\mathrm{Var}(\varepsilon^i)\le M^2\)。等权组合的特质项 \(\varepsilon^p=\frac1N\sum_{i=1}^N\varepsilon^i\)：
Assume each idiosyncratic variance is bounded, \(\mathrm{Var}(\varepsilon^i)\le M^2\). The portfolio's idiosyncratic term is \(\varepsilon^p=\frac1N\sum_{i=1}^N\varepsilon^i\):
$$\mathrm{Var}(\varepsilon^p)=\frac{1}{N^2}\sum_{i=1}^N\mathrm{Var}(\varepsilon^i)\le\frac{1}{N^2}NM^2=\frac{M^2}{N}\ \xrightarrow{N\to\infty}\ 0.\quad\blacksquare$$
当组合的特质冲击消失时，称该组合充分分散。再由一价定律，\(p(R^i)=\mathbb E[R^i]p(1)+\boldsymbol\beta_i'\mathbf p(\tilde{\mathbf f})+p(\varepsilon^i)\)。
命题 6.4。 当 \(\varepsilon^i\to0\) 时 \(p(\varepsilon^i)\to0\)。
故对充分分散的组合，APT 定价近似成立，特质风险定价约等于零。
Remark 6.1。 APT 是一个经验性表示而非经济上自洽的模型：因子由数据驱动、缺乏经济基础。它告诉我们若组合充分分散则可由 SDF 定价，但不解释因子为何存在、风险溢价从何而来。
When the portfolio's idiosyncratic shock vanishes, the portfolio is well-diversified. By the Law of One Price, \(p(R^i)=\mathbb E[R^i]p(1)+\boldsymbol\beta_i'\mathbf p(\tilde{\mathbf f})+p(\varepsilon^i)\).
Proposition 6.4. As \(\varepsilon^i\to0\), \(p(\varepsilon^i)\to0\).
So APT pricing holds approximately for well-diversified portfolios, with idiosyncratic risk priced near zero.
Remark 6.1. APT is an empirical representation, not an economically grounded model: the factors are data-driven and lack economic foundations. It says a well-diversified portfolio can be priced by an SDF, but does not explain why the factors exist or where the risk premia come from.
6.2 CAPM as a Linear Factor Model
CAPM 可写成以市场组合为唯一因子的单因子模型。其经济基础是：在若干偏好假设下，投资者的问题退化为均值方差最优化，于是市场（财富）组合就是定价因子。设市场（总财富）组合总收益为 \(R^m\)，则消费 \(c_t=R^m(w-c_0)\) (6.15)，SDF 与 \(R^m\) 线性相关 \(R^*=\alpha+\beta R^m\)。
CAPM can be written as a one-factor model with the market portfolio as the sole factor. Its economic basis: under several preference assumptions the investor's problem reduces to mean-variance optimization, so the market (wealth) portfolio is the pricing factor. With market (total-wealth) gross return \(R^m\), consumption is \(c_t=R^m(w-c_0)\) (6.15), and the SDF is linear in \(R^m\), \(R^*=\alpha+\beta R^m\).
6.2.1 Quadratic Utility
设两期、\(t=0,1\)，投资者用二次效用 \(u(c)=c-\frac b2 c^2\ (b>0)\)，求解
Two periods \(t=0,1\); investor with quadratic utility \(u(c)=c-\frac b2 c^2\ (b>0)\) solves
$$\max_{c_0,c_1}\ u(c_0)+\beta\,\mathbb E[u(c_1)]\tag{6.16}$$
约束条件 (subject to)：
subject to:
$$c_0\le e_0-m,\tag{6.17}\qquad$$
$$c_1\le c_2+m\tilde R,\tag{6.18}$$
其中 \(m\) 是投入风险组合的金额，\(\tilde R\) 是其随机总收益。目标函数 (6.19) 关于 \(\mathbb E[\tilde R]\) 递增、关于 \(\mathrm{Var}(\tilde R)\) 递减——投资者只关心均值与方差，故 CAPM 成立。
二次效用有饱和点：当 \(c>1/b\) 时边际效用变负。它只在 \(\tilde R\) 取合理值时才有意义，且隐含的绝对风险厌恶随财富上升（不合现实），但作为推出均值方差的最简设定仍很有用。
where \(m\) is the amount invested in the risky portfolio and \(\tilde R\) its random gross return. The objective (6.19) increases in \(\mathbb E[\tilde R]\) and decreases in \(\mathrm{Var}(\tilde R)\) — the investor cares only about mean and variance, so CAPM holds.
Quadratic utility has a bliss point: marginal utility turns negative for \(c>1/b\). It is meaningful only for reasonable values of \(\tilde R\), and implies absolute risk aversion rising in wealth (unrealistic) — but as the simplest setup delivering mean-variance it is still useful.
6.2.2 CARA Utility with Normal Consumption Growth
设 CARA 效用 \(u(c)=\frac{1-e^{-ac}}{a}\)，且消费增长服从正态 \(\frac{c_2}{c_0}\sim\mathcal N(\mu,\sigma^2)\)。投资者求解
With CARA utility \(u(c)=\frac{1-e^{-ac}}{a}\) and normal consumption growth \(\frac{c_2}{c_0}\sim\mathcal N(\mu,\sigma^2)\), the investor solves
$$\max_{c_0}\ \frac{1-e^{-ac_0}}{a}+\beta\,\mathbb E\!\left[\frac{1-e^{-ac_1}}{a}\right].$$
正态下 \(\mathbb E[e^{-ac_1}]\) 化为 \(e^{-\mu a+\frac12 a^2\sigma^2}\)，目标 (6.20) 关于 \(\mu\) 递增、关于 \(\sigma^2\) 递减——同样退化为均值方差，CAPM 成立。
Under normality \(\mathbb E[e^{-ac_1}]\) becomes \(e^{-\mu a+\frac12 a^2\sigma^2}\), so the objective (6.20) increases in \(\mu\) and decreases in \(\sigma^2\) — again reducing to mean-variance, so CAPM holds.
6.2.3 Log Utility
设对数效用在无穷期可加可分 \(u(\{c_t\}_{t=0}^\infty)=\sum_{j=0}^\infty\beta^j\ln c_j\)。投资者求解 \(\max u\) s.t. \(\sum_{j=0}^\infty\beta^j p_j c_j\le w\)。Lagrangian \(\mathcal L=\sum\beta^j\ln c_j+\sum\lambda_j(w-\sum\beta^j p_j c_j)\)，一阶条件 (6.21)、(6.22)：
Log utility, additively separable over infinite periods, \(u(\{c_t\}_{t=0}^\infty)=\sum_{j=0}^\infty\beta^j\ln c_j\). The investor solves \(\max u\) s.t. \(\sum_{j=0}^\infty\beta^j p_j c_j\le w\). With Lagrangian \(\mathcal L=\sum\beta^j\ln c_j+\sum\lambda_j(w-\sum\beta^j p_j c_j)\), the FOCs (6.21), (6.22):
$$\frac{1}{c_j}=\lambda_j p_j,\tag{6.21}$$
$$\frac{1}{c_{j+1}}=\lambda_{j+1}p_{j+1}.\tag{6.22}$$
由 (1.7)，\(m=\dfrac{\beta u'(c_{t+1})}{u'(c_t)}=\beta\dfrac{c_j}{c_{j+1}}\)。定义财富组合总收益 \(R^w=\dfrac{c_{j+1}}{c_j}\)（合理，因总量上 \(c_t=D_t\)，总消费等于总红利）。市场出清下红利全被消费，故 SDF 线性于 \(R^w\)：
By (1.7), \(m=\dfrac{\beta u'(c_{t+1})}{u'(c_t)}=\beta\dfrac{c_j}{c_{j+1}}\). Define the wealth-portfolio gross return \(R^w=\dfrac{c_{j+1}}{c_j}\) (sensible since at the aggregate level \(c_t=D_t\), aggregate consumption equals aggregate dividends). Market clearing has all dividends consumed, so the SDF is linear in \(R^w\):
$$m=\beta\frac{1}{R^w}=\beta\cdot 1-\frac{1}{1^2}(R^w-1)+o(R^w-1)\approx 1+\beta-R^w.$$
对数效用的短视性 (myopia)。 对数投资者把财富的固定比例投入风险资产，与预期收益无关——故消费/储蓄决策与投资决策可分离，市场组合即定价因子。
Myopia of log utility. A log investor puts a fixed fraction of wealth into risky assets regardless of expected returns — so the consumption/saving decision separates from the investment decision, and the market portfolio is the pricing factor.
6.3 Intertemporal CAPM (ICAPM): Merton (1973)
Merton (1973) 用连续时间动态规划研究 SDF。核心新意：引入状态变量 (state variable) \(x(t)\) 刻画投资机会集的变动，得到比静态 CAPM 更丰富的结构——除均值方差需求外还出现对冲需求。SDF 形如 \(\frac{dZ_x}{}\)：\(m=\mu dt+s\,dZ_x(t)\)。
Merton (1973) studies the SDF in continuous-time dynamic programming. The key novelty: a state variable \(x(t)\) captures shifts in the investment opportunity set, yielding a richer structure than the static CAPM — a hedging demand appears alongside the mean-variance demand. The SDF takes the form \(m=\mu\,dt+s\,dZ_x(t)\).
6.3.1 The General Maximization Problem
投资者最大化生命期效用加遗赠：
The investor maximizes lifetime utility plus a bequest:
$$\max\ \mathbb E_0\!\left[\int_0^T e^{-\rho t}u\big(c(t)\big)\,dt+e^{-\rho T}B\big(W(T)\big)\right]\tag{6.23}$$
其中 \(u'>0,u''<0\)，\(T\) 为终止时刻，\(B\) 为遗赠函数。设状态向量 \(\mathbf x(t)\)、\(n+1\) 个资产，价值函数 \(J(W(t),\mathbf x(t),t)\) (6.24)。资产价格服从
with \(u'>0,u''<0\), termination \(T\), bequest \(B\). With state vector \(\mathbf x(t)\), \(n+1\) assets, and value function \(J(W(t),\mathbf x(t),t)\) (6.24), asset prices follow
$$\frac{dP_i(t)}{P_i(t)}=\alpha_i\,dt+\sigma_i\,dZ_i(t),\qquad i=0,1,\dots,n.\tag{6.25}$$
\(\mathrm{Cov}(dP_i/P_i,dP_j/P_j)=\rho_{ij}\sigma_i\sigma_j dt\)。取标量状态 \(x(t)\)：\(dx=\mu\,dt+s\,dZ_x\)，\(\mathrm{Cov}(dP_i/P_i,dx)=\rho_{ix}\sigma_i s\,dt=\sigma_{ix}dt\)。财富演化 (6.26) 由预算约束得到。代入 Bellman 方程并用 Itô 引理展开 (6.27)，得目标 (6.28)，组合权重满足 \(\sum_{i=0}^n\omega_i=1\) (6.29)。一阶条件：
\(\mathrm{Cov}(dP_i/P_i,dP_j/P_j)=\rho_{ij}\sigma_i\sigma_j dt\). For a scalar state \(x(t)\): \(dx=\mu\,dt+s\,dZ_x\), \(\mathrm{Cov}(dP_i/P_i,dx)=\rho_{ix}\sigma_i s\,dt=\sigma_{ix}dt\). Wealth evolution (6.26) follows from the budget constraint. Plugging into the Bellman equation and expanding by Itô's lemma (6.27) gives the objective (6.28), with portfolio weights \(\sum_{i=0}^n\omega_i=1\) (6.29). The FOCs:
$$e^{-\rho t}u'\big(c(t)\big)=J_W,\tag{6.31}$$
$$(\alpha_i-r)J_W W(t)+J_{WW}W^2(t)\sum_{j=1}^n\omega_j(t)\sigma_{ij}+J_{Wx}W(t)\sigma_{ix}=0.\tag{6.32}$$
把 (6.32) 对 \(i=1,\dots,n\) 合并为矩阵形式并解出最优权重：
Combining (6.32) for \(i=1,\dots,n\) into matrix form and solving for the optimal weights:
$$\boldsymbol\omega^*=-\Sigma^{-1}(\boldsymbol\alpha-r\mathbf 1)\,\frac{J_W}{J_{WW}W(t)}-\Sigma^{-1}\boldsymbol\sigma_x\,\frac{J_{Wx}}{J_{WW}W(t)}.\tag{6.33}$$
$$\omega_i^*=-\sum_{j=1}^n\nu_{ij}(\alpha_j-r)\frac{J_W}{J_{WW}W(t)}-\sum_{j=1}^n\nu_{ij}\sigma_{jx}\frac{J_{Wx}}{J_{WW}W(t)}.\tag{6.34}$$
(6.33) 的两项分别是均值方差（短视）需求与对冲需求。
Remark 6.2。 \(u''<0\)（风险厌恶）保证第一项方向正确（\(-J_W/J_{WW}W>0\)）。若 \(u''=0\)（风险中性）投资者只看期望，需求退化。
The two terms of (6.33) are the mean-variance (myopic) demand and the hedging demand.
Remark 6.2. \(u''<0\) (risk aversion) makes the first term well-signed (\(-J_W/J_{WW}W>0\)). If \(u''=0\) (risk neutrality) the investor cares only about means and the demand degenerates.
6.3.2 Special Case: Constant Investment Opportunity Set
当投资机会集恒定（\(x(t)\) 与收益无关）时无对冲需求，CAPM 作为线性因子表示成立。设单一风险资产 \(\frac{dP}{P}=\alpha\,dt+\sigma\,dZ\)，CRRA 效用，猜测价值函数
When the investment opportunity set is constant (\(x(t)\) independent of returns) there is no hedging demand, and CAPM holds as a linear factor representation. With a single risky asset \(\frac{dP}{P}=\alpha\,dt+\sigma\,dZ\), CRRA utility, guess the value function
$$J(W(t),t)=h^{-\gamma}e^{-\rho t}\frac{W(t)^{1-\gamma}}{1-\gamma},\qquad \omega^*(t)=\frac{\alpha-r}{\gamma\sigma^2},\quad c^*(t)=hW(t),\tag{6.36}$$
$$h=\frac{\rho}{\gamma}-\frac{1-\gamma}{\gamma}\left[r+\frac12\frac{(\alpha-r)^2}{\gamma\sigma^2}\right].$$
代入一阶条件 (6.37) \(e^{-\rho t}u'(c)=J_W\) 与 (6.38) \((\alpha-r)J_W W+J_{WW}W^2\omega\sigma^2=0\) 验证，均平凡成立——猜测正确。最优风险权重 \(\omega^*=\frac{\alpha-r}{\gamma\sigma^2}\) 不含对冲项。
Substituting into the FOCs (6.37) \(e^{-\rho t}u'(c)=J_W\) and (6.38) \((\alpha-r)J_W W+J_{WW}W^2\omega\sigma^2=0\) verifies them — both hold trivially, so the guess is correct. The optimal risky weight \(\omega^*=\frac{\alpha-r}{\gamma\sigma^2}\) has no hedging term.
6.3.3 General Case: Stochastic Investment Opportunity Set
当投资机会集随 \(x(t)\) 变动时，出现两基金分离：最优组合是均值方差组合与对冲组合的组合。由 (6.33)，
When the opportunity set moves with \(x(t)\), a two-fund separation appears: the optimal portfolio combines a mean-variance fund and a hedging fund. From (6.33),
$$\boldsymbol\omega^*=-\Sigma^{-1}(\boldsymbol\alpha-r\mathbf 1)\frac{J_W}{J_{WW}W(t)}-\Sigma^{-1}\boldsymbol\sigma_x\frac{J_{Wx}}{J_{WW}W(t)}.\tag{6.39}$$
重写为切点组合 \(\boldsymbol\omega_m\) 与对冲组合 \(\boldsymbol\omega_x\) 的加权 (6.42)：
Rewrite as a weighting of the tangency fund \(\boldsymbol\omega_m\) and the hedging fund \(\boldsymbol\omega_x\) (6.42):
$$\boldsymbol\omega_m=\frac{\Sigma^{-1}(\boldsymbol\alpha-r\mathbf 1)}{\mathbf 1'\Sigma^{-1}(\boldsymbol\alpha-r\mathbf 1)},\qquad \boldsymbol\omega_x=\frac{\Sigma^{-1}\boldsymbol\sigma_x}{\mathbf 1'\Sigma^{-1}\boldsymbol\sigma_x}.\tag{6.42}$$
- \(\boldsymbol\omega_m\) 是切点组合（静态均值方差），与 (2.19) 一致。
- \(\boldsymbol\omega_x\) 是模拟状态变量 \(x(t)\) 的对冲组合——它通过最大化与 \(x(t)\) 的相关性构造 (6.43)：
- \(\boldsymbol\omega_m\) is the tangency portfolio (static mean-variance), consistent with (2.19).
- \(\boldsymbol\omega_x\) is the hedging portfolio mimicking the state variable \(x(t)\) — constructed by maximizing its correlation with \(x(t)\) (6.43):
$$\max_{\{\omega_i\}_{i=1}^n}\ \frac{\big(\boldsymbol\omega'\boldsymbol\sigma_x\big)^2}{\boldsymbol\omega'\Sigma\boldsymbol\omega}\ \Rightarrow\ \boldsymbol\omega=\frac{\Sigma^{-1}\boldsymbol\sigma_x}{\mathbf 1'\Sigma^{-1}\boldsymbol\sigma_x}=\boldsymbol\omega_x.\tag{6.43}$$
(6.42) 的直觉：\(\boldsymbol\omega^*=\boldsymbol\omega_m n_m+\boldsymbol\omega_x n_x\)，无风险资产持有 \(1-\mathbf 1'\boldsymbol\omega^*=1-n_m-n_x\)。
Remark 6.3。 称 \(\boldsymbol\omega_x\) 为对冲组合，因投资者总想消除 \(x(t)\) 带来的不确定。
Remark 6.4。 对冲组合的存在意味着投资者不再纯粹做均值方差有效——他愿意牺牲一点均值方差效率去对冲状态变量风险。
Intuition for (6.42): \(\boldsymbol\omega^*=\boldsymbol\omega_m n_m+\boldsymbol\omega_x n_x\), with risk-free holding \(1-\mathbf 1'\boldsymbol\omega^*=1-n_m-n_x\).
Remark 6.3. \(\boldsymbol\omega_x\) is called the hedging portfolio because the investor always wants to remove the uncertainty brought by \(x(t)\).
Remark 6.4. The presence of the hedging portfolio means the investor is no longer purely mean-variance efficient — he sacrifices some mean-variance efficiency to hedge state-variable risk.
6.3.4 ICAPM
以代表性投资者 (representative agent) 加总，记市场组合权重 \(\omega_j^M\) (6.44)–(6.45)，市场 beta \(\beta_{iM}\) (6.46) 与对冲组合 beta \(\beta_{ix}\) (6.47)：
Aggregating with a representative agent, denote market weights \(\omega_j^M\) (6.44)–(6.45), market beta \(\beta_{iM}\) (6.46), and hedging-portfolio beta \(\beta_{ix}\) (6.47):
$$\beta_{iM}=\frac{\mathrm{Cov}\big(\frac{dP_i}{P_i},\frac{dP^M}{P^M}\big)}{\mathrm{Var}\big(\frac{dP^M}{P^M}\big)},\tag{6.46}$$
$$\beta_{ix}=\frac{\mathrm{Cov}\big(\frac{dP_i}{P_i},dx\big)}{\mathrm{Var}(dx)}=\frac{\sigma_{ix}}{s^2}.\tag{6.47}$$
把 (6.46)、(6.47) 代入 (6.45)，得连续时间的 ICAPM Beta 表示：
Plugging (6.46), (6.47) into (6.45) gives the continuous-time ICAPM beta representation:
$$\alpha_i-r=\lambda_M\beta_{iM}+\lambda_x\beta_{ix},\qquad \lambda_M=\frac{-J_{WW}^R W^R}{J_W^R}\sigma_M^2,\quad \lambda_x=\frac{-J_{Wx}^R}{J_W^R}s^2.\tag{6.48}$$
\(\lambda_M,\lambda_x\) 不依赖具体资产 \(i\)，故 (6.48) 是一个 CAPM 式的超额收益 beta 表示。当 \(\beta_{Mx}=0\)（市场与状态变量正交）时，\(\lambda_M\) 恰是市场风险溢价——即 ICAPM 退化为 CAPM。
无代表性投资者时。 (6.38) 的一阶条件对每个投资者 \(h\)（共 \(N\) 个）成立 (6.49)。用各投资者在总投资中的权重 \(q_h\)（\(\sum_h q_h=1,\ q_h\ge0\)）与 \(d_h\equiv q_h/(-J_{WW}^h W^h/J_W^h)\)、\(d\equiv\sum_h d_h\) 加总 (6.50)，得市场层面的市场 beta (6.51) 与对冲 beta (6.52)，最终
\(\lambda_M,\lambda_x\) do not depend on the specific asset \(i\), so (6.48) is a CAPM-like beta representation of excess return. When \(\beta_{Mx}=0\) (market orthogonal to the state variable), \(\lambda_M\) is exactly the market risk premium — ICAPM collapses to CAPM.
Without a representative agent. The FOC (6.38) holds for every investor \(h\) (out of \(N\)) (6.49). Aggregating with each investor's weight \(q_h\) in total investment (\(\sum_h q_h=1,\ q_h\ge0\)), \(d_h\equiv q_h/(-J_{WW}^h W^h/J_W^h)\) and \(d\equiv\sum_h d_h\) (6.50), one obtains the market-level market beta (6.51) and hedging beta (6.52), and finally
$$\alpha_i-r=\lambda_M\beta_{iM}+\lambda_x\beta_{ix},\tag{6.53}$$
与代表性投资者情形同形：当 \(\beta_{Mx}=0\) 时 \(\lambda_M\) 即市场风险溢价。
ICAPM 的核心讯息。 投资者不仅关心财富的瞬时波动（市场风险），还关心未来投资机会会不会变差（状态变量风险）。能对冲"机会变差"的资产更受青睐、要求更低收益，于是出现第二个定价因子。这为多因子模型（如 Fama–French）提供了理论动机：因子应代表状态变量。
Same form as the representative-agent case: when \(\beta_{Mx}=0\), \(\lambda_M\) is the market risk premium.
The key message of ICAPM. Investors care not only about the instantaneous volatility of wealth (market risk) but also about whether future investment opportunities deteriorate (state-variable risk). An asset that hedges "worsening opportunities" is more desirable and demands a lower return, producing a second pricing factor. This gives a theoretical motivation for multi-factor models (e.g. Fama–French): the factors should proxy for state variables.
References
- Cochrane, J. H. (2005). Asset Pricing (Revised Edition). Princeton University Press.
- Merton, R. C. (1973). An Intertemporal Capital Asset Pricing Model. Econometrica 41(5), 867–887.
- Ross, S. A. (1976). The Arbitrage Theory of Capital Asset Pricing. Journal of Economic Theory 13(3), 341–360.
- Fama, E. F. and K. R. French (1993). Common Risk Factors in the Returns on Stocks and Bonds. Journal of Financial Economics 33(1), 3–56.