Beta and an investor's holding period.

Do you have long-term or short-term investment goals? For example, is your investment objective to take a special vacation, buy a home, fund a college education or prepare for retirement? How much risk are you willing to accept in meeting your investment goals and, more importantly, how should risk be measured and interpreted? If you invest in a portfolio of securities (such as a mutual fund) because you want the advantages associated with diversification, how should the performance of the portfolio be measured? How should the results of current investment research that question the appropriate measure of risk be translated into an appropriate investment strategy for an individual investor? Should the proportion of wealth invested in stocks of small firms, which are generally believed to out-perform those of larger companies, be increased or decreased? As the findings discussed below will show, an individual investor's expected holding period--his/her investment horizon--can play a significant role in answering these and similar questions. Specifically, it is shown that investors need to construct and evaluate their portfolios in a manner consistent with their expected holding periods. Failure to do so can result in inappropriate portfolio construction and evaluation.

Measuring Risk and Return

It is generally accepted that investors are, for the most part, risk averse. This simply means that a disproportionately higher expected return is demanded for a security with greater risk. While the process of measuring a security's return is universally accepted, defining "risk," let alone determining the appropriate measure of risk, is not as clear. If we define risk as the variability (or differences) of returns from those that are expected, the total risk of a security can be measured by its standard deviation. In turn, this total risk can be separated into two components. The first component exists due to the risk associated with the market as a whole (e.g., uncertainty about general economic conditions, such as inflation, interest rates, or GNP). This component is called systematic risk and is common to all securities to a greater or lesser extent. Furthermore, since all securities are influenced by or related to the same systematic risks (e.g., inflation), their returns will also be related. The measure which captures the influence of systematic risk is called "beta."

The second component of risk is called unsystematic risk. This is the unique or firm-specific risk associated with an individual security. For example, a labor strike at Acme Industries' sole manufacturing plant may well affect Acme Industries alone or maybe a few other firms. Unlike a factor such as inflation, however, the strike will not affect all firms. An important feature of unsystematic risk is that it can virtually be eliminated. Indeed this is the principle underlying most mutual funds. By investing in a wide range of securities, the negative performance of one security (or group of securities) can be offset by the positive performance of other securities. Since unsystematic or firm-specific risk can be eliminated through diversification, investors are not rewarded for this risk. That is, an investor cannot expect to receive a higher return for taking on risk that can be eliminated.

Systematic risk (also called market risk), on the other hand, cannot be avoided through diversification. Because of this, beta is argued to be a relevant measure of a security's risk. Beta measures the volatility of a security relative to the volatility of the market. The precise relation between expected return and systematic risk, as well as the valuation of securities that follows, is the essence of the Capital Asset Pricing Model (abbreviated CAPM). The CAPM was developed in the 1960s by William Sharpe, 1990 Nobel laureate in economics. In this model, beta serves as an index of a security's systematic risk. Several investor services such as Value Line Investment Survey, Standard and Poor's Stock Reports, Merrill Lynch Investment Service, and Bloomberg Investment Service provide betas on companies whose common stocks are actively traded.

The appeal of beta as a measure of risk stems from its logical and theoretical foundation. Essentially, it maintains that investors require a higher expected return on assets which have greater nondiversifiable risk. Currently, however, beta as the primary variable explaining returns is being debated among academics. While debate regarding beta is not new, the current debate stems largely from a study done in 1990 by Professors Eugene Fama and Kenneth French of the University of Chicago.(1) Professors Fama and French studied all common stocks traded on the NYSE and AMEX from June 1963 to December 1990 (common stocks traded on NASDAQ are added to the sample in 1973). They provide evidence that the ratio of a firm's book-to-market value and firm size explain stock returns far better than beta. The empirical methods used in their study, the overlapping time periods they analyzed, as well as the logic and interpretation of their results continue to be debated.(2) Unfortunately, Professors Fama and French do not offer an alternative measure of risk, but rather provide a different explanation of historical stock returns. Earlier research on the CAPM has also concentrated on extending it to include other variables in addition to beta. Most of these empirical studies conclude that beta is the dominant factor in determining a security's return.(3) Interestingly, however, little justification for the particular holding period assumed (e.g., daily versus weekly versus annual returns) and the possible impact of the chosen holding period on beta, is seldom discussed.(4)

Before the effect of an investor's holding period on beta is examined, another aspect regarding the use of beta bears mentioning. There are three common indices used to evaluate the performance of portfolios (primarily mutual funds). These are the Treynor, Sharpe, and Jensen Indices. Of these three measures, the Treynor and Jensen Indices are more general and can be used to measure the performance of both individual assets as well as portfolios. The more general use of these indices stems from the fact that they use beta as the appropriate measure of risk and are formulated in the context of the CAPM.(5) Since the Sharpe Index measures risk by using the standard deviation of the portfolio, it is only appropriate for evaluating portfolios. Let's now consider the effect of the holding period on beta.

Beta and the Holding Period

Professors Frank Reilly and David Wright(6) investigate the relation between the betas published by Value Line Investment Survey and Merrill Lynch Investment Service and find that betas do in fact differ when different return intervals--holding periods--are used. Specifically, both investment services employ a five-year total time period for calculating beta, but Merrill Lynch uses monthly stock returns (60 observations) whereas Value Line uses weekly returns (260 observations). The substantial differences in the beta estimates of these two services were concluded to be attributed to the differences in the length of the return intervals selected (i.e., monthly versus weekly). More recently, Professors Haim Levy and Deborah Gunthorpe(7) examine how beta and empirical tests of the CAPM are affected by the holding period selected. They examine the effect of the holding period on a random sample of two-hundred stocks traded on the NYSE and AMEX from January 1963 to December 1990. Their study concentrates on defensive stocks which are considered less risky than the market as a whole (stocks with betas less than one), and aggressive stocks which are considered more risky than the market as a whole (stocks with betas greater than one). To calculate the holding period, it is assumed that an investor purchases a security at one point in time and then sells it at another point in time.(8) For example, for a daily holding period it is assumed that the investor purchases the security at the closing price on say Monday, holds it one day, and then sells it at the closing price on the following Tuesday. Beta is the slope coefficient in a regression of the given holding period return of the stock on the return of the market, which is assumed to the a value-weighted index of all stocks traded on the NYSE and AMEX for the same holding period assumed. Interestingly, Professors Levy and Gunthorpe find that when the investment horizon for estimating beta increases from a daily to an annual holding period, a stock initially thought to be defensive when daily returns are used may be reclassified as aggressive when an annual horizon is used. The reverse occurs for stocks thought to be aggressive when daily returns are considered--they are defensive when annual returns are used.

The precise empirical results found by Professors Levy and Gunthorpe are shown in Table 1. A total of six different. holding periods are examined by calculating daily, weekly, monthly, quarterly, semiannual, and annual rates of returns. Value-weighted portfolios are formed by dividing the full sample of two-hundred stocks into ten equal-sized classes (called deciles). The division criteria is based on an annual holding period (i.e., using betas calculated using annual returns). Frame A of Table 1 shows the results when the sample is first analyzed by dividing the full sample into those stocks with betas greater than one and less than one. As shown, when the holding period for aggressive stocks decreases, the resulting betas are found to consistently decrease from 1.354 for an annual investment horizon to 1.006 for a daily horizon. For defensive stocks, there is an overall, although not perfect, increase in beta from 0.715 for an annual horizon to 0.792 for a daily horizon.

Frame B provides a further refinement of the sample in terms of ten deciles or portfolios ranked by their betas calculated using annual returns. For example, the first decile portfolio has the highest average beta (1.890) whereas the last decile has the lowest average beta (0.404). In examining the ten portfolio deciles under all six alternative investment horizons, note that the results also hold for the top three and bottom three deciles. That is, the average beta of the first decile consistently decreases as the investment horizon is shortened from an annual horizon to a daily horizon (i.e., 1.890 to 1.214).(9)

What are the Implications for Investors?

The findings by Professors Levy and Gunthorpe imply that the way in which the returns are partitioned to estimate beta--how the data is sliced so to speak--can lead to very different interpretations and evaluations of the risks and expected returns of securities. This result can be quite significant for individual investors and highlights the following point:

Investors need to construct and evaluate their portfolios in a manner consistent with their expected holding period.

To further illustrate the implications, consider the following examples. Suppose you wish to construct a somewhat conservative portfolio--one that will not expose you to as much risk as the market as a whole. You decide on a defensive portfolio with a beta of 0.80. (The beta of the market is by definition 1.0 since it is perfectly correlated or related to itself.) To construct this defensive portfolio, suppose you use a time series of daily stock returns to estimate beta (or you use the daily betas published by an investment service). Given the findings of Professors Levy and Gunthorpe, if your investment horizon is one year (and not one day as implied by your use of daily returns to estimate beta), your defensive portfolio may in fact turn out to be an aggressive portfolio (with a beta greater than 1.0)! Therefore, the subsequent evaluation of the performance of your portfolio will also be affected since you have in essence constructed a portfolio TABULAR DATA OMITTED with greater risk than you thought. The opposite effect can occur for stocks with betas greater than one.

Another interesting implication is related to the small-firm effect--the fact that the stocks of small firms on average out-perform the stocks of large firms. Interestingly, one explanation for this effect might be the interval used to estimate beta. Consider the following example: using daily data suppose you estimate the beta of a small firm to be 1.2 (an aggressive stock exhibiting more risk than the market). Using an annual holding period (and thus annual data), however, the beta is only 0.80. Since with daily data the stock is considered riskier than the market, the stock may appear to have performed better than expected. Because the interval chosen can affect the estimate of a security's risk (beta), it can affect both the construction and evaluation of a portfolio. Indeed, it may be that some of the documented stock-price anomalies (such as the small-firm effect) are substantially affected by the holding period selected.

Conclusion

The implication of the results found by Professors Levy and Gunthorpe for individual investors is straightforward: Measurement and evaluation of the return on a security or portfolio requires that the interval used to estimate the risk of the security--beta--should match the investor's expected holding period. Failure to do so may result in an undesirable investment strategy and an inappropriate evaluation of the performance of one's investment. Thus, for example, individual investors who revise their portfolios once a year for tax planning purposes, should probably use annual data to calculate portfolio returns and betas.

When Professor Sharpe was interviewed by the New York Times(10) regarding the results of Professors Fama and French who cast doubt on the use of beta, he was quoted as saying "... I am not willing to make investment decisions based on the theory that there is no relationship between beta, properly measured, and expected returns." Indeed, given the results discussed above, the news may not be that beta is dead, but rather the measurement of beta requires greater consideration.

Endnotes

1. The interested reader should see "The Cross-Section of Expected Stock Returns," by Eugene Fama and Kenneth French, Journal of Finance, June 1990, pp. 427-466.

2. The interested reader should, for example, see the following articles: "Beta and Return -- Announcements of the 'Death' of Beta Seem Premature," by Fischer Black, The Journal of portfolio Management, Fall 1993, pp. 8-18; "If Beta is Dead, Where is the Corpse?," by Peter L. Bernstein, Forbes, July 20, 1992, page 343; and "Is Beta Dead Again?," by Richard C. Grinold, Financial Analysts Journal, July-August 1993, pp. 28-34.

3. It is well-known that beta can be affected by serially correlated returns (for example, if the return yesterday is correlated with the return today) or if the distribution of returns is not stable over time.

4. While the study by Professors Fama and French employ monthly stock returns, numerous other empirical tests of the CAPM that employ daily, weekly, monthly, quarterly, semiannual, and annual holding period returns have also been conducted.

5. When the Treynor and Sharpe Indices are used to rank portfolios (mutual funds), the rankings of the portfolios will be essentially identical as long as the portfolios being examined are well diversified. This assures that the portfolios are essentially the market portfolio or a good proxy for the market.

6. See "A Comparison of Published Betas" by Frank K. Reilly and David J. Wright, Journal of portfolio Management, Spring 1988, pp. 64-69.

7. See the current working paper by Professors Levy and Gunthorpe entitled "Portfolio Composition and the Investment Horizon."

8. Transactions costs are ignored.

9. Table 1 also reports the results from a matched-pair T-test of the difference between the daily-return holding period and the five alternative holding periods. As can be seen in Frame A, for aggressive stocks the difference in beta is always statistically significant at the 1% level. For the defensive stocks, there is a statistically significant difference between daily and weekly horizons as well as daily and monthly, and daily and annual horizons.

10. See the New York Times, February 18, 1992, page C1.

Haim Levy is Graduate Research Professor of Finance, University of Florida, Gainesville, Florida and Hebrew University, Jerusalem, Israel. Deborah Gunthorpe is Assistant Professor of Finance, The University of Tennessee, Knoxville, TN. John Wachowicz, Jr. is Associate Professor of Finance, The University of Tennessee, Knoxville, TN.