Asset Pricing：Equity Premium Puzzle_综合

Asset Pricing：Equity Premium Puzzle

Consider a security with price $p_t$ ? at time $t$ that pays off an amount $x_{t+1}$ ?? at time $t + 1$ . Using the SDF representation of the price we can write:
$p_t=\mathbb E_t[m_{t+1}x_{t+1}]=\mathbb E_t[m_{t+1}]\mathbb E_t[x_{t+1}]+Cov_t[m_{t+1},x_{t+1}]$
for a given $m_{t+1}$ , $R^f_t=\dfrac{1}{\mathbb E_t[m_{t+1}]}\to p_t=\dfrac{\mathbb E_t[x_{t+1}]}{R^f_t}+Cov_t[m_{t+1},x_{t+1}]$ ?

Where the first term is the discounted expected payoff, and the second term is a risk adjustment.

In returns we have:
$\mathbb E_t[m_{t+1}R_{t+1}]=1,R^f_t=\dfrac{1}{\mathbb E_t[m_{t+1}]}\\\to\mathbb E_t[m_{t+1}(R_{t+1}-R_t^f)]=0\\\to\mathbb E_t[m_{t+1}]\mathbb E_t[R_{t+1}-R_t^f]+Cov_t[m_{t+1},R_{t+1}-R_t^f]=0\\\to\mathbb E_t[R_{t+1}-R_t^f]=-\dfrac{Cov_t[m_{t+1},R_{t+1}]}{\mathbb E_t[m_{t+1}]}$
for a generic portfolio $h\in\mathbb R^J$ ：
$\mathbb E_t[(R_{t+1}-R_t^f)h]=-\dfrac{Cov_t[m_{t+1},R_{t+1}h]}{\mathbb E_t[m_{t+1}]}=-\dfrac{\rho_t(m_{t+1},R_{t+1}h)\sigma_t(m_{t+1})\sigma_t(R_{t+1}h)}{\mathbb E_t[m_{t+1}]}$
Note that all results hold also for the unconditional expectation $\mathbb E[·]$ ?, so we can drop the expectation subscript:
$\mathbb E[(R_{t+1}-R_t^f)h]=-\dfrac{Cov[m_{t+1},R_{t+1}h]}{\mathbb E[m_{t+1}]}=-\dfrac{\rho(m_{t+1},R_{t+1}h)\sigma(m_{t+1})\sigma(R_{t+1}h)}{\mathbb E[m_{t+1}]}\\-\dfrac{\sigma(m_{t+1})}{\mathbb E[m_{t+1}]}\rho(m_{t+1},R_{t+1}h)=\frac{\mathbb E[(R_{t+1}-R_t^f)h]}{\sigma(R_{t+1}h)}\\\dfrac{\sigma(m_{t+1})}{\mathbb E[m_{t+1}]}|\rho(m_{t+1},R_{t+1}h)|=|\frac{\mathbb E[(R_{t+1}-R_t^f)h]}{\sigma(R_{t+1}h)}|$
since for any $h\in\mathbb R^J,\rho(·)\in[-1,1]$ :
$\dfrac{\sigma(m_{t+1})}{\mathbb E[m_{t+1}]}\geq\sup_{h\in\mathbb R^J}|\frac{\mathbb E[(R_{t+1}-R_t^f)h]}{\sigma(R_{t+1}h)}|$
So we prove the following theorem:

Theorem (Hansen-Jagannathan Bound): The ratio of the standard deviation of a stochastic discount factor to its mean exceeds the Sharpe Ratio attained by any portfolio.

可以用这个theorem来对提出的stochastic discount factor进行可行性分析。 如果提出的 $m$ 满足上述不等式，说明 $m$ 可行。

另一方面，在给定的SDF $m$ ? 下，这个定理也可以帮助我们理解资产组合Sharpe Ratio的上界。

Finance theory explains the expected excess return on any risky asset over the riskless interest rate as the quantity of risk times the price of risk. Such as Hansen-Jagannathan Bound:
$E(R_x)\leq\frac{\sigma(m)}{E(m)}\sigma(R_x)$
in which : $\sigma(R_x)$ is the quantity of risk, and $\dfrac{\sigma(m)}{E(m)}$ ? is the market price of risk.

Equity Premium Puzzle:

for any asset j :
$\mathbb E[R^j-R^f]=-\dfrac{Cov[m,R^j]}{\mathbb E[m]},R^f=\frac{1}{\mathbb E[m]}\\\mathbb E[R^j]-R^f=-R^f\times Cov[m,R^j]\\\mathbb E[R^j]-R^f=-R^f\times\frac{Cov[\partial_1u,R^j]}{E[\partial_0u]}$
so Hansen-Jagannathan Bound can be rewritted as:
$\sigma(m)\geq\frac{1}{R^f}|\frac{\mathbb E[R^j-R^f]}{\sigma(R^j)}|\\\sigma(\frac{\partial_1u}{\mathbb E[\partial_0u]})\geq\frac{1}{R^f}|\frac{\mathbb E[R^j-R^f]}{\sigma(R^j)}|$
However:

The right hand side of this inequality, the Sharpe Ratio of any asset, can be very high for some portfolios.
Consumption volatility is generally low.

Together, to be consistent with the Hansen-Jagannathan bound these two facts imply that the curvature of the utility function $u$ ? must be very high. But curvature is synonimous for risk aversion, and to justify the empirical data we should accept that agents have an unrealistically high level of risk aversion. This inconsistency between theory and empirical facts is commonly referred to as the Equity Premium Puzzle.

Equity premium review:

Simple bound：
$R^e=R^j-R^f\\\frac{|\mathbb E(R^e)|}{\sigma(R^e)}\leq\frac{\sigma(m)}{\mathbb E(m)}\approx\gamma\sigma(\Delta c)$
把数据带进去：
$\frac{|\mathbb E(R^e)|}{\sigma(R^e)}\leq\frac{\sigma(m)}{\mathbb E(m)}\approx\gamma\sigma(\Delta c)\\0.5=\frac{0.08}{0.16}\leq\gamma\times0.015\\\gamma\geq33$
This seems like a lot of risk aversion.
Correlation puzzle：把 $\rho=0.4$ 引入：
$\frac{\mathbb E(R^e)}{\sigma(R^e)}=\rho\frac{\sigma(m)}{\mathbb E(m)}\approx\rho\gamma\sigma(\Delta c)\\\frac{0.5}{0.4}=\gamma\times0.015\\\gamma=83$
Risk free rate puzzle：If we accept huge $\gamma$ , then $E(\beta(c_{t+1}/c_t)^{-\gamma})=1/E(m)$ goes nuts. With $m=e^{-\delta-\gamma\Delta c}$ and normality or in continuous time
$r^f=\delta+\gamma E(\Delta c)-\frac{1}{2}\gamma(\gamma+1)\sigma^2(\Delta c)\\r^f=\delta+\gamma\times0.01-\frac{1}{2}\gamma(\gamma+1)(0.015)^2\\0.01=0.01+\gamma\times0.01-\frac{1}{2}\gamma(\gamma+1)(0.015)^2\\\gamma=87.889$
High variance causes people to save more (precautionary). This effect can offset the fact that a high mean causes then to want to consume more now.

Comments and Finance

The puzzle is quantitative not qualitative. Consumption growth is positively correlated with stock returns.
The puzzle requires us to integrate financial and macro data. You can’t see it from prices alone.
Traditional asset pricing was about the CAPM, $E(R^{ei})=\beta_iE(R^{em})$ . This takes $ER^{em})$ for granted and doesn’t ask where it came from. For example, the log utility iid CAPM predicts $C_{t+1}/C_t=R_m^{-1}$ ??. The whole point of CAPM is to tie consumption to the market return, and substitute out for consumption growth in marginal utility.
Traditional portfolio theory predicts
$\omega=\frac{1}{\gamma}\frac{E(R^e)}{\sigma^2(R^e)}$
Thus, you hold all equity if
$1=\frac{1}{\gamma}\frac{0.08}{0.16^2}\\\gamma=3.125$
This seems perfectly sensible, and 4 years of portfolio theory has happily used this formula. What’s wrong? The portfolio theory also says $\Delta C=R$ . Why take portfolio advice and not consumption advice? If you don’t like consumption advice this is the wrong portfolio model!