Asset Pricing：Equity Volatility Puzzle

Asset Pricing：Equity Volatility Puzzle

Advances in Return Predictability

Effcient Market Hypothesis

Bachelier (1900): behaviour of prices使得 speculation(投机)应该是一个公平的游戏；股票价格遵循random walk。

Samuelson (1965): In an informationally efficient market price changes 一定是不可预测的 if they fully incorporate expectations of all market participants.

Fama (1970): 价格总是完全反映所有available information的market是有效的。

Malkiel (1992): 如果证券价格不受向所有参与者披露这些信息的影响，那么资本市场是有效的。此外，efficiency with respect to an information set意味着，不可能通过根据该信息集进行交易来获得经济利润。

Assumptions

1. Market equilibrium可以用expected returns的形式表示：
   $$E[p_{j,t+1}|{\Omega }_t]=(1+E[r_{j,t+1}|{\Omega }_t])p_{j,t}$$

2. Equilibrium expected returns are projected on the information set ${\Omega }_t$?.

Market efficiency rules out the possibility of trading systems based solely on ${\Omega }_t$?? that have expected returns in excess of equilibrium returns.（不相关）
$$\begin{gathered} z_{j,t+1}=r_{j,t+1}?E[r_{j,t+1}|{\Omega }_t] \\ E[m_{t+1}?z_{j,t+1}|{\Omega }_t]=0 \end{gathered}$$
The above suggests three approaches to testing market efficiency:

1. Testing whether prices fully reflect all available information: Empirically meaningless. No content. 无法测试
2. By revealing information to market participants and measuring price response: Empirically unfeasible.
3. By measuring profits generated by trading on information: Testable!!

Empirical Strategy

上述第三种方法主要用于两种方式：

1. Researchers have studied the profits generated by market professionals. If superior profits are achieved (after adjusting for risk) then markets cannot be efficient.
2. One can ask whether hypothetical trading rules based on specified information sets earn superior returns. This approach requires a clearly defined information set + a model for risk.

Taxonomy of Information Sets

In order to implement trading based tests an information set must be defined.

1. Weak Form Efficiency: The information set includes only the history of the prices or returns themselves
2. Semi-Strong Form Efficiency: The information set includes all information known to all market participants (publicly available information)
3. Strong Form Efficiency: The information set includes all information known to any market participant (private information)

Abnormal Returns：
$$R_{t+1}^A\equiv R_{t+1}?E_{t+1}^M[R_{t+1}]$$
The null of market efficiency is then :（零假设）
$$H_0:E[m_{t+1}?R_{t+1}^A|{\Omega }_t]=0$$
If abnormal returns are unforecastable then the hypothesis of market efficiency is not rejected.

Joint Hypothesis Problem

The null of market efficiency contains an implicit joint hypothesis that:

1. Markets are efficient

2. The correct model for risk has been specified.

   $m_{t+1}=\frac{u^{\prime }(c_{t+1})}{u^{\prime }(c_t)}\to$ Risk Aversion

   for example：$u(c_t)=a{c}_t+b,m_{t+1}\equiv 1\to$? risk neutral

The debate between rational expectations models Vs irrational behavioural models is captured by the tension implicit in the joint hypothesis problem.

一些研究表明：Abnormal returns exist if there are costs of gathering and processing information.

Modern Taxonomy of the Efficient Market Hypothesis

Tests for predictability: (Weak Form)

1. Time-Series
2. Cross-Sectional

Event studies: (Semi-Strong Form)

1. Investigate information based studies after release of public information (see Malkiel’s definition) to test for abnormal returns.
2. Since time horizons are so short risk adjustment is unimportant so we avoid the joint hypothesis problem.

Tests for private information / superior performance: (Strong Form)

1. Mutual fund / Hedge fund performance（共同基金/对冲基金）
2. Insider Trading（内部交易）

Do Stock Prices Move Too Much to be Justified by Subsequent Changes in Dividends?, Shiller (1981)

将总额（real）回报定义为：
$$R_{i,t+1}=\frac{D_{i,t+1}+P_{i,t+1}}{P_{i,t}}$$
Taking expectations and rearranging
$$P_{i,t}=\frac{E_t[D_{i,t+1}]+E_t[P_{i,t+1}]}{E_t[R_{i,t+1}]}$$
Prices vary because conditional expected dividends vary or conditional expected returns vary.

Imposing constant expectations over time, assuming no-bubbles, and iterating forward:
$$P_{i,t}=\sum _{k=1}^{\infty }\frac{E[D_{i,t+k}]}{E[R{]}^k}$$
假设该结果也适用于总体股票市场：
$$P_t=\sum _{k=1}^{\infty }\frac{E[D_{t+k}]}{E[R{]}^k}$$
Shiller(1981) shows that the variance of the LHS is higher than the plausible variance of the RHS. Stock market volatility is too high to accord with rational expectations.

Prices and dividends appear to grow at a constant exponential rate $\lambda$?. Detrend both series by this rate:
$$\begin{gathered} {\lambda }^t{d}_t=D_t,var(d_t)<\infty \\ {p}_t\equiv P_t/{\lambda }^t=\sum _{k=1}^{\infty }\frac{E[D_{t+k}]}{{\lambda }^{t}E[R{]}^k}=\sum _{k=1}^{\infty }\frac{E[{\lambda }^{t+k}{d}_{t+k}]}{{\lambda }^{t}E[R{]}^k}=\sum _{k=1}^{\infty }E[R{]}^{?k}{\lambda }^{k}E[d_{t+k}] \\ {p}_t=\sum _{k=1}^{\infty }E[\bar{R}{]}^{?k}E[d_{t+k}] \end{gathered}$$
Idea : ex-post rational prices
$$p_t^{?}=\sum _{k=1}^{\infty }E[\bar{R}{]}^{?k}{d}_{t+k}$$
Decompose dividends into expected and unexpected components:（分解股息）
$$d_{t+k}=E_t[d_{t+k}]+{\tilde{d}}_{t+k}$$
Then ex-post rational prices are related to actual prices via:
$$\begin{gathered} p_t^{?}=\sum _{k=1}^{\infty }E[\bar{R}{]}^{?k}{d}_{t+k}=\sum _{k=1}^{\infty }E[\bar{R}{]}^{?k}(E_t[d_{t+k}]+{\tilde{d}}_{t+k}) \\ =\sum _{k=1}^{\infty }E[\bar{R}{]}^{?k}{E}_t[d_{t+k}]+\sum _{k=1}^{\infty }E[\bar{R}{]}^{?k}{\tilde{d}}_{t+k} \\ =p_t+\sum _{k=1}^{\infty }E[\bar{R}{]}^{?k}{\tilde{d}}_{t+k} \end{gathered}$$
方差：$var[p_t^{?}]\ge var[p_t]$

1. deflate series by CPI to get real prices and dividends.

2. Estimate $\lambda$:
   $$\log (P_t)=a+bt+{\eta }_t,\lambda =e^b$$

3. detrend $P_t,D_t$?.

4. Taking unconditional expectations to estimate the discount rate:
   $$E[p_t]=\frac{1}{E[\bar{R}]?1}E[d_t]$$

5. Construct ex-post rational prices using terminal condition
   $$p_t^{?}=\sum _{k=1}^{\infty }E[\bar{R}{]}^{?k}{d}_{t+k}+E[\bar{R}{]}^{?T}{p}_T^{?}$$

Results:

1. $var[p_t]\ge var[p_t^{?}]?$? ratio of 5 to 13.
2. A single big picture delivers the punchline: a key moment condition is violated.
3. Problems: de-trended dividends are non-stationary so $var(p_t)$ does not even exist.

Fama and French (1988): Permanent and Temporary Components of Stock Prices

Fama and French ran the following long-horizon forecasting regression:
$$R_{t,t+T}=b_{0,T}+b_{1,T}{R}_{t?T,t}+{\epsilon }_{t,t+T}$$
Statistical Issues :

1. Finite sample bias in AR(1) regressions - coefficients negatively biased.
2. OLS standard errors are wrong because of overlapping observations.
3. None-the-less robust standard errors are computed and Monte Carlo simulations are used to correct unbiased coefficents.

Fama and French do some data mining to forecast excess returns on stocks and bonds at various horizons:
$$ExRet(t,t+T)=\alpha (T)+\beta (T)X(t)+\epsilon (t,t+T)$$
No strong theoretical motivation.

They identify forecasting variables that have been used extensively in subsequent work:

1. dividend / price ratio : $D/P$.
2. default premium : $Baa?Aaa$ corporate bond spread.
3. Slope of the term structure : $y(n,t)?y(1,t)$.

Present Value Identity

Campbell & Shiller’s (1988) decomposition.
$$\begin{gathered} 1=R_{t+1}^{?1}{R}_{t+1}=R_{t+1}^{?1}\frac{P_{t+1}+D_{t+1}}{P_t} \\ \frac{P_t}{D_t}=R_{t+1}^{?1}\frac{P_{t+1}+D_{t+1}}{D_t}=R_{t+1}^{?1}(1+\frac{P_{t+1}}{D_{t+1}})\frac{D_{t+1}}{D_t} \end{gathered}$$
对数线性化：
$$\begin{gathered} p_t?d_t=?r_{t+1}+\Delta d_{t+1}+\log (1+e^{p_{t+1}?d_{t+1}}) \\ P/D=\exp (p?d) \\ f(x)=\log (1+e^x) \\ \rho =\frac{1}{1+e^{p?d}} \\ f(x)\approx f(p?d)+f^{\prime }(p?d)(x?(p?d))=\log (1+P/D)+\frac{P/D}{1+P/D}(x?(p?d)) \\ =\log (1+P/D)?\frac{P/D}{1+P/D}(p?d)+\frac{P/D}{1+P/D}x \\ =?\log \rho ?\frac{P/D}{1+P/D}(p?d)+(1?\rho )x \\ k=?\log \rho ?\frac{P/D}{1+P/D}(p?d) \\ {p}_t?d_t=?r_{t+1}+\Delta d_{t+1}+k+(1?\rho )(p_{t+1}?d_{t+1}) \end{gathered}$$
Iterating 1 forward and taking conditional expectations we get:
$$p_t?d_t=const.+{\mathbb{E}}_t[\sum _{j=1}^{\infty }(1?\rho {)}^{j?1}(\Delta d_{t+j}?r_{t+j})]$$
Equation 2 is obtained by ruling out the explosive behaviour of stock prices where ${\lim }_{j\to \infty }(1?\rho {)}^j(p_{t+j}?d_{t+j})=0$???. This is equivalent to ruling out bubbles.

1. Price-dividend ratios can move if and only if there is news about current dividends, future dividend growth or future returns.

2. If $\Delta d_t,r_t$ are totally unpredictable, i.e. if $E_t(\Delta d_{t+j}),E_t(r_{t+j})$ are the same for every time t, then $p_t?d_t$ must be constant (which we know isn’t true!).

The Variance of Price / Dividend Ratios

If we forget the constant i.e. treat variables as deviations from the mean（均值取0）：
$$\begin{gathered} \mathbb{E}[(p_t?d_t)(p_t?d_t)]=\mathbb{E}[(p_t?d_t)\times \sum _{j=1}^{\infty }(1?\rho {)}^{j?1}(\Delta d_{t+j}?r_{t+j})] \\ Var(p_t?d_t)\approx Cov[p_t?d_t,\sum _{j=1}^{\infty }(1?\rho {)}^{j?1}\Delta d_{t+j}]?Cov[p_t?d_t,\sum _{j=1}^{\infty }(1?\rho {)}^{j?1}{r}_{t+j})] \end{gathered}$$
这说明：$p?d$?? varies if and only if it either dividend growth is predictable or that future returns are predictable!

两边同除 $Var(p_t?d_t)$：
$$1\approx \sum _{j=1}^{\infty }(1?\rho {)}^{1?j}{b}_d^{(j)}?\sum _{j=1}^{\infty }(1?\rho {)}^{1?j}{b}_r^{(j)}$$
where $b^{(j)}$? means the j-year ahead regression coefficient:
$$\begin{gathered} r_{t+j}=a_r^{(j)}+b_r^{(j)}(p_t?d_t)+{?}_{t+j}^r \\ \Delta d_{t+j}=a_d^{(j)}+b_d^{(j)}(p_t?d_t)+{?}_{t+j}^d \end{gathered}$$
一期模型中：
$$1\approx b_d?b_r$$
Suppose $b_r=0$, thus $b_d\approx 1$ and:
$$\Delta d_{t+1}=a_d+1.0(p_t?d_t)+{?}_{t+1}^d$$
It seems that all variation in P/D ratios is due to the discount channel and none due to the cashflow channel!