Impact of estimation period and return interval on estimated beta

Capital asset pricing model is still widely applied in financial industry nowadays. Although the idea behind is more than 50 years old, financial analysts still like to use it in practice, since it encompasses theoretically-derived relationship between required return and systematic risk which has been a subject to frequent empirical research. The key factor inside the CAPM is beta, as a measure of asset’s sensitivity to non-diversifiable risk. It has been established in the previous article that beta is not all that difficult to estimate. However it is important to understand that different estimations of beta can arise depending on the data that is used during the estimating procedure.

Beta analysis – Length of the estimation period

We will focus on three factors that can have a significant impact on the estimated beta coefficient. First is the length of the estimation period, second is the return interval and finally the proxy used for market portfolio. We will illustrate with an example how different decisions impact the beta estimations.

In this example we focused on the most famous index in the world S&P 500. The goal was to estimate betas for all the companies that are included in the index. At this point it should be mentioned that exactly 25 companies were excluded from analysis because of their short term presence on the stock market, which consequently means a lack of available data.

Two variations of estimation were completed. In both cases betas were estimated for 475 companies based on weekly returns and S&P 500 was used as a proxy for market portfolio. However first estimation was performed on the 5-year period and the second using 2-year period. The histograms below display the distribution of beta estimates for 475 companies.

Beta distribution

It can be immediately observed that estimations over 2-year period are shifted a bit to the right compared to the estimates over the 5-year period. The cumulative distribution function shows that 58.3% of the companies have estimated beta smaller or equal to 1 in the case with estimating over 5-year period. On the other hand only 34.1% of the companies have estimated beta lower or equal to 1 when applying shorter time period.

This example demonstrates how different methods can yield significantly diverse estimations. Maybe the two averages are a bit misleading, since one might think that the difference is not exactly large. But when considering a single company the discrepancy can be enormous. In one extreme case company’s beta estimated using 5-year horizon is 0.66, while beta estimated for the same company over a 2-year period is 1.67. In our example the estimations of betas for individual companies over the shorter time period were on average 25.2% higher than betas that were estimated over the longer time span.

Beta analysis – Return interval

The second key factor that influences the estimation result is the return interval. There are a lot of different possibilities to choose from, but we focused on the two return intervals that are most commonly used in beta estimation, namely weekly and monthly returns. The procedure remained the same as in the previous example. Once more 475 betas were estimated for each company using S&P 500 as a proxy for market portfolio. In both cases betas were estimated over the same time period (5 years) but in the first case using monthly returns and in the second example using weekly returns. The histograms below show the distribution of estimated beta parameters applying two different procedures.

Beta distribution

It is clear that there is a significant difference between the two distributions. Histogram that represents the estimated betas based on monthly returns is shifted considerably to the left compared to the histogram of betas estimated using weekly returns. Cumulative distribution function demonstrates that 71.6% of the companies have beta estimations lower or equal 1 when monthly returns are used in estimation procedure. Remember that 58.3% of the estimated betas were below or equal to 1 using shorter return interval.

Recall that the same 475 companies were analyzed and the same 5-year time period was used to estimate betas. In both cases the estimation was done by regressing individual company’s returns against the returns of the S&P 500 index. The only difference between the two estimating procedures was the return interval applied. Yet the estimated beta factors are evidently different. Estimated betas for individual company are on average 35% lower using the monthly returns for estimation compared to methodology with shorter return intervals.

Beta analysis – Proxy used for market portfolio

The final factor that influences the estimated coefficients is the proxy used for market portfolio. There are several possibilities that can be used to imitate market portfolio. One option is take the most relevant local stock index, but difficulties can occur within some less liquid or emerging markets. In some cases the financial markets are not developed enough and consequently provide a poor proxy for market portfolio. Furthermore, it can happen that one or a few companies represent a major part of the stock index, and this has some strange implications. It essentially means that company’s returns are regressed against the returns of the few leading companies, rather than a diversified stock index. The right index to use in the analysis should be determined by the marginal investor inside the analyzed company. No one can really tell who the marginal investor is, but it is reasonable to assume that it is represented by the largest holders of stock in the company.

Once more we return to our example of beta estimation for 475 companies inside the well-known S&P 500 index. Two variations of betas will be estimated. In the first example betas are estimated against S&P 500, which is what was done in the previous examples. The second estimation is done with regressing stock returns against NYSE composite index. The aforementioned index is an index that covers all common stock listed on the New York Stock Exchange, including some other investable assets. It represents a wider market compared to S&P 500 which includes only 500 largest companies based on market capitalization. Betas in both instances were estimated using weekly returns and 5-year estimation period. The histogram below displays the dissimilarity between the two methods.

Beta distribution

As expected alternative market portfolio proxy produced different estimates of regression coefficients betas. The histogram containing betas that were estimated against NYSE composite index are shifted a tad to the left compared to the distribution of betas estimated against S&P 500. Cumulative distribution function indicates that 67.8% of the estimated betas in the first example are lower or equal to 1, while the S&P 500 example displayed 58.3% of such values of estimated betas.

The estimated beta for individual company was on average 10.7% lower when NYSE composite index was used as a proxy for market portfolio.

Conclusion

The purpose of this analysis was to examine how various data inputs affect the estimated parameters. It is clear that all three factors play an important role in estimation process. The choice regarding the data applied throughout analysis ultimately comes down to the characteristics of the examined company.