1. Consumption-Based Theory
消费基础理论 (consumption-based theory) 是资产定价的出发点。核心只有一条：任何资产今天的价格，等于其未来收益用随机贴现因子 (stochastic discount factor, SDF) 加权后的期望——而 SDF 正是消费的边际替代率。本章从最简单的两期、单一风险资产出发，逐步推广到多期、连续时间与组合选择。
Consumption-based theory is the starting point of asset pricing. One idea drives everything: the price of any asset today equals the expectation of its future payoff weighted by a stochastic discount factor (SDF) — which is the marginal rate of substitution in consumption. This chapter builds up from the simplest two-period, single-asset case to multiple periods, continuous time, and portfolio choice.
1.1 Simple Two-Period, One Risky Asset
1.1.1 Setup
考虑一个消费跨两期的个体，效用是时间可分的 (time-separable)：
Consider an individual consuming over two periods with time-separable utility:
$$u(c_0,\tilde c_1) = u(c_0) + \beta\,\mathbb{E}\big[u(\tilde c_1)\big],$$
其中 \(c_0\) 是确定的 0 期消费、\(\tilde c_1\) 是随机的 1 期消费、\(\beta\in(0,1)\) 是贴现因子。三条假设：消费时间可分、期望效用、跨期效用函数相同。个体有禀赋 \(e_0,e_1\)，可买入 \(w\) 股、每股价格 \(p_0\)、随机收益 \(\tilde x\)，故两期预算约束为
where \(c_0\) is deterministic period-0 consumption, \(\tilde c_1\) is random period-1 consumption, and \(\beta\in(0,1)\) is the discount factor. Three assumptions: time-separable consumption, expected utility, and the same utility function across periods. The agent has endowments \(e_0,e_1\) and buys \(w\) shares of a security at price \(p_0\) with random payoff \(\tilde x\), so the two budget constraints are
$$c_0 = e_0 - p_0\,w, \qquad \tilde c_1 = e_1 + \tilde x\,w.$$
1.1.2 Investor's Problem
把预算约束代入目标，问题变成对持股 \(w\) 的无约束最大化：
Substituting the budget constraints, the problem becomes an unconstrained maximization over the share holding \(w\):
$$\max_{w}\; u(e_0 - p_0 w) + \beta\,\mathbb{E}\big[u(e_1 + \tilde x\,w)\big].$$
证明 / Proof：一阶条件 → 定价方程
对 \(w\) 求一阶条件 (f.o.c.)：
The first-order condition with respect to \(w\):
$$u'(c_0)\,p_0 = \beta\,\mathbb{E}\big[\tilde x\,u'(\tilde c_1)\big] \;\Longrightarrow\; p_0 = \frac{\beta\,\mathbb{E}[\tilde x\,u'(\tilde c_1)]}{u'(c_0)} = \mathbb{E}\!\left[\beta\frac{u'(\tilde c_1)}{u'(c_0)}\,\tilde x\right]. \quad\blacksquare$$
这给出基于投资者最优消费决策的「正确」价格 \(p_0\)。
This gives the "correct" price \(p_0\) implied by the investor's optimal consumption decision.
1.1.3 Stochastic Discount Factor (SDF)
把上式写成压缩形式——基本定价方程 (fundamental pricing equation)：
Write it in compact form — the fundamental pricing equation:
无风险资产。 设有一种资产在 \(t=1\) 恒定支付 1，价格记为 \(p_0^*\)。由基本方程对任意资产成立，代入得 \(p_0^*=\mathbb{E}[m\cdot 1]=\mathbb{E}[m]\)。记无风险收益率 \(r_f\)，则
Risk-free asset. Suppose an asset pays a constant 1 at \(t=1\), with price \(p_0^*\). Since the pricing equation holds for any asset, \(p_0^*=\mathbb{E}[m\cdot 1]=\mathbb{E}[m]\). Denoting the risk-free rate \(r_f\),
$$p_0^* = \frac{1}{1+r_f} \quad\Longrightarrow\quad \mathbb{E}[m] = \frac{1}{1+r_f}.$$
1.1.4 Risk Correction and the Risk Premium
对风险资产，用协方差恒等式展开 \(p_0=\mathbb{E}[m\tilde x]\)，并代入 \(\mathbb{E}[m]=1/(1+r_f)\)：
For a risky asset, expand \(p_0=\mathbb{E}[m\tilde x]\) with the covariance identity and use \(\mathbb{E}[m]=1/(1+r_f)\):
$$p_0 = \mathbb{E}[m]\,\mathbb{E}[\tilde x] + \operatorname{Cov}(m,\tilde x) = \underbrace{\frac{\mathbb{E}[\tilde x]}{1+r_f}}_{\text{risk-neutral price}} + \underbrace{\operatorname{Cov}(m,\tilde x)}_{\text{correction}}.$$
修正项的符号。 \(\mathbb{E}[\tilde x]/(1+r_f)\) 是用无风险利率折现的「错误」基准，\(\operatorname{Cov}(m,\tilde x)\) 加以修正：\(\operatorname{Cov}(m,\tilde x)>0\) → 实际折现率低于 \(r_f\) → 风险溢价为负；\(\operatorname{Cov}(m,\tilde x)<0\) → 折现率高于 \(r_f\) → 溢价为正；无风险资产 \(\operatorname{Cov}(m,1)=0\)，溢价为零。Sign of the correction. \(\mathbb{E}[\tilde x]/(1+r_f)\) is the "wrong" benchmark discounted at \(r_f\); \(\operatorname{Cov}(m,\tilde x)\) corrects it: \(\operatorname{Cov}(m,\tilde x)>0\) → effective discount rate below \(r_f\) → negative premium; \(\operatorname{Cov}(m,\tilde x)<0\) → above \(r_f\) → positive premium; the risk-free asset has \(\operatorname{Cov}(m,1)=0\), hence zero premium.
单一风险资产恒为正溢价。 因为 \(m=\beta u'(\tilde c_1)/u'(c_0)\)，更高的 \(\tilde x\) 带来更高的 \(\tilde c_1\)、更低的 \(u'(\tilde c_1)\)、更低的 \(m\)，故 \(\operatorname{Cov}(m,\tilde x)<0\)，溢价为正。多资产时：与消费负相关（坏状态付得少）的资产溢价为正；能在坏状态对冲消费风险（\(\operatorname{Cov}(m,\tilde x)>0\)）的资产更值钱、溢价为负。写成总收益 \(\tilde R=\tilde x/p_0\)：
A single risky asset always carries a positive premium. Since \(m=\beta u'(\tilde c_1)/u'(c_0)\), a higher \(\tilde x\) raises \(\tilde c_1\), lowers \(u'(\tilde c_1)\), and lowers \(m\), so \(\operatorname{Cov}(m,\tilde x)<0\) and the premium is positive. With multiple assets: an asset that pays little in bad states earns a positive premium; one that hedges consumption risk (\(\operatorname{Cov}(m,\tilde x)>0\)) is more valuable and earns a negative premium. In gross-return form \(\tilde R=\tilde x/p_0\):
$$\mathbb{E}[\tilde R] - R_f = -R_f\,\operatorname{Cov}(m,\tilde R) = -\frac{\operatorname{Cov}\big(u'(\tilde c_1),\tilde R\big)}{\mathbb{E}[u'(\tilde c_1)]}, \qquad R_f \equiv 1+r_f.$$
证明 / Proof：风险溢价公式
基本方程的收益形式为
In return form the fundamental equation reads
$$1 = \mathbb{E}[m\tilde R] = \mathbb{E}[m]\,\mathbb{E}[\tilde R] + \operatorname{Cov}(m,\tilde R).$$
无风险资产给出 \(\mathbb{E}[m]=1/R_f\)；代入并乘 \(R_f\) 整理：
The risk-free asset gives \(\mathbb{E}[m]=1/R_f\); substituting and multiplying by \(R_f\):
$$\mathbb{E}[\tilde R] - R_f = -R_f\,\operatorname{Cov}(m,\tilde R). \quad\blacksquare$$
1.1.5 Risk-Neutral Pricing
除了「加法式」风险修正，还可以直接调整 \(\tilde x\) 的概率分布，使其期望收益按无风险利率折现即可。设 \(\mathbb{E}^*[\cdot]\) 是某分布 \(F^*\) 下的期望，满足
Instead of an additive risk correction, we can adjust the probability distribution of \(\tilde x\) so that its expected payoff is simply discounted at the risk-free rate. Let \(\mathbb{E}^*[\cdot]\) be the expectation under a distribution \(F^*\) with
$$p_0 = \mathbb{E}[m\tilde x] = \frac{\mathbb{E}^*[\tilde x]}{1+r_f}.$$
Remark 1.1. \(F^*\) 称为该资产的风险中性概率 (risk-neutral probability)——它把真实概率人为调整，使得贴现价格如同风险中性者所计算，于是可直接套用无风险利率。它只是人为构造的新分布，与真实分布的「对错」无关。Remark 1.1. \(F^*\) is the asset's risk-neutral probability — the true probabilities are adjusted so the discounted price is as if the agent were risk-neutral, letting us apply the risk-free rate. It is an artificial construct, with no bearing on the "correctness" of the true distribution.
1.1.6 Simple Parametric Example on the Risk-Free Rate
取常相对风险厌恶 (constant relative risk aversion, CRRA) 的幂效用：
Take constant relative risk aversion (CRRA) power utility:
$$u(c) = \frac{c^{1-\gamma}-1}{1-\gamma}, \qquad \gamma>0.$$
证明 / Proof：CRRA 的导数、相对风险厌恶系数与对数效用极限
一、二、三阶导数为 \(u'(c)=c^{-\gamma}\)，\(u''(c)=-\gamma c^{-\gamma-1}\)，\(u'''(c)=\gamma(\gamma+1)c^{-\gamma-2}\)，故相对风险厌恶系数
The derivatives are \(u'(c)=c^{-\gamma}\), \(u''(c)=-\gamma c^{-\gamma-1}\), \(u'''(c)=\gamma(\gamma+1)c^{-\gamma-2}\), so the relative risk aversion coefficient is
$$-\frac{u''(c)}{u'(c)}\,c = \frac{\gamma c^{-\gamma-1}}{c^{-\gamma}}\,c = \gamma.$$
当 \(\gamma\to1\)，由洛必达法则 (L'Hôpital) \(u(c)\to\ln c\)（对数效用是 CRRA 的极限特例）：
As \(\gamma\to1\), by L'Hôpital's rule \(u(c)\to\ln c\) (log utility is the limiting CRRA case):
$$\lim_{\gamma\to1}\frac{c^{1-\gamma}-1}{1-\gamma} = \lim_{\gamma\to1}\frac{-\ln c\cdot e^{(1-\gamma)\ln c}}{-1} = \ln c. \quad\blacksquare$$
把 CRRA 代入 SDF 定义得 \(m=\beta\,\tilde c_1^{-\gamma}/c_0^{-\gamma}=\beta(\tilde c_1/c_0)^{-\gamma}\)。再假设对数消费增长服从正态：\(\ln(\tilde c_1/c_0)\sim\mathcal{N}(\mu_c,\sigma_c^2)\)。代入 \(\mathbb{E}[m]=1/R_f\)：
Substituting CRRA into the SDF gives \(m=\beta\,\tilde c_1^{-\gamma}/c_0^{-\gamma}=\beta(\tilde c_1/c_0)^{-\gamma}\). Assume lognormal consumption growth \(\ln(\tilde c_1/c_0)\sim\mathcal{N}(\mu_c,\sigma_c^2)\). Substituting into \(\mathbb{E}[m]=1/R_f\):
$$r_f^{\log} \equiv \ln R_f = -\ln\beta + \gamma\,\mu_c - \tfrac{1}{2}\,\gamma^2\,\sigma_c^2.$$
证明 / Proof：对数正态下的（对数）无风险利率
对数正态的矩公式 \(\mathbb{E}[e^{z}]=e^{\mu+\frac12\sigma^2}\)。因 \(-\gamma\ln(\tilde c_1/c_0)\sim\mathcal{N}(-\gamma\mu_c,\gamma^2\sigma_c^2)\)，
With the lognormal moment \(\mathbb{E}[e^{z}]=e^{\mu+\frac12\sigma^2}\), and since \(-\gamma\ln(\tilde c_1/c_0)\sim\mathcal{N}(-\gamma\mu_c,\gamma^2\sigma_c^2)\),
$$\mathbb{E}[m]=\beta\,\mathbb{E}\!\left[e^{-\gamma\ln(\tilde c_1/c_0)}\right]=\beta\,e^{-\gamma\mu_c+\frac12\gamma^2\sigma_c^2}, \qquad \mathbb{E}[m^2]=e^{-2\gamma\mu_c+2\gamma^2\sigma_c^2}.$$
由 \(1/R_f=\mathbb{E}[m]=\beta e^{-\gamma\mu_c+\frac12\gamma^2\sigma_c^2}\)，取对数并整理即得 \(r_f^{\log}\)。\(\blacksquare\)
From \(1/R_f=\mathbb{E}[m]=\beta e^{-\gamma\mu_c+\frac12\gamma^2\sigma_c^2}\), taking logs and rearranging gives \(r_f^{\log}\). \(\blacksquare\)
三项含义。 \(-\ln\beta\)：不耐心，越急于消费利率越高；\(\gamma\mu_c\)：消费平滑动机，预期增长越快越想借未来钱，推高利率；\(-\frac12\gamma^2\sigma_c^2\)：预防性储蓄 (precautionary saving)，不确定性越大越想存钱，压低利率。这正是「无风险利率之谜」的出发点。The three terms. \(-\ln\beta\): impatience (raises \(r_f\)); \(\gamma\mu_c\): consumption-smoothing motive (faster expected growth → borrow → raises \(r_f\)); \(-\frac12\gamma^2\sigma_c^2\): precautionary saving (more uncertainty → save → lowers \(r_f\)). This is the starting point for the "risk-free rate puzzle."
1.2 Multi-Period, Multiple Assets: Discrete Time
1.2.1 Setup
推广到无限期、多资产。投资者最大化终生效用 \(\mathbb{E}_t\sum_{s\ge0}\beta^s u(c_{t+s})\)。资产 \(i\) 现价 \(p^i_t\)，下一期支付红利加价格 \(d^i_{t+1}+p^i_{t+1}\)。
Extend to an infinite horizon with many assets. The investor maximizes lifetime utility \(\mathbb{E}_t\sum_{s\ge0}\beta^s u(c_{t+s})\). Asset \(i\) trades at \(p^i_t\) and pays dividend-plus-price \(d^i_{t+1}+p^i_{t+1}\).
1.2.2 Investor's Problem
用值函数写成贝尔曼方程，对每种资产求一阶条件，得到与两期完全相同形式的欧拉方程：
Writing the value function as a Bellman equation and taking FOCs for each asset gives an Euler equation of exactly the same form as the two-period case:
$$p^i_t = \mathbb{E}_t\big[m_{t+1}\,(d^i_{t+1}+p^i_{t+1})\big], \qquad m_{t+1}=\beta\frac{u'(c_{t+1})}{u'(c_t)}.$$
证明 / Proof：贝尔曼方程、包络定理 → 欧拉方程
设值函数 \(V(W_t)=\max\,u(c_t)+\beta\mathbb{E}_t V(W_{t+1})\)，其中 \(W_{t+1}=\sum_i(d^i_{t+1}+p^i_{t+1})\xi^i_t\)。对 \(\xi^i_t\) 的一阶条件为
Let the value function be \(V(W_t)=\max\,u(c_t)+\beta\mathbb{E}_t V(W_{t+1})\) with \(W_{t+1}=\sum_i(d^i_{t+1}+p^i_{t+1})\xi^i_t\). The FOC for \(\xi^i_t\) is
$$-p^i_t\,u'(c_t) + \beta\,\mathbb{E}_t\big[V'(W_{t+1})\,(d^i_{t+1}+p^i_{t+1})\big] = 0.$$
由包络定理 (envelope theorem) \(V'(W_t)=u'(c_t)\)，代入即消去值函数导数，得欧拉方程。\(\blacksquare\)
By the envelope theorem \(V'(W_t)=u'(c_t)\), substituting eliminates the value-function derivative and yields the Euler equation. \(\blacksquare\)
1.2.3 Beta Pricing Representation (C-CAPM)
线性化 SDF，把基本方程写成 beta 定价——消费 CAPM (C-CAPM)：
Linearizing the SDF rewrites the equation as beta pricing — the Consumption CAPM (C-CAPM):
证明 / Proof：从 SDF 到 beta 定价
由 \(\mathbb{E}_t[R^i]-R_f=-R_f\operatorname{Cov}_t(m,R^i)\)；CRRA 下 \(m\approx\text{const}\,(1-\gamma\Delta c_{t+1})\)，代入得
From \(\mathbb{E}_t[R^i]-R_f=-R_f\operatorname{Cov}_t(m,R^i)\) and, under CRRA, \(m\approx\text{const}\,(1-\gamma\Delta c_{t+1})\), substituting gives
$$\mathbb{E}_t[R^i_{t+1}]-R_f \approx \gamma\,\operatorname{Cov}_t(R^i_{t+1},\Delta c_{t+1}) = \beta_{i,c}\,\lambda_c, \qquad \lambda_c=\gamma\,\operatorname{Var}_t(\Delta c_{t+1}). \quad\blacksquare$$
注释。 C-CAPM 把溢价拆成「数量 × 价格」：\(\beta_{i,c}\) 是资产承担的消费风险数量，\(\lambda_c\) 是单位消费风险的价格（所有资产共用）。它是标准 CAPM 的消费版——市场组合被消费增长取代。Comment. C-CAPM splits the premium into quantity × price: \(\beta_{i,c}\) is the consumption risk carried, \(\lambda_c\) the price per unit (common to all assets). It is the consumption analogue of the CAPM, with the market portfolio replaced by consumption growth.
1.2.4 Equity Volatility Puzzle
Shiller (1981) 的方差界：若价格是未来红利的理性折现 \(p_t=\mathbb{E}_t[p^*_t]\)（\(p^*_t\) 为事后完美预见价格），则由 \(p^*_t=p_t+\varepsilon_t\)、\(\operatorname{Cov}(p_t,\varepsilon_t)=0\) 得 \(\sigma(p_t)\le\sigma(p^*_t)\)。
Shiller's (1981) variance bound: if price is the rational discounted value of future dividends \(p_t=\mathbb{E}_t[p^*_t]\) (with \(p^*_t\) the ex-post perfect-foresight price), then since \(p^*_t=p_t+\varepsilon_t\) with \(\operatorname{Cov}(p_t,\varepsilon_t)=0\), we get \(\sigma(p_t)\le\sigma(p^*_t)\).
$$\operatorname{Var}(p^*_t) = \operatorname{Var}(p_t) + \operatorname{Var}(\varepsilon_t) \;\Longrightarrow\; \sigma(p_t)\le\sigma(p^*_t).$$
谜题。 数据里股价波动 \(\sigma(p_t)\) 远超这个上界——「波动过度 (excess volatility)」，平滑的红利/消费基本面无法解释。它与股权溢价之谜并列。The puzzle. In the data, \(\sigma(p_t)\) far exceeds this bound — "excess volatility" that smooth dividend/consumption fundamentals cannot explain. It is the twin of the equity premium puzzle.
1.3 Multi-Period, Multiple Assets: Continuous Time
1.3.1 Setup
转入连续时间。消费增长与风险资产价格服从扩散过程，\(W_t\) 为布朗运动 (Brownian motion)：
Move to continuous time. Consumption growth and the risky-asset price follow diffusions, with \(W_t\) a Brownian motion:
$$\frac{dc_t}{c_t}=\mu_c\,dt+\sigma_c\,dW_t, \qquad \frac{dS_t}{S_t}=\mu\,dt+\sigma\,dW_t.$$
1.3.2 Investor's Problem
由 HJB 方程求 CRRA 投资者最优，得定价核 (pricing kernel) \(\Lambda_t=e^{-\delta t}u'(c_t)\) 的动态（由伊藤引理作用于 \(c_t^{-\gamma}\)）：
Solving the CRRA investor's optimum via the HJB equation yields the pricing kernel \(\Lambda_t=e^{-\delta t}u'(c_t)\) dynamics (Itô's lemma on \(c_t^{-\gamma}\)):
$$\frac{d\Lambda_t}{\Lambda_t} = -r\,dt - \underbrace{\gamma\,\sigma_c}_{\theta}\,dW_t,$$
其中 \(\theta=\gamma\sigma_c\) 是风险的市场价格 (market price of risk)。
where \(\theta=\gamma\sigma_c\) is the market price of risk.
1.3.3 Continuous-Time Equity Risk Premium
无套利要求贴现价格 \(\Lambda_t S_t\) 是鞅 (martingale)，\(\mathbb{E}_t[d(\Lambda_t S_t)]=0\)，展开得
No-arbitrage requires the discounted price \(\Lambda_t S_t\) to be a martingale, \(\mathbb{E}_t[d(\Lambda_t S_t)]=0\), giving
$$\mu - r = \gamma\,\sigma_{S,c} = \gamma\,\rho\,\sigma\,\sigma_c.$$
证明 / Proof：定价核鞅条件 → 风险溢价
由伊藤乘法 \(d(\Lambda S)=\Lambda\,dS+S\,d\Lambda+d\Lambda\,dS\)，除以 \(\Lambda S\) 令漂移项为零：
By Itô's product rule \(d(\Lambda S)=\Lambda\,dS+S\,d\Lambda+d\Lambda\,dS\); dividing by \(\Lambda S\) and setting the drift to zero:
$$\mu + (-r) + (-\theta)\,\sigma\,\rho = 0 \;\Longrightarrow\; \mu - r = \theta\,\sigma\,\rho = \gamma\,\sigma_c\,\sigma\,\rho. \quad\blacksquare$$
1.3.4 Equity Premium Puzzle
由上式得夏普比率上界——Hansen–Jagannathan 界：
This implies a Sharpe-ratio bound — the Hansen–Jagannathan bound:
$$\frac{\mu-r}{\sigma} = \gamma\,\rho\,\sigma_c \;\le\; \gamma\,\sigma_c.$$
谜题（Mehra–Prescott, 1985）。 历史上 \(\sigma_c\approx1\%\)–\(2\%\)，股票夏普比率约 \(0.4\)，要拟合需 \(\gamma\gtrsim25\)–\(50\)，远超合理范围。这就是股权溢价之谜，催生 Habit、Long-Run Risks、Disaster Risks 等模型。The puzzle (Mehra–Prescott, 1985). Historically \(\sigma_c\approx1\%\)–\(2\%\) and the equity Sharpe ratio ≈ \(0.4\); matching it needs \(\gamma\gtrsim25\)–\(50\), far beyond plausible values. This is the equity premium puzzle, motivating Habit, Long-Run Risks, and Disaster-Risk models.
1.4 Portfolio Choice
1.4.1 Setup
换一个视角：给定收益分布，投资者如何配置？设无风险利率 \(r\)、风险资产超额收益 \(R^e\)、投资比例 \(\omega\)，期末财富 \(W=W_0(1+r+\omega R^e)\)。
A dual view: given the return distribution, how should the investor allocate? With risk-free rate \(r\), risky excess return \(R^e\), and weight \(\omega\), terminal wealth is \(W=W_0(1+r+\omega R^e)\).
1.4.2 The Investor's Problem
最大化期末财富的期望效用 \(\max_\omega\mathbb{E}[u(W)]\)，一阶条件为 \(\mathbb{E}[u'(W)R^e]=0\)。
Maximize \(\max_\omega\mathbb{E}[u(W)]\); the FOC is \(\mathbb{E}[u'(W)R^e]=0\).
1.4.3 Special Case: CARA Utility and Normal Returns
取 CARA 效用 \(u(W)=-e^{-aW}\)、\(R^e\sim\mathcal{N}(\mu_e,\sigma_e^2)\)，则最优风险持仓有闭式解：
With CARA utility \(u(W)=-e^{-aW}\) and \(R^e\sim\mathcal{N}(\mu_e,\sigma_e^2)\), the optimal risky position has a closed form:
$$\phi^{*} = \omega^{*} W_0 = \frac{\mu_e}{a\,\sigma_e^2}.$$
证明 / Proof：CARA-正态下的均值-方差需求
设投入风险资产的金额 \(\phi=\omega W_0\)，则 \(W=W_0(1+r)+\phi R^e\) 服从正态。CARA 下期望效用为
Let \(\phi=\omega W_0\) be the dollar amount in the risky asset, so \(W=W_0(1+r)+\phi R^e\) is normal. Under CARA, expected utility is
$$\mathbb{E}[-e^{-aW}] = -\exp\!\Big(-a\,\mathbb{E}[W] + \tfrac{a^2}{2}\operatorname{Var}(W)\Big).$$
最大化它等价于均值-方差权衡，一阶条件给出最优持仓：
Maximizing it is a mean–variance trade-off, whose first-order condition gives the optimal position:
$$\max_{\phi}\;\; \phi\,\mu_e - \tfrac{a}{2}\,\phi^2\,\sigma_e^2 \;\Longrightarrow\; \phi^{*}=\frac{\mu_e}{a\,\sigma_e^2}. \quad\blacksquare$$
注释。 持仓正比于预期超额收益、反比于风险厌恶与方差——经典「均值-方差」需求。CARA 下金额 \(\phi^*\) 与财富无关；CRRA 下则是比例 \(\omega^*\) 与财富无关。Comment. Holdings rise with the expected excess return and fall with risk aversion and variance — classic mean–variance demand. Under CARA the dollar amount \(\phi^*\) is wealth-independent; under CRRA the fraction \(\omega^*\) is.
1.4.4 Stock Market Participation Puzzle
在 \(\omega=0\) 处求导，\(\frac{\partial}{\partial\omega}\mathbb{E}[u(W)]\big|_{\omega=0}=u'(W_0(1+r))\,\mathbb{E}[R^e]\)。只要股权溢价 \(\mathbb{E}[R^e]>0\)，导数为正——任何风险厌恶者都应持有正数量股票。
Evaluating the derivative at \(\omega=0\): \(\frac{\partial}{\partial\omega}\mathbb{E}[u(W)]\big|_{\omega=0}=u'(W_0(1+r))\,\mathbb{E}[R^e]\). As long as \(\mathbb{E}[R^e]>0\), this is positive — every risk-averse investor should hold some stock.
谜题。 但现实中大量家庭完全不参股。常见解释：固定参与成本 (fixed participation cost) \(K\)、背景风险 (background risk)、信息与信任摩擦。The puzzle. Yet many households hold no stock. Common explanations: a fixed participation cost \(K\), background risk, and information/trust frictions.
References
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