本讲导读 本讲（Toda 第 4 讲）讲消费基础资产定价 (cCAPM)。§1 Lucas (1978) 模型：由无套利 \(P_t=\mathbb E_t[m_{t+1}(P_{t+1}+D_{t+1})]\) 与边际效用过程 \(\Lambda_t\) 得价格 \(P_t=\frac1{\Lambda_t}\sum_n\mathbb E_t[\Lambda_{t+n}D_{t+n}]\)；可加效用下 SDF \(m=\beta u'(c')/u'(c)\)；CRRA 下价格-红利比 = 对数消费/红利增长的 MGF 贴现和 (1)、(2)。§2 闭式解（对数正态、Gaussian VAR(1) Burnside 1998、有限状态马尔可夫链 \(v=(I-\beta P\,\mathrm{diag}(e^{X\alpha}))^{-1}\beta Pe^{X\alpha}\)）。§3 资产定价之谜：无风险利率 \(R_f=1/(\beta M(-\gamma))\)、\(\mathbb E[R]=M(1)/(\beta M(1-\gamma))\)、对数股权溢价 \(=\log\frac{M(1)M(-\gamma)}{M(1-\gamma)}\)；Prop 1（溢价非负，由 \(\log M\) 凸）；对数正态下溢价 \(=\gamma\sigma^2\) ⟹ 股权溢价之谜（\(\gamma=48.5\)）与 无风险利率之谜（Weil 1989）。§4 解释：4.1 罕见灾难（Rietz 1988/Barro 2006）、4.2 不完全市场（Constantinides-Duffie 1996）、4.3 习惯形成（Campbell-Cochrane 1999）、4.4 长期风险（Bansal-Yaron 2004）、4.5 异质偏好（Gârleanu-Panageas 2015）。含每讲参考文献。

由无套利与边际效用过程定价 / Pricing via no-arbitrage and the marginal-utility process 与追逐套利的从业者不同，经济学家关心资产价格为何如此。Lucas (1978) 给出由经济基本面计算资产价格的框架，即 cCAPM。考虑多期经济 \(t=0,1,\dots\)，资产在 \(t\) 付红利 \(D_t\)（适应随机过程），\(P_t\) 为除息价。设 \(m_t\) 为 SDF，由无套利Unlike practitioners chasing arbitrage, economists care why asset prices are what they are. Lucas (1978) provides a framework to compute asset prices from the economy's fundamentals — the cCAPM. Consider a multi-period economy \(t=0,1,\dots\); an asset pays dividend \(D_t\) (an adapted stochastic process) at \(t\), with ex-dividend price \(P_t\). Let \(m_t\) be the SDF; by no-arbitrage

$$P_t=\mathbb E_t[m_{t+1}(P_{t+1}+D_{t+1})].$$

定义边际效用过程 \(\{\Lambda_t\}\)：取任意 \(\Lambda_0\)，\(\Lambda_t=\Lambda_0\prod_{s=1}^t m_s=\Lambda_{t-1}m_t\)，则 \(\Lambda_t P_t=\mathbb E_t[\Lambda_{t+1}(P_{t+1}+D_{t+1})]\)。迭代、用重期望律、并设无泡沫条件 \(\lim_{T\to\infty}\mathbb E_t[\Lambda_T P_T]=0\)，得Define the marginal-utility process \(\{\Lambda_t\}\): take any \(\Lambda_0\), \(\Lambda_t=\Lambda_0\prod_{s=1}^t m_s=\Lambda_{t-1}m_t\), so \(\Lambda_t P_t=\mathbb E_t[\Lambda_{t+1}(P_{t+1}+D_{t+1})]\). Iterating, using the law of iterated expectations, and assuming the no-bubble condition \(\lim_{T\to\infty}\mathbb E_t[\Lambda_T P_T]=0\),

$$\Lambda_t P_t=\sum_{n=1}^\infty\mathbb E_t[\Lambda_{t+n}D_{t+n}]\iff P_t=\frac1{\Lambda_t}\sum_{n=1}^\infty\mathbb E_t[\Lambda_{t+n}D_{t+n}].$$

SDF 与 CRRA 价格-红利比 / The SDF and the CRRA price-dividend ratio 设投资者有可加效用 \(\mathbb E_0\sum_{t=0}^\infty\beta^t u(c_t)\)。此时 SDF 是什么？取两期 \(t=0,1\)，\(\pi_s\) 为状态 \(s\) 客观概率、\(p_s\) 为状态-\(s\) Arrow 证券价格。投资者问题 \(\max u(c_0)+\sum_s\pi_s\beta u(c_s)\) s.t. \(c_0+\sum_s p_s c_s\le w\)。Lagrangian 的一阶条件 \(0=u'(c_0)-\lambda\)、\(0=\pi_s\beta u'(c_s)-\lambda p_s\)，消去 \(\lambda\) 并用 SDF 定义得Suppose the investor has additive utility \(\mathbb E_0\sum_{t=0}^\infty\beta^t u(c_t)\). What is the SDF? Take two periods \(t=0,1\), with \(\pi_s\) the objective probability of state \(s\) and \(p_s\) the price of the state-\(s\) Arrow security. The investor's problem is \(\max u(c_0)+\sum_s\pi_s\beta u(c_s)\) s.t. \(c_0+\sum_s p_s c_s\le w\). The Lagrangian first-order conditions \(0=u'(c_0)-\lambda\), \(0=\pi_s\beta u'(c_s)-\lambda p_s\), eliminating \(\lambda\) and using the SDF definition, give

$$m_s=\frac{p_s}{\pi_s}=\beta\frac{u'(c_s)}{u'(c_0)}.$$

故取 \(\Lambda_0=u'(c_0)\) 则 \(\Lambda_t=\beta^t u'(c_t)\)，资产价格 \(P_t=\mathbb E_t\sum_{n=1}^\infty\beta^n\frac{u'(c_{t+n})}{u'(c_t)}D_{t+n}\)。若 \(u(c)=\frac{c^{1-\gamma}}{1-\gamma}\)（CRRA），记 \(X_t=(\log c_t,\log D_t)'\)、\(\alpha=(-\gamma,1)'\)，则So taking \(\Lambda_0=u'(c_0)\), \(\Lambda_t=\beta^t u'(c_t)\), and the asset price \(P_t=\mathbb E_t\sum_{n=1}^\infty\beta^n\frac{u'(c_{t+n})}{u'(c_t)}D_{t+n}\). If \(u(c)=\frac{c^{1-\gamma}}{1-\gamma}\) (CRRA), writing \(X_t=(\log c_t,\log D_t)'\), \(\alpha=(-\gamma,1)'\),

$$\frac{P_t}{D_t}=\mathbb E_t\sum_{n=1}^\infty\beta^n\left(\frac{c_{t+n}}{c_t}\right)^{-\gamma}\frac{D_{t+n}}{D_t}=\mathbb E_t\sum_{n=1}^\infty\beta^n e^{\alpha'(X_{t+n}-X_t)}=\sum_{n=1}^\infty\beta^n M_{X_{t+n}-X_t}(\alpha),\tag{1}$$

其中 \(M_{X_{t+n}-X_t}\) 是 \(X_{t+n}-X_t\) 的矩母函数 (MGF)。给定 \(\beta,\gamma\) 与对数消费/红利的随机过程，原则上即可算出价格-红利比。where \(M_{X_{t+n}-X_t}\) is the moment generating function (MGF) of \(X_{t+n}-X_t\). Given \(\beta,\gamma\) and the stochastic process of log consumption/dividend, in principle one can compute the price-dividend ratio.

代表性主体与 i.i.d. 例 / Representative agent and the i.i.d. example 要可操作，需知个体消费。常用（强）假设：所有主体有相同 CRRA 效用、市场完全 ⟹ 可视作代表性主体消费总禀赋，用总消费 \(C_t\) 代替个体 \(c_t\)；并设红利等于总消费 \(C_t=D_t\)。记 \(M_{t,n}\) 为对数消费增长 \(\log(C_{t+n}/C_t)\) 的 MGF，则To be operational, one needs individual consumption. The usual (heroic) assumptions: all agents have identical CRRA utility and markets are complete ⟹ treat the economy as a representative agent consuming the aggregate endowment, using aggregate consumption \(C_t\) instead of individual \(c_t\); and assume the dividend equals aggregate consumption, \(C_t=D_t\). Letting \(M_{t,n}\) be the MGF of log consumption growth \(\log(C_{t+n}/C_t)\),

$$\frac{P_t}{C_t}=\sum_{n=1}^\infty\beta^n M_{t,n}(1-\gamma).\tag{2}$$

例：设对数消费增长 \(\Delta c_{t+1}=\log(C_{t+1}/C_t)\) i.i.d.，\(M(s)=\mathbb E[e^{s\Delta c}]\)，则（设 \(\beta M(1-\gamma)<1\)，否则价格-红利比无穷）Example: if log consumption growth \(\Delta c_{t+1}=\log(C_{t+1}/C_t)\) is i.i.d. with \(M(s)=\mathbb E[e^{s\Delta c}]\), then (assuming \(\beta M(1-\gamma)<1\), else the ratio is infinite)

$$\frac{P_t}{C_t}=\sum_{n=1}^\infty\beta^n(M(1-\gamma))^n=\frac{\beta M(1-\gamma)}{1-\beta M(1-\gamma)}.$$

对数正态与 Gaussian VAR(1) / Lognormal and Gaussian VAR(1) (1)、(2) 把价格-红利比表为对数消费/红利增长 MGF 的贴现和。对特定增长过程可得闭式解。例如 \(N(\mu,\sigma^2)\) 的 MGF 为 \(M(s)=e^{\mu s+\frac12\sigma^2 s^2}\)，故消费增长对数正态时可闭式求解。Burnside (1998) 给出对数消费/红利增长服从 Gaussian VAR(1) 时的解（用于评估 VAR 离散化精度，见 Farmer and Toda 2017）；Tsionas (2003) 推广到非高斯冲击、de Groot (2015) 推广到随机波动率。(1), (2) express the price-dividend ratio as a discounted sum of MGFs of log consumption/dividend growth. For specific growth processes one can get closed forms. E.g. the MGF of \(N(\mu,\sigma^2)\) is \(M(s)=e^{\mu s+\frac12\sigma^2 s^2}\), so the ratio is closed-form if consumption growth is lognormal. Burnside (1998) solved the case where log consumption/dividend growth follows a Gaussian VAR(1) (useful for assessing the accuracy of VAR discretization, see Farmer and Toda 2017); Tsionas (2003) generalized to non-Gaussian shocks and de Groot (2015) to stochastic volatility.

有限状态马尔可夫链的闭式解 / Closed form for a finite-state Markov chain 设对数消费/红利增长服从 \(S\) 状态马尔可夫链，转移矩阵 \(P=(p_{ss'})\)，\(X_t=(\log(C_t/C_{t-1}),\log(D_t/D_{t-1}))'\)。由 Euler 方程 \(P_t=\mathbb E_t[\beta(C_{t+1}/C_t)^{-\gamma}(P_{t+1}+D_{t+1})]\) (3)，除以 \(D_t\) 得Suppose log consumption/dividend growth follows an \(S\)-state Markov chain with transition matrix \(P=(p_{ss'})\), \(X_t=(\log(C_t/C_{t-1}),\log(D_t/D_{t-1}))'\). The Euler equation \(P_t=\mathbb E_t[\beta(C_{t+1}/C_t)^{-\gamma}(P_{t+1}+D_{t+1})]\) (3), divided by \(D_t\), gives

$$V_t=\beta\,\mathbb E_t[\exp(\alpha'X_{t+1})(V_{t+1}+1)],\qquad\alpha=(-\gamma,1)'.\tag{4}$$

因增长是马尔可夫链，记 \(x_s\) 为状态 \(s\) 的对数增长向量，(4) 变为 \(v_s=\beta\sum_{s'=1}^S p_{ss'}e^{\alpha'x_{s'}}(v_{s'}+1)\) (5)，\(v_s\) 为状态 \(s\) 的价格-红利比。令 \(v=(v_1,\dots,v_S)'\)、\(X=(x_1,\dots,x_S)'\)（\(S\times2\)），则 (5) 等价于线性方程Since growth is a Markov chain, letting \(x_s\) be the log-growth vector in state \(s\), (4) becomes \(v_s=\beta\sum_{s'=1}^S p_{ss'}e^{\alpha'x_{s'}}(v_{s'}+1)\) (5), with \(v_s\) the price-dividend ratio in state \(s\). Letting \(v=(v_1,\dots,v_S)'\), \(X=(x_1,\dots,x_S)'\) (\(S\times2\)), (5) is equivalent to the linear equation

$$v=\beta P\,\mathrm{diag}(e^{X\alpha})(v+1)\iff v=(I-\beta P\,\mathrm{diag}(e^{X\alpha}))^{-1}\beta Pe^{X\alpha},$$

即得闭式解（Mehra and Prescott 1985、Cecchetti et al. 1993、Bonomo et al. 2011 等使用）。a closed-form solution (used by Mehra and Prescott 1985, Cecchetti et al. 1993, Bonomo et al. 2011, among others).

无风险利率、股票收益与股权溢价 / Risk-free rate, stock return, equity premium 在 CRRA 代表性主体、i.i.d. 消费增长假设下计算。SDF 为 \(\beta(C_{t+1}/C_t)^{-\gamma}\)，毛无风险利率 \(R_f\) 满足Compute under CRRA representative agent and i.i.d. consumption growth. The SDF is \(\beta(C_{t+1}/C_t)^{-\gamma}\), and the gross risk-free rate \(R_f\) satisfies

$$\frac1{R_f}=\mathbb E[\beta(C_{t+1}/C_t)^{-\gamma}]=\beta M(-\gamma)\iff R_f=\frac1{\beta M(-\gamma)}.$$

毛股票收益 \(R_{t+1}=\frac{P_{t+1}+C_{t+1}}{P_t}=\frac{C_{t+1}}{C_t}\frac{P_{t+1}/C_{t+1}+1}{P_t/C_t}=\frac{C_{t+1}}{C_t}\frac1{\beta M(1-\gamma)}\)，故期望股票收益 \(\mathbb E[R]=\frac{M(1)}{\beta M(1-\gamma)}\)。对数股权溢价为The gross stock return \(R_{t+1}=\frac{P_{t+1}+C_{t+1}}{P_t}=\frac{C_{t+1}}{C_t}\frac{P_{t+1}/C_{t+1}+1}{P_t/C_t}=\frac{C_{t+1}}{C_t}\frac1{\beta M(1-\gamma)}\), so the expected stock return is \(\mathbb E[R]=\frac{M(1)}{\beta M(1-\gamma)}\). The log equity premium is

$$\log\mathbb E[R]-\log R_f=\log\frac{M(1)M(-\gamma)}{M(1-\gamma)}.$$

命题 1（股权溢价非负）及证明 / Proposition 1 (nonnegative equity premium) and proof 命题 1：设 \(M\) 是矩母函数，则 \(\log M(1)+\log M(-\gamma)\ge\log M(1-\gamma)\)。证明：设 \(X\) 为对数消费增长，任取 \(s_1,s_2\) 与 \(t\in(0,1)\)，令 \(p=\frac1{1-t}\)、\(q=\frac1t\)（\(1/p+1/q=1\)）。由 Hölder 不等式Proposition 1: if \(M\) is a moment generating function, then \(\log M(1)+\log M(-\gamma)\ge\log M(1-\gamma)\). Proof: let \(X\) be log consumption growth, take any \(s_1,s_2\) and \(t\in(0,1)\), and let \(p=\frac1{1-t}\), \(q=\frac1t\) (\(1/p+1/q=1\)). By Hölder's inequality

$$M((1-t)s_1+ts_2)=\mathbb E[e^{(1-t)s_1 X}e^{ts_2 X}]\le\mathbb E[e^{s_1 X}]^{1-t}\mathbb E[e^{s_2 X}]^t=M(s_1)^{1-t}M(s_2)^t.$$

取对数知 \(\log M\) 凸（\(M\) 对数凸）。记 \(f=\log M\)，因 \(\gamma>0\)，取 \(t=\frac\gamma{\gamma+1}\) 得 \(0=(1-t)(-\gamma)+t\)，取 \(t=\frac1{\gamma+1}\) 得 \(1-\gamma=(1-t)(-\gamma)+t\)。由凸性Taking logs shows \(\log M\) is convex (\(M\) is log-convex). Let \(f=\log M\). Since \(\gamma>0\), \(t=\frac\gamma{\gamma+1}\) gives \(0=(1-t)(-\gamma)+t\), and \(t=\frac1{\gamma+1}\) gives \(1-\gamma=(1-t)(-\gamma)+t\). By convexity

$$f(0)\le\frac1{\gamma+1}f(-\gamma)+\frac\gamma{\gamma+1}f(1),\qquad f(1-\gamma)\le\frac\gamma{\gamma+1}f(-\gamma)+\frac1{\gamma+1}f(1).$$

两式相加并用 \(f(0)=\log M(0)=0\)（\(M(0)=1\)），得 \(f(1-\gamma)\le f(-\gamma)+f(1)\)，即结论。\(\blacksquare\)Adding these and using \(f(0)=\log M(0)=0\) (\(M(0)=1\)) gives \(f(1-\gamma)\le f(-\gamma)+f(1)\), the conclusion. \(\blacksquare\)

股权溢价之谜 / The equity premium puzzle 设消费增长对数正态 \(\Delta c_{t+1}\sim N(\mu,\sigma^2)\)，\(M(s)=e^{\mu s+\frac12\sigma^2 s^2}\)，则对数无风险利率If consumption growth is lognormal \(\Delta c_{t+1}\sim N(\mu,\sigma^2)\), \(M(s)=e^{\mu s+\frac12\sigma^2 s^2}\), then the log risk-free rate

$$\log R_f=-\log\beta+\mu\gamma-\tfrac12\sigma^2\gamma^2,\tag{6}$$

对数股权溢价 \(\log\mathbb E[R]-\log R_f=(\mu+\tfrac12\sigma^2)+(-\mu\gamma+\tfrac12\sigma^2\gamma^2)-(\mu(1-\gamma)+\tfrac12\sigma^2(1-\gamma)^2)=\gamma\sigma^2\)。用 Mehra-Prescott (1985) 数据：消费增长波动 \(3.57\%\)/年、股权溢价 \(6.18\%\)/年，则 \(0.0618=\gamma\times0.0357^2\iff\gamma=48.49\)，过高。这就是股权溢价之谜：Mehra-Prescott 认为相对风险厌恶合理上界为 \(10\)，此界下模型无法生成现实的股权溢价。The log equity premium is \(\log\mathbb E[R]-\log R_f=(\mu+\tfrac12\sigma^2)+(-\mu\gamma+\tfrac12\sigma^2\gamma^2)-(\mu(1-\gamma)+\tfrac12\sigma^2(1-\gamma)^2)=\gamma\sigma^2\). Using Mehra-Prescott (1985) data: consumption-growth volatility \(3.57\%\)/yr, equity premium \(6.18\%\)/yr, then \(0.0618=\gamma\times0.0357^2\iff\gamma=48.49\), far too high. This is the equity premium puzzle: Mehra-Prescott argue a reasonable upper bound for relative risk aversion is \(10\), under which the model cannot generate a realistic equity premium.

无风险利率之谜 / The risk-free rate puzzle 为何 \(\gamma\le10\) 合理？由 (6) 改写 \(\log R_f=-\log\beta+\frac{\mu^2}{2\sigma^2}-\frac12\sigma^2(\gamma-\frac\mu{\sigma^2})^2\)，是 \(\gamma\) 的二次函数，在 \(\gamma=\mu/\sigma^2\) 取最大。故调高 \(\gamma\) 以增大股权溢价时，也会（在某点前）抬高无风险利率；但历史上 \(R_f\) 很低（约 \(1\%\)）——调高风险厌恶无法解释低无风险利率，即 无风险利率之谜（Weil 1989）。再调高 \(\gamma\) 越过顶点后 \(\log R_f\) 转降，但 \(\frac{\partial\log R_f}{\partial\mu}=\gamma\)，\(\gamma=100\) 时 \(1\%\) 的增长变化会引起 \(100\%\) 的无风险利率变化，而历史上 \(R_f\) 相当稳定（参 Kocherlakota 1996）。要使之成谜，需维持诸多假设（CRRA、代表性主体、消费=红利、完全市场、i.i.d.、对数正态）；放松其一即得各种解释。Why is \(\gamma\le10\) reasonable? Rewriting (6), \(\log R_f=-\log\beta+\frac{\mu^2}{2\sigma^2}-\frac12\sigma^2(\gamma-\frac\mu{\sigma^2})^2\), a quadratic in \(\gamma\) maximized at \(\gamma=\mu/\sigma^2\). So raising \(\gamma\) to enlarge the equity premium also raises \(R_f\) (up to a point); but historically \(R_f\) is low (~\(1\%\)) — raising risk aversion cannot explain the low risk-free rate, the risk-free rate puzzle (Weil 1989). Raising \(\gamma\) past the peak makes \(\log R_f\) decreasing, but \(\frac{\partial\log R_f}{\partial\mu}=\gamma\), so \(\gamma=100\) means a \(1\%\) growth change implies a \(100\%\) change in \(R_f\), while \(R_f\) is historically quite stable (see Kocherlakota 1996). For it to be a puzzle one must maintain many assumptions (CRRA, representative agent, consumption $=$ dividend, complete markets, i.i.d., lognormal); relaxing one gives various explanations.

4.1 罕见灾难 / 4.1 Rare disasters 或许之谜源于高斯假设。最早的解释之一是罕见灾难模型（Rietz 1988），曾被弃为不现实，后经 Barro (2006) 用国际数据校准复兴。设无灾难时总消费按常数率 \(\mu\) 增长，灾难时大降（\(0Perhaps the puzzle arises from the Gaussian assumption. One of the earliest explanations is the rare disasters model (Rietz 1988), initially dismissed as unrealistic, then revived by Barro (2006) with internationally calibrated parameters. With no disaster, aggregate consumption grows at constant rate \(\mu\); with a disaster, it drops sharply (\(0

$$\frac{C_{t+1}}{C_t}=e^\mu\times\begin{cases}1&\text{(no disaster, prob }1-p)\\ b&\text{(disaster, prob }p)\end{cases}$$

对数消费增长 MGF 为 \(M(s)=e^{\mu s}(1-p+pb^s)\)。对数股权溢价The MGF of log consumption growth is \(M(s)=e^{\mu s}(1-p+pb^s)\). The log equity premium

$$\log\frac{\mathbb E[R]}{R_f}=\log(1-p+pb)+\log(1-p+pb^{-\gamma})-\log(1-p+pb^{1-\gamma})\approx p(1-b)(b^{-\gamma}-1),$$

用 \(\log(1+pa)\approx pa\)（\(p\ll1\)，连续时间下精确）。对数消费增长方差 \(\sigma^2=p(1-p)(\log b)^2\)。Barro (2006) 取 \(p=0.017\)，设 \(\sigma=0.0357\) 得 \(b=0.759\)（灾后产出损失 \(24\%\)）；\(\gamma=10\) 时对数股权溢价 \(0.0608\)（\(6.08\%\)），非常接近数据。（Gourio 2012、Gabaix 2012、Wachter 2013。）using \(\log(1+pa)\approx pa\) (\(p\ll1\), exact in continuous time). The variance of log consumption growth is \(\sigma^2=p(1-p)(\log b)^2\). Barro (2006) takes \(p=0.017\); setting \(\sigma=0.0357\) gives \(b=0.759\) (a \(24\%\) output loss after disaster); with \(\gamma=10\) the log equity premium is \(0.0608\) (\(6.08\%\)), very close to the data. (Gourio 2012, Gabaix 2012, Wachter 2013.)

4.2 不完全市场（Constantinides-Duffie 1996）/ 4.2 Incomplete markets 或许之谜源于完全市场假设。现实中市场不完全（现有资产不张成所有状态——若完全，你本可卖光未来劳动收入，从而丧失工作激励）。Constantinides and Duffie (1996) 是典型：给定无套利资产集与总消费增长过程，可构造与资产价格相容的个体禀赋过程。设连续统主体 \(i\in[0,1]\) 均为贴现 \(\beta\)、相对风险厌恶 \(\gamma\) 的可加效用，\(c_{i0}=C_0\)，对 \(t\ge1\) 定义Perhaps the puzzle arises from the complete-market assumption. In reality markets are incomplete (existing assets do not span all states — if complete, you could sell off all future labor income and lose the incentive to work). Constantinides and Duffie (1996) is typical: given a no-arbitrage asset set and an aggregate consumption-growth process, one can construct individual endowment processes consistent with asset prices. With a continuum of agents \(i\in[0,1]\) all having additive utility with discount \(\beta\) and relative risk aversion \(\gamma\), \(c_{i0}=C_0\), define for \(t\ge1\)

$$\frac{c_{it}}{C_t}=\frac{c_{i,t-1}}{C_{t-1}}e^{\sigma_t\eta_{it}-\frac12\sigma_t^2},$$

\(\sigma_t>0\) 为个体（相对总量）消费增长标准差、\(\eta_{it}\) 跨主体与时间 i.i.d. 标准正态。则 \(\mathbb E_t[c_{it}]=C_t\)（截面平均即总消费）。个体 SDF \(M_{it}=\beta(c_{it}/c_{i,t-1})^{-\gamma}=\beta(C_t/C_{t-1})^{-\gamma}e^{-\gamma\sigma_t\eta_{it}+\frac12\gamma\sigma_t^2}\)。因个体消费增长含与所有资产支付独立的特质成分，对特质冲击取期望得新 SDF\(\sigma_t>0\) is the standard deviation of individual (relative to aggregate) consumption growth, and \(\eta_{it}\) is i.i.d. standard normal across agents and time. Then \(\mathbb E_t[c_{it}]=C_t\) (the cross-sectional average is aggregate consumption). The individual SDF \(M_{it}=\beta(c_{it}/c_{i,t-1})^{-\gamma}=\beta(C_t/C_{t-1})^{-\gamma}e^{-\gamma\sigma_t\eta_{it}+\frac12\gamma\sigma_t^2}\). Since individual consumption growth has an idiosyncratic component independent of all asset payoffs, taking expectations over the idiosyncratic shock gives a new SDF

$$\frac{\Lambda_t}{\Lambda_{t-1}}=\mathbb E_t[M_{it}]=\beta\left(\frac{C_t}{C_{t-1}}\right)^{-\gamma}e^{\frac12\gamma(\gamma+1)\sigma_t^2},$$

解出 \(\sigma_t=\sqrt{\frac2{\gamma(\gamma+1)}}\left(-\log\beta+\gamma\log\frac{C_t}{C_{t-1}}+\log\frac{\Lambda_t}{\Lambda_{t-1}}\right)^{1/2}\) 即可令其成立。这是可能性定理；特质冲击能否解释资产价格是实证/数量问题（Brav et al. 2002、Cogley 2002、Storesletten et al. 2007、Krueger and Lustig 2010；另见 Toda and Walsh 2015, 2017）。solving \(\sigma_t=\sqrt{\frac2{\gamma(\gamma+1)}}\left(-\log\beta+\gamma\log\frac{C_t}{C_{t-1}}+\log\frac{\Lambda_t}{\Lambda_{t-1}}\right)^{1/2}\) makes this hold. This is a possibility theorem; whether the idiosyncratic shock explains asset prices is an empirical/quantitative question (Brav et al. 2002, Cogley 2002, Storesletten et al. 2007, Krueger and Lustig 2010; see also Toda and Walsh 2015, 2017).

4.3 习惯形成 / 4.3 Habit formation 或许之谜源于 CRRA 假设。Campbell and Cochrane (1999) 的代表性主体效用 \(\mathbb E_0\sum_t\beta^t\frac{(C_t-X_t)^{1-\gamma}}{1-\gamma}\)，\(X_t\) 为"习惯"水平（参考点）。令剩余消费比 \(S_t=\frac{C_t-X_t}{C_t}\)，则 SDF \(M_{t+1}=\beta\left(\frac{C_{t+1}-X_{t+1}}{C_t-X_t}\right)^{-\gamma}=\beta\left(\frac{S_{t+1}}{S_t}\right)^{-\gamma}\left(\frac{C_{t+1}}{C_t}\right)^{-\gamma}\)。如 Constantinides-Duffie，通过摆弄 \(S_t\) 可复制任意 SDF（Pohl 2016 形式证明）。弱点：\(S_t\) 不可观测，模型不可证伪。Perhaps the puzzle arises from the CRRA assumption. Campbell and Cochrane (1999) use a representative-agent utility \(\mathbb E_0\sum_t\beta^t\frac{(C_t-X_t)^{1-\gamma}}{1-\gamma}\), with \(X_t\) a "habit" level (reference point). Letting the surplus consumption ratio \(S_t=\frac{C_t-X_t}{C_t}\), the SDF \(M_{t+1}=\beta\left(\frac{C_{t+1}-X_{t+1}}{C_t-X_t}\right)^{-\gamma}=\beta\left(\frac{S_{t+1}}{S_t}\right)^{-\gamma}\left(\frac{C_{t+1}}{C_t}\right)^{-\gamma}\). As with Constantinides-Duffie, playing with \(S_t\) can replicate any SDF (Pohl 2016 proves this formally). Weakness: \(S_t\) is unobservable, so the model is not falsifiable.

4.4 长期风险 与 4.5 异质偏好 / 4.4 Long-run risk and 4.5 Heterogeneous preferences 4.4 长期风险：或许之谜源于 i.i.d. 假设。Bansal and Yaron (2004)、Bansal et al. (2012) 的长期风险模型结合小而持久的风险（长期风险）与对不确定性早期解决的偏好，生成大的股权溢价。4.5 异质偏好：或许之谜源于代表性主体（相同偏好）假设。Gârleanu and Panageas (2015) 考虑 i.i.d. 消费增长但两类主体（一类风险容忍、一类风险厌恶）的模型。（脚注：经济学家永不会就之谜的解释达成一致——若达成，就无法再写论文了。）4.4 Long-run risk: perhaps the puzzle arises from the i.i.d. assumption. The long-run risk model of Bansal and Yaron (2004), Bansal et al. (2012) combines small but persistent risk (long-run risk) with a preference for early resolution of uncertainty to generate large equity premia. 4.5 Heterogeneous preferences: perhaps the puzzle arises from the representative-agent (identical-preferences) assumption. Gârleanu and Panageas (2015) consider i.i.d. consumption growth but two types of agents (one risk-tolerant, one risk-averse). (Footnote: economists will never agree on the explanation of puzzles — if they did, they could no longer write papers on the topic.)