Calculating Stock Beta, Volatility, and More

Beta, Alpha and R-squared

Beta of a stock is a measure of relative risk of the stock with respect to the market. The convention (though not a rule) is to use S&P 500 index as the proxy for market. A beta value of greater than 1 means that the stock returns amplify the market returns on both the upside and downside. On the contrary, a beta value of less than 1 means that the stock returns are subdued in comparison to the market returns. It is very important to understand that beta is a relative measure of risk, and not an absolute measure of risk. That means that we are only saying how risky the stock is vis-a-vis the market. If the stock market itself is overheated and volatile, then a beta of 1 means that the stock is equally volatile, and equally risky.

The beta is calculated using regression analysis. In simple terms if you plot the returns of stock as a function of the returns of the market benchmark (such as S&P 500) and fit it with a straight line, then beta is nothing but the slope of the fitted line. The regression equation is given as shown below.

$$ R^{stock}_i = \alpha + \beta \times R^{market}_i + \epsilon_i $$

In Capital Asset Pricing Model, the returns of the stock $R^{stock}$ and that of the market $R^{market}$ are adjusted for the risk-free rate. Here for simplicity I leave that out. Given the current low interest rate environment, this adjustment will add little value.

Alpha of a stock gives you a measure of the excess return with respect to the benchmark. A positive alpha for a stock or portfolio gives you sense of how well your asset outperformed a benchmark.

The $R^2$ is a measure of how well the the returns of a stock is explained by the returns of the benchmark. If your investment goal is to track a particular benchmark, then you should chose stocks that show a high $R^2$ with respect to the benchmark. The $R^2$ value of $1$ means that the benchmark completely explains the stock returns, while a value of $0$ means that the benchmark does not explain the stock returns.

R-squared is defined as:

$$ R^2 = 1 - SS_{res}/SS_{tot} $$

where

$$ SS_{res} = \sum_i (R^{stock}_i - f^{stock}_i)^2 $$

$$ SS_{tot} = \sum_i (R^{stock}_i - <R^{stock}>)^2 $$

Volatility and Momentum

The measures discussed in the earlier section are what I would call relative measures, i.e., they are with respect to a proxy that is a representation of market. Time series measures such as volatility and momentum are what I would call innate measures.

Volatility is nothing but the standard deviation of the returns of the stock. It gives us a sense of how much the stock returns fluctuate and how risky it is. High volatile stocks have high risk, and also have the potential to offer higher returns. A 3-year history of 1-month returns can be a good sample to calculate volatility.

Momentum is a measure of the past returns over a certain period. The 1-year momentum will be the 1-year return of the stock, where as a 3-year momentum will be the 3-year return of the stock.

Python Code

One can use data from yahoo finance to calculate the stock beta as shown:

from pandas.io.data import DataReader
from datetime import date
import numpy as np
import pandas as pd

# Grab time series data for 5-year history for the stock (here AAPL)
# and for S&P-500 Index
sdate = date(2008,12,31)
edate = date(2013,12,31)
df = DataReader('WFM','yahoo',sdate,edate)
dfb = DataReader('^GSPC','yahoo',sdate,edate)

# create a time-series of monthly data points
rts = df.resample('M',how='last')
rbts = dfb.resample('M',how='last')
dfsm = pd.DataFrame({'s_adjclose' : rts['Adj Close'],
                        'b_adjclose' : rbts['Adj Close']},
                        index=rts.index)

# compute returns
dfsm[['s_returns','b_returns']] = dfsm[['s_adjclose','b_adjclose']]/\
    dfsm[['s_adjclose','b_adjclose']].shift(1) -1
dfsm = dfsm.dropna()
covmat = np.cov(dfsm["s_returns"],dfsm["b_returns"])

# calculate measures now
beta = covmat[0,1]/covmat[1,1]
alpha= np.mean(dfsm["s_returns"])-beta*np.mean(dfsm["b_returns"])

# r_squared     = 1. - SS_res/SS_tot
ypred = alpha + beta * dfsm["b_returns"]
SS_res = np.sum(np.power(ypred-dfsm["s_returns"],2))
SS_tot = covmat[0,0]*(len(dfsm)-1) # SS_tot is sample_variance*(n-1)
r_squared = 1. - SS_res/SS_tot
# 5- year volatiity and 1-year momentum
volatility = np.sqrt(covmat[0,0])
momentum = np.prod(1+dfsm["s_returns"].tail(12).values) -1

# annualize the numbers
prd = 12. # used monthly returns; 12 periods to annualize
alpha = alpha*prd
volatility = volatility*np.sqrt(prd)

print beta,alpha, r_squared, volatility, momentum

Some caveats about the sample code. The returns are calculated using the adjusted close from Yahoo finance data. This is because the adjusted close accounts for dividends and splits etc. In my personal experience I have found the returns calculated this way to be a reasonably close estimate but not always accurate. The volatility is calculated here as a simple standard deviation of the returns. From an option-pricing model perspective volatility is calculated assuming a log-normal distribution for the returns.

The alpha shown above is annualized by scaling by a factor of 12, the periodicity of returns. The same goes for volatility, which is scaled by $\sqrt{12}$ in order to annualize.