1. Motivation

This note develops an information-based asset pricing model based on Tirole (2009) where thinking through market contingencies is costly and fear of missing an important detail restrains trading behavior. For example, think about a statistical arbitrageur who decided not to release the throttle on an otherwise profitable trading strategy because she noticed that it had an unexplained industry . Alternatively, consider a value investor who didn’t fully invest in a seemingly undervalued conglomerate because of his unfamiliarity with every single one of its business lines. In both of these examples, traders weren’t necessarily afraid of releasing too much information to the market while building their position; instead, they were afraid of taking on a large position and then being held-up by “Mr. Market” after the fact.

To see how the model works, imagine you’re a market neutral statistical arbitrageur who usually puts together a position of type to exploit the momentum anomaly at the monthly horizon. This position has a price of dollars, usually generates a payout of dollars, and costs Mr. Market dollars to put together. You and Mr. Market split the gains to trade with you getting a fraction, , and Mr. Market getting a fraction, , so that:

(1)  

In a more general model the distribution of these bargaining positions would be an equilibrium object, but for now I take them as given.

Portfolio position is the best position you could put together using the available information, but you realize that something may well go wrong. The problem is that you simply can’t write out every single possible contingency. There are just too many. With probability , your boiler-plate portfolio position will only deliver a value of dollars where . In this situation, you actually need to put together the position to exploit momentum while staying market neutral. Portfolio is different from but impossible to specify ahead of time. Rebalancing your position ex post will cost dollars. For example, during the Quant Crisis of August 2007 the market suddenly and unexpectedly went sideways for quantitative traders in long-short equity positions. According to Khandani and Lo (2007), some of the most consistently profitable quant funds in the history of the industry reported “month-to-date losses ranging from to ” of assets under management.

Before trading, you can exert cognitive effort to find out about what may go wrong and how to put together your portfolio accordingly. I assume that once you start trading the position rather than , Mr. Market immediately knows that rather than is optimal. Put differently, changing your usual behavior is an eye-opener. The entire reason for search for the correct portfolio position is to avoid the ex post rebalancing costs. I denote this cognitive effort by:

(2)  

where denotes the probability that you discover the correct portfolio position conditional on not being the right position. Thus, in contrast to the Veldkamp 2006 model, here you invest cognitive effort until your marginal thinking costs equal the change in your expected ex post hold up costs.

2. Agents and Assets

Suppose that you are thinking of putting together a portfolio position denoted by the -dimensional vector, . To do this, you have to buy and sell stocks from Mr. Market subject to a fixed transaction cost of . For example, imagine you’re a market neutral statistical arbitrageur who usually puts together a position of type to exploit the momentum anomaly. Initially, you believe that the price of this portfolio position is dollars:

(3)  

where denotes the -dimensional vector of random payouts of the assets in the market. For example, in the classic Jegadeesh and Titman (1993) setting, you would repurchase th of your portfolio holdings each month at a total price of dollars when using a momentum holding period of months.

However, with probability your initial ideas about how the market will play out turn out to be wrong, and the portfolio position won’t deliver the required payouts. Instead, the position will only be worth dollars where . For example, perhaps lots of other statistical arbitrageurs are also putting on a similar position. In such a world, your seemingly well-hedged position would be anything but. You could easily lose a large chunk of your assets under management if you didn’t quickly rebalance your portfolio in the event of sudden fire sales as documented in Khandani and Lo (2007). There is a different portfolio that delivers your desired payout, but after you put on the initial position it will take an adjustment cost of dollars to switch over to the portfolio . I assume that it is worth it for you to enter into the market even if you know that you will never discover the correct portfolio position ahead of time:

(4)  

3. Information Structure

You and Mr. Market can agree to transact the portfolio position . This portfolio position may or may not be what suits your needs as the buyer. If it doesn’t, an initially unknown portfolio position will deliver the desired payouts provided that you can return to the market and rebalance your position. At the initial stage, though, both you and Mr. Market are aware only of , although you both know that it may not be the right one. You both may, before contracting, incur a cognitive cost to think about alternatives to , and whomever finds out that portfolio position is the right one can decide whether to immediately trade on this information or not. The key idea here is that the discovery of the correct position is an “eye-opener”.

I assume that you have bargaining power and Mr. Market has bargaining power where . For example, if you are the only trader in the market trying to sell the stocks required by portfolio and there are lots of other traders lining up to buy them, then your bargaining power, , will be close to . Conversely, if you are one of many traders trying to short a hard to locate stock, then your bargaining power, , will be close to .

Before trading, you can incur thinking costs of . Here, denotes the probability that you will discover the correct portfolio position conditional on not being the right one. For example, if you wanted to know when lots of other statistical arbitrageurs were also putting on a similar position and thus destroying your market neutrality of the time, then and you would be dollars in cognitive costs to maintain this level of informativeness. I assume both that:

(5)  

as well as that:

(6)  

The first assumption reads that learning nothing costs you nothing while it is prohibitively expensive to always know when you need to deviate from the standard position. The second assumption guarantees a unique solution to the cognitive effort optimization problem described in Equation (14) below.

4. Asset Pricing

In this world, how much effort should you expend trying to figure out if is the correct portfolio position? At what point is it worth it to just trade and then clean up any mess after the fact? Well, first let’s consider the case where you don’t discover the correct portfolio position ahead of time. Conditional on not finding an alternative portfolio, , the posterior probability that is not correct conditional on searching with intensity is given by:

(7)  

The numerator is probability that is the correct portfolio, , times the probability that you didn’t discover this fact during your search, . The denominator is the probability that you didn’t find an alternative portfolio, . If is the appropriate portfolio then Mr. Market captures a fraction of the renegotiation gain creating a hold-up:

(8)  

where dollars is the surplus available to be split after realizing ex post that is the correct portfolio. Let denote your equilibrium level of search. The ex ante price for portfolio accounts for the possible hold-up so that:

(9)  

This equation reads that the price you are willing to pay for the portfolio given your search equilibrium intensity minus the transaction cost and expected hold up costs must equal Mr. Markets share of the gains to trade. Rewriting this equation to isolate the price function yields:

(10)  

Next, let’s consider the case where you find out that is the correct portfolio after expending some cognitive effort. You now how choices:

1. You can trade portfolio and disclose its correctness to Mr. Market.
2. You can still trade portfolio and then rebalance your position ex post.

By disclosing , you will realize a fraction of the gains to trade, . If you conceal and continue to trade position anyways, you will get where the middle term comes from the fact that you know you will have to rebalance your portfolio and the price is given by Equation (10) above. Combining these two expressions and simplifying then says that revealing the correct portfolio position yields an efficiency gain of:

(11)  

The first term in this equation says that you avoid paying your share of the rebalancing costs. The second term in this equation says that you capture Mr. Market’s expected share of the gains to rebalancing. After all, if you start trading portfolio immediately, then the price no longer has to account for the fact that Mr. Market might hold you up later for his share of the gains to rebalancing, . The important thing about Equation (11) is that it’s always positive. Thus, you always want to trade on when you find out that this is the right portfolio.

5. Cognitive Effort

Note that even before getting into any mathematical details, it’s clear that a social planner would ask you to look for new contingencies until the marginal cost of looking for the next important detail would just offset of the expected rebalancing costs, :

(12)  

Moreover, in the absence of any rebalancing costs, , any investment in cognition is purely rent-seeking!

So what happens in the model? You choose your level of cognitive effort by solving the optimization problem:

(13)  

What are the terms in this equation? Well, there are possible outcomes: could be the right portfolio and you could discover it right away, could be the right portfolio and you might not discover it until it’s too late, and could be the right portfolio all along. The first terms represent your payouts in each of these states weighted by the probabilities that they occur. The final term is just your cognitive costs.

Differentiating with respect to your cognitive effort level, , and observing that in equilibrium is has to be that yields:

(14)  

This equation is pretty interesting. It says that you will search until your marginal cost of mulling over your portfolio equals your expected rebalancing costs plus a pair of additional terms. The first extra term says that you will increase your cognitive efforts in order to avoid being held up by Mr. Market after the fact. The second additional term says that you won’t fully account for this hold up problem since you can adjust the price ex ante. The sum of these additional terms will always be greater than ; thus, you will always expend too much cognitive effort. Put differently, this model is populated by Woody Allen traders who neurotically search for negative contingencies.

6. Discussion

This model delivers a couple of interesting implications. First, spontaneous cognition is a natural source of noise in financial markets. This is an attractive feature since noise traders are akin to theoretical dark matter. Without them, informed traders would be unable to exploit their informational advantage in existing noisy rational expectations models. Nevertheless, these traders are inherently difficult to identify in the data and make welfare analysis problematic. How should the social planner weight noise trader utility?

Second, traders spend too much time trying to identify the perfect portfolio position. If you are a statistical arbitrageur, you obviously can’t sit on the sidelines until you have a sure fire strategy. You have to trade even when you are not completely certain about your strategy’s payouts. No one will pay you fees to sit on their money. Interestingly, I find that traders might well choose to expend too much cognitive effort looking for holes in your strategy in order not to be held-up by Mr. Market. I have never seen another model predict this.

Third and finally, traders specialize in identifying bad news for themselves. A trader has little motivation to look for confirming evidence that position is correct. After all, this would be his portfolio position if he exerted no effort whatsoever and spent the morning in sweatpants on his couch. In fact, the adverse selection problem can be severe enough that traders will not go through with the trade unless they find something wrong with the boiler-plate portfolio position.

1. Introduction

Fama and French (1993) found that both a firm’s size and its book-to-market ratios are highly correlated with its average excess return as illustrated in Figure 1 below. For instance, the center panel says that stocks with low book-to-market ratios (i.e., the portfolios at the bottom linked with an orange line) have too high a on the market when considering their paltry realized excess returns. For some reason, it doesn’t take much to get traders to hold growth stocks.

This post reviews the analysis in Daniel and Titman (1997) which asks the natural follow up question: Why? The original explanation proposed in Fama and French (1993) was that these additional excess returns earned by small firms with high book-to-market ratios were due to exposures to latent risk factors. e.g., a stock with a high book-to-market ratio will tend to do poorly when the entire economy suffers from a financial crisis and precisely when you need cash the most. As a result, you are willing to pay less in order to hold this risk. However, Daniel and Titman (1997) suggest an alternative explanation: some omitted variable both causes value stocks to earn higher excess returns (i.e., have a high ) and comove with one another (i.e., have a high ).

Daniel and Titman (1998) highlight a nice parallel between the causal inference problem outlined above, and the inference problem facing an econometrician when trying to figure out the causal effect of going to college on a student’s future earnings. We all know that people with college degrees earn more over their lifetime than people without college degrees (e.g., see Card (1999)). Just as above, the main question is: Why? On one hand, it could be that the process of getting a degree raises your earning power (analogous to the “factor model”). However, it could also be that IQ really drives everyone’s lifetime earnings and on average people with higher IQs are more likely to get college degrees (analogous to the “characteristics model”). In this situation, finding that college graduates earn more than non-graduates says nothing about the relative value of person ‘s IQ or her degree in determining her salary:

(1)  

Similarly, finding that stocks with high book-to-market ratios realize higher excess returns says nothing about where these excess returns are coming from. The only real conceptual difference between the two inference problems is in the case of graduation vs. IQ, the inputs to the regression are data; by contrast, in the case of factors vs characteristics, the inputs to the regression are estimated coefficients:

(2)  

where is the monthly abnormal return to holding stock/portfolio and is stock ‘s loading on the high-minus-low book-to-market factor from Fama and French (1993).

I begin in Section 2 by describing Fama and French (1993)‘s interpretation of the size and value premia. Then, in Section 3, I outline the alternative interpretation of these effects given by Daniel and Titman (1997). The authors propose a test to determine if some of the effect of the size and value premia flow through a channel other than the factor loadings. In Section 4, I describe this test and replicate their empirical analysis suggesting that there is indeed a component to the size and value premia that cannot be explained by factor loadings. Finally, in Section 5, I conclude with a short discussion of Daniel and Titman (1997)‘s results. All of the code used to create the figures in this post can be found on GitHub.

2. Distress Factor Loading

This section describes Fama and French (1993)‘s interpretation of the value premia—i.e., the higher excess returns earned by stocks with a high book-to-market ratio. A stock with a high book-to-market ratio has lots of tangible assets on its books in accounting terms (i.e., a high book value); however, the market does not value the equity in this company very highly (i.e., a low market capitalization). These stocks are in financial distress. Define as the abnormal return to stock after accounting for its comovement with the market return:

(3)  

The Figure 2 below shows that firms with high book-to-market ratios have really high returns and firms with low book-to-market ratios have really low returns on average.

If a financial crisis comes along it will hit all of the firms already in financial distress the hardest. Fama and French (1993) point out that the outsized excess returns earned by high book-to-market stocks is consistent with the idea that traders don’t want to find out that their stocks have become worthless in the middle of a financial crisis. Thus, in order to hold these stocks, they must be rewarded with higher average excess returns. If this story is true, then these higher average excess returns will result from a larger term in the intercept to the regression equation:

(4)  

One way to test this hypothesis would be to create a group of test assets, run versions of the time series regression specified in Equation (4) above to collect the and coefficients, and test to see if a nice linear relationship holds between the realized excess returns and each stock/portfolio’s loading on the HML factor:

(5)  

Figure 3 below shows that causal diagram assumed in Fama and French (1993) linking the each stock’s average excess returns, , to its loading on the HML factor, . Figure 4 then shows that controlling for exposure to size and book-to-market ratio explains away much of the residual variation in the excess returns of the test assets in Fama and French (1993) that isn’t explained by their comovement with the market.

3. Characteristics-Based Pricing

In this section I describe Daniel and Titman (1997)‘s alternative interpretation of the value premium. These authors start with a similar first stage regression model:

(6)  

but replace the unconditional expectation with the conditional expectation . i.e., they propose that there is an omitted variable related to the fundamental “distressed-ness” of each firm . Under this hypothesis, as a firm gets more and more financially distressed, its average excess returns must rise by an amount in order to induce traders to hold the stock. Thus, the time series regression in Equation (4) becomes:

(7)  

with the second stage regression:

(8)  

Figure 5 below shows the causal diagram assumed in Daniel and Titman (1997) linking the each stock’s average excess returns, , to its loading on the HML factor, , and its distressed-ness, . The dotted line linking and captures the idea that distressed firms are likely to have larger loadings on the factor in the same way that people with higher IQs are more likely to go to college.

The natural way to break this logjam and determine whether the value premium is due to a factor loadings or characteristic-based explanation would be to use an instrument. e.g., find some variable that is correlated with each firm’s factor loading, , but uncorrelated with its distress status, ; or, find some variable that is correlated with each firm’s distress status but uncorrelated with its factor loading. Similarly, to solve the graduation vs. IQ debate from the introduction, you would need either an instrument that randomly assigns people with the same IQ to college and non-college groups or an instrument that randomly shocks people’s IQs once they have made their college decision one way or another.

Daniel and Titman (1997) instrument for each firm’s level of distress, . Note that the analogy to the instrumental variables approach here is imprecise since we can’t actually observe each firm’s level of distress directly. e.g., it would be impossible to predict the variable in a regression. Within each size and book-to-market bucket, Daniel and Titman (1997) use a firm’s exposure to the factor prior to the portfolio formation period as the instrument:

(9)  

The logic behind this instrument is the following: If characteristics drive expected returns, there should be firms with characteristics that do not match their factor loadings. All the stocks in the same size and book-to-market deciles will have the same loading on the factor. However, within each of the size and book-to-market buckets, there will be firms whose returns have been highly correlated with the factor in the past as well as firms whose returns have been weakly correlated with the factor in the past. Daniel and Titman (1997) think about this within group historical variation as exogenous and use it to instrument for each firm’s true level of distress.

I use to denote the firms with the highest historical correlation with the factor and to denote the firms with the lowest historical correlation. To empirically estimate whether or not more distressed firms earn higher average excess returns independent of their factor loading, Daniel and Titman (1997) first sort stocks into size and book-to-market buckets to create a residual that captures the excess returns not explain by firms’ factor loadings:

(10)  

They then compute:

(11)  

which captures the mean effect of being more distressed, , times the average level of additional distressed experienced by firms with a high historical correlation with the factor:

(12)  

4. Empirical Analysis

This section replicates the main empirical results in Daniel and Titman (1997). I calculate each stock’s book equity using COMPUSTAT data as the stock holder’s equity plus any deferred taxes and any investment tax credit, minus the value of any preferred stock. I calculate each stock’s market equity using CRSP data as the number of shares outstanding times its share price. To compute the book-to-market ratio in year , I use the book equity value from any point in year , and the market equity on the last trading day in year . The market equity value used in forming the size portfolios is the last trading day of June of year . I exclude firms that have been listed on COMPUSTAT for less than years or have a book-to-market ratio of less than . I demand that firms have prices available on CRSP in both December of and June of year . See Figure 6 below for a summary of the timing.

I use the size and book-to-market ratio data to create the Fama and French (1993) and factors as follows. For the factor, big stocks are above the median market equity of NYSE firms and small stocks are below the median. For the factor, low book-to-market ratio stocks are below the th percentile of the book-to-market ratios of NYSE firms, medium book-to-market ratio stocks are in the middle percent, and high book-to-market ratio stocks are in the top . Using these buckets, I then form value-weighted portfolios and then estimate the and factors as the intersection of these portfolio returns:

(13)  

To create the size and book to market portfolio returns, I use cutoffs at , , , and for both the size and book-to-market ratio dimensions. To create the size and book to market portfolio returns, I use cutoffs at and for both the size and book-to-market ratio dimensions.

To estimate a firm’s historical exposure to the factor, I take all of the firms in each of the size and book-to-market ratio buckets as of July each year . For each of these firms, I then estimate the following time series regression from January of to December of for a total of months:

(14)  

I harvest the regression coefficients and sort the stocks into buckets based on the realized loadings to assign a value of to each firm using cutoffs at , , , and . Thus, a firm in the bucket in July had a loading from January to December that was among the highest within its size and book-to-market grouping. I drop the month period between July and December because it appears that the returns to stocks in the portfolio behave abnormally over this sample period as illustrated in Figure 7 below.

Now comes the punchline of the paper: a portfolio that is long firms in the high distress group, , and short firms in the low distress group, , within each of the size and book-to-market buckets generates abnormal returns relative to the Fama and French (1993) factor model. To see this, first take a look at Figure 8 below. Just as in Figure 2, it’s clear that a stock’s average excess returns rise as it becomes smaller and its book-to-market ratio gets larger. i.e., the average height of the numbers increases as you move northwest across the panels. However, Figure 8 also shows that, within each of the size and book-to-market portfolios, firms with higher historical loadings on the factor tend to earn higher excess returns. i.e., the average height of the numbers increases as you move from left to right within each of the panels. What’s more, moving to Figure 9 reveals that this effect is robust to the Fama and French (1993) factor model. Figure 9 plots the coefficient estimates and standard errors to the time series regressions:

(15)  

All of the estimated s are positive except for , are statistically significant at the level, and more are quite close to this threshold. By contrast, a purely factor model explanation would predict that all of these s should be .

5. Discussion

Daniel and Titman (1997) is a really nice paper that makes a very simple and insightful point: factor loadings do not imply a causal relationship. They support this point by giving evidence that even after controlling for factor exposure, firm’s which are more distressed prior to portfolio formation (i.e., have a distress characteristic) earn higher returns. However, there is a big caveat that comes with the findings. Namely, any characteristics-based model of stock returns necessarily admits arbitrage. After all, a characteristics-based explanation for the value premium says that by choosing stocks with different characteristics, you can change your portfolio’s average return without adjusting its risk loadings. i.e., you can create an arbitrage opportunity. This fact makes it difficult to interpret the phrase “characteristics-based explanation.” As Arthur Eddington (1934) wrote, “it is a good rule not to put overmuch confidence in the observational results that are put forward until they have been confirmed by theory.”

1. Introduction

This post reviews the analysis in Campbell, Lettau, Malkiel, and Xu (2001) who find that firm level volatility has been rising over the period from July 1962 to December 1997. I’ve posted the code I used here. What does this mean? The authors look at the day-to-day variations in the stock returns of all publicly listed firms on the NYSE, Amex, and NASDAQ exchanges in each month, , during this sample. They then show how to decompose the variance of the daily returns each month for a typical firm (e.g., a firm selected randomly with probability proportional to its market cap) into a market-specific component, an industry-specific component, and a firm specific component. Campbell, Lettau, Malkiel, and Xu (2001) find that this firm-specific variance has been steadily rising as plotted in the figure below.

I begin by detailing how Campbell, Lettau, Malkiel, and Xu (2001) estimate their market-specific, industry-specific, and firm-specific (i.e., idiosyncratic) volatility components. A natural first approach would be to estimate industry-level regressions:

(1)  

and firm-level regressions:

(2)  

where denotes the value-weighted excess return on the market, denotes the value-weighted excess return on industry , and denotes the excess return on stock . You could then just insert the realized and terms into expressions for the variation in and to get the desired result, no?

(3)  

The problem here is that and are hard to estimate and may well vary over time. Thus, a -independent procedure is necessary. After describing this procedure in Section 2, I then replicate the variance time series used in Campbell, Lettau, Malkiel, and Xu (2001) in Section 3. Finally, in Section 4 I conclude by extending the sample period to December 2012 and discussing the interpretation of the results.

The analysis in Campbell, Lettau, Malkiel, and Xu (2001) ties in closely with numerous other findings in asset pricing, macroeconomics, and behavioral finance. In the empirical asset pricing literature, Ang, Hodrick, Xing, and Zhang (2006) find a puzzling result that firms with high idiosyncratic volatility “have abysmally low average returns.” e.g., stocks with the highest idiosyncratic volatility have lower excess returns than stocks with the lowest idiosyncratic volatility. In a macroeconomic context, this analysis is directly supports the granular origins theory of Gabaix (2011) which proposes that the key source of aggregate macroeconomic fluctuations is idiosyncratic firm-specific shocks to large firms. Finally, in a behavioral finance setting, the fact that market and industry models explain so little of the variation in firm-level stock returns (somewhere between and ) suggests a new kind of problem for traders: scarce attention. “What this information consumes is rather obvious,” writes Herbert Simon. “It consumes the attention of its recipients. Hence a wealth of information creates a poverty of attention, and a need to allocate that attention efficiently among the overabundance of information sources that might consume it.” As highlighted in Chinco (2012), it takes time to sift through all of the competing information about each and every firm and subset of firms.

2. Statistical Model

In this section I explain how Campbell, Lettau, Malkiel, and Xu (2001) compute the market-wide, industry-level, and idiosyncratic contributions to the variation in a firm’s daily returns. The challenge to doing this is that estimating the firm-specific s empirically is noisy and unreliable. To get around this challenge, Campbell, Lettau, Malkiel, and Xu (2001) decompose the daily return variance of a “typical” firm rather than every firm. e.g., they think about the determinants of the daily return variance of a firm selected at random each month from the market with probability proportional to its relative market capitalization.

To see why looking at a typical firm might be helpful here, define the relative market capitalization of an entire industry, , and the relative market capitalization of a particular firm, , in month as follows:

(4)  

where and . I use the notation that denotes a particular firm in the market, and use to denote the subset of firms in industry : . The key insight is that the value weighted sums of the industry-level and firm-specific s have to sum to unity:

(5)  

This observation follows mechanically from the fact that the market return is just the value-weighted sum of its industry constituents and the industry returns are just the value-weighted sums of their firm constituents:

(6)  

By sampling appropriately, you can eliminate the pesky s from Equation (3) by converting their weighted sums into s.

How would you go about decomposing the variance of a typical firm in practice? First, consider estimating the market-wide and industry-specific variance components in a -independent fashion. Instead of running the regression in Equation (1), consider computing the difference between each industry’s value weighted return, , and the value weighted return on the market, :

(7)  

where now lacks the tilde and is connected to via the relationship:

(8)  

Since is not a regression residual, it will not be orthogonal to the market return, . Thus, when computing the the variance of the value weighted industry return, , we get:

(9)  

Applying the sampling trick described above then allows us to remove the s by averaging over all industries:

(10)  

This result says that if you select an industry with probability , then the expected variance of this industry’s daily returns in month will consist of a market component, , and an industry-specific component, . When the market component is big:

(11)  

then most of the variation in value weighted industry returns tends to come from broad market-wide shocks. Conversely if the industry-specific component is relatively large, then most fo the variation in value weighted industry returns tends to come from different industry-specific shocks which are only felt in their particular corner of the market.

Next consider estimating the firm-specific variance component in a -independent way using the same procedure. Instead of running the regression in Equation (2), I compute the difference between each firm’s excess returns and the value weighted excess returns on its industry:

(12)  

Since is not a regression residual, it will no longer be orthogonal to the value weighted industry return, , and thus:

(13)  

However, the same sampling trick means that the expression for the value weighted average variance over all stocks within each industry will be -independent:

(14)  

The interpretation of this equation is similar to the interpretation of the market-to-industry decomposition above. Putting both of these pieces together then gives the full decomposition as follows:

(15)  

3. Trends in Volatility

I now replicate these variance measures from Campbell, Lettau, Malkiel, and Xu (2001) using daily and monthly CRSP data on NYSE, AMEX, and NASDAQ stocks over the sample period form July 1962 to December 1997. I restrict the data to include only common stocks with share prices above . The riskless rate corresponds to the day T-Bill rate. First, to compute the empirical analogue of the market variance component in Equation (15), I compute the mean and variance of daily excess returns on the value-weighted market:

(16)  

where denotes the number of days in month . I plot the annualized market component of the variance of daily firm returns in the figure below.

Next, I compute the industry-specific contribution to the variance of a typical firm’s daily excess returns using the Fama and French (1997) industry classification codes. There are industries in this classification system, and I code all stocks without a specified industry as their own group leading to total industries. Clicking on the image below gives a plot of the number of firms in each of these industries.

The industry-specific contribution to the variance in daily excess returns of a typical firm is then given by:

(17)  

I plot the resulting time series in the figure below.

Finally, I compute the idiosyncratic contribution to the variance of the daily excess returns of a typical firm as follows:

(18)  

This is the time series I plotted in the introduction. Empirically, it seems that the overwhelming majority of the daily variation in firm-level excess returns comes from idiosyncratic shocks. One ways to quantify this statement is to look at the “model” fit each month:

(19)  

where the model corresponds to a market model with industry factors. The term captures the fraction of the daily variation in firm-level excess returns that is explained by model and industry factors in each month and is consistently below . i.e., more than of the daily variation is explained by firm-specific shocks!

4. Discussion

There are a couple of interesting take away facts from this analysis. First, the nature of the variance time series dramatically changes after December 1997 when the original sample period in Campbell, Lettau, Malkiel, and Xu (2001) ends. Specifically, the post 1997 time period is dominated by a pair of volatility spikes which are not firm-specific: the dot-com boom and the financial crisis. When compared to these events, the slow run up in the firm-specific variance component looks relatively minor.

Nevertheless, even with these large macroeconomic shocks, the majority of the variation in daily firm-level excess returns is driven by firm-specific information. e.g., even during this later period the model fit, , scarcely crosses the threshold in spite of all of the systemic risk in the market! What’s more, there is substantially cyclicality in the model fit at roughly the horizon. i.e., once every the predictive power of the value weighted market and industry factors grows and then shrinks by roughly or around -to- of its baseline. One way to interpret this finding is that there should be year cycles in the profitability of technical analysis using market-wide factors.

1. Introduction

One of the most astonishing things about financial markets is that there is interesting economics operating at so many different scales. Yet, no one would ever guess this fact by looking at standard asset pricing theory. To illustrate, take a look at the canonical Euler equation:

(1)  

Here, and denote the ex-dividend price and dividend payout of the th asset in the economy at time , denotes the prevailing stochastic discount factor, and denotes the conditional expectations operator given time information. Equation (1) says that the price of the th asset in the current period, , is equal to the expected discounted value of the asset’s price and dividend payout in the following period, . At first glance this formulation seems perfectly sensible, but a closer look reveals two striking features:

1. Time is dimensionless. i.e., Equation (1) is written in sequence time not wall clock time. Each period could equally well represent a millisecond, an hour, a year, a millenium, or anything in between. We usually think of the stochastic discount factor, , as a function of traders’ utility from aggregate consumption. Thus, as Cochrane (2001) points out, if “stocks go up between 12:00 and 1:00, it must be because (on average) we all decided to have a big lunch…. this seems silly.”
2. The total number of stocks doesn’t show up anywhere in Equation (1). Not only do traders have to know when there is a profitable arbitrage opportunity somewhere out there in the market, they also have to find out exactly where this opportunity is and deploy the necessary funds and expertise to exploit it. Where’s Waldo? puzzles are hard for a reason. Identifying and trading into arbitrage opportunities is a fundamentally different activity when searching through rather than predictors. More is different. This is the key insight highlighted in Chinco (2012).

In this post, I start by writing down a simple statistical model of returns in Section 2 which allows for shocks at different time horizons and across asset groupings of various sizes. Then, in Sections 3 and 4, I show how shocks at vastly different scales are difficult for traders to spot (…let alone act on). Such shocks can look like noise to “distant” traders in a mathematically precise sense. In Section 5, I conclude with a discussion of these observations. The key take away is that financial theories do not necessarily need to be globally applicable to make effective local predictions. e.g., a theory governing the optimal behavior of a high frequency trader may not have any testable predictions at the quarterly investment horizon where institutional investors operate.

2. Statistical Model

I start by writing down a statistical model of returns that allows for shocks at different time scales and across asset groupings of different sizes. e.g., Apple’s stock returns might be simultaneously affected by not only bid-ask bounce at the investment horizon but also momentum at the investment horizon. Alternatively, at the Apple might realize both an earnings announcement shock as well as a national economic shock felt by all US firms.

Let denote the smallest investment horizon, so that all other time scales are indexed by an :

(2)  

For concreteness, you might think about in modern asset markets. Thus, for a monthly investment horizon meaning that asset market investment horizons span somewhere between and orders of magnitude from high frequency traders to buy and hold value investors. This is a similar ratio to the ratio of the height of human to the diameter of the sun.

Let denote the log price change of the th stock from time through time :

(3)  

where denotes whether or not stock has attribute , denotes the mean growth rate in the price of all stocks with attribute from time through time , and denotes idiosyncratic noise in stock ‘s percent return from time through time . e.g., suppose that the mean growth rate of all technology stocks from January st, 1999 through the end of January st, 1999 was or . Then, I would write that:

(4)  

and Intel, Inc would realize a boost in its January, 1999 returns since:

(5)  

The price shocks, , take on the form:

(6)  

The summation captures the idea that all shocks occur in a particular instant and then cumulate over time. e.g., there is a particular time interval, , during which a news release hits the wire or a market order flashes across the screen. Changes over time intervals longer than reflect the accumulation of changes across these tiny time intervals. The parameters and control the size and frequency of the th shock. Each attribute’s size parameter has units of percent per , and the bigger the the bigger the impact of the th shock on the returns of all stocks with that attribute. Each attribute’s frequency parameter has units of shocks per , and the bigger the the more often all stocks with attribute realize a shock of size . The idiosyncratic return noise is the summation of Gaussian shocks at each interval:

(7)  

3. Time Series

Very different financial theories can operate at vastly different time scales. e.g., attributes that are relevant at the millisecond time horizon will completely wash out by the monthly horizon and vice versa. In this section, I look at only the time series properties of one stock, so I suppress the subscript and write Equation (3) as:

(8)  

To see why, consider the problem of a value investor, Alice, operating at the monthly investment horizon. Suppose that she wants to know whether or not her arch nemesis Bill, a high frequency trader operating at the millisecond investment horizon, is actively trading in her asset. e.g., suppose that she is worried that Bill might have found some really clever new predictor that flits in and out of existence before she can take advantage of it. From Alice’s point of view, the random variable has the unconditional distribution:

(9)  

Let denote the cumulative distribution function of . e.g., governs the cumulative distribution of the average of the shocks that Bill sees over the length of each period from Alice’s perspective. Then, via the Berry-Esseen theorem we have that at the monthly investment horizon:

(10)  

Equation (10) says that the maximum vertical distance between the CDF of the monthly mean of the variable fluctuating at the time scale is identical to the normal distribution to within one part in one-hundred thousand.

There are a couple of ways to put this figure in perspective. First, note that trading strategies have to generate well above abnormal returns per month in order to outpace trading costs. Second, note that Alice would need around of data to distinguish between a variable drawn from the standard normal distribution and at this level of granularity via the Kolmogorov–Smirnov test. Thus, Bill’s behavior at the investment horizon is effectively noise to Alice when looking only at monthly data. In order to figure out what Bill is doing, she has to stoop down to his investment horizon.

4. Cross Section

In the same way that different financial theories can operate at different time scales, different financial theories can also operate at vastly different levels of aggregation. On one hand, this statement is a bit obvious. After all, modern financial theory is built on the idea of risk minimization through portfolio diversification, and traders talk about strategies being “market neutral”. On the other hand, diversification is not the only force at work. Financial markets have many assets and traders use a vast number of predictors. What’s more, only a few of these predictors are useful at any point in time. As Warren Buffett says, “If you want to shoot rare, fast-moving elephants, you should always carry a loaded gun.” Pulling the trigger is easy. Finding the elephant is hard. Traders face a difficult search problem when trying to parse new shocks.

Suppose that Alice is a value investor specializing in oil and gas stocks and now wants to figure out where her other arch nemesis, Charlie, is trading in her market. Even if she knows that he is trading at roughly her investment horizon, it may still be hard for her to spot his price impact due to the vast number of possible strategies that he could be employing. In this section I study the returns of stocks with attributes:

(11)  

where I suppress all the time horizon arguments since I am concerned with the cross-section. For simplicity, suppose that Alice knows that Charlie is making a bet on only of the attributes so that:

(12)  

where if , then for all . e.g., Alice is worried that Charlie’s spotted the one way of sorting all oil and gas stocks so that all the stocks with that attribute (e.g., operations in the Chilean Andes) have high returns and all of the stocks without the attribute have low returns. How many stocks does Alice have to follow in order for her to spot the sorting rule—i.e., the non-zero entry in ?

It turns out that Alice only needs to examine stocks so long as she gets to pick exactly which ones:

1. Stock : Has attributes , , ,
2. Stock : Has attributes , , ,
3. Stock : Has attributes , , ,

The fact that Alice can identify the correct attribute even though she has fewer observations than possible attributes, , is known as compressive sensing and was introduced by Candes and Tao (2005) and Donoho (2006). See Terry Tao’s blog post for an excellent introduction. For example, suppose that only the first stock had high returns of :

(13)  

then Alice can be sure that Charlie has been sorting using the first of the stock attributes. The interesting part is Alice can’t identify Charlie’s strategy using any less than stocks since:

(14)  

e.g., stocks gives Alice just enough combinations to answer yes or no questions.

What’s more, this result generalizes to the case where the data matrix, , is stochastic rather than deterministic. i.e., in real life Alice can’t decide how many oil and gas stocks with each attribute are traded each period in order to make it easiest to decipher Charlie’s trading strategy. Donoho and Tanner (2009) show that in a world where is a random matrix with Gaussian entries, , there is a maximum number of predictors, , above which it is impossible for Alice to spot relevant attributes from among possibilities using only stocks given by:

(15)  

and is summarized in the figure below replicated from Donoho and Stodden (2006). The -axis runs from to and gives values for summarizing the relative amount of data available to Alice. The -axis also runs from to and gives values for summarizing the level of sparsity in the model. The underlying model is:

(16)  

where , is zero everywhere except for entries which are , and each with columns normalized to unit length. The forward stepwise regression procedure enters variables into the model in a sequential fashion, according to greatest -statistic value. The procedure iteratively takes the single regressor with the highest -statistic until reaching the threshold (i.e., the Bonferroni threshold) which is roughly when . The threshold given by Donoho and Tanner (2009) then corresponds to the white diagonal line cutting through the phase space above which linear regression procedure fails and below which it succeeds.

The interesting part about this result is that this bound on comes from a deep theorem in high-dimensional geometry which relates both compressive sensing and error correcting code as suggested by the deterministic example above. It is not due to any knitty gritty details of Alice’s search problem. Notice how the original bound in the and example has an information theoretic interpretation! Thus, Charlie can hide behind the sheer number of possible explanations in the cross section in the same way that Bill can hide behind the sheer number of observations in the time series.

5. Discussion

The speed at which traders interact has greatly increased over the past decade. e.g., Spread Networks invested approximately in a new fiber optic cable linking New York and Chicago via the straightest possible route saving about miles and shaving off their delay. Table 5 in Pagnotta and Philippon (2012) documents the many investments in speed made by exchanges around the world. What’s more, trading behavior at this time scale seems to be decoupled from asset fundamentals. i.e., it’s unlikely that a stock’s value truly follows any of the patterns found in one of Nanex’s crop-circle-of-the-day plots. Motivated by events such as the flash crash there has been a great deal of discussion in recent years about the impact of high frequency trading on asset prices and welfare.

However, the rough calculations above suggest that traders with a monthly investment horizon might not even care about second-to-second fluctuations in asset prices. e.g., think of how high and low frequency bands of the same radio wave can carry rock and classical music to your FM radio receiver without interfering with one another. High frequency trading may be revealing nothing about the fundamental value of the companies in the market place, but just because these traders make short-run returns behave strangely doesn’t mean that they will ruin the market for institutional investors trading at a longer horizon. In this light, perhaps the canonical Euler equation needs to have some additional input parameters, , and :

(17)  

which define the range over which the theory is effective?

1. Introduction

Where do macroeconomic fluctuations come from? Does common variation in firms’ output necessarily come from a single source? In this post, I work through a model which suggests that productivity “factors” might be the result of weak interactions between lots of otherwise independent production decisions. I start with a real world example which has little to do with macroeconomics, then develop the model with this concrete example in mind, and finally conclude by relating this work back to the original questions.

Example (Millenium Bridge):
On June 10th 2000 the Millennium footbridge over the River Thames in London opened to thousands of excited visitors. While engineers carefully designed the bridge to support the total mass of people and survive high wind speeds, the bridge nevertheless started vibrating uncontrollably. The source of this vibration was clear. As shown in the video below at the second mark, all of the people on the bridge locked step. Of course, the visitors didn’t do this on purpose, so how did this happen? How did many random footsteps turn into a coherent left-right-left-right march?

Here is the rough story:

1. Oscillators: A person’s center of mass which drifts back and forth from right to left as she walks.
2. Scale: There were so many people that a small group was really likely to lock step by pure chance.
3. Weak Interaction: Each step makes the bridge wobble slightly and forces others to compensate by shifting their weight.
4. Positive Feedback: People respond to an unexpected bridge shift by moving their center of mass in the opposite direction.

With that many people on the bridge, it was really likely that a handful of people standing right next to each other would happen to lock step for a few strides by pure chance. When this happened, the entire bridge shifted a little bit from right to left. This effect was at first imperceptible. However, because the bridge moved a little bit to the left, the people around this initial cluster shifted their weight a bit to the right in the following instant to compensate. Thus, this initial imperceptible shift got amplified until the entire bridge was vibrating violently and everyone on the bridge was marching in lock step.

Strogatz (2005) shows how to study this phenomenon more generally as a group of weakly interacting oscillators using the Kuramoto Model. In Section , I begin by laying out the basic mathematical framework. Then, in Section , I solve this model up to a constant parameter. In Section , I then show how to solve for this constant. Finally, in Section , I conclude by discussing some potential applications of this idea to macrofinance.

2. Framework

How could I model this idea? In this section, I lay out the mathematical framework for the Kuramoto model which captures these key elements. Consider a world with oscillators, and let denote the phase of the oscillator at time in units of radians. For instance, when people walk they shift their weight slightly from left to right and then back again. Thus, the side to side movement in a person’s center of mass while they walk can be thought of as a simple oscillator. might then correspond to person leaning all the way to the left while might correspond to person leaning all the way to the right. The time derivative of oscillator ‘s phase, , is known as its frequency and has units of radians per second. I use to denote the natural frequency at which the phase of the oscillator changes in units of radians per second. For example, corresponds to the rate at which person would shift her weight from side to side in the absence of any other walkers.

I then look only at the following functional form for the interactions of each oscillator:

(1)  

This formulation means that the frequency of the oscillator at time equals its natural frequency, , plus a constant which has units of radians per second times the average of the sine of the distances between the phase of oscillator and the phase of every other oscillator. Thus, if most of the other people on the bridge just got done taking their left step when your right foot hits, then you will slow down your walking speed. Conversely, if most of the other people on the bridge just picked up their right foot when yours hits the ground, then you will increase your walking speed. The coupling constant, , then determines exactly how much you are going to change your walking speed in each of these instances. I assume each oscillator’s natural frequency is drawn from a symmetric unimodal distribution, , centered around :

(2)  

In summary, this section proposes a simple model which captures the key components of the introductory story. The key state variables, , represent the phases of oscillators. The rates at which each of the oscillators changes its phase are weakly interacting and governed by a coupling constant, . The sinusoidal functional form implies a positive feedback effect. “Solving” the model means determining if there are values of the coupling constant such that, for any collection of oscillators with natural frequencies selected independently and identically from the distribution , all of the oscillators will lump together and take on the similar frequencies. i.e., if there are a bunch of people with different baseline walking speeds on a bridge, will they all end up locking step? This problem is hard because the phase of every single oscillator is pulling on the frequency of every other oscillator at the same time. It is not obvious that a coherent lump will emerge.

3. Solution Strategy

Is this model tractable? In this section, I now show how to “solve” this model up to the determination of a threshold value of the coupling constant . i.e., I show that if the bridge is sufficiently wobbly, all of the people on the bridge will lock step; whereas, if it is stiffer than this threshold value, no synchronization will occur. I leave the task of solving for this threshold value to the next section.

The trick is look for a change of variables that simplifies the problem. I do this by introducing the order parameter :

(3)  

What is this equation saying? is a scalar quantity on the interval. In the first line, the expression then maps this value onto the circumference of the unit circle. Thus, the order parameter denotes the average of a group of unit vectors starting. i.e., captures the average amount that each person on the bridge has shifted their weight. In the second line, denotes the phase coherence and denotes the average phase—i.e., ‘s polar coordinates. Where does the name “phase coherence” come from? Intuitively, if all of the unit vectors are pointing in the same direction, then ; conversely, if each of the unit vectors is uniformly spaced around the circle, then . Thus, for instance, at the second mark in the video, everyone on the bridge is leaning in the same direction so that . then denotes the direction which people are leaning towards. By embedding the phases of each of the oscillators on a unit circle, I can characterize their average phase with a pair of order statistics.

I now show how to simplify the problem by using these order statistics and the sinusoidal functional form of the link between each pair of oscillator’s phases. First, I multiply both sides of the equation by to give:

(4)  

Then, I apply Euler’s formula to rewrite the expression:

(5)  

Equating the imaginary components then yields:

(6)  

Notice that the left hand side depends only on the order statistics, and , as well as the phase of oscillator . Importantly, it does not depend on the phase of any of the other oscillators. Thus, I can substitute this simplified form into Equation (1) to get:

(7)  

This new equation says that rather than depending on the phase of every single other oscillator, the frequency of oscillator depends on its natural frequency, its current phase, and order statistics characterizing the average phase of all of the other oscillators. In the language of the Millenium bridge example, this equation says that my walking speed is determined by my natural walking speed, my current weight distribution, and the average weight distribution of everyone else on the bridge. I don’t have to worry about whether Bob from Hertfordshire who is standing in front of me happens to be leaning a bit more to his left. All I care about is the average weight distribution for everyone on the bridge.

I then look for solutions where and . By choosing the right reference frame, I can then set without loss of generality to get:

(8)  

These assumptions are necessary to solve the model in closed form, but are somewhat at odds with the introductory example. constant means that people on the bridge are not transitioning from incoherent steps (start of the video) to synchrony (the second mark). Rather, their level of phase coherence is constant. means that their average weight distribution is evolving at the mean natural frequency. There will be some fast walkers who shift their weight quickly from left to right; there will be some slow walkers who shift their weight gradually from left to right; but, the average location of everyone’s weight must change according to the average natural walking speed. In this setting, the oscillators can be split into groups: locked and drifting. Oscillators in the first group are phase locked at the original frequency . Oscillators in the second group, on the other hand, drift around the unit circle in a non-uniform way.

(9)  

The conditions above show why I refer to the constant as the model’s coupling strength. As increases, more and more of the oscillators will become locked and rotate around the circle at the natural frequency . i.e., as the bridge becomes more wobbly, more and more people will start to deviate from the natural walking speed and synchronize their steps with the rest of the people on the bridge.

I then look only for solutions where the drifting oscillators form a stationary distribution on the circle. This stationarity condition means that the probability that a fast walker is leaning left or right does not change over time. Let denote the fraction of oscillators with the natural frequency that lie in the interval . Then, stationarity implies that:

(10)  

In summary, in this section solves this model up to the determination of the coupling constant . I first propose a pair of order parameters and , and then show that sinusoidal link between the frequencies of each pair of oscillators means that each oscillator’s frequency only depends on these order statistics. The only free parameter that I have not used yet is the coupling constant, . I tie my hands and propose that any solution must be such that both order parameters are constants, and , and also that the distribution of frequencies, , must be stationary.

4. Coupling Threshold

In this section, I ask the question: “Does there exist a which solves the model above subject to these constraints?” I do this in steps. First, I solve for the stationary density. Then, I enforce the conditions and . Finally, I look for satisfactory solutions of .

Step 1: What does look like? is pinned down by the fact that all oscillators have to lie somewhere on the circle:

(11)  

where by definition for all of the drifting oscillators. This constraint basically says that everyone who is walking on the bridge has to be leaning in some direction. It may be right; it may be left; it may be right down the center; but, it is somewhere. is then given by:

(12)  

Step 2: What are the consequences of imposing and ? To answer this question, I return to Equation (3):

(13)  

Since , then in order to have be constant it must be the case that:

(14)  

So, in order to to be constant, either both of these summations have to be constant, or their changes have to exactly offset each other as time marches forward. Let’s examine each of these summations in turn. Looking first at the locked state, I have that for all such that :

(15)  

Thus for all locked oscillators is an implicit function of rather than . Since the distribution of locked phases is symmetric about by assumption, I have that:

(16)  

Looking at the drifting state, I have that:

(17)  

However, this integral has to vanish since and . Thus, in order for to be a constant, I have to have that:

(18)  

Step 3: So, what values of solve this equation? An obvious solution is that . This solution corresponds to the completely incoherent state with . A second set of solutions comes from the solution to the equation:

(19)  

This equation bifurcates continuously from at a value obtained by letting . Thus, there is a critical value, , such that for all the drifting oscillators lump together. I can compute as follows:

(20)  

In other words, when the bridge is sufficiently stiff there is only one outcome: incoherence. People with randomly selected natural walking speeds will place their steps at random. However, at some critical level of “wobbliness” there suddenly exists a second possibility. As a result of the weak interaction of each person’s leaning from left to right as they walk, people on the bridge might synchronize their steps. An aggregate behavior might emerge from seemingly independent private decisions.

5. Conclusion

What does any of this have to do with macroeconomics? Instead of people leaning left to right on a bridge as they walk, think about firms making lumpy investment decisions between complimentary inputs to their production technology. For instance, think about a firm hiring people (labor) and purchasing equipment (capital). Because these are complimentary inputs, firms want to always have them in roughly equal proportions. Thus, they oscillate between increasing their stock of labor and capital.

First, think about a world where the firms’ decisions to invest are completely independent. Suppose that on average of all firms increase either their labor or their capital stock in any given period. If each addition adds to the firm’s output , then growth in aggregate output, , is given by:

(21)  

where is an indicator variable if a firm makes an investment in either labor or capital. Thus, the standard deviation of output growth would be:

(22)  

Thus, if firm volatility is on the order of per year and there are firms in the economy, the aggregate volatility of the economy should be on the order of per year as discussed in Gabaix (2011).

With this benchmark in mind, now think about a different world where each firm’s investment decision’s weakly interact. If I am running a gas station and you are running a convenience store across the street, I will be quicker to add an extra pump to my station in the months after you add a few more aisles to your store. After all, this means that more people will be parked out front of your store, and each of these cars needs gas to run. This weak interaction will exist even if I don’t know about your extra capacity. I don’t need to understand exactly why more drivers are stopping by my station.

In this world, suppose that a fraction of all firms are phase locked to make investments at the same time. Then, the standard deviation of output growth would be:

(23)  

where the approximation holds for grows large. The idea is that, as the number of firms grows large, the dominant component of aggregate volatility will come from common investment decisions made by the phase locked firms. These decisions will happen in out of every periods. Nevertheless, none of these phase locked firms need to be actively trying to coordinate investment decisions in the exact same way that none of the people on Millenium bridge was trying to march in step.