I. Module 2 Homework 1 [practice]: A Little Model of Time-Varying Expected Returns

Due No due date
Points 18
Questions 10
Time Limit None
Allowed Attempts 2

Instructions

Expected returns vary over time. Here is a nice structure we use to represent this idea:

$LaTeX: \begin{align*} x_{t} & =\phi x_{t-1}+\varepsilon_{t}\\ r_{t+1} & =x_{t}+\delta_{t+1} \end{align*}$
$LaTeX: x_{t}$ denotes the expected return, and $LaTeX: r_{t+1}$ denotes the actual log return. (We usually run these in logs, I showed levels in class because it's easier.) Actual returns are expected returns $LaTeX: x_{t}$ plus unpredictable noise $LaTeX: \delta_{t+1}$ . $LaTeX: \varepsilon_{t+1}$ and $LaTeX: \delta_{t+1}$ can be correlated -- good returns can be positively or negatively associated with good news about expected returns. In fact, $LaTeX: \varepsilon_{t+1}$ and $LaTeX: \delta_{t+1}$ are negatively correlated -- when prices go up we have a good actual return $LaTeX: \delta_{t+1}>0$ but it's bad news for subsequent expected returns $LaTeX: \varepsilon_{t+1}<0$ . In the lecture, I used $LaTeX: x_{t}=a+b\times dp_{t}$ , but we more generally think of expected returns as following a latent (we can't observe it directly) state variable of this form, and then prices reveal $LaTeX: x_{t}$ to us.

We use this sort of time series model widely in finance -- for example, all the term structure models are built this way. It's worth getting familiar with it.

When the problem calls for numerical values, use $LaTeX: \sigma_{\varepsilon} =0.018$ , $LaTeX: \phi=0.94$ , $LaTeX: \sigma_{\delta}=0.18$ and the correlation between $LaTeX: \varepsilon_t$ and $LaTeX: \delta_t$ shocks $LaTeX: \rho=-\frac{\phi}{1-\phi^{2}}\frac {\sigma_{\varepsilon}}{\sigma_{\delta}}=-0.80756$ . These numbers are close to what we see in dividend-yield regressions, and the latter number reverse-engineers a very nice special case.

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