In the discussion in Chapter 6 there appeared evidence that divergence of opinion could be interpreted as a risk measure or was correlated with a risk measure. Let’s look at why divergence of opinion may be a surrogate for risk, and what evidence there is that divergence of opinion and risk are cor- related. The Qu et al. model has already been discussed, in which volatility (an element of risk) results from investors trading as they observe prices that imply that others have information they lack.1
People usually disagree most when there is little solid information, and they are most uncertain. Disagreements about the true value of a security increase with the uncertainty about its value. Risk is, in turn, correlated with uncertainty. Consider different types of securities. Most observers would say there was the least uncertainty about the value of a bond issued by a company with a high credit rating. Next would be a utility stock with highly predictable earnings. Then there would be a typical industrial company whose earnings could fluctuate widely.
Finally, there would be a developmental stage company with only a new product idea. There is considerable risk to investment in such a com- pany, and considerable uncertainty about its future. In general, it is the companies about whose future there is the greatest uncertainty that are considered the riskiest and about whose value there is the greatest diver- gence of opinion. Thus, it is to be expected that there will be a positive correlation between risk and divergence of opinion.
Such a relationship has been found by Daley et al.2 They showed that the disperdion of analysts’ beliefs (as measured by coefficient of variation)
1Shiseng Qu, Laura Starks, and Hong Yan, “Risk, Dispersion of Analysts Forecasts and Stock Returns,” working paper, University of Texas, September 30, 2003. Pre- sented at the FMA meeting in 2003.
2Lane A. Daley, David W. Senkow, and Robert L Vigeland, “Analysts’ Forecasts, Earnings Variability, and Option Pricing: Empirical Evidence,” Accounting Review (October 1988), pp. 563–585.
was correlated with the magnitude of the unexpected earnings when next reported. The correlation was 0.347 with the absolute value of the unex- pected earnings and 0.201 with the square of the expected earnings (both were statistically significant). Unexpected earnings are the difference between the mean analysts’ forecasts of earnings and the earnings actually reported. Prices usually respond when company earnings are other than expected. Thus unexpected earnings can be considered a measure of risk.
It is not hard to come up with reasons to explain why the dispersion of analysts’ predictions and unexpected earnings should be correlated.
Imagine that a company’s earnings depend on a factor that varies (such as the state of the economy) and analysts have different predictions for this factor. If the company’s earnings are only a little affected by this fac- tor, the analysts’ estimates of its impact on earnings will be similar, and if the factor proves to be different than estimated by the typical analyst, there will be little impact on earnings. However, if the company’s earn- ings are very sensitive to this factor, the same analysts’ divergence of opinions about this factor will lead to a wider dispersion in earnings esti- mates. Also, whenever the factor is different from what typical analysts expected (say, there is an unexpected decline in the economy), earnings will differ from the mean of the estimates. Thus, one would expect the companies for which there was considerable divergence of opinion among the analysts to also be the ones most likely to produce disap- pointing earnings (or unexpectedly good earnings).
There is another reason for a positive correlation between divergence of opinion and earnings variability. There are usually a large number of factors and potential events that could affect a company significantly.
Due to limitations of time and human brain-processing capacity, no ana- lyst or investor can take into account all of these. Much of the diver- gence of opinion among analysts and investors probably arises from differences in which factors they explicitly consider. For instance one may consider new competition in a particular product, but not the state of the business cycle in different markets, while another considers the business cycles, but not competition in that product. If a company is exposed to a large number of such factors, they can produce large varia- tions in earnings, as well as large variations in analysts’ forecasts.
Ajinkya, Atiase, and Gift also found a strong correlation between the divergence of opinion as measured by analysts’ earnings forecasts (stan- dardized by the mean forecast earnings per share) and the month to month changes in the mean of analysts’ estimates.3 The Spearson correlations
3Bipin B. Ajinkya, Rowland K. Atiase, and Michael J. Gift, “Volume of Trading and the Dispersion in Financial Analysts’ Earnings Forecasts,” Accounting Review (April 1991), pp. 389–401.
ranged from 0.467 (for 1978) to 0.519 (for 1981) and the Pearson correla- tions from 0.550 (for 1979) to 0.605 (for 1981). Since the standardization for size in the analysts’ forecast was achieved by dividing by the mean forecast, and the standardization of the change in analysts’ mean forecast was achieved by dividing by price, the correlation was not merely because price was used in calculating both variables, which would happen if the standardization of divergence of earnings forecasts was done by the alter- native procedure of dividing by price, as is sometimes done.4 Since most researchers would agree that revisions in earnings estimates were corre- lated with risk, this finding shows that the divergence of analysts’ opinions is both a measure of risk and of pure divergence of opinion.
If stocks with high divergence of opinion among analysts frequently report earnings different than expected, their stocks will also be more volatile. Ajinkya, Atiase, and Gift5 and Ajinkya and Gift6 also docu- mented a risk and divergence of opinion correlation. They measured investors’ divergence of opinion by the divergence in analysts’ opinions and risk by the variability in returns as measured by the standard devia- tion of returns implied by option prices. This correlation between the dispersion of analysts’ forecasts and the volatility held—whether the volatility was measured historically, or whether it was with the expected volatility calculated from option prices.
Daley et al. found a correlation of 0.554 between the variance of analysts’ forecasts and the variance of return (calculated over 30 days).7 They also used option data to calculate the implied volatility. For implied volatilities calculated for options expiring after the next earn- ings reporting date, the correlation was positive and statistically signifi-
4Since not all analysts’ forecasts are reported at the same time, some of the correla- tion may be due to this. Imagine there was no divergence of opinion among analysts.
This would mean that at any one time they were in agreement. However, if only some reported this month, and in the database for this month some analysts were represented by their estimates as of last month, there would be a positive correlation between the dispersion in analysts’ forecasts and the change in the mean of analysts’
forecasts. This artificial correlation could be reduced if the change in mean forecasts was calculated for a pair of months that did not include the month over which the dispersion in analysts forecasts was calculated.
5Ajinkya, Atiase, and Gift, “Volume of Trading and the Dispersion in Financial An- alysts’ Earnings Forecasts.”
6Bipin B. Ajinkya and Michael J. Gift, “Dispersion of Financial Analysts’ Earnings Forecasts and the (Option Model) Implied Standard Deviations of Stock Returns,”
Journal of Finance (December 1985), pp. 1353–1365.
7Lane A. Daley, David W. Senkow, and Robert L Vigeland, “Analysts’ Forecasts, Earnings Variability, and Option Pricing: Empirical Evidence,” Accounting Review (October 1988), pp. 563–585.
cant. For options expiring before the next earnings announcement, the correlation was positive, but nonsignificant. The latter result was what they predicted from a simple model in which a new earnings announce- ment affected stock prices upon announcement, but not before. How- ever, given that much information relevant to earnings appears before the earnings announcement (industry sales, macro-economic data, prod- uct introductions, other firms’ earnings reports, etc.) and that both divergence of analysts’ estimates and volatility are serially correlated, I would have expected a positive correlation also before the earnings were announced. I suspect a larger sample over a longer period would have shown significance.
A more recent study of analysts’ estimates of earnings is by Ander- son, Ghysels, and Juergens.8 They find for 1991–1997 that not only does the divergence in analysts’ estimates of earnings (unstandardized) forecast variance in return for the next year, but that a model using it alone provides better forecasts of variance than other models tested.
Malkiel concluded “the best single risk proxy is not the traditional beta calculation but rather the dispersion of analysts’ forecasts.”9
Barry and Gultekin show that betas increase with their measure of analysts’ dispersion of opinion.10 The beta increases from 0.770 for the lowest coefficient of variation groups to 1.136 for the groups with the highest coefficient of variation. Barron and Stuerke also found a positive correlation between beta and the log of the dispersion in analysts’ fore- casts and between beta and the log of analysts forecasts updated within 30 days of the release of earnings.11 They also showed that these mea- sures correlated with the variance of daily returns over the year preced- ing the earnings announcement and with the absolute value of the cumulative abnormal return around the next earnings announcement.
Not surprisingly, beta also correlated with the variance of daily returns over the year preceding the announcement.
8Evan W. Anderson, Eric Ghysels and Jennifer L. Juergens, “Do Heterogenous Be- liefs and Model Uncertainty Matter for Asset Pricing?” working paper, June 13, 2003.
9Burton Malkiel, “Risk and Return: A New Look,” in Benjamin M. Friedman (ed.), The Changing Role of Debt and Equity in Financing U.S. Capital Formation (Chi- cago, IL: University of Chicago Press, 1982), pp. 27–45.
10Christopher B. Barry and Mustafa N. Gultekin, “Differences of Opinion and Ne- glect: Additional Effects on Risk and Return,” Table 4 in John B. Guerard and Mustafa N. Gultekin (eds.), Handbook of Security Analyst Forecasting and Asset Allocation (Greenwich, CT, JAI Press Inc., 1992).
11Orie E. Barron and Pamela S. Stuerke, “Dispersion in Analysts’ Earnings Forecasts as a Measure of Uncertainty,” Journal of Accounting, Auditing, & Finance (Summer 1998), pp. 245–269.
Thus divergence of opinion appears useful as an indicator of risk.
Variances of return estimates are useful for investors. Even if this risk is non-systematic risk, active investors may take large enough positions in a stock they believe will outperform the market for that stock’s variance to make an appreciable contribution to their portfolio’s variance.
This correlation between risk and divergence of opinion creates econometric problems for any one trying to test the effects of divergence of opinion on asset prices or on returns because the pure divergence of opinion effect is to reduce returns while the risk effect is to increase them.
There appear to be several reasons for investors to avoid stocks with high divergence of opinion. One is the “winner’s curse” effect.12 The opin- ions the “winner” invests in may be the wrong set of opinions (and win- ner’s curse theory suggests that if he (or she) chooses to invest in them, he has an above average chance of acting on a wrong set of opinions). Then he will be disappointed. A second factor is that such stocks tend to be riskier, and most investors should wish to avoid risky stocks.
A third factor applies to those investing other people’s money (and perhaps a few individual investors who are worried about their spouse’s opinions or their own self-esteem). Investing in high divergence of opin- ion firms is risky for a manager’s career. There are likely to be analysts’
reports implying, or even stating, that the investment should not have been made. In the event the investment loses money, his superiors and those who hire managers will have something to point to implying he was imprudent or even stupid. If the reports were actually recommenda- tions to sell or to hold (i.e., not to buy), you need to explain why you acted contrary to them. If the most pessimistic reports merely indicated that earnings were likely to be much lower than others expected, it can be argued that you should have believed analyst X and anticipated the disappointing earnings and the price decline that followed the earning announcement. It is safer to fail conventionally, that is, to buy a stock in a company that did worse than anyone predicted.
In theory, if some investors are avoiding a stock for any of these rea- sons, the price should be lower and the expected return higher. The win- ner’s curse effect has had relatively little discussion anywhere (it appears in no textbook or popular investment book for instance). Thus, it is implausible that prices adjust for it.
Modern financial theory divides risk into that which can be easily diversified away and that which cannot (systematic risk). It will be argued here that divergence of opinion is correlated with both. This, in
12Edward M. Miller (Principal investigator and author of most), Study of Energy Fuel Resources, vol. 1 (Cambridge, MA: Abt Associates, 1969).
the presence of restrictions on short selling, has interesting implications for the security markets and for investment policy.
Price and Diversifiable Risk
Most investors are aware of reasons for avoiding risky stocks in their portfolios. They understand that higher risk should only be accepted if there is also higher return. All things being equal, this implies a risk-return trade-off among securities. Emphasis is put on “all things being equal”
condition, because the divergence of opinion effect is excluded in most discussions. Conventional wisdom is that prices have adjusted so that there is such a tradeoff between risk and return. The textbook version of this wisdom is that returns should only be related to systematic risk because investors can and have diversified away all nonsystematic risk.
The most popular model among academics of portfolio building is Markowitz optimization. This results in a fully diversified portfolio (the textbook market portfolio) only if the returns put into the models are those predicted by the capital asset pricing model. In this case, some of every asset is held and there are no short positions. If one puts in expected returns that differ appreciably from those predicted by the capital asset pricing model, the portfolios no longer resemble the market portfolio. If short sales are permitted, typically there will be a position in virtually all stocks, but many of these will be short positions. However, the typical investor constrains weights to be nonnegative, because he is either legally unable to go short, or unwilling to go short (or believes the obstacles to short selling make such positions undesirable). For such investors, Markowitz optimization (with noncapital asset-pricing-model-predicted rates of return) will typically produce zero holdings for most stocks.
If the investor has a reasonably high acceptance of risk (variability in final value of portfolio) and believes some stocks will have much higher risk-adjusted returns than others, the optimization process will produce a nondiversified portfolio. Such an optimized portfolio reflects a tradeoff between risk and return such that putting any more money into the stocks expected to have the highest returns will increase risk to an unacceptable degree. The relevant measure of risk for each investor is a form of “systematic risk” since it reflects the correlation of each stock with the optimized portfolio.
In Markowitz optimization, which seems a reasonable model for many investors, the limit to the stock’s weight is set by a risk-return trade-off. At the optimum weight for the stock, the loss of utility from increased risk caused by further increases in the weight will exceed the gain in utility from a higher portfolio return. All things equal, the higher a stock’s standard deviation in returns, the lower the optimal weight for any given estimated
expected return. The optimal weight also decreases as the covariance of the stock with the rest of the portfolio increases. In turn, this covariance increases with the stock’s standard deviation. With the high divergence of opinion stocks also having high return variances (standard deviation squared), the limit to holdings of a stock is reached at a lower weight in a particular portfolio. This means that the stock has to be priced lower in order to be included in more portfolios (a requirement for markets to clear). This somewhat reduces the price raising effect of high divergence of opinion in the absence of a correlation with return variance or volatility.
For instance, if an investor believes Yahoo will have high returns, the optimization program will cause him to buy Yahoo. It will stop adding Yahoo only when adding any more Yahoo to the portfolio produces less utility than adding another, lower return stock whose return is less corre- lated with this portfolio. Since the optimized portfolio heavily weights Yahoo, the alternative stock will typically be one that whose return has a low correlation with Yahoo. The optimization model rejects additional holdings of Yahoo because the return on Yahoo has a high correlation with the returns of this portfolio, which is one that overweights Yahoo.
The higher the variance of the return to Yahoo, the less of Yahoo the computer buys. Thus, the optimal purchases of a stock depend not only on the beta (calculated relative to the portfolio as a whole) of the stock, but also on the nonsystematic or diversifiable risk of the stock. In a stock market where there is high diversity of opinion and investors are risk averse, it follows that “nonsystematic risk” could be priced. Since the typical investor will be less than fully diversified, they should ratio- nally require higher returns from stocks with high “diversifiable risk.”
Thus, each investor will purchase less of stocks with a high diversifiable risk (for a given forecast return). When the market aggregates the demand curves of all investors, these lower purchases imply a higher average return for stocks with high diversifiable risks (all things equal).
This effect could produce a tendency for nondiversifiable risk to be priced. This is in addition to the recognized tendency for “systematic”
risk to be priced. Since high divergence of opinion stocks tend also to be high diversifiable risk stocks, this is an effect opposite in direction to the pure divergence of opinion effect.
This is a way that recognition of divergence of opinion (along with restricted short selling) changes financial theory. When investors dis- agree about the merits of securities, investors will concentrate their portfolios on the securities they value highly. This will lead to diversifi- able risk being priced, while it is not priced in models with rational investors and unrestricted short selling.
Xu and Malkiel document that there is a strong tendency for the stocks in the S&P 500 with a high idiosyncratic risk (diversifiable risk)