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:

Commodity futures prices can also be downloaded from the CBOE.

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

Project suggestion:

  1. Download a set of indices and individual stock prices from your favorite source. For example download the csv files at the bottom of the following websites:
    1. S&P 500 index
    2. VIX (click historical prices on left and download from bottom left)
    3. Your favorite company, for example Apple: type AAPL or Apple on the top left bar, then historical prices, then download
    4. A currency, for example, type EURUSD=X for the Euro-USD exchange rate
    5. 30 year Treasury Yield (^TYX)

Here are some suggestions:

  • Calculate the log-returns of all these assets
  • Make a histogram of the asset return distribution: what do you notice. Do some returns look normally distributed, are some asymmetrical, skewed?
  • Try to fit a distribution. First find the sample mean and the standard deviation. Then take a random subsample of returns and test if it has the same mean, standard deviation that the whole period of returns (assuming this is the population).
  • Find a well fitting distribution and use a contingency table to check for goodness of fit.
  • Take the VIX and S&P returns and regress the VIX on the S&P returns. Calculate the correlation. Notice that the VIX is a measure of implied volatility of the S&P 500 stock index, which usually goes up when stock prices go down fast. Also regress the interest rate on the S&P. What can you conclude about the behavior of interest rates relative to equity (S&P).
  • 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:
(2)
\begin{align} \triangle y_{t+1}=\frac{y_{t+1}}{y_t} \end{align}
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