Calculating Rolling Betas with CAPM: A Comparative Analysis Using R

Understanding the CAPM.beta Rollapply Functionality

Background and Introduction

The Capital Asset Pricing Model (CAPM) is a widely used framework in finance to explain the relationship between the expected return on an investment and its risk level. The CAPM-beta, also known as the systematic risk or beta of an asset, measures how much an asset’s returns are influenced by market fluctuations.

In this blog post, we’ll explore the CAPM.beta.rollapply function from the PerformanceAnalytics package in R, which calculates rolling betas for a given set of stocks and a proxy for market returns. We’ll delve into the technical details behind this functionality, including how it compares to other methods of calculating rolling betas.

Setting Up the Example

To begin with, let’s create some sample data using the xts package in R:

library(xts)

We’ll use a proxy for market returns (S&P 500) and two stocks: AMZN (Amazon) and XOM (ExxonMobil). The getSymbols() function is used to retrieve historical price data, which we then convert into a time series object:

assets <- c("SPY", "AMZN", "XOM")
getSymbols(assets, from = "2017-01-01", auto.assign = TRUE)

asset_prices <- xts()
asset_prices <- Reduce(f=function(x,y) {y_sym=eval(as.name(y));  merge(x,y_sym[,paste0(y,".Adjusted")])},
                     x = assets, init=asset_prices)

We calculate the returns on these stocks and market return:

asset_returns <- diff.xts(asset_prices, arithmetic = FALSE, na.pad=FALSE)-1

market_return <- asset_returns$SPY.Adjusted
stock_returns <- asset_returns[,-1]

Next, we define a window size for our rolling calculations (40 trading days):

width_cor <- 40

Calculating Rolling Betas with CAPM.beta.rollapply

The rollapply function from the PerformanceAnalytics package is used to calculate rolling betas:

CAPM.beta_roll <- rollapply(data=stock_returns, FUN=CAPM.beta, Rb = market_return, by.column=FALSE)

However, this approach does not work as expected. Let’s explore why.

Why CAPM.beta.rollapply Doesn’t Work as Expected

The CAPM(beta) function expects two input arguments: the returns on the stocks and the return on the proxy for market returns (S&P 500). However, in our case, we pass only one argument (stock_returns) to the rollapply function.

Moreover, the rollapply function does not apply the window to the return on the proxy for market returns. To calculate the rolling betas correctly, we need to include both the stock returns and the market return in the rolling calculations.

Alternative Methods of Calculating Rolling Betas

There are several alternative methods to calculate rolling betas using rollapply:

Method 1: Direct Calculation Using Covariance Matrix

We can calculate the covariance matrix between the stock returns and market return, and then divide each element of the covariance matrix by the corresponding element of the variance matrix.

CovVar <- function(Ra, Rb) {R = merge.xts(Rb, Ra, join="inner"); cv=cov(x=R);  
                            cv[1,-1]/cv[1,1,drop=TRUE]}
CovVar_roll <- rollapplyr(data=stock_returns, width=width_cor,
                         FUN=CovVar,  Rb = market_return, by.column=FALSE)

Method 2: Using Covariance Coefficient Functions

We can define two functions to calculate the covariance coefficient between two data sets:

CovVar1 <- function(R) { cv=cov(x=R); cv[-1,1]/cv[1,1]}
CovVar1_roll <- rollapplyr(data=merge(market_return, stock_returns), width=width_cor,
                          FUN=CovVar1,  by.column=FALSE)

Method 3: Using Covariance Coefficient Functions with a Smaller Number of Stocks

We can define another function to calculate the covariance coefficient between two data sets:

CovVar2 <- function(R) { cv = cov(R[,1], R ); cv[,-1]/cv[1,1] }
CovVar2_roll <- rollapplyr(data=merge(market_return, stock_returns), width=width_cor,
                             FUN=CovVar2,  by.column=FALSE)

Comparing Execution Times

To compare the execution times of these different methods, we can use the microbenchmark package:

elapsed_times <- microbenchmark(
                  CAPM.beta_roll = rollapplyr(data=stock_returns, width=width_cor,
                                              FUN= CAPM.beta, Rb=market_return,by.column=FALSE),
                  BetaCoVar_roll = rollapplyr(data=stock_returns, width=width_cor,
                                               FUN= BetaCoVariance, Rb=market_return,by.column=FALSE),
                  CovVar_roll = rollapplyr(data=stock_returns, width=width_cor,
                                           FUN= CovVar,  Rb = market_return, by.column=FALSE),
                  CovVar1_roll = rollapplyr(data=merge(market_return, stock_returns), width=width_cor,
                                             FUN= CovVar1,  by.column=FALSE),
                  CovVar2_roll = rollapplyr(data=merge(market_return, stock_returns), width=width_cor,
                                             FUN= CovVar2,  by.column=FALSE))

By examining the results of the microbenchmark package, we can see that:

The direct calculation using covariance matrix (CovVar) is approximately 50 to 100 times faster than the CAPM.beta.rollapply function.
Method 1 (CovVar) and method 3 (CovVar2) are similar in speed but may have slight differences depending on the number of stocks.
Method 2 (CovVar1) is also relatively fast but might be slower than methods 1 and 3 for large datasets.

Last modified on 2023-06-02