Time Series Properties of Stock Returns, Comparative Analysis of High- and Low-Frequency Data

Introduction

Fama (1970) suggests that stock returns exhibit random walks through time. This means that one cannot observe any trend in the movement of returns on assets and that returns cannot be forecasted based on information about past returns. Following Fama (1965) and Mandelbrot (1963), the unconditional distribution of returns has been found to be Lepturokurtic, skewed and exhibits volatility clustering. This means that returns are not normally distributed as one may expect them to be and using them in models that assume normality can lead to misleading findings. Returns have been widely used to determine the risks of a company as well as determine the cost of capital to be used in making investment decisions. The question then arises. How frequent should the data used in estimating risk and determining the cost of capital be observed? Should it be monthly, daily, weekly or annually?
The objective of this paper is to determine whether there are differences between the time series properties of high-frequency time-series data and low-frequency time series data. To achieve this objective, the paper employs stock price data of Tesco Plc to determine whether there are differences in the movement of the returns of the stock over time. The time period considered for the study is 1st January 2005 to 31st December 2007. The period 2008 and above is excluded because of the recent global financial crisis. The data is first observed over a daily basis covering the said period and over a monthly basis covering the same period. The data is retrieved from the yahoo finance (www.yahoofinance) data base. Returns are calculated using stock price data and their statistical properties are analysed on a comparative basis. The rest of the paper is organised as follows: section 2 provides a discussion of the importance of returns in decision making; section 3 explores the statistical properties of the returns of the two data sets; section 4 provides a discussion of how primary data could be useful.

Importance of Stock Returns

The properties of the returns are important for a number of reasons. Firstly, returns are often used to predict future performance. Returns are also widely used to study important phenomena about the company. For example, they are used in event studies such as mergers and acquisitions (Fama et al., 1969), as well as in estimating the cost of equity capital using the capital asset pricing model (CAPM) and testing the CAPM (Bodie et al., 2005).
It can be observed from table 1 that the mean value of the returns is .013356 on a monthly basis and .0005607 on a daily basis. The monthly standard deviation is lesser than the daily standard deviation.
Figure one above shows the movement in the daily stock returns of Tesco Plc over the period 1st January 2005 to December 2007. It can be observed that consistent with the random walk hypothesis, the returns exhibit random walks through time. One cannot determine a real trend in the returns series. Moreover, one can observe that the returns are very volatile but they appear to be experiencing some volatility clustering. The daily returns exhibit persistence in volatility over time. Beltratti and Morana (1999) suggest that using high-frequency data presents new insights on the volatility structure of returns especially its persistence. This is consistent with Baillie and Bollerslev (1989) who use daily exchange rate data to show that there is integration in variance. In addition, Baillie et al. (1996) argue that persistence can be attributed to Fractionally Integrated Generalised Autoregressive Conditional Heteroscedesticity (FIGARCH). The histogram below shows the distribution of the daily returns. It can be observed that the distribution is approximately normal.
Figure 3 above shows the time series movement of the monthly returns. It can be observed that the returns exhibit some time-series trends although the trends die out quickly. The monthly returns look a bit more predictable as opposed to the daily returns. Figure 4 shows a histogram of the frequency distribution of the returns. It can be observed that the returns are far from being normally distributed. As shown by the density curve it can be observed that the returns distribution has fatter tails and is leptokurtic. That is, it is less peaked than the normal distribution.

Primary Data

As earlier mentioned above, one of the main reasons why a researcher may want to use stock returns is to estimate the cost of equity capital for a company or to assess the risk of the company. It is obvious from the above analysis that the cost of capital based on each of the above data sets will be different. This indicates that using secondary data in this case can be misleading. In contrast to collecting secondary data as has been done above, primary data could be collected from Financial Analysts and Stock Brokers who have in-depth knowledge of the company, as well as from the financial manager of the company. One could simply ask them to give their opinions on what they think the company cost of capital could be. An average figure can then be calculated from the different responses given by analysts. Primary data can be somehow more appealing instead of secondary data because of the difficulties with deciding on the best frequency For example, the CAPM has a number of short comings that may lead to inaccuracies in the cost of equity capital estimated from the model. (Bodie et al., 2005) For example, the market index used in the model is only a proxy and not the true market index.

Conclusion

Based on the above analysis, one can conclude that the statistical properties of daily returns are different from those of monthly returns. Moreover, returns observed over a high frequency are approximately normally distributed and thus have better statistical properties. However, to make better decisions, it is important to combine both primary and secondary data.