11. Suggested Explanation: Rare Events
罕见事件 (rare events) 思路：即使灾难极少发生，持有风险资产是对"灾难来临时一起暴跌"的补偿，这种尾部风险就能在合理风险厌恶下产生高股权溢价。Barro (2006) 在完全市场、代表性代理人框架下，给消费增长加一个小概率的灾难性向下跳跃，用历史灾难分布校准，能同时匹配股权溢价与无风险利率。但该思路也有争议：灾难窗口与溢价度量的"窗口错配"、期权隐含的灾难概率远低于 Barro 校准值、以及罕见灾难让所有股票同跌从而削弱横截面解释力。
The rare-events idea: even if disasters almost never happen, holding risky assets compensates for "crashing together when disaster strikes," and this tail risk can generate a high equity premium under reasonable risk aversion. Barro (2006), in a complete-market representative-agent framework, adds a small-probability disastrous downward jump to consumption growth and, calibrating to the historical disaster distribution, matches both the equity premium and the risk-free rate. The approach is debated, though: a "window mismatch" between the disaster window and the premium measurement, option-implied disaster probabilities far below Barro's calibration, and rare disasters making all stocks fall together, which weakens cross-sectional explanatory power.
11.1 Disastrous Aggregate Shocks: Barro (2006)
11.1.1 Setup
完全市场、代表性代理人。单一证券支付红利 \(A_t\)（即消费品）；\(t+1\) 期支付 \(A_{t+1}\) 作为消费交给代表性代理人。CRRA 偏好：
Complete market, representative agent. A single security pays dividend \(A_t\) (the consumption good); the period-\(t+1\) payoff \(A_{t+1}\) is delivered as consumption to the representative agent. CRRA preferences:
$$U_t=\mathbb E_t\!\left[\sum_{s=0}^\infty e^{-\rho s}u(C_{t+s})\right],\qquad u(C)=\frac{C^{1-\theta}-1}{1-\theta}.$$
11.1.2 Asset Price
由代表性代理人的最优化（如 (1.20)），\(t\) 期资产价格 (11.1)：
From the representative agent's optimization (e.g. (1.20)), the period-\(t\) asset price (11.1):
$$P_t=\mathbb E_t\!\left[e^{-\rho}\Big(\frac{A_{t+1}}{A_t}\Big)^{-\theta}A_{t+1}\right]\ \Rightarrow\ P_t=(A_t)^\theta\,\mathbb E_t\!\left[e^{-\rho}(A_{t+1})^{1-\theta}\right].\tag{11.1}$$
11.1.3 Disastrous Aggregate Shock
对数聚合红利（消费）\(A_{t+1}\) 服从带跳跃的随机游走 (11.2)：
The log aggregate dividend (consumption) \(A_{t+1}\) follows a random walk with a jump (11.2):
$$v_{t+1}=\begin{cases}0 & \text{with probability }e^{-p},\\[2pt]\ln(1-b) & \text{with probability }1-e^{-p}.\end{cases}\tag{11.2}$$
- \(u_{t+1}\) 为正态扰动。
- \(b\) 是待校准的随机变量，刻画消费向下跳跃的幅度：\(b\) 接近 1 时灾难极其严重，\(A_{t+1}\) 趋于零。
- \(p\) 接近 0 时，\(v_{t+1}=0\) 的概率接近 1，灾难 \(v_{t+1}=\ln(1-b)\) 罕见发生。
- \(u_{t+1}\) is a normal disturbance.
- \(b\) is a random variable to be calibrated, capturing the magnitude of the downward consumption jump: when \(b\) is close to 1 the disaster is extremely severe and \(A_{t+1}\) goes to zero.
- When \(p\) is close to 0, \(v_{t+1}=0\) with probability close to 1, so the disaster \(v_{t+1}=\ln(1-b)\) is rare.
11.1.4 Calibration
由 (11.2) 改写 (11.1)。因 \(u_{t+1},v_{t+1}\) 独立，\(\mathbb E_t[f(u_{t+1})g(v_{t+1})]=\mathbb E_t[f(u_{t+1})]\,\mathbb E_t[g(v_{t+1})]\)。记股权（总）收益 \(R^e_{t+1}=\frac{A_{t+1}}{P_t}\)、无风险利率 \(R^f_{t+1}\)。展开得股权溢价 (11.3) 与无风险利率 (11.4)：
Rewrite (11.1) using (11.2). Since \(u_{t+1},v_{t+1}\) are independent, \(\mathbb E_t[f(u_{t+1})g(v_{t+1})]=\mathbb E_t[f(u_{t+1})]\,\mathbb E_t[g(v_{t+1})]\). Let the equity (gross) return be \(R^e_{t+1}=\frac{A_{t+1}}{P_t}\) and the risk-free rate \(R^f_{t+1}\). Expanding gives the equity premium (11.3) and the risk-free rate (11.4):
$$\ln\mathbb E_t[R^e_{t+1}]=\rho+\theta\gamma-\frac{\theta^2\sigma^2}{2}+\theta\sigma^2+\ln\frac{e^{-p}\big(1-\mathbb E[1-b]\big)+\mathbb E[1-b]}{e^{-p}\big(1-\mathbb E[1-b]^{1-\theta}\big)+\mathbb E[1-b]^{1-\theta}}.\tag{11.3}$$
$$\ln R^f_{t+1}=\rho+\theta\gamma-\frac{\theta^2\sigma^2}{2}-\ln\mathbb E\!\left[e^{-p}+(1-e^{-p})\,\mathbb E[1-b]^{-\theta}\right].\tag{11.4}$$
(11.3)、(11.4) 给出股权溢价与无风险利率的表达式。方法论：把灾难概率 \(p\)、幅度 \(b\) 等更多对象显式写出用于校准，再用 (11.3)、(11.4) 拟合数据。
(11.3), (11.4) give the equity-premium and risk-free-rate expressions. Methodology: write out more objects explicitly (the disaster probability \(p\), magnitude \(b\), etc.) for calibration, then fit the data with (11.3), (11.4).
11.1.5 Summary
- 贡献： 提出一个聚合层面小概率灾难模型；用聚合灾难的经验分布校准，灾难带来的风险溢价 + 由校准匹配的无风险利率，能同时解释双重之谜。
- 批评：
- 窗口错配。 聚合消费的灾难（峰谷跌幅）若用四年窗口校准，则股权溢价也应按四年窗口度量（约 24%），而非按一年（6%）——否则产生错配偏误。Barro (2006) 之后许多论文沿用这一错配传统，具有误导性。
- Backus et al. (2011) 用股指期权反推隐含灾难参数，发现投资者感知的灾难概率远低于 Barro (2006) 的校准值。
- Julliard and Ghosh (2012) 指出：用罕见灾难拟合股权溢价的唯一办法是假设极高的灾难概率，不现实；且罕见灾难让所有股票同跌，削弱了 C-CAPM 拟合横截面超额收益的能力（横截面消费风险离散度被压缩）。
- Contribution: propose an aggregate-level small-probability disaster model; calibrating to the empirical distribution of aggregate disasters, the disaster-driven risk premium plus a calibration-matched risk-free rate jointly explain the dual puzzle.
- Critiques:
- Window mismatch. If the aggregate-consumption disaster (peak-to-trough decline) is calibrated over a four-year window, the equity premium should also be measured over four years (about 24%), not one year (6%) — otherwise a mismatch bias arises. Many papers after Barro (2006) follow this misleading tradition.
- Backus et al. (2011) back out the implied disaster parameter from equity-index options and find the disaster probability perceived by investors is much lower than Barro's (2006) calibration.
- Julliard and Ghosh (2012) point out that the only way to fit the equity premium with rare disasters is to assume an extremely high disaster probability, which is unrealistic; and rare disasters make all stocks fall together, weakening the ability of C-CAPM to fit cross-sectional excess returns (the cross-sectional dispersion of consumption risk is compressed).
References
- Backus, D., M. Chernov and I. Martin (2011). Disasters Implied by Equity Index Options. The Journal of Finance 66(6), 1969–2012.
- Barro, R. J. (2006). Rare Disasters and Asset Markets in the Twentieth Century. The Quarterly Journal of Economics 121(3), 823–866.
- Julliard, C. and A. Ghosh (2012). Can Rare Events Explain the Equity Premium Puzzle? The Review of Financial Studies 25(10), 3037–3076.
- Rietz, T. A. (1988). The Equity Risk Premium: A Solution. Journal of Monetary Economics 22(1), 117–131.