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Beta 1948-54

Figure 7.2

to forecast the Beta for any stock for the period 1962-1968. We then compute (via regression analysis) its Beta for the years 1955-1961. To determine how this Beta should be modified, we substitute it for (3a in the equation. We then compute pc from the foregoing equation and use it as our forecast.

Notice the effect of this on the Beta for any stock. If p;l were 2.0, then our forecast would be 0.343 + 0.677(2) = 1.697, rather than 2.0. If P(1 were 0.5, our forecast would be 0.343 + 0.677(0.5) = 0.682 rather than 0.5. The equation lowers high values of Beta and raises low values. One more characteristic of this equation should be noted. It modifies the average level of Betas for the population of stocks. Since it measures the relationship between Betas over two periods, if the average Beta increased over these two periods, it assumes that average Betas will increase over the next period. Unless there is reason to inspect a continuous drift in Beta, this will be an undesirable property. If there is no reason to expect this trend in the average Beta to continue, then the estimates can be improved by adjusting the forecasted Betas so that their mean is the same as the historical mean.

To make this point more concrete, let us examine an example. Assume that in estimating the equation Blume found the average Beta in 1948-1954 was one and the average Beta in 1955-1961 was 1.02. These numbers are consistent with his results, though there are other sets of numbers that would also be consistent with his results. Now, to determine v hat the average forecasted Beta should be for the period 1962-1968, we simply substitute 1.02 in the right-hand side of the estimating equation. The answer is 1.033. As discussed earlier, Blume's technique results in a continued extrapolation of the upward trend in Betas observed in the earlier periods.

If there is no reason to believe that the next period's average Beta will be more than this period's, then the forecasts should be improved by adjusting the forecast Beta to have the same mean as the historical mean. This involves subtracting a constant from all Betas after adjusting them toward their mean. In our example, this is achieved by subtracting 1.033 from each forecast of Beta and adding 1.02.

Measuring the Tendency of Betas to Regress Toward One— Vasicek's Technique

Recall that the actual Beta in the forecast period tends to be closer to the average Beta than is the estimate obtained from historical data. A straightforward way to adjust for this tendency is to simply adjust each Beta toward the average Beta. For example, taking one-half of the historical Beta and adding it to one-half of the average Beta moves each historical Beta halfway toward the average. This technique is widely used.15

It would be desirable not to adjust all stocks the same amount toward the average but rather to have the adjustment depend on the size of the uncertainty (sampling error) about Beta. The larger the sampling error, the greater the chance of large differences from the average, being due to sampling error, and the greater the adjustment. Vasicek [92] has suggested the following scheme that incorporates these properties: If we let pt equal the average Beta, across the sample of stocks, in the historical period, then the Vasicek procedure involves taking a weighted average of Pj and the historical Beta for security i. Let a^ stand for the variance of the distribution of the historical estimates of Beta over the sample of stocks. This is a measure of the variation of Beta across the sample of stocks under consideration. Let ct|, stand for the square of the standard error of the estimate of Beta for

■For example, Merrill Lynch has used a simple weighting technique like this to adjust its Betas.

security i measured in time period 1. This is a measure of the uncertainty associated with the measurement of the individual securities Beta. Vasicek [92] suggested weights of a- a2

Note that these weights add up to 1 and that the more the uncertainty about either estimate of Beta, the lower the weight that is placed on it. The forecast of Beta for security i is o2 ct2

This weighting procedure adjusts observations with large standard errors further toward the mean than it adjusts observations with small standard errors. As Vasicek has shown, this is a Bayesian estimation technique.16

While the Bayesian technique does not forecast a trend in Betas as does the Blume technique, it suffers from its own potential source of bias. In the Bayesian technique, the weight placed on a stock's Beta, relative to the weight on the average Beta in the sample, is inversely related to the stock's standard error of Beta. High Beta stocks have larger standard errors associated with their Betas than do low Beta stocks. This means that high Beta stocks will have their Betas lowered by a bigger percentage of the distance from the average Beta for the sample than low Beta stocks will have their Betas raised. Hence, the estimate of the average future Beta will tend to be lower than the average Beta in the sample of stocks over which Betas are estimated.

Unless there is reason to believe that Betas will continually decrease, the estimate of Beta can be further improved by adjusting all Betas upward so that they have the same mean as they had in the historical period.

Accuracy of Adjusted Beta

Let us examine how well the Blume and the Bayesian adjustment techniques worked as forecasters, compared to unadjusted Betas. Klemkosky and Martin [57] tested the ability of these techniques to forecast over three 5-year periods for both one-stock and ten-stock portfolios. As would be suspected, in all cases both the Blume and Bayesian adjustment techniques led to more accurate forecasts of future Betas than did the unadjusted Betas. The average squared error in forecasting Beta was often cut in half when one of the adjustment techniques was used. Klemkosky and Martin used an interesting decomposition technique to search for the source of the forecast error. Specifically, the source of error was decomposed into that part of the error due to a misestimate of the average level of Beta, that part due to the tendency to overestimate high Betas and underestimate low Betas, and that part that is unexplained by either of the first two influences. As might be expected, when the Blume and Bayesian techniques were compared with the unadjusted Betas, almost all of the decrease in eiTor came from the reductions in the tendency to overestimate high Betas and underestimate low Betas. This is not surprising because this is exactly what the two techniques were designed to achieve. Klemkosky and Martin found that the Bayesian technique had a slight tendency to outperform the Blume technique. However,

16The reader should note that this is just one of an infinite number of ways of forming prior distributions. For example, priors could have been set equal to 1 (the average for all stocks market weighted) or to an average Beta for the industry to which the stock belongs, and so on.

the differences were small and the ordering of the techniques varied across different periods of time.

Most of the literature dealing with Betas have evaluated Beta adjustment techniques by their ability to better forecast Betas. However, there is another, and perhaps more important criterion by which the performance of alternative Betas can be judged. At the beginning of Shis chapter we discussed the fact that the necessary inputs to portfolio analysis were expected returns, variances, and correlations. We believe that analysts can be asked to provide estimates of expected returns and variances, but that correlations will probably continue to be generated from some sort of historical model.17 One way Betas can be used is to generate estimates of the correlation between securities. The correlations between stocks (given the assumptions of the single-index model) can be expressed as a function of Beta.

Another way to test the usefulness of Betas, as well as the performance of alternative forecasts of Betas, is to see how well Betas forecast the correlation structure between securities.

Betas as Forecasters of Correlation Coefficients

Hlton, Gruber, and Urich [31] have compared the ability of the following models to forecast the correlation structure between securities:

1. The historical correlation matrix itself

2. Forecasts of the correlation matrix prepared by estimating Betas from the prior historical period

3. Forecasts of the correlation matrix prepared by estimating Betas from the prior two periods and updating via the Blume technique

4 Forecasts prepared as in the third model but where the updating is done via the Vasicek Bayesian technique

One of the most striking results of the study was that the historical correlation matrix Itself was the poorest of all techniques. In most cases it was outperformed by all of the Beta forecasting techniques at a statistically significant level. This indicates that a large part of the observed correlation structure between securities, not captured by the single-index model, represents random noise with respect to forecasting. The point to note is that the i>ingle-index model, developed to simplify the inputs to portfolio analysis and thought to lose information due to the simplification involved, actually does a better job of forecasting than the full set of historical data.

The comparison of the three Beta techniques is more ambiguous. In each of two five-year samples tested, the Blume adjustment technique outperformed both the unadjusted Uetas and the Betas adjusted via the Bayesian technique. The difference in the techniques was statistically significant. However, the Bayesian adjustment technique performed better than the unadjusted Beta in one period and worse in a second. In both cases, the results were statistically significant. This calls for some further analysis. The performance of any forecasting technique is, in part, a function of its forecast of the average correlation between all stocks and, in part, a function of its forecast of previous differences from the

"It is possible that analysts will be used to subjectively modify historical estimates of Beta to improve their accuracy. Several firms currently use analysts' modified estimates of Beta.

mean. We might stop for a moment and see why each of the Beta techniques might produce forecasts of the average correlation coefficient between all stocks that are different from the average correlation coefficient in the data to which the technique is fitted.

Let us start with the unadjusted Betas. This model assumes that the only correlation between stocks is one due to common correlation with the market. It ignores all other sources of correlation such as industry effects. To the extent that there are other sources of correlation that are, on the whole, positive, this technique will underestimate the average correlation coefficient in the data to which it is fitted. This is exactly what Elton, Gruber, and Urich [31] showed happened in both periods over which the model was fitted.

The Blume technique suffers from the same bias, but it has two additional sources of bias. One is that the Blume technique adjusts all Betas toward one. This tends to raise the average correlation coefficient estimated from the Blume technique. The correlation coefficient is the product of two Betas. To the extent that Betas are reduced to one symmetrically (with no change in mean), the cross products between them will tend to be larger. For example, the product of 1.1 and 0.9 is larger than the product of 1.2 and 0.8. There is another source of potential problems in the Blume technique. Remember that the Blume technique adjusts the Betas in period two for the changes in Betas between period one and two. If the average change in Beta between periods one and two is positive (negative), the Blume technique will adjust the average Beta for period two up (down).18 In the Elton, Gruber, and Urich study there was an upward drift in Betas over the period studied and this, combined with the tendency of the Blume technique to shrink all Betas toward one, resulted in forecasts of an average correlation coefficient well above the average correlation coefficient for the sample to which the model was fitted.

The Bayesian adjustment to Betas, like the Blume adjustment, has some upward forecast bias because of its tendency to shrink Betas toward one, but it does not continue to project a trend in Betas and, hence, correlation coefficients as the Blume technique does. However, as pointed out earlier, it has a new source of bias: one that tends to pull Betas and correlation coefficients in a downward direction. This occurs because high Beta stocks are adjusted more toward the mean than low Beta stocks.

Short of empirical tests, it is difficult to say whether, given any set of data, the alternative sources of bias, which work in different directions, will increase or decrease the forecast accuracy of the result. We do know that unless there are predictable trends in average correlation coefficient, the effect of these biases on forecast accuracy will be random from period to period. This source of randomness can be eliminated. One way to do it is to force the average correlation coefficient, estimated by each technique, to be the same and to be equal to the average correlation coefficient that existed in the period over which the model was fitted. If correlation coefficients do not have stable trends, this will be an efficient forecast procedure. It uses only available data and is also easy to do.

When the adjustments were made, the Bayesian adjustment produced the most accurate forecasts of the future correlation matrix. Its difference from the Blume technique, the unadjusted Beta, and the historical matrix was statistically significant in all periods tested. The second-ranked technique varied through time with the Blume adjustment, outperforming the unadjusted Beta in one period and the unadjusted Beta outperforming Blume in one period.19

ÏSThis would be a desirable property if trends in average correlation coefficients were expected to persist over time, but we see no reason to expect them to do so.

I9In addition, tests were made that forced the average correlation coefficient from each technique to be the same and equal to the average correlation coefficient that occurred in the forecast period. This is equivalent to perfect foresight with respect to the average correlation coefficient. The rankings were the same as those discussed above when this was done.

The forecasts from the three Beta techniques were compared with the forecasts from a fourth Beta estimate, Beta equals one for all stocks, as well as with the historical correlation matrix, as a forecast of the future. The mean forecast was adjusted to be the same for all techniques. The performance of the historical correlation matrix and the Beta-equals-one model was inferior to the performance of all other models at a statistically significant level.

Let us stop a minute and review the work on estimating Betas. There are two reasons for estimating Betas: The first is in order to forecast future Betas. The second is to generate correlation coefficients as input to the portfolio problem. Empirical evidence strongly suggests that to forecast future Betas one should use either the Bayesian adjustment or the Blume adjustment rather than unadjusted Betas. The evidence on the choice between the Blume and Bayesian adjustment is mixed, but the Bayesian adjustment seems to work slightly better.

If the goal is estimating the future correlation matrix as an input to the portfolio problems, things get more complex. Unadjusted Betas and adjusted Betas, both by the Bayesian and the Blume techniques, all contain potential bias as forecasters of future correlation matrices.20 The forecasts from all of these techniques can be examined directly or the forecasts can be adjusted to remove bias in the forecast of the average correlation coefficient. The first fact to note is that each of these three estimates of Beta outperforms the historical correlation matrix as a forecast of the future correlation matrix. Second, note that when compared to a Beta of one, all produce better forecasts. The ranking among these three techniques is a function of whether we make the adjustment to the average forecast. Since we believe it is appropriate to do so, we find that the Bayesian adjustment technique performs best.

Recently, attempts have been made to incorporate more data than past return information into the forecasts of Betas. We will now take a brief look at some of the work that has been done in this area.

Fundamental Betas

Beta is a risk measure that arises from the relationship between the return on a stock and the return on the market. However, we know that the risk of a firm should be determined by some combination of the firm's fundamentals and the market characteristics of the firm's stock. If these relationships could be determined, they would help us both to better understand Betas and to better forecast Betas.

One of the earliest attempts to relate the Beta of a stock to fundamental firm variables was performed by Beaver, Kettler, and Scholes [7]. They examined the relationship between seven firm variables and the Beta on a company's stock. The seven variables they used were

1. Dividend payout (dividends divided by earnings),

2. Asset growth (annual change in total assets).

3. Leverage (senior securities divided by total assets).

4. Liquidity (current assets divided by current liabilities).

5. Asset size (total assets).

6. Earning variability (standard deviation of the earnings price ratio).

7. Accounting Beta (the Beta that arises from a time series regression of the earnings of the firm against average earnings for the economy, often called the earnings Beta).

°As discussed earlier, a smaller set of potential biases is present when Betas are estimated.

An examination of these variables would lead us to expect a negative relationship between dividend payout and Beta under one of two arguments:

1. Since management is more reluctant to cut dividends than raise them, high payout is indicative of confidence on the part of management concerning the level of future earnings.

2. Dividend payments are less risky than capital gains; hence, the company that pays out more of its earnings in dividends is less risky.

Growth is usually thought of as positively associated with Beta. High-growth firms are thought of as more risky than low-growth firms. Leverage tends to increase the volatility of the earnings stream, hence to increase risk and Beta.

A firm with high liquidity is thought to be less risky than one with low liquidity and, hence, liquidity should be negatively related to market Beta. Large firms are often thought to be less risky than small firms, if for no other reason than that they have better access to the capital markets. Hence, they should have lower Betas. Finally, the more variable a company's earning stream and the more highly correlated it is with the market, the higher its Beta should be.

Table 7.5 reports some of the results from the Beaver et al. [7] study. Note all variables had the sign that we expected.

The next logical step in developing fundamental Betas is to incorporate the effects of relevant fundamental variables simultaneously into the analysis. This is usually done by relating Beta to several fundamental variables via multiple regression analysis. An equation of the following form is estimated:

0( = a0 + a,X, + a2X2 + ■ • • + aNXN + e¡ (7.6)

where each X¡ is one of the N variables hypothesized as affecting Beta. Several studies have been performed that link Beta to a set of fundamental variables, such as that studied by Beaver et al. [7].21 The list of variables that has been studied and linked to Betas is too long to review here. For example, Thompson [91] reviews 43 variables while Rosenberg [79]

Table 7.5 Correlation between Accounting Measures of Risk and Market Beta

Period 1 Period 2

1947-1956 1957-1965

Table 7.5 Correlation between Accounting Measures of Risk and Market Beta

Period 1 Period 2

1947-1956 1957-1965

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