Do the Ignorant Accumulate the Money?

For the 1997 Eastern Finance Association meeting

Abstract

Because the less informed misestimate asset returns, they anticipate higher returns from their risky investments. They thus over-invest in risky securities. They are rewarded by higher total portfolio returns. Too high a proportion of less informed investors lowers the return on risky assets. Equilibrium requires that the less informedís rate of return equal the informedís. The less informed control a stable equilibrium percentage of total wealth. Because an individualís recent investment experience correlates with his terminal wealth, learning need not reduce the less informedís risky asset exposure. Implications exist for the slope of the return versus systematic risk curve.

*I thank Robert Graber for serving as my research assistant while this paper was being prepared.

TABLE OF CONTENTS

Milton Friedman's statement of this position is widely quoted (1953, p. 175), ìPeople who argue that speculation is generally destabilizing seldom realize that this is largely equivalent to saying that speculators lose money, since speculation can be destabilizing in general only if speculators on the average sell when the currency is low in price and buy when it is high.î

There is only one problem with this argument. In an efficient, or a mildly inefficient market, the informed may make money at the expense of the less informed. The reason for this is that financial theory predicts that risk taking is rewarded (see Reilly 1989, Bodie, Kane, & Marcus 1993, or any other text). Those taking greater risks can earn a higher rate of return. Poor information and inadequate security analysis often lead to taking greater risks. If the return to the extra risks taken by the less informed exceeds the losses due to poorer security selection, the less informed will experience a higher rate of return on their total portfolios than the better informed. The wealth of the less informed will grow more rapidly than the wealth of the informed. Of course, this argument assumes that all other things than information are being held constant. In particular, the less informed and the informed are presumed to have the same risk preferences.

As a simple example of this argument, consider the numbers reported in the widely used Reilly text (1989, p. 46), where stocks have expected earnings of roughly 10% (the reported 9.9% rounded) and a large standard deviation of 21.1%, and risk-free securities and bank accounts earn 4% (the reported 3.5% rounded). As any finance text explains, investors choose the asset mix that maximizes their utility. Those that choose to buy more stock (because they are more tolerant of risk) earn more on average than those who are averse to risk and stay primarily, or even entirely, in the risk-free asset.

What determines the actual return earned by classes of investors, such as the less informed? I would suggest that it is their asset mix, the fraction of their assets put in stocks and other risky assets relative to the fraction in low risk assets, including bank accounts and bonds. Brinson, Hood, and Beebower (1986) found that for large pension plans, asset mix explained 93.6% of the variation in total portfolio return. Asset mix is likely to be similarly important for classes of individual investors. To determine whether the less informed earn more than the informed, we must ask whether the less informed are more likely to invest in risky asset classes, such as stocks. It will be argued that they are.

How should we model the behavior of the less informed? I would suggest that ignorance leads to mistakes. These mistakes could arise from the widely discussed overreaction (De Bondt & Thaler 1985, 1987, Jegadeesh & Titman 1993) or from a tendency to hold losers too long (Shefrin & Statman 1985). It could be from the errors that lie behind any of the numerous anomalies that have been discovered (see Jacobs and Levy 1988 for a list and for references). However, the argument here doesnít really depend on the nature of the mistakes, merely that they occur. The simplest model is that their mistakes are random, equally likely to raise or lower the returns from the stock or bond being considered. Such errors might be thought of as being distributed in the usual bell shaped manner (probably in a close to normal distribution). The investors, while not all-knowing, and not perfectly informed, will be presumed to be rational, and to act rationally, given their knowledge. In particular, the reader can think of them as using Markowitz optimization, either formally or informally (i.e., making judgements that approximate optimization without formally going through an optimization process).

With random errors, a few securities will appear to provide extremely profitable opportunities. Suppose the less informed investors believe they have identified stocks that will yield 50% per year. These investors believe the potential return to taking the risk of investing in such a stock to be an extra 46%, rather than the true 6%. Naturally, the less informed will invest more of their wealth in such high return opportunities than will the better informed investors.

Investors concentrate on the securities they estimate to have the highest returns, and reduce their holdings of the others to zero. If the returns from a stock are underestimated, the holdings are typically reduced to zero. The asymmetry arises because few investors are willing to make short sales. Among individuals, the less informed are unlikely to even understand the mechanics of short selling. Most institutional investors are forbidden to make short sales.

Not surprisingly, the risky part of the less informed investorsí portfolio is less diversified than that of the better informed investors. This need not imply that they are undiversified. Given the textbook assumptions (efficient market that is pricing consistent with the capital asset pricing model, lending rates equal to borrowing rates, no transactions costs or indivisibilities), each investor holds some of every stock. If less informed investors make errors, they will believe some stocks are over priced, and others under-priced, causing them to purchase more of the latter, and probably none of the former. However, if they are rational (as is here assumed) portfolios will still be designed to provide adequate diversification (see discussion below).

In at least one simple utility function, the fraction of the investorís wealth put into risky assets is proportional to the risk premium he anticipates earning on these risky assets.¹ Other plausible utility functions would yield a similar conclusion. Admittedly, arguing that the less informed and less sophisticated investors go through a formal optimization process is unrealistic, but the resulting conclusion appears well founded.

The above argument does not depend on less informed investors' underestimating risk (although they probably do so).² All that is required is that they overestimate the returns from the stocks included in their stock portfolios. Such overestimation is quite consistent with unbiased estimates of the returns from each individual stock, since the estimated returns conditional on selection for inclusion in the portfolio can be expected to be biased upwards.

Random overestimation implies that the less informed will invest a substantially greater proportion of their wealth in risky assets than the better informed. This outcome is especially likely if the investor believes he has identified several unusually good current opportunities (chances of a lifetime), but does not expect similar future opportunities.

One implication of the above is that (given the same wealth and risk preference) the less informed will play a greater role in the market for risky assets, such as stocks, than would otherwise be expected given their share of the population and wealth.

Given the traditional efficient market assumption, in which prices fully reflect available information, the expected rates of return on assets reflect only their systematic risk, as any standard textbook explains. It also follows that the expected return on any portfolio will depend only on its systematic risk. It then follows that the portfolios of the less informed, reflecting a higher level of systematic risk because of a higher proportion of stocks and other risky assets, should earn higher rates of return. Thus, the surprising conclusion is that the less informed investors should do relatively better than the informed investors, contrary to the speculations of Friedman and others.

Some readers may be bothered by the idea of markets being efficient in spite of the presence of less informed investors. The difficulty probably arises because the usual textbook treatment of efficient markets assumes everyone has perfect information. However, a belief in an efficient market does not require such an unrealistic assumption. Many uninformed trades cancel out other uninformed trades. More importantly, with many well informed investors, as well as many poorly informed investors, minor divergences from theoretically correct prices can persuade the better informed investors to hold the stocks that the less informed neglect. The result is close to an efficient market, even if not a perfectly efficient one.³

Black (1986) has pointed to the importance of noise in understanding financial markets. The above argument not only incorporates noise traders, but more importantly provides an explanation for why they may persist over time. De Long, Shleifer, Summers, and Waldman⁴ in a series of papers made the very important point (on which this paper builds), that less informed investors may earn a higher rate of return on their total portfolios (stocks plus risk free assets) because they believe they have a more favorable risk-return opportunity and hence invest in securities with a higher return. Thus, the wealth of the less informed may grow more rapidly than the wealth of the informed.

The above demonstration that the less informed investors should earn higher rates of return than better informed investors creates problems for efficient market theory. If they consistently earn more than the informed investors, their wealth should come to be a steadily greater percentage of the funds invested in the market, and the informed investorís percentage of the market should steadily shrink, approaching zero if given enough time. After a long enough period of time, the informed investors must control such a small share of total wealth that their trades (offsetting those of the less informed investors) can no longer insure efficient markets.

Notice this implies a logical inconsistency in mainstream theory. Markets are argued to be efficient because informed investorsí trading keeps them efficient. This requires that informed investors control some appreciable fraction of the wealth. However, it was just shown that, in the long run, the share of the wealth of informed investors in equilibrium tends towards zero. Such near zero shares of the wealth held by informed investors implies that markets will not be efficient.

Actually, the situation is worse than that discussed above. Becoming well informed costs money (information and security analysis are not free). This should reduce the rate of return earned by those investors who become informed. This effect serves to pull the rate of return earned by informed investors even further below that earned by the less informed, making the textbook theory even less defensible.

Admittedly, a belief in the survival of the less informed or less rational traders does not require the absence of short selling. In the long run the highest growth rates for a portfolio are obtained by those that maximize the geometric mean of their returns over time. This is achieved by investors with a logarithmatic utility function. However, Samuelson (1971) has pointed out that rational investors need not have the logarithmic utility function that maximizes long run wealth growth. There may be individuals operating with less than perfect information whose strategies produce portfolios with a long run geometric average rate of return that exceeds those of more rational or better informed investors. The ratio of the wealth of one of these investors to a maximally rational, well informed individual increases the further one looks into the future. Given the wide variety of poorly informed strategies or irrational ones, it is highly likely that at least one of these has a higher expected long run geometric average rate of return than any of the strategies that rational individuals might adopt. In a long run (leaving aside saving and dissaving issues) if new investors do not enter or leave the market, these less informed investors will come to dominate the market, as has been pointed out by Shefrin & Statman (1994) and Blume & Easley (1992).

Usually, when a logical application of a theory leads to ridiculous conclusions (or ones demonstrably false), one looks to find an assumption behind the theory that is either incorrect, or a poor approximation of reality. One then replaces this assumption with a better one.

I. WHY LESS INFORMED INVESTORS EVENTUALLY COME TO AFFECT PRICES

As the percentage of wealth controlled by the informed investors decreases, eventually a point is reached where the buying of the less informed bids up some stock's price. Normally, this will occur after the informed have reduced their holdings to zero (i.e. are constrained by the relevant short selling restriction), and the few willing to go short have the largest positions they are comfortable with, or are legally allowed to hold.

Once this position is reached, further buying by the less informed must raise the price. This increased price draws forth additional selling, either from less informed investors, or from informed investors who are willing to take a larger short position if offered a higher price. At this point, the assumption of a flat market demand curve becomes untenable.

For what percentage of the wealth controlled by informed investors this state is reached depends on several factors. One is the extent to which different less informed investors are purchasing the same stocks. If all less informed investors are making the same mistake, they will all be putting the same stocks into their portfolio in more than the market proportion, and the stocks will more quickly reach the level of holdings at which most informed investors have sold out.

If the less informed investors differ among themselves, some of the buying will be matched by selling by other less informed investors. The net impact on the market will be reduced, and the wealth of the informed investors can shrink to a lower level before the buying by the less informed investors impacts appreciably any stock prices. However, given the assumptions stated above, sooner or later the relative wealth of the informed investors will decline to such a small level that the assumption of efficient markets no longer holds.

The only exception to the above argument is if, through incredible luck, the less informed investors are buying stocks in exact proportion to the market portfolio. In this case, the share of the wealth held by the uninformed can keep growing faster than that of the informed. Thus, the share of the total wealth held by the informed investors will decline steadily towards zero. Only then can prices stay at the level predicted by efficient market theory even as the informed's share of the wealth approaches zero. This outcome is sufficiently unlikely that it need not be taken seriously.

Thus, let us drop the unrealistic assumption made earlier that informed investors can keep prices at the level predicted by efficient market theory.⁵ Instead, the less informed investors will be presumed to be able to bid prices up. Such bidding up of prices lowers the rate of return earned on these securities. For instance, one might imagine that the less informed could bid up by 25% the prices of the stocks found most attractive, before the lowered return discouraged any further purchasing. If this happens, the set of stocks that the less informed think promise high returns might actually yield only 8% (this is the 10% a typical stock would have earned, divided by 1.25). Such lower returns on their stocks obviously reduce the returns on their total portfolios, but still leave it possible for the aggregated portfolios of the less informed to outperform the aggregated portfolios of the better informed. Fortunately, it is easy to calculate whether with this lower return the total (risk free assets included) portfolios of the less informed would still outperform the total portfolios of the informed.⁶ If the less informed's aggregate portfolios would still outperform that of the better informed, their share of wealth would keep growing.

II. AN EQUILIBRIUM WEALTH DISTRIBUTION?

The above suggests an equilibrium distribution of wealth between the less informed and the informed (or across different ability levels). The reason is that unless the wealth distribution is such that the wealth of the informed and the less informed grow at an equal rate, the fraction of total wealth controlled by one group will be changing. As explained below increasing the share of wealth controlled by either group lowers that groupís rate of return, thus slowing the growth of that groupís wealth. Conversely, a decrease in the share of wealth of either group increases that groupís rate of return, thus increasing that groupís rate of wealth growth.

If the relative rates of return on total portfolios are the same for two groups (and if both groupsí net consumption are equal percentages of their total wealth), the two portfolios will grow at equal rates, leaving the equilibrium wealth distribution unchanged. The proof is trivial. Let Wi be the informed initial wealth, Wu be the less informed initial wealth, ri be the rate of return over one year earned by the informed on their total portfolio, and ru be the rate earned by the less informed. After one year informed wealth is (1+ri)Wi, and less informed wealth is (1+ru)Wu. The wealth ratio after one year R is given by

R=(1+ri)Wi/(1+ru)Wu

If ri=ru, it follows that the new wealth ratio is Wi/Wu which was the original ratio. Thus, if the equilibrium distribution of wealth between the informed and less informed investors is somehow achieved, it will persist.

The basic argument can be seen on Figure 1 below, which plots the difference in total portfolio returns versus the wealth ratios for the informed and the less informed investors. Remember the total portfolio is defined to include both the risk-free and the risky assets. Line ABC shows how the differential return declines as the wealth ratio increases. A horizontal line is drawn through the value of 0 for the difference in the total portfolio rates of return. Only when there is no difference (i.e. the difference is 0) does the wealth ratio not change. The equilibrium value for the wealth ratio occurs where this horizontal line is cut by the line ABC. Of course, an equilibrium value for this ratio implies an equilibrium share of total wealth for the less informed.

If the income distribution differs from that required for equal total portfolios, rates of return will change so as to move the wealth distribution towards this equilibrium. If the informed have less than their equilibrium share of wealth, the difference of the total return ratio will exceed 0. This situation is shown by a position on the line AB. Their wealth will increase relative to that of the less informed. This is shown by the direction of the arrow. This movement will continue until the equilibrium at B is reached.

If the ratio of the informed wealth to the less informed wealth exceeds the equilibrium ratio, the differences in their total rates of return will be below 0. This situation corresponds to a position on line BC. The ratio of the informedís wealth to the less informedís wealth will decline, causing a movement along line BC towards B. This is shown by the arrows. Again, the movement will continue until the equilibrium at B is reached.

Thus, if the ratio of the informedís wealth to the less informedís wealth is other than D, forces arise in the form of a differential rates of return on the respective total portfolios, which tend to return the ratio to its equilibrium value. Thus, the equilibrium situation requires that the total wealth of the two classes be growing at the same rate.

The argument can be put in symbols. Let the subscript i refers to the informed, and the subscript u to the less informed, w to the fraction of wealth in risky assets, and Rf to the risk free rate of return, Ri is the return to risky investment for the informed, and Ru the return for the less informed. If wiRi+(1-wi)Rf > wuRu+(1-wu)Rf, then the fraction of total wealth in the informedís portfolios will increase. This will cause Ri to decrease until wiRi+(1-wi)Rf= wuRu+(1-wu)Rf. If wiRi+(1-wi)Rf < wuRu+(1-wu)Rf, then the fraction of wealth controlled by the less informed will increase, causing Ru to decrease until again wiRi+(1-wi)Rf= wuRu+(1-wu)Rf.

Thus, the distribution of wealth between the less informed and the informed will be such that wiRi+(1-wi)Rf = wuRu+(1-wu)Rf, regardless of the initial wealth distribution.

III. SOME SIMPLE NUMBERS

To see if such an equilibrium is plausible, let us look at some simple numbers. If the informed divide their wealth half to stocks earning 10%, and half to the risk-free asset earning 4%, their average return would be 7%. If the less informed have bid certain stocks up so that they earn only 8%, this could be offset by their having three quarters of their wealth in stocks, and only a quarter in the risk-free asset. They would also earn 7%. The wealth of both groups would then be growing at 7% per year, and there would be an equilibrium.

What would be required to induce the less informed investors to put three quarters instead of half of their assets in stocks? Suppose both groups of investors have a simple quadratic utility function that causes one believing the risk premium to be 6% to invest half of their assets in stocks. All that would be necessary for the less informed to invest three quarters of their assets in stocks, would be for them to be convinced that the set of stocks they intended to invest in would earn 13%.⁷ Such an outcome seems possible.

Imagine the less informed have a portfolio building rule of investing in the 25 best stocks (a number believed adequate for diversification). Suppose the universe of stocks contains a thousand stocks, whose average return is expected to be 10%. However, the less informed make small errors in estimating the returns from these thousand stocks, and conclude they have identified 500 they think will outperform the market (i.e. have positive alpha). Since they desire a portfolio with 25 stocks, they choose the top 5% of these for inclusion in their portfolio. It seems very plausible that their errors could be large enough to convince themselves that this 25 stock portfolio will outperform the market by 3%. (Unbiased estimates of expected returns whose standard error is 1.5% would be more than adequate to produce the hypothesized overestimation, since a 3% excess return is approximately 2 standard deviations away from the mean). They would anticipate an expected return of 13% from their chosen portfolio. This would be adequate to induce them to keep three quarters of their wealth in stocks.

Indeed, it seems implausible for a market to be so perfect that the informed never err about the expected returns from stocks, and the less informed have a standard error of only 1.5 percentage points. Yet, such a minor imperfection in forecasting of means by some investors can sustain a market in which some stocks are priced to yield 8% and others 10%. Larger errors could yield markets with greater imperfections.

The author does not find it very implausible that it would take many years (even using sophisticated statistical methods) for the investors who thought they were good enough to pick a 25 stock portfolio (out of an universe of a thousand stocks) that would have an expected return of 13% to discover that their stock portfolios only earned 8%, and that a shift to a new forecasting technique (or a new advisor) would not solve the problem.

Boldt and Arbit (1984, p. 30) have pointed out that it would take 30 years of data to confirm at the 95% probability level that a manager able to earn an excess return of 7.1% was not merely benefitting from measurement error. This point, which has been made repeatedly in different forms, is usually used to argue against selecting managers on the basis of historical success. However, the same problem affects individuals trying to evaluate their own relative skill. Even if they keep accurate records of their investment performance, it will take many years of experience for them to have statistically significant evidence that they have, or do not have, skill.

It is easy to imagine death eliminating investors before they learn that they, or their advisors, lack expertise. Their place in the less informed group would be taken by new investors who thought they could beat the market by 3%.

For simplicity, the analysis has been done with only a single group of less informed investors. There could be several groups of less informed investors. For instance, some may trade excessively and experience much reduced returns. Another group may be very gullible and fall regularly for frauds. Such investors have low, or even negative returns. Over time such investors' share of the total wealth declines. Thus, the less informed investors that drive this model are not grossly uninformed (although such may exist), but those whose errors have the primary effect of causing them to purchase more of the risky assets than they otherwise would. It is these investors who are most likely to affect prices.

Some believe that a showing that the average uninformed or ignorant investor loses money, or earns less than a competitive rate, implies that the role of the less informed in the market can be ignored. However, because there are multiple types of less informed investors, such a showing is inadequate to exclude the possibility that the wealth of some group of less informed investors will eventually grow to a sufficiently large fraction of the total for their mistakes to influence prices.

IV. DIFFERENCES IN SAVINGS AND CONSUMPTION RATES

It was pointed out earlier that for the informed and less informed to have equal growth rates for total wealth, the additions and subtractions from their portfolios due to saving and dissaving must be the same percentages of their wealth. This assumption simplified the exposition.

However, the assumption is easily relaxed. Suppose, the net (of withdrawals for consumption) additions to the informedís portfolio, as a percentage of their original portfolio, is 1% less than the corresponding percentage for the additions being made by the less informed. For there to still be an equilibrium wealth distribution, all that is required is for the informed to earn 1% more on their total portfolio than the less informed.

This requires merely that the horizontal line on Figure 1 be replaced by a horizontal line at 1%. As long as the informed can earn a rate of return of 1 percentage point more than the less informed, an equilibrium is still possible.

Figure 2 (below) can be used to visualize a situation where there is a decrease in savings among the informed, equivalent to 1% of their wealth. If there were no adjustments in rates of return, the effect would be a continuing decrease in the informedís share of total wealth. This would result in their bidding less for their desired stocks, and a raising of these stocksí returns. A new equilibrium would be established when the informed total portfolio rate of return exceeded that of the less informed by 1%. The new return difference reflects lower prices for stocks preferred by the informed. This increases their rates of return. The new return difference is shown in the figure by a new horizontal line at 1%, and a new equilibrium at F. The new equilibrium ratio for the wealth of the informed to that of the less informed is E. Notice the fall in the ratio of the informed's wealth to that of the less informed.

It might have been thought that a permanent lowering of the informed's propensity to save would have resulted in a continuous decline in their share of total wealth. However, this is not so. Once the difference in rates of return has increased from 0% to 1%, the share of the wealth held by the informed will remain constant. Each year lower saving causes the wealth of the informed to fall behind that of the less informed by an amount equal to 1% of the informedís portfolios. However, the informed are gaining an amount equivalent to 1% of their portfolio due to higher portfolio rates of return. The higher return cancels the lower savings. Thus, the wealth of both groups grow at the same percentage rate after the change in relative savings, just as it did before.

It should be noticed that learning that takes the form of some previously less informed investors becoming classified as informed can be accommodated. Suppose such learning transfers an amount equal to 1% of the wealth of the informed from the less informed to the informed each year. This can be offset by the rate of return advantage of the informed declining by 1%. The analysis is essentially the same as if the contribution of savings to portfolio growth in the informed group increased by 1%.

There are presumably limits to how much portfolio additions from other non-investment sources could differ before an equilibrium becomes impossible. Fortunately, there are no strong reasons to believe that the informed and the less informed will differ in non-investment income.⁸

V. IMPLICATIONS FOR INVESTORS

Notice the equilibrium described here differs from the standard efficient markets equilibrium. In an efficient market there is no opportunity to identify securities that offer higher than normal risk adjusted returns using publicly available information. Thus the best that can be done is to diversify widely, minimize the resources devoted to researching securities, and minimize turnover by a buy and hold strategy.In contrast, in the model outlined here, there could be some overvalued and undervalued securities that can be identified with publicly available information. Securities that are popular with the less informed investors are bid up, causing them to have lower rates of return. Other securities, whose merits the less informed donít recognize, have higher returns. In the example discussed initially, the less informed had bid certain securities up by 25%, causing the expected return (as estimated by the informed investors that made the effort to estimate it) to fall to 8%, when other investments were earning 10%. If analysis could separate the 8% stocks from the 10% ones, the increase in returns could justify the increased expense of an active investment policy, especially for investors that have substantial sums to invest (such as institutional investors).

Such a market has been called a bounded efficient one (Miller 1987). Investment analysis covers its cost not by finding winners, but by avoiding losers (Miller 1978). The optimal strategy becomes to study a small portion of the universe of stocks intensely, hoping to identify overpriced ones that the less informed have bid up. Once these have been identified, and enough stocks have been purchased to provide the desired diversification, additional search brings few gains (and has costs). When the stocks the informed buy are earnings 10% (if of average risk), identifying more 10% stocks does not raise the average return earned by the portfolio. Thus, informed investors do not take a position in all available stocks, but only in those correctly priced for the risk.

Finally, notice that the above model provides a clear rationale for security analysis. In the academic efficient market model, there is not reason to do security analysis, and hence no reason to hire security analysts. In the above model, with a large number of less informed investors bidding certain stocks up, it is possible for the better informed investors to earn more than the less informed through security analysis. This provides a rationale for the existence of security analysts.

VI. DIFFERENT LEVELS OF SYSTEMATIC RISK

The above examples were worked out in a simple model where investments were either risk free or risky, and there was no difference in risk between the different risky investments. Of course, risk is really a continuous variable measureable in several ways, and different investments have different risks. Fortunately, this complication is easily introduced into the argument.

As the author has argued elsewhere (Miller 1977), it is likely that the high market beta securities are also those with a high standard deviation of returns, and high difficulty in estimating future returns. The less informed are likely to make only moderate errors in estimating future returns from such stable companies as public utilities and dividend paying industrial companies.

Now consider the so called ìstoryî stocks, perhaps a high technology start up. It is very hard for anyone to estimate such venturesí true returns. The less informed are prone to believe the stories, and conclude that such stocks offer unusually high risk adjusted returns. Thus, they will put more of their wealth into such investments than will better informed investors. In equilibrium, these stocks should offer higher returns than standard industrial stocks. If the stocks are bid up well beyond the price justified by their greater risk, the less informed will experience low returns relative to the better informed, and the fraction of wealth they control will decline. This will cause the prices of these stocks to decline, and their rates of return to rise, restoring equilibrium.

If these stocks offer returns proportional to their betas, the less informed investors will invest relatively more in these high return investments, and their wealth will grow relative to that of the better informed. This will cause their prices to be bid up, and their rates of return to decline, again resulting in the equilibrium situation. Thus, in equilibrium one would expect the rates of return on these high beta securities to be lower than would be predicted by the capital asset pricing model. The argument is illustrated by Figure 3 (below).

These lower returns cause the well informed investors to invest less than they otherwise would in these high beta stocks. However, this is more than offset by the higher than proportionate investing by the less informed, lured by their perception of extraordinarily high returns. Again, an equilibrium should emerge in which the wealth of the less informed grows at the same rate as that of the informed. However, the informed experience, on average, less utility because of more variable investment outcomes.

The above argument depends on high systematic risk stocks generally having higher returns than less risky stocks. Fortunately, Shukla and Trzcinka (1991) have shown that this held not only when risk was measured by beta over a twenty year period, but also for systematic risk measures derived from arbitrage pricing theory.

Thus, a testable prediction of this model is that the slope of the line connecting risk and return will be less than predicted by the capital asset pricing model. As is well known this prediction is realized, with the return on high beta stocks exceeding that on low beta stocks (for an early test see Black, Jensen, and Scholes 1972, see Copeland and Weston 1988, p. 215, and Jacobs and Levy 1988 for other references).

VII. THE EFFECT OF LEARNING

Many believers in textbook efficient market models will argue that investors could not consistently deceive themselves, convincing themselves that they can pick stocks well, while experiencing normal or even subnormal returns. Admittedly, a long series of disappointments may cause investors to shift towards a more conservative strategy. However, security returns are highly variable, and it is easy to rationalize away failures and attribute success to genius. A failure can be attributed to a mistake that has now been learned from, and hence will not be repeated.

Unfortunately, many individuals do not realize the power of compound interest. A stock investor who earns the average 10% per year will double his money in a little more than 7 years. Doubling your money is impressive, and it is easy for such an investor to convince himself that he has unusual skill which will prevent the losses others experience.

It is important to remember that high risk portfolios have highly variable outcomes. Many holders of such portfolios do very well. And many do very poorly. Let us look at the plausible reactions of each of these groups of individuals to their experiences. Those who have done very well attribute their success to skill rather than luck. They are likely to conclude that they (or their advisors) indeed have the ability to select high return stocks. As a minimum they will continue with their high risk, ignorance-based strategy. If anything, they are likely to increase the fraction of their wealth invested in risky assets. Possibly, they will borrow to increase the amount invested.

Admittedly, there will be many investors who are unlucky as well as less informed, or imprudent. Some of these may learn from their experience. Many of them will merely decide what their mistake was, and resolve to avoid it in the future. Unfortunately, there are many possible mistakes to be made, and they are free to make another. Again, if their errors are random, these errors will lead them to believe they have some very high return opportunities available, if they will only accept the risk. Again, they may be lucky or unlucky. If lucky, they act as described in the previous paragraph, and continue investing heavily in risky assets. If unlucky, they again identify their mistakes, decide to avoid making them a second time, and proceed to make new ones.

Admittedly, this sequence may continue long enough so that the investors eventually learn. What they have to learn is the psychologically difficult fact that they are not as smart, or not as informed, as they would like to think. Many people will not admit this to themselves. This tendency to believe good things about ourselves actually serves useful psychological purposes. For one thing, it is generally in our interest to have others believe good things about ourselves. In turn, it is hard to persuade them to hold these opinions if we do not hold them about ourselves. Evolutionary psychologists currently believe that such abilities to deceive ourselves were probably selected for during evolution.

The alternative is to master the extremely sophisticated idea that when there are random errors in estimation, one is likely to systematically overestimate the returns from a project if one acts as if oneís best estimates were truly the expected returns from the project. This is true even if oneís estimates are, on average, unbiased estimates of the expected returns. This is a very difficult concept to master (judging from the authorís difficulty in communicating it to PhD and MBA students) which was introduced to the professional finance literature only in the late seventies. Since the idea is still not in the MBA textbooks, few of the less informed investors will discover it on their own.

In considering whether investors will continue with less than optimal strategies (such as investing on their own estimates of the potential returns), it is important to remember what psychologists have discovered in animal conditioning experiments. Rats have been found to continue longer in a particular behavior pattern if they are rewarded on a random schedule (a variable ratio schedule) than if they are rewarded every time the desired action is performed. Gambling is argued to be reinforced on such a variable ratio schedule (Kalat 1990, p. 264).

The theory applied to gambling probably also extends to investing in risky assets. If less informed investors are occasionally rewarded for investing in risky assets by large winnings, they are likely to persist in their investment behavior for a long time, probably longer than if they were consistently rewarded with small winnings. Indeed, being consistently rewarded with small winnings is more likely to be the experience of the well informed investor. After making accurate estimates of the returns from different assets, the well-informed investor diversifies widely and then puts much of his wealth into a risk-free asset.

However, it is plausible that investors learn, and eventually those who have a long run of money losing stocks will reduce the percentage of their wealth invested in stocks. However, the important thing to notice is that bad luck and ignorance may have already reduced their wealth to the level where altering their asset allocation will have little market impact.

Consider two less informed investors who each invests $20,000, convinced he has exceptional skill. After several years, the unlucky one has reduced his money to a fifth of what he had, or $4,000. He finally concludes he lacks skill, and reduces the fraction of his wealth invested in stocks from 50% to zero. There is now $2000 less invested in stocks.

What of the other investor, the lucky one? To be symmetrical with the one who has lost all but a fifth of his money, imagine he has multiplied his money five fold. His $20,000 is now $100,000. He also is capable of learning from experience. Seeing his stunning investment success, he decides he is a good stock picker. He increase the fraction of his wealth invested in risky assets moderately, from 50% to 60%. This decision raises his commitment to risky assets by $10,000.Notice what learning has done. The net effect of investor learning is that the lucky one increases the amount committed to risky assets by $10,000, and the unlucky one decreases it by $2,000. The net effect is an $8,000 increase.

Yes, investors do learn from their experience. However, the interaction of luck and learning from experience implies that, all other things being equal, those who have earned high rates of return have more money than those who have earned low rates of return. If learning leads to equal percentage shifts in wealth committed to risky assets (or similar effects through adjusting estimated future returns), those with favorable experience will have more wealth, and will shift more money into risky assets than the unlucky investors with poor recent experience will shift away from risky assets. Thus, allowing for learning increases the tendency for the less informed class of investors to invest a higher fraction of their wealth in risky assets, and to earn a higher return on their total portfolios, increasing the fraction of the marketís wealth they control.

Because of the correlation of wealth with historical experience, reevaluations of skills based on investment experience may easily increase the stock commitment of the less informed investors. These are the ones who are most likely to evaluate their expertise by their recent experience, rather than doing a correct Bayesian assessment that included correct prior probabilities (based on the expertise and motivation of competitors) for them actually having unusual expertise.

VIII. CONCLUSIONS

Because the less informed tend to make more errors in evaluating returns on risky assets, they anticipate higher returns from the risky assets they choose. This causes them to invest more of their total wealth in risky securities. In a perfectly efficient market, where all securities have the same risk adjusted returns, this will cause the less informed investors to earn higher total returns.

However, a higher return (not offset by differences in saving rates) on one groupís portfolio implies that this groupís share of total wealth will increase. A perpetually increasing share of wealth for the less informed group is inconsistent with the continuance of efficient markets.

For equilibrium with multiple groups, the rate of growth of the wealth of each group must be the same. In the absence of saving differences, this implies that differences in the fraction of wealth invested in risky assets must be offset by differences in the average risk premiums earned on risky assets. This equilibrium, and the forces maintaining it, can be shown with a simple graph.

Differences in saving behavior between groups can be easily incorporated into the model.

Because errors in return estimation likely correlate with systematic risk, the slope of the security market line is reduced because less informed investors bid up the prices of securities whose systematic risk is high.

It is unlikely that learning will eliminate this effect for several reasons. Some are psychological. However, because an individualís investment experience is correlated with his end of period wealth, equal percentage shifts in the allocation to risky assets (or equivalent changes in beliefs) increase total funds allocated to risky assets.

IX. APPENDIX

DIVERSIFICATION CONSIDERATIONS THAT MAY WORRY SOME

Possibly reduced diversification in the portfolios of the less informed increases the standard deviation of their returns. This in turn could reduce the long term geometric average rates of return, causing them to steadily lose wealth share. To explain the nature of this concern, imagine an less informed investor who made but one error, but this was one that led him to believe he had identified a stock that would double in a year. This would naturally lead him to put much of his money into this single stock, reducing his diversification while also increasing the percentage of his wealth in risky stocks.

However, the lower diversification would increase the standard deviation over time of his portfolio. It is well known that the arithmetic average return is below the geometric average, and that it is the geometric average of a series of returns that determines the value of a portfolio. Increasing the standard deviation of the returns of a portfolio will reduce the long run average return from the portfolio. A useful approximation is:
geometric mean = arithmetic mean - variance/2.

This effect has the potential for reducing the long run average return of a single investor who makes errors and hence becomes grossly undiversified. This could plausibly reduce the returns from such an investor's portfolio by so much that, in spite of being willing to invest more in risky assets, their wealth actually grows at a lower rate. For instance, an investor who was so impressed by a stock's potential that he put all his money in it would have a standard deviation of .49236 using one set of estimates (Statman 1987, Table 1). By the above formula, his geometric average return would be reduced by 12.12%. This reduction exceeds the average annual return of about 10%, and such an investor's wealth would be expected to decline over the long run.

A question sometimes asked about this model is whether the increased systematic risk of a portfolio could increase the variability sufficiently to lower the geometric average return. This is not a serious worry for the informed investors, who if rational only choose to increase the systematic risk of their portfolios (usually by buying a higher percentage of risky assets) only if they anticipate that returns will be higher, and sufficiently higher to justify the increased risk. In their decision making, the estimated relevant long run increase in the rate of return should have already been reduced for the effect of the induced greater variability.

However, for the less informed in this model, where investors take on a higher percentage of stocks because they perceive higher returns which are not really there, it is conceivable that the extra variability that results from a higher systematic risk actually reduces their returns.

Consider the simple case where the less informed are too few to prevent all securities from being priced in accordance with the capital asset pricing model. It was argued above that the less informed would put more of their wealth into stocks. This admittedly increases the variability of their portfolios, and this effect reduces the geometric average rate of return, offsetting part of the increase in average rate of return from owning a higher fraction of stocks. However, this effect is not large enough to cause the less informed investors to actually have a lower rate of return.

The basic reason is that the return to incurring higher systematic risk must be large enough to induce the informed investors to incur the higher systematic risk, which requires that the added return, after adjustment for the greater variability, still be positive. For the added returns from increasing risk to be positive is all that is required to insure (in an efficient market) that the less informed investors (taken as a group) also have their expected returns raised from errors that cause them to take on a higher systematic risk.

Of course, if the less informed investors take on a higher systematic risk as a consequence of their errors, their utility may actually be lower. This lower utility is implicit in a model where they have failed to achieve the optimum portfolio (from the definition of an optimum portfolio).

Even in the extreme case where the less informed investors purchase all stocks, and informed all risk free assets, the historical average geometric return for the less informed investors is 10%, versus 4% for the informed. The direction of the effect is the same for intermediate values. The point, of course, is that the historical data is already expressed as geometric average returns, and already includes the return depressing effect of the higher variability of stocks. Thus, we can be confident that even if the less informed have their returns reduced by the effects of greater variance, that as long as the effect of errors is to induce them to invest more in stocks, or in other higher return assets, that this should raise their expected return.

An apparently harder problem is the possibility that errors might lead investors to increase the non-systematic risk of their portfolios. In theory, this should lower the geometric average return from their portfolios and would moderate the return increasing effect of their errors. While this would lower the investors' utilities from their portfolios (another disadvantage of not choosing the optimum portfolios), this should not produce a large enough lowering of their rates of return to eliminate the effect being discussed. There are several reasons for this.

One is that even the less informed investors will normally have diversified portfolios. A rational investor (as is assumed here), even if less informed about certain stocks, will be well diversified. In particular, the possibility that a less informed investor would irrationally put so much of his wealth into stocks as to actually reduce his long run geometric average return is excluded by this assumption (even if it were not excluded by the assumption that markets are close enough to efficient for higher systematic risk to bring higher geometric average returns).

Suppose the rational but less informed investor purchases 25 different stocks. The empirical evidence is that this number of stocks provides almost the same reduction in non-systematic risk as buying all available stocks. For instance, one study (Statman 1987) estimates that perfect diversification would give an estimated annual standard deviation of 19.138%, while a 25 stock portfolio would have a standard deviation of 21.196%, or roughly 2% higher. Since a one stock portfolio was estimated to have a standard deviation of 49.236%, almost all (over 93%) of the obtainable benefits from diversification are obtained by holding 25 stocks.

The above formula can be used to estimate the impact of the less than complete diversification that might result from rational investors coming to believe some stocks have higher risk adjusted yields than others. If the long run standard deviation of a perfectly diversified portfolio (i.e., one that holds every stock in proportion to the market portfolio) is .19158 (Statman 1987), half of the variance is 1.835%. Now suppose the less informed investors believe (even if incorrectly) that stocks differ in their risk adjusted rates of return, and thus choose to hold only 25 different stocks (in equal proportions). This should reduce their rate of return by 2.246%. This is only slightly greater than the 1.835% reduction with perfect diversification. Less than perfect diversification reduces the geometric rate of return by .411%, a relatively minor reduction. The above illustrative calculation overstates the effect of variance on geometric average returns. This is because both informed and less informed investors are likely to keep some funds in the risk free assets, lowering the difference in the variance of their portfolios.

In the real world, factors such as the lower costs of buying in round lots, the costs of record keeping, and interpretations of the prudent man rule which require keeping informed about stocks held, prevent investors from following the theoretically correct policy of buying some of every stock. In practice, informed and less informed investors may not differ appreciably in diversification. In the extreme case, these factors may have limited the portfolios of the informed investors to the same number of stocks as the uninformed might choose, say 25. Then the only difference being uninformed makes is that it changes the choice of these stocks.

There could be a minor difference in non-systematic risk if the informed were able to use an optimization program (such as Markowitz optimization) to choose these stocks for minimum risk, while the less informed, believing some stocks to have higher risk-adjusted returns, reached a solution somewhat further from optimal. However, this effect is likely to be minor.

Furthermore, the above argument is in terms of classes of investors, with the less informed investors being treated as a class. The less informed considered as a group have more diversification than single investors do. Even the portfolio represented by a large number of individual investors, each of whom holds the single stock that he thinks has the best return to risk ratio, will be well diversified (unless the less informed investors concentrate on a small number of stocks). Given that different investors study different stocks (depending partially on where they live, what they read, and who advises them), the portfolio of holdings by all single stock investors is likely to be well diversified.

Thus the variance in the aggregate stock portfolios of the less informed group should be only slightly greater than in the portfolios of the informed. The small difference in diversification should have little impact on the geometric returns earned. Admittedly, among less informed investors diversification effects should be greater if the errors made by the less informed are different than if their errors correlate with each other.

It should also be realized that the category of less informed investors is composed of a number of different investors, each with his own portfolio, and that there is not the rebalancing that would occur if a single investor held a portfolio identical to the total of all of the less informed investors' portfolios. Because the less informed are less diversified, some members of the group should experience more rapid growth in wealth than others, and the degree of diversification across persons in this group should tend to shrink over time.

If there were no new entry into the group of less informed investors, then as some of the less informed investors experience great success and others less success or actual loses, the combined portfolio of all the less informed investors would tend to increasingly resemble the portfolios of the more successful within the less informed group. Over an infinite period, the percentage of the aggregate wealth of the less informed controlled by the most successful less informed investors would tend towards unity.

However, in reality there are always people entering and leaving (often by death) the market. Hence, this prevents the theoretical possibility of the less informed group's wealth becoming concentrated in just a few hands, which would lower its collective diversification.

Finally, if less diversification among the less informed really does occur, its effect is to lower slightly the geometric average of returns earned by this group of investors, thus lowering the share of wealth they control in equilibrium. As the previous section showed, a slight lowering of the rate of return earned by a group need not prevent an equilibrium from existing, but will merely change the equilibrium share of wealth.

Footnotes

U=E(Rt)-Ast2 (1)

where U = utility, E(r) = the expected return, st is the standard deviation of his total portfolio, and A is a coefficient of risk aversion. If the investor has a risk free asset available with a rate of return of Rf, and a risky portfolio with a return of Rp and a standard deviation of sp, the investor's total portfolio return Rt is given by

Rt=Rf+y(Rp-Rf) (2)

and the standard deviation of the total portfolio is given by

st2=y2sp2 (3)

It then follows (from substitution in 1) that the utility is given by

U=Rf+y(Rp-Rf) - y2Asp2. (4)

and dU/Dy= (Rp-Rf) - 2yAsp2. (5)

For the investor to maximize his utility, he must equate dU/Dy to zero, which implies

y = (Rp-Rf)/2Asp2. (6)

The fraction of the investor's wealth put into risky assets is proportional to the risk premium he anticipates earning on these risky assets. Notice that the proportion of his assets invested in risky assets could exceed one, indicating that he would borrow to invest in risky assets, or would obtain leverage through options or futures. Other plausible utility functions would yield a similar conclusion.

It is possible to construct utility functions such that higher rates of return on risky assets would not lead to a higher fraction of assets invested in risky assets. This can happen if the investor's goal is to merely achieve a certain wealth level, and exceeding this level brings little additional utility. In this case, higher anticipated returns on risky assets might not raise the proportion of assets invested in such assets. However, for realistic utility functions, higher anticipated returns from risky assets should result in investing more in such assets.

2. There are several reasons for suspecting that estimation errors may overestimate both returns and risks.

One reason is that there is a large number of things that can go wrong (lawsuits, entry of competitors, new government regulations, product recalls, etc.) which are individually of low probability, but collectively important. To simplify the analysis investors often place the probability of these events at zero, rather than at the correct low number. Failure to consider the possibility of such adverse events both raises returns and lowers risks.

Also, both firms and brokers have incentives to promulgate good news and information about the possibility of favorable events. Thus, most less informed investors are likely to have heard about possible good news (say a new product in development, or the potential cost saving from a new management). However, there is less incentive to publicize bad news, and the less informed investors may not have discovered some adverse facts.

3. Certainly there is considerable evidence that markets are close to efficient (for a recent review, see Fama 1991). The consistent failure of institutional investors to outperform the averages shows that markets are close to efficient, even if not perfectly efficient (Dunn and Theisen 1983, Brinson, Hood, and Beebower 1986, Bogle 1991, Figures 4 and 5, Lakonishok, Shleifer, and Vishny 1992). As long as mistakes do not appreciably change the rates of return on the securities bought because of these mistakes (which requires merely that mistake-inspired trades be offset by better informed investors' trades), the portfolios of the mistake prone, being riskier, should earn higher rates of return than the portfolios of the informed. Of course, this assumes that the uninformed do not trade so often as to lose most of their income through spreads and commissions.

Admittedly, Ippolito (1989) reported that mutual funds slightly outperformed the indices, a fact which he interpreted as showing some stock picking ability. However, most of this advantage was consumed by management fees and other expenses, as theory would predict. Thus, even conceding this degree of skill, the wealth in these mutual funds would not be growing faster than that of less expert individual investors who did not have the mutual fundsí management expenses.4. De Long, Shleifer, Summers, and Waldman, (1991) have argued that some investors frequently make errors in estimating variability of security returns, normally being overconfident about their predictions, and have shown that such investors can survive and possibly come to dominate in markets. They give very little consideration to misestimates of rates of return, saying (p. 5), "Moreover we assume the misperceptions of different noise traders about a particular asset are correlated, for if all traders confused about the returns on a stock have different misperceptions, their trades will cancel out." I explicitly do not make this assumption here. While De Long et al. (1990b, p. 731) assert that equity underpricing "is a necessary condition for noise traders to earn higher expected returns," this argument shows they can earn higher returns simply because they put more of their total portfolio into risky assets, and more into risky assets with higher systematic risks, regardless of whether equities as a whole are underpriced. A earlier statement by the same authors (De Long et al. 1990a, p. 731) states, "Finally, as we showed in earlier work, if trader's mistakes cause them to take positions that carry more market risk than rational investor's positions, they can earn higher return in the market even if they make judgement errors." However, in their 1991 paper these authors clearly recognize and show that investors who err about individual securities can earn higher returns. The earlier work referred to, while not explicitly referenced, is apparently an earlier draft of the noise paper (here referenced as De Long et al. 1990b). Finally, in a summary paper, Shleifer and Summers recognize that noise traders can earn higher expected returns in the market through undertaking higher risks, but they make the puzzling comment that "Almost for sure, they fail to affect demand in the long run" (1990, p. 25).

5. Readers may ask why the case of uninformed investors forcing prices down is not discussed. There are several reasons. One is that the arguments are symmetric, and if the case of prices being forced up is understood, the case of prices being forced down is the reverse.

However, there are several reasons for concentrating on the case where the uninformed bid prices up. One is that most uninformed investors are probably unwilling or unable to go short. Many individual investors do not understand short selling, and hence will not engage in it. The uninformed investors are likely to be disproportionately represented in this group. Thus, if they underestimate returns from a stock, they are likely to reduce their holdings to zero, and then stop selling. This limits the amount of downward pressure on prices their mistakes cause.

Consider, a case where investors make errors of plus or minus 20% in the expected returns from stocks in a market where the average stock is believed to yield 10%. Most investors making downwards errors will reduce their holdings to zero. Few will sell short. Hence those who do not hold the stock (typically a majority) will not be willing to go short, and will do no selling. The effective impact on the market will be limited, and most likely offset by buying from the informed investors, leaving the price unchanged.

However, every investor who overestimates the return by 20% is likely to be a buyer (even if holdings had previously been zero). This makes it much more likely that the error-induced buying will be large enough to change the price. Thus, the concentration in the body of the text on error-induced buying rather than error-induced selling. The total earnings of the informed investors Rti is given by

Rti=Rf + wi(Ri-Rf) (7)

and for the uninformed investors by

Rtu=Rf + wu(Ru-Rf)(8)

where w is the fraction of wealth in risky assets, Rf to the risk free rate of return, Ri is the return to risky investment for the informed, and Ru the return for the uninformed. The subscript i refers to the informed investors, and the subscript u to the uninformed ones. The total rate of return on the portfolios of the uninformed will exceed that of the informed if

wu/wi > (Ri-Rf)/(Ru-Rf) (9)

In words, the necessary condition for the uninformed investors to earn more on their total portfolio is for the fraction of their portfolio invested in risky assets to rise by a higher percentage than the reciprocal of the fraction by which the risk premium they earn on risky assets is reduced. For this to happen seems quite plausible.

7.. See footnote 1 for the demonstration that the percentage of assets in the risky assets is proportional to the anticipated risk premium for such a utility function. If the less informed investors believe the risk premium on the set of stocks they will purchase is 9% (13%-4%), rather than 6% (10%-4%), they will invest 50% more in stocks than the fully informed investors.

8. It is interesting to speculate about what direction any differences might take. It was shown that the less informed will normally overestimate the returns available from risky assets. Thus, they will overestimate the returns from savings. While there is no way of proving absolutely what the effect of changing the rate of return on invested savings is, it is plausible that investors anticipating a greater reward in future consumption will save more. As an example, an investor who thinks he has identified a stock that will double by year end might forgo a vacation or a new car in order to invest. Arguments do exist that individuals might save less if the rate of return on savings were higher. These often depend on a hidden assumption that the higher rate of return will persist for a long period. For instance, the goal of obtaining a specified wealth at retirement requires less saving if rates of return are higher. De Long, Shleifer, Summers, and Waldman (1991) show that consumption should be higher and savings lower for a specific utility function if investors consistently overestimated returns. However, investors who believe that an extraordinarily good investment project has been identified need not expect to again find such a good project. They may very well believe that the long run average rate of return they can earn is much lower, perhaps only the rate that is readily available on risk-free investments, such as offered by the local bank. A belief that a very high return is only temporarily available is a condition in which someone, rational or not, should increase his saving. The type of person who fails to devote time and effort to becoming fully informed about an investment is the same type of impulsive person who is likely to dip into his portfolio to finance an item of consumption. In experiments with children it was found that the ability to defer gratification increased with intelligence (Mischel and Metzer 1962). Since our less informed investors are also likely to be less intelligent, they are likely also to be less able to defer gratification, and to do less saving and more dissaving, hence making smaller net additions to their portfolios.

It is not clear from these a priori considerations whether the informed or the uninformed investors are likely to add more to their portfolios from net savings. Some arguments go one way, and others the opposite way. Most likely, the difference between the two groups is not large enough to prevent an equilibrium wealth distribution from existing.

9. See Miller 1978 for an exposition (the first to my knowledge), with a worked out example.

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