I. Module 1 Homework 1 [final]: Horizon Effects in Returns

Due No due date
Points 33
Questions 29
Time Limit None

Instructions

This is a final version of this assignment. You may attempt it once only.

This assignment gives you some practice and review in discrete-time time-series manipulations. You'll also answer some interesting questions -- how should returns behave at different horizons, daily, weekly, monthly, annual, etc. You'll learn about the standard errors of returns, and why it's so hard to measure premiums in the stock market. And you'll investigate whether correlation properties of returns make stocks more or less attractive to long-run investors.

Suppose first that one-year log returns $r_{t}=\log(R_{t})$ are not correlated over time $cov(r_{t},r_{t+j})=0$ and have mean $E(r_{t})=\mu$ and variance $\sigma^{2}\left\{r_{t}\right\}=\sigma^{2}$ that are constant over time.

The compound log long-horizon return is $\log(R_{t+1}R_{t+2}..R_{t+k})=r_{t+1}+r_{t+2}+..+r_{t+k}$ and the annualized compound log long-horizon return is

$LaTeX: \log\left[ (R_{t+1}R_{t+2}..R_{t+k})^{\frac{1}{k}}\right] =\frac{1}{k}\left[ r_{t+1}+r_{t+2}+..+r_{t+k}\right].$ Now, let's think about how these returns scale with horizon. The big question underlying this analysis is: are returns in some sense "safer" for long-horizon investors?

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