Learning from the price does not come for free

Stock charts and any patterns arising from them can be extremely complex and difficult to explain. Investors spend millions of dollars to better understand these patterns and hopefully analyze market data for profit (for example, through technical analysis and algorithmic trading), but standard economic theory ignores this reality. Obtained literature (e.g. Grossman and Stiglitz 1980, Hellwig 1980, Vives 2008, Banerjee 2011) assumes that market participants are highly sophisticated and know the exact relationship between origin and price. Investors can fully understand the price function and thus read the value of the asset pricelessly to uncover price-relevant information (Corsetti et al. 2019 argues that high-frequency trading makes the information that traders can extract from the price). This literature further assumes that all investors are equally sophisticated, with no differences in their experience, ability or knowledge of the market. The fact that real-world investors not only differ, but also expend large amounts of resources to better understand the market implies that the notion that market participants fully understand the market environment does not represent the real world (see Duffie et al. 2022 analysis of data prices in financial markets). What if price information is expensive to interpret and investors make mistakes in the estimation process? How can we determine the sophistication of investors when explaining the value of an asset? How do investors’ sophistication affect market value and trading volume?

In a recently published research paper (Mondria et al. 2022), we have developed a framework to relax these economic projections and answer the important questions raised above. The framework allows investors to monitor prices for free (actually like Google or Yahoo Finance) but investors need to make an effort to understand exactly how these prices relate to the fundamentals of companies. These efforts reveal the underlying relationships that investors can use to guide their trading strategies. Efforts to learn, analyze, or generally better understand the relationship between value and fundamentals can be broadly interpreted as resource expenditure. Investors choose the level of effort to apply intuitively and then work on the information they acquire.

Our structure can be explained from a behavioral perspective. Each investor has two rights. There is a self who creates ideas about how the market works and chooses the sophisticated level of the investor. Then there’s the other self who adds noise when reasonably bound and explains the information in the price. Although throughout our paper we follow this behavioral explanation based on individual investors, we can alternatively interpret two of our investors as the research department and trading desk of an investment organization. The trading desk is responsible for asset transactions, and it is up to the research department of the organization to improve the understanding of how to create information from value. The research departments describe the method of how to extract the best signal from the price in the form of research report, but add noise to the understanding of the methods of trading desk report.

Modern markets have shown inconsistencies that should not exist if standard economic theory is true. These inconsistencies include price movement (future returns positively depend on current prices), excessive return volatility, and excessive trading volume. These have been consistently observed empirically, but under standard economic theory they cannot be easily explained theoretically (e.g., Zegadish and Titman 1993, Odeon 1999, Barbar and Odeon 2000, Moskovitz et al. 2012). Our framework provides an explanation for these effects and shows that inconsistencies can come from costly explanations of asset prices.

Let’s use price momentum as an example to explain how our mechanism works. In particular, when investors have limited funds to analyze price data, they will not be able to acquire all the information and understand the full relationship between price and fundamentals. Partial understanding means that investors are not fully integrating all available information. When their transactions are combined, prices will react less to the data, which will lead to price movement inconsistencies.

In addition, our structure allows our investors to test sophisticated level choices. The trade-offs associated with sophistry are as follows. Increasing sophistication improves investors’ ability to interpret market data, but acquiring sophistication is expensive. The choice of the sophisticated level leads to strategic complementarity, so that the incentives that investors face to gain more information are positively related to how sophisticated the average investor is. The more sophisticated the average investor, the greater the incentive for a new investor to acquire information. Intuitively, sophisticated investors who spend millions of dollars to analyze market data have a greater incentive to do so when their peers spend millions of dollars to analyze the market.

The strategic complementarity of our paper and other results are largely driven by the ‘noise’ added to the price signal when investors interpret the information contained in the price. To illustrate this noise, consider the following two extreme examples. First, if investors can be the least sophisticated, their trading will be similar to white noise (i.e. randomness) because nothing is informing their decision. The randomness of trading will be combined with prices that do not reflect the fundamentals, which is the extreme case of noisy prices. Second, if investors can be as sophisticated as possible, the prices will provide all the information about the basics and there will be no noise. In reality, investors are in a spectrum between these two extremes, and this noise has strategic complementarity. As investors create better algorithms to interpret price data, price ‘noise’ will decrease as investors become better at understanding the relationship between price and fundamentals. This small noise implies that prices are more informative than before, which in turn will encourage investors to dedicate more resources to studying prices, as price data is now more valuable.

A major contribution we make is a framework that combines a financial theory with long-term empirical results. This framework allows the theory to adapt and then explain the effects that were previously seen as inconsistencies that should not be subject to standard economic theory. The structure provides the theoretical basis for some of the most well-known inconsistencies in money, such as the momentum pattern of asset returns. Our paper also touches on how the structure can be integrated to explain trends in the asset management industry, such as animal husbandry and increasing levels of sophistication through strategic complementary effects. Future research could use the framework to better understand financial markets and add subtlety to empirical results rather than writing their existence as abnormal.


Banerjee, S. (2011), “Learning from Value and Scattering in Faith”, Review of Financial Studies 24: 3025-3068.

Barber, B and T Odean (2000), “Trading is Hazards to Your Wealth: The Common Stock Performance of Individual Investors”, Journal of Finance 55: 773-806.

Corsetti, G, R Lafarguette and A Mehl (2019), “The Quality of Fast Business and Entropy”, VoxEU.org, 13 August.

Duffy, D. T. Foucault, L. Weldcamp and X. Vives (2022), “The Impact of Technology on Finance”, VoxEU.org, 27 May.

Grossman, S. and J. Stiglitz (1980), “On the Impossibility of an Informationally Efficient Market”, American Economic Review 70: 393-408.

Hellwig, M (1980), “On aggregation of information in competitive markets”, Journal of Economic Theory 22: 477-498.

Jagadesh, N. and S. Titman (1993), “Winners and sellers return to buying: Impact for Stock Market Efficiency”Journal of Finance 48: 65-91.

Mondria, J, X Vives and L Yang (2021), “Expensive Interpretation of Asset Value”, Management science 68: 52-74.

Moskowitz, T, YH Ooi and LH Pedersen (2012), “Time Series Momentum”, Journal of Financial Economics 104: 228-250.

Odean, T (1999), “Do Investors Trade Too Much?”, American Economic Review 89: 1279-1298.

Vives, X (2008), Market Information and Education: Impact of Market MicrostructurePrinceton University Press.

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