Time for action – analyzing stock returns

Perform the following steps to analyze stock returns:

  1. First, let's calculate simple returns. NumPy has the diff function that returns an array built up of the difference between two consecutive array elements. This is sort of like differentiation in calculus. To get the returns, we also have to divide by the value of the previous day. We must be careful though. The array returned by diff is one element shorter than the close prices array. After careful deliberation, we get the following code:
    returns = np.diff( arr ) / arr[ : -1]

    Notice that we don't use the last value in the divisor. Let's compute the standard deviation using the std function:

    print "Standard deviation =", np.std(returns)

    This results in the following output:

    Standard deviation = 0.0129221344368
  2. The log return is even easier to calculate. We use the log function to get the log of the close price and then unleash the diff function on the result.
    logreturns = np.diff( np.log(c) )

    Normally, we would have to check that the input array doesn't have zeroes or negative numbers. If it did, we would have got an error. Stock prices are, however, always positive, so we didn't have to check.

  3. Quite likely, we will be interested in days when the return is positive. In the current setup, we can get the next best thing with the where function, which returns the indices of an array that satisfies a condition. Just type the following code:
    posretindices = np.where(returns > 0)
print "Indices with positive returns", posretindices

    This gives us a number of indices for the array elements that are positive.

    Indices with positive returns (array([ 0,  1,  4,  5,  6,  7,  9, 10, 11, 12, 16, 17, 18, 19, 21, 22, 23, 25, 28]),)
  4. In investing, volatility measures price variation of a financial security. Historical volatility is calculated from historical price data. The logarithmic returns are interesting if you want to know the historical volatility—for instance, the annualized or monthly volatility. The annualized volatility is equal to the standard deviation of the log returns as a ratio of its mean, divided by one over the square root of the number of business days in a year, usually one assumes 252. Calculate it with the std and mean functions. See the following code:
    annual_volatility = np.std(logreturns)/np.mean(logreturns)
annual_volatility = annual_volatility / np.sqrt(1./252.) 
print annual_volatility
  5. Take note of the division within the sqrt function. Since, in Python, integer division works differently than float division, we needed to use floats to make sure that we get the proper results. Similarly, the monthly volatility is given by:
    print "Monthly volatility", annual_volatility * np.sqrt(1./12.)

What just happened?

We calculated the simple stock returns with the diff function, which calculates differences between sequential elements. The log function computes the natural logarithms of array elements. We used it to calculate the logarithmic returns. At the end of the tutorial we calculated the annual and monthly volatility (see returns.py).

import numpy as np

c=np.loadtxt('data.csv', delimiter=',', usecols=(6,), unpack=True)

returns = np.diff( c ) / c[ : -1]
print "Standard deviation =", np.std(returns)

logreturns = np.diff( np.log(c) )

posretindices = np.where(returns > 0)
print "Indices with positive returns", posretindices

annual_volatility = np.std(logreturns)/np.mean(logreturns)
annual_volatility = annual_volatility / np.sqrt(1./252.)
print "Annual volatility", annual_volatility

print "Monthly volatility", annual_volatility * np.sqrt(1./12.)
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