Project 1: Stock return distribution

Stock prices fluctuate daily due to analyst expectations, positive and negative surprises affecting the company's bottom line and even due to buying and selling pressure by people buying and selling the stock.

The random distribution of the stock price is difficult to discern, but the stock price returns follow a roughly normal distribution, particularly when the return is defined as a log-return.

If the stock price of a company on day $t$ is defined by $S_t$, then the log-return of the stock on day $t+1$ is

(1)
\begin{align} R_{t+1}=\ln\left(\frac{S_{t+1}}{S_t}\right) \end{align}

Stock returns have many statistically interesting properties:

1. log-returns are (roughly) normally distributed
2. some stocks and stock indices are more skewed (VIX for example) and/or have fat tails
3. some assets have a high correlation over time
4. stock (and option) prices can be modeled using mathematical models like Brownian motions

The first 3 of the above statements can be tested by doing hypothesis tests and goodness of fit/contingency tables. Stock prices can easily be downloaded and saved to csv from the following sources:

Other economic indicators can be obtained from the federal reserve website.

Project suggestion:

• For the treasury yield, instead of searching for the distribution of the log-returns, analyze the distribution of the yield differences. So if $y_t$ is the yield (interest rate) on day $t$, then the yield change (difference) is: