Roy Mehta's  Instablog

Introduction

This paper will explore how best to predict beta in the Capital Asset Pricing Model (CAPM) over the next six months. In present day financial theory, we use the relationship between a stock and market in the past to figure out what that relationship might be in the future. That relationship is captured in the CAPM, but there are still nuances within the CAPM that can be altered for better or worse results. One of these attributes is how much data, in terms of time, is used to calculate the beta of a stock. Here, we firstly figure out what time frame from the stock’s past prices might be best to predict the stock’s future beta over the next six months.

                Though the findings in this paper pinpoint what time horizon can be used to forecast the beta of a stock or ETF over the next six months, it still remains to be seen whether those betas were significantly similar. I tested this using three well-known stocks. The results were positive and show there is value in this exploration.  

Background

                Modern portfolio theory tells us there are two types of risk, systematic risk and unsystematic risk. Unsystematic risk can be reduced to zero using diversification, but systematic risk, in general, cannot be. Beta is a measure of this non-diversifiable risk. The beta tells the portfolio manager what will happen to the expected return of the asset if the market moves. It is a single risk factor that shows what a large run-up or drop could do to an asset’s value. A large beta means a larger return when the market increases, but a more negative return when the market decreases. It encompasses the direction and magnitude of the asset’s return based on the market. Higher beta stocks are more volatile, and should then be considered more risky.

                With these ideas in mind, it is clear that knowing the beta of an asset, can be valuable. It can give risk managers an idea of how assets they manage will move and how much risk their portfolios actually have. Portfolio managers have views of the market in general, and can create portfolios to take advantage of their views by constructing an assortment of stocks with targeted betas. If portfolio manager X is bullish, he will want a portfolio that is at the very least positive, and maybe even above one to beat the market.

                It should also be briefly noted that the CAPM does have its disadvantages, mainly the assumptions on which it is based. These assumptions include that investors hold diversified portfolios, single holding periods are used, anyone can borrow and lend at the same risk-free rate, all the securities in the market are valued efficiently, there are no taxes or transaction costs, and all asset returns can be plotted on the securities market line(SML). The SML shows the relationship between risk or beta and asset return. Though some of these assumptions are significant, they are made in order to focus closely on the relationship between risk and return.

Analysis 1: Choosing a Time-Frame

                Generally, people find the beta between the market and a stock over the last year, and consider that number to be the beta going forward. This paper challenges that notion, and instead tests whether we should use a different time-frame to increase the accuracy of that beta.

                To get beta, you take the excess returns of a stock and index over a certain amount of time and use linear regression to find the coefficients in the CAPM. The risk-free is usually some relatively safe bond rate. In the calculations for this paper, I used the 10-year US Bond found on Yahoo Finance.com. The S&P500 is a popular candidate for the market portfolio, so I used that for my market returns. When running the regression (see Table 1 for code), we get two coefficients, the beta and alpha. The Alpha is the return that cannot be attributed to the market returns and beta. In regression terms, alpha is the intercept, while beta would be considered the slope.

I tested three time frames — 6 months, 1 year, and 2 years. I assumed each year is about 250 days (the approximate amount of trading days in a year).  So, I took the log returns for nine different ETFs, the sector SPDRs, over about 10 years, from 4/12/2000 to 3/23/2010, and divided them into four parts with 625 days in each part. The prices were taken from Yahoo Finance as well and were adjusted for splits and dividends. In each part, I found the betas of the 375^th to 500^th day (six months), 250^th to the 500^th day(one year), and finally the 1^st to 500^th day(two year). This gave me three betas. Then, I found the beta over day 501 to 625. This gave me a “real beta” to use for comparison. The idea was to compare each of the three betas to the “real beta” and see which was the closest.  Below shows the results for each part. Part 1 shows data for the first 2.5 years or 625 days, part 2 shows the next 2.5 years, and so on.