A Peer-based Model of Fat-tailed Outcomes
AA Peer-based Model of Fat-tailed Outcomes
Ben KlemensUS Census [email protected]. ∗ April 3, 2013
Abstract
It is well known that the distribution of returns from various financialinstruments are leptokurtic, meaning that the distributions have “fattertails” than a Normal distribution, and have skew toward zero. This paperpresents a graceful micro-level explanation for such fat-tailed outcomes,using agents whose private valuations have Normally-distributed errors,but whose utility function includes a term for the percentage of otherswho also buy.
Many researchers have pointed out that day-to-day returns on equities have “fattails,” in the sense that extreme events happen much more frequently than wouldbe predicted by a Normal distribution, and have skew toward zero, meaningthat extreme negative returns are more likely than extreme positive returns.This has been re-verified by many of the sources listed below. The fat tailsof actual equity return distributions is far from academic trivia: if extremeevents are more likely than predicted by a Normal distribution, models basedon Normally-distributed returns can systematically under-predict risk.Here, I present an explanation for the non-Normality of equity returns usinga micro-level model where agents observe and emulate the behavior of others.There are several reasons for rational agents to take note of the actions of otherrational agents; the model here is agnostic as to which best describes real-worldagents, but given some motivation to emulate others, I show that the wider-than-Normal distribution of equity returns follows.From the tulip bubble of 1637 to the housing bubble of 2007, herding behav-ior has been used to explain extreme market movements [Mackay, 1841, Schiller, ∗ This paper originated as work done at Caltech, under the guidance of Matt Jackson, KimBorder, and Peter Bossaerts. The agent-based modeling work was done at the BrookingsInstitution’s Center for Social and Economic Dynamics, and the author thanks the CSED’smembers for their support. Thanks also to Josh Tokle and Taniecea Arceneaux of the UnitedStates Census Bureau. a r X i v : . [ q -f i n . T R ] A p r grow it?’”Section 3 will demonstrate that once we take emulative behavior as given, itis easy to grow fat-tailed outcomes. Section 4 concludes, pointing out that,because situations where outcomes are fat-tailed but not entirely off the chartsare common, we may be able to use emulative preferences to explain more thanthey have been used for in the past. This section gives an overview of two threads of the economics literature thatdo not quite meet. The first is an overview of the existing literature on thedistribution of equity returns; the second is a survey of the situations positedin the finance literature where individuals gain utility from emulating others.
The second central moment, also known as the variance, is defined as: µ = σ = (cid:90) ∞−∞ ( x − µ ) f ( x ) dx, where x ∈ R is a random variable, f ( x ) is the probability distribution function(PDF) on x , and µ is the mean of x (cid:16)(cid:82) ∞−∞ xf ( x ) dx (cid:17) .One could similarly define the third and fourth central moments: µ = (cid:90) ∞−∞ ( x − µ ) f ( x ) dx, and µ = (cid:90) ∞−∞ ( x − µ ) f ( x ) dx. skew is sometimes the third central moment, S ≡ µ , and sometimes S ≡ µ /σ . The kurtosis may be κ ≡ µ , κ ≡ µ /σ , or κ ≡ µ /σ − . In this paper, I will use S ≡ µ , and κ ≡ µ /σ . I will refer to κ as normalized kurtosis to remind the reader that it is dividedby variance squared.The more elaborate normalizations make it easy to compare these momentsto a Normal distribution, because for a Normal distribution with mean µ andstandard deviation σ , µ /σ = 3 . A Normal distribution is symmetric andtherefore has zero skew (whether normalized or not). One can use these factsto check empirical distributions for deviations from the Normal.Fama [1965] ran such a test on equity returns, and found that they wereleptokurtic, meaning that µ (cid:29) σ , and were skewed. However, he is not thefirst to notice these features—Mandelbrot [1963, footnote 3] traces awarenessof the non-Normality of return distributions as far back as 1915. Many of thepapers cited in the following few paragraphs reproduce the results using theirown data sets. Bakshi et al. [2003] gathered data on several index and equityreturns, and (with few exceptions) found a skew toward zero (i.e., negative skew,meaning that extreme downward events are more likely than extreme upwardevents).Most of the explanations for the deviation from the Normal have focusedon finding a closed-form PDF that better fits the data. Mandelbrot [1963]showed that a stable Paretian (aka symmetric-stable) distribution fit betterthan the Normal. Blattberg and Gonedes [1974] showed that a renormalizedStudent’s t distribution fit better than a symmetric-stable distribution. Kon[1984] found that a mixture of Normal distributions fit better than a Student’s t .The mixture model produces an output distribution by summing a first Normaldistribution, N ( µ , σ ) , with an independent second Normal distribution, N ( µ , σ ) . Depending on the values of the five input parameters (two means, twostandard deviations, and a mixing parameter), the distribution produced bysumming the two can take on a wide range of mean, standard deviation, skew,and kurtosis.The mixture model raises a few critiques. Kon found that the sum of twodistributions satisfactorily matches only about half of the equity return distri-butions he tests. Others require as many as four input distributions—and thuseleven input parameters—to explain the four moments of the distribution to bematched. Barbieria et al. [2010, pp 1095–96] tested a set of four broad equityindices (MSCI’s USA, Europe, UK, and Japan indices) against a comparablemodel claiming Normality with variances changing over time, and rejected themodel for all four indices.As with all of the distribution models, the use of a sum of several distribu-tions raises the question of how the given distributions go beyond being a good3t to being a valid explanation of market behavior. After all, one could fit aFourier sequence to a data series to arbitrary precision, but it is not necessarilyan explanation of market behavior. This brings us to the second thread of theliterature, covering the micro-level behavior of market actors. The literature provides many rational motivations for emulating others, vari-ously termed herding, information cascades, network effects, peer effects, spill-overs—not to mention simple questions of fashion. This section provides asample of some of the theoretical results for such models, and a discussion ofherding in the finance context. None of these models were written with thestated intention of describing an observed leptokurtic distribution, but this sec-tion will calculate the kurtosis of the output distributions implied by some ofthese models to see how they fare.
The restaurant problem
Among the most common of the models whereagents emulate others are the herding or information cascade models, e.g. Baner-jee [1992] or Bikhchandani et al. [1992]. In these models, agents use the priorchoices of other agents as information when making decisions.A sequence of agents chooses to eat at restaurant A or B . The first will useits private information to choose. The second will use its private information,plus the information revealed by the observable choice made by the first agent.The third agent will add to its private information the information provided byobserving where the first two entrants are eating. Thus, if the first two agentsare eating at restaurant A , the third may ignore a preference for restaurant B and eat at A . Once the preponderance of prior choices leans toward restaurant A , we can expect that all future arrivals will choose it as well. The next day,both restaurants start off empty again, and early arrivals in the sequence mighthave private information that restaurant B is better, so subsequent arrivalswould all go to restaurant B .Network externalities are a property of goods where consumption by othersincreases the utility of the good, such as a social networking web site whoseutility depends on how many others are also subscribed, computer equipmentthat needs to interoperate with others’ equipment, or coordination problems likethe choice of whether to drive on the right or left side of the road. The typicalanalysis (e.g., that of Choi [1997]) matches that of the restaurant problem.Both the information and the direct utility stories can be shown to pro-duce a bifurcated distribution of results with probability one: over many days,restaurant A will show either about 0% attendance or about 100% attendanceevery day. Many goods show such a blockbuster/flop dichotomy, such as movies[de Vaney and Walls, 1996].But for our purposes, a sharply bimodal distribution is not desirable. First,one would be hard-pressed to find an equity whose returns are truly bimodal.More importantly, such a bifurcated outcome distribution is typically platykur-tic , the opposite of the leptokurtosis we seek. Consider an ideal bimodal dis-4ribution with density r ∈ (0 , at a and density − r at b (for any values of a, b ∈ R , a (cid:54) = b ). The distribution has normalized kurtosis equal to r − r − . For a symmetric distribution, r = 0 . , the normalized kurtosis is one, and itremains less than three for any r ∈ ( . , . . Thus, a model that predicts abifurcated distribution can only show a large fourth moment if the distributionis lopsided, which is not sustainable for equity returns. Distribution models
Brock and Durlauf [2001] specify a model similar to theone presented here. In the first round, a prior percentage of actors is given, andpeople act iff that percentage would be large enough to give them a positiveutility from acting. In subsequent rounds, individuals use the percentage ofpeople who chose to act in the prior round to decide whether to act or not.The specific details of Brock and Durlauf’s assumptions lead to two possibleoutcomes. One is a bifurcation, much like the outcomes for the restaurantproblem models above. The other, due to the specific form of the assumptions,is that the output distribution is the input distribution transformed via thehyperbolic tangent. The tanh transformation reduces the normalized kurtosis,and is therefore inappropriate for deriving leptokurtic equity returns.Glaeser et al. [1996] point out that the more people emulate others, themore likely are extreme outcomes, which they measure via “excess variance.”They do this via a Binomial model: if being the victim of a crime is a drawfrom a Bernoulli trial with probability p , then the mean of n such trials is np , and the variance is np (1 − p ) . Thus, given n and the sample mean (orequivalently, n and p ) we can solve for the expected variance, and if the observedvariance is significantly greater, then we can reject the hypothesis of independentBernoulli trials. However, this process says nothing about whether the observedvictimization rates are Normally distributed or not: excess variance is not excesskurtosis or skew. Finance
Within the theoretical finance literature, papers abound regardingherding behavior (e.g., Grossman [1976, 1981], Radner [1979], Choi [1997], Mine-hart and Scotchmer [1999]), although they concern themselves not with explain-ing herding, but with the information aggregation issues entailed by herding.Many stories regarding the emulation of others apply to the situation of therational, self-interested manager of an asset portfolio: • Pricing is partly based on the value of the underlying asset and partly onwhat others are willing to pay for the asset. At the extreme, people willbuy a stock which pays zero dividends only if they are confident that thereare other people who will also buy the stock; as more people are willingto buy, the value of the stock to any individual rises.5
It has long been a lament of the fund manager that if the herd does badlybut he breaks even, he sees little benefit; but if the herd does well andhe breaks even, then he gets fired. Therefore, behaving like others mayexplicitly appear in a risk-averse fund manager’s utility function. • Since an undercapitalized company is likely to fail, the success of a publicoffering may depend on how well-subscribed it is, providing another jus-tification for putting the behavior of others in the fund manager’s utilityfunction. • If a large number of banks take simultaneous large losses, then they maybe bailed out; since a bail-out is unlikely if only one bank takes a loss, thismay also serve as an incentive for financiers to take risks together. • Simply following the herd: “[. . . ] elements such as fashion and sense ofhonour affected the banks’ decision to take part in a syndicated loan.Banks are certainly not insensitive to prevailing trends, and if it is ‘the inthing’ to take part in syndicated loans[. . . ], people sometimes consent tooreadily.” [Jepma et al., 1996, p 337]The model of this paper is a reduced form model which simply assumes thata financier’s expected utility from an action is increasing with the percentage ofother people acting. I make no effort to explain which of the above motivationsare present at any time, but assert that given these effects, the model below isapplicable.Empirical studies of analyst recommendations find that they do indeedherd. For example, Graham [1999] finds evidence of herding among investmentnewsletter recommendations, and finds that the more reputable ones are morelikely to herd. Meanwhile, Hong et al. [2000] finds evidence of herding amonginvestment analysts, and finds that inexperienced analysts are “more likely tobe terminated for bold forecasts that deviate from consensus,” and therefore less reputable analysts are more likely to herd. Welch [2000] finds that an an-alyst recommendation has a strong impact on the next two recommendationsfor the same security by other analysts, and that this effect is uncorrelated withwhether the recommendations prove to be correct or not. Although these pa-pers disagree in the details, they all find empirical evidence that analysts areinclined to behave like other analysts (and therefore the people who listen toanalysts are likely to also behave alike), so the model below is apropos.
One run of the model below finds an output equilibrium demand given an inputdistribution of individual preferences. Repeating a single run thousands of timesgives a distribution of equilibrium outcomes, which will have large kurtosis andskew under certain conditions.One run of the model consists of a plurality of agents (the simulations belowuse 10,000), each privately deciding whether to purchase a good. Each has6n individual taste for consuming, t ∈ R , where t ∼ N ( (cid:15), and (cid:15) is a smallnon-negative offset, fixed at zero or 0.05 in the simulations to follow.Let the proportion of the population consuming be k ∈ [0 , , and let thedesire to emulate others be represented by a coefficient α ∈ [0 , ∞ ) . Then theutility from consuming is U c = t + αk. (1)The utility from not consuming is U nc = α (1 − k ) . (2)That is, agents who do not consume get utility from emulating the − k agentswho also do not consume, but have a taste for non-consumption normalized tozero. One can show that this normalization is without loss of generality. Agentsconsume iff U c > U nc .A Bayesian Nash equilibrium is a set of acting agents, comprising the pro-portion k a percent of the population, where all acting agents have U c > U nc given k a percent acting, and all agents outside the acting set have U c ≤ U nc given k a percent acting.It can be shown that, given the assumptions here, the game has a cutoff-typeequilibrium, where there is a cutoff value T such that every agent with privatetastes greater than T acts and every agent with t ≤ T does not act. An agentwith private taste t equal to the cutoff T will have U c = U nc .One could embed this model of the distribution of demand into a largermodel, such as a simple supply-demand model where supply remains fixed anddemand shifts as per the model here, and prices thus vary with demand. Tomaintain focus on the core concept, this paper will cover only the core modeldescribing the distribution of ˆ k . Recall the restaurant problem, where we measured the turnout to restaurant A every day for a few weeks or months. Each day gave us another draw ofdiners from the population, and it was the aggregate of turnouts over severaldays that added up to the bimodal distribution. Similarly, the literature onequities did not claim that if we surveyed willingness to pay by all membersof the market at some instant in time, the distribution would be leptokurtic;rather, the claim is that every day there is a new distribution of willingness topay, which produces a single outcome for the day, and tallying those outcomesover time generates a leptokurtic distribution. This model draws a sampledistribution (which one could think of as today’s market, and which will beclose to a Normal distribution), finds the equilibrium value ˆ k , and then repeatsuntil there are enough samples of ˆ k that we can estimate the moments of ˆ k ’sdistribution.We must first solve for the equilibrium percent acting for a single run. Brieflyswitching from the equilibrium percent acting k to the equilibrium cutoff taste T , one can find the equilibrium for a single distribution by finding the value of7Fix N and (cid:15) .–Generate a new population of agents:–Set the initial value of k = .–For each agent:–Draw a taste t from a N ( (cid:15), distribution.–While k this period is not equal to k last period:–For each agent:–Consume iff U k ≥ U nk .–Recalculate k .–Record the equilibrium percent acting k .Figure 1: The algorithm for finding the equilibrium level of consumption forone run. T such that an agent with that value is indifferent between action and inaction,given that the cutoff is at that value (that is, U c in Equation 1 equals U nc inEquation 2). Write the proportion not acting given cutoff T as CDF ( T ) (i.e.the cumulative distribution function of the empirical distribution of tastes upto the cutoff T ); then any value of T that satisfies T = α (1 − CDF ( T )) (3)is an equilibrium.There are typically no closed-form solutions for T , so the work will requirea numeric search.I use an agent-based simulation to organize the draws. For each step, thesimulation draws 10,000 agents from the fixed distribution, then the simula-tion algorithm solves for equilibrium via tatônnement, as detailed below. Theequilibrium reached via market simulation is a Bayesian Nash equilibrium as inEquation 3. There are other search strategies for finding the equilibrium giventhe draws of t , but the agent-based model has the advantages of always findingthe equilibrium and providing a realistic story of what happens in the market.Repeating the process for thousands of draws from the fixed distribution,starting each simulation with a new set of random draws of tastes t from thesame distribution, will produce a distribution of the statistics ˆ T and ˆ k , includingmultiple modes when there are multiple equilibria.The algorithm for a single run of the simulation is displayed in Figure 1. Ineach step, agents consume or do not based on the value of k from the last step,and the process repeats until the value of k no longer changes. The output ofthe process is the equilibrium value of ˆ T and the equilibrium percent acting ˆ k .With a sufficiently large number of runs (in the simulations here, 20,000), itis possible to calculate the moments S (ˆ k ) and κ (ˆ k ) .The code itself is a short script written in C using the open source Apophenialibrary [Klemens, 2008], and is available upon request.80.0050.010.0150.020.0250.030.035 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 α = 1 α = 1 . α = 1 . Figure 2: Above, three distributions of the equilibrium percent acting k , for20,000 runs with α equal to 1 (unimodal), 1.3 (bimodal with modes near 0.3and 0.7), and 1.6 (bimodal with modes near 0.1 and 0.9). Below, a full sequenceof such distributions, for α = 0 . in front up to α = 2 at the back. Vertical axisis the percent of runs (out of 20,000 per α ) whose equilibrium is in the givenhistogram bin. The three slices in the 2-D plot are indicated by a line on thefloor of the 3-D plot. 9 N o r m a li ze d k u r t o s i s ( κ / σ ) Emulation parameter ( α ) Figure 3: The normalized kurtosis reveals the narrow range of transition fromNormal-type distribution ( κ/σ = 3 ) to bimodal-type distribution ( κ/σ = 1 ). It is instructive to begin with the symmetric case, where (cid:15) = 0 , so agents’ privatetastes are drawn from a N (0 , distribution.Figure 2 shows a sequence of distributions of the equilibrium percent acting k , from the distribution given α = 1 up to the distribution for α = 2 . , withdistributions for three specific values of α highlighted. Small values of α (whereutility is mostly private valuation) result in a Normal output distribution ofprices, while large values of α (where utility is mostly public) give a coordination-game style bifurcation.As α goes from the Normal range to the bifurcated range, there is a smallrange of α where the transition occurs, and the distribution is neither fullybifurcated nor Normal.At large α , the value of k between the sink that sends the simulation to thelower equilibrium and the sink that sends the simulation to the higher equilib-rium (near 0.5) is an unstable equilibrium; in theory it occurs with probabilityzero, but in a finite simulation it occurs with small probability. Below, we willsee that these distributions with a small middle mode behave like a bifurcateddistribution, so I will refer to them as such.The small transition range is especially clear when we look at the normalizedkurtosis of each α ’s distribution, which is not at all a uniform shift. As in Figure3, the normalized kurtosis is consistently three for small values of α (as for aNormal distribution), is consistently one for large values of α (as for a symmetric The figures are the aggregate of 20,000 runs of the simulation. If an equilibrium wasreached even once, then it appears as a mark in the 3-D plot. The 2-D plots have lowerresolution, and unlikely events may blend with the axes. α ≈ and α ≈ . . Figure 4 shows the sequence of distributions where (cid:15) = 0 . . For α ≈ ,the distribution is roughly equivalent to the (cid:15) = 0 situation but shifted upwardslightly; for α ≈ and above, where the outcome distribution is bifurcated,the slight shift in the distribution’s center causes positive outcomes to be morelikely than negative outcomes.However, between these two outcomes lies a range of α where the (cid:15) = 0 case would have led to a bifurcation, but the lower tail of the distribution issuppressed because the nobody-acts equilibrium is not feasible. In this range,we have an asymmetric but unimodal distribution.Figure 5 plots normalized kurtosis for each α ’s distribution. The neighbor-hood of α ≈ . is again salient, because the normalized kurtosis in that rangeis an order of magnitude larger than three. The model’s exceptional success ingenerating a leptokurtic outcome makes the plot’s vertical scale rather large, soit may be difficult to discern that the kurtosis up to the peak is three, and afterthe peak is one, as in the (cid:15) = 0 case.The bottom plot of Figure 5 shows that normalized skew follows the samestory relative to α as did kurtosis: it spikes around 1.3, where the distributionof equilibrium percent acting has heavily negative skew.Thus, given a realistic value of (cid:15) (i.e., anything but exactly zero), and avalue of α that is not too small to be equivalent to the private preferences caseand not too large to be equivalent to the full herding case, the distribution ofoutcomes is unimodal, leptokurtic, and has a negative skew. There are several explanations for why rational agents would choose to emulateothers, all of which advise that a utility function meant to describe a trader inthe finance markets should include a term for the desire to emulate others.Meanwhile, we know that equity return distributions show certain consis-tent deviations from the Normal distribution implied by naïve application ofa Central Limit Theorem. Adding a term for the emulation of others to indi-vidual utilities produces aggregate outcome distributions that show these samedeviations from Normal: extreme outcomes happen more often, and do so asym-metrically.However, the story is not quite as simple as saying that people tend toimitate others. The type of distribution observed in equity returns appears in amiddle-ground between two extreme types of utility function. With α small, thedistribution of cutoffs is more-or-less that of a situation of purely private utility.With α large, the distribution follows the story of agents that simply follow theherd. But between these two situations, there is a transition range where the The units on α are utils per percent acting, so exact values of α are basically meaningless.Rescaling t (by changing its variance) would produce entirely different values of α , but thequalitative effects described here would still hold. α = 1 α = 1 . α = 1 . Figure 4: Two views of the α -to-cutoff-frequency relation. PDFs of cutoffsfor three given levels of α (1=unimodal near center, 1.3=unimodal to right,1.6=bimodal) are displayed in 2-D form at top. At bottom is a series of PDFsfor a range of values of α from α = 0 . at front to α = 2 at rear.1250100150200250 0 0.5 1 1.5 2 2.5 N o r m a li ze d k u r t o s i s ( κ / σ ) Emulation parameter ( α )-12-10-8-6-4-202 0 0.5 1 1.5 2 2.5 N o r m a li ze d s k e w ( S / σ ) Emulation parameter ( α )Figure 5: The relationship between n (on the horizontal axis) and κ/σ (on thevertical axis, top) or S /σ (on the vertical axis, bottom).13istribution of cutoffs has the desired characteristics of being unimodal, havinglarge kurtosis, and skew toward zero. Thus, the model explains this type ofdistribution via an interplay between private and emulative utility.This paper has shown that peer effects can generate leptokurtic outcomesunder certain conditions. This creates the possibility that an observed leptokur-tic distribution can be explained by peer effects. For example, Jones et al. [2003]found leptokurtic outcomes in Congressional actions such as budget allocations;I suggest in this paper that a model of Congressional representatives who em-ulate each other can generate such an outcome distribution. When outcomeshave a blockbuster/flop bimodality, there is little doubt that peer effects are atplay, but the model here shows that even more subtle outcomes, with unimodaldistributions but fat tails, may also be the result of agents who gain direct orindirect utility from emulating each other. References
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