How Market Ecology Explains Market Malfunction
HHow Market Ecology Explains Market Malfunction
Maarten P. Scholl a,b,1 , Anisoara Calinescu b,1 , and J. Doyne Farmer a,b,2 a Institute for New Economic Thinking, Oxford Martin School, University of Oxford; b Computer Science Department, University of OxfordThis manuscript was compiled on September 22, 2020
Standard approaches to the theory of financial markets are based onequilibrium and efficiency. Here we develop an alternative based onconcepts and methods developed by biologists, in which the wealthinvested in a financial strategy is like the population of a species. Westudy a toy model of a market consisting of value investors, trendfollowers and noise traders. We show that the average returns ofstrategies are strongly density dependent, i.e. they depend on thewealth invested in each strategy at any given time. In the absence ofnoise the market would slowly evolve toward an efficient equilibrium,but the large statistical uncertainty in profitability makes this noisyand uncertain. Even in the long term, the market spends extendedperiods of time far from perfect efficiency. We show how coreconcepts from ecology, such as the community matrix and foodwebs, apply to markets. The wealth dynamics of the market ecologyexplains how market inefficiencies spontaneously occur and givesinsight into the origins of excess price volatility and deviations ofprices from fundamental values. market ecology | market efficiency | agent-based modeling W hy do markets malfunction? According to the theoryof market efficiency, markets always function perfectly:Prices always reflect fundamental values and they only changewhen there is new information that affects fundamental values.Thus, by definition, any problems with price setting are causedby factors outside the market. Empirical evidence suggestsotherwise. Large price movements occur even when thereis very little new information (1) and prices often deviatesubstantially from fundamental values (2). This indicates thatto understand how and why markets malfunction we need togo beyond the theory of market efficiency.Here we build on earlier work (3–7) and develop the theoryof market ecology, which provides just such an alternative.This approach borrows concepts and methods from biology andapplies them to financial markets. Financial trading strategiesare analogous to biological species. Plants and animals arespecialists that evolve to fill niches that provide food; similarly,financial trading strategies are specialized decision makingrules that evolve to exploit market inefficiencies. Tradingstrategies can be classified into distinct categories, such astechnical trading, value investing, market making, statisticalarbitrage and many others. The capital invested in a strategyis like the population of a species. Trading strategies interactwith one another via price setting and the market evolves asthe wealth invested in each strategy changes through time,and as old strategies fail and new strategies appear.The theory of market ecology builds on the inherentcontradictions in the theory of market efficiency. A standardargument used to justify market efficiency is that competitionfor profits by arbitrageurs should cause markets to rapidlyevolve to an equilibrium where it is not possible to makeexcess profits based on publicly available information. Butif there are no profits to be made, there are no incentivesfor arbitrageurs, so there is no mechanism to make marketsefficient. This paradox suggests that, while markets may be efficient in some approximate sense, they cannot be perfectlyefficient (8). In contrast, under the theory of market ecology,trading strategies exploit market inefficiencies but, as newstrategies appear and as the wealth invested in each strategychanges, the inefficiencies change as well. To understand howthe market functions, it is necessary to understand how eachstrategy affects the market and how the interactions betweenstrategies cause market inefficiencies to change with time. Thetheory of market ecology naturally addresses a different setof problems than the theory of market efficiency, and can beviewed as a complement rather than a substitute for it.Here we study a stylized toy market model with threetrading strategies. We approach the problem in the same waythat an ecologist would study three interacting species. Westudy how the average returns of the strategies depend on thewealth invested in each strategy and how their wealth evolvesthrough time under reinvestment, and how this endogenoustime evolution causes the market to malfunction.Unlike previous studies that use a market impact rule forprice setting (3, 9), here we use market clearing. This providesa better model and in some cases leads to substantially differentresults. In contrast to market impact, under market clearing,all the properties of the market ecology depend strongly onthe wealth invested in each strategy.We show that evolution toward market efficiency is veryslow. The expected deviations from efficiency are typicallysubstantial, even in the long term, and cause extendeddeviations from fundamental values and excess volatility(which in extreme cases becomes market instability). Ourstudy provides a simple example of how analyzing markets inthese terms and tracking market ecologies through time couldgive regulators better insight into market behavior. Significance Statement
We develop the mathematical analogy between financial tradingstrategies and biological species and show how to applystandard concepts from ecology to financial markets. Weanalyze the interactions of three stereotypical trading strategiesin ecological terms, showing that they can be competitive,predator-prey or mutualistic, depending on the wealth investedin each strategy. The deterministic dynamics suggest thatthe system should evolve toward an efficient state where allthree strategies make the same average returns. However, thishappens so slowly and the evolution is so noisy that there arelarge fluctuations away from the efficient state, causing burstsof volatility and extended periods where prices deviate fromfundamental values. This provides a conceptual frameworkthat gives insight into the reasons why markets malfunction.
The authors declare no conflict of interest. E-mail: [email protected]
September 22, 2020 | vol. XXX | no. XX | a r X i v : . [ q -f i n . GN ] S e p ig. 1. The three trading strategies correspond to noise traders, value investors andtrend followers. They invest their capital in a stock and a bond. The mixture for eachstrategy changes with time as strategies accumulate wealth based on their historicalperformance. bondstock noise tradervalue investortrend follower
A. Model Description.
The structure of the model is schemati-cally summarized in Figure 1. There are two assets, a stockand a bond. The bond trades at a fixed price and yields r = 1% annually in the form of coupon payments that arepaid out continuously. The stock pays a continuous dividend D ( t ) that is modeled as an autocorrelated geometric Brownianmotion, of the form d D ( t ) = g D ( t ) dt + σ D ( t ) d U ( t ) ,d U ( t ) = (1 − ω ) d Z ( t ) + ωd U ( t − θ ) , [1]where g is the average rate of dividend payments per unitof time t , σ is the variance, ω is the autocorrelation of theprocess, and Z and U are standard Wiener processes. Weapproximate the continuous processes by discrete processeswith a time step equal to one day. We use estimates frommarket data by Lebaron (10), taking g = 2% per year forthe growth rate of the dividend with a volatility of σ = 6%.(See reference (11), for example, for a review of the empiricalevidence on dividends).We use market clearing to set prices. Each asset a has afixed supply Q a , but the excess demand E ( t ) for the stock byeach trading strategy varies in time. We allow the tradingstrategies to take short positions and to use leverage (i.e. toborrow in order to take a position in the stock that is largerthan their wealth). We impose a strategy-specific leveragelimit λ ∗ . Because we use leverage and because the strategiescan have demand functions with unusual properties, marketclearing is not always straightforward – see Materials andMethods.The size of a trading strategy is given by its wealth W ( t ),i.e. the capital invested in it at any given time. In ecology thiscorresponds to the population of a species, which is also calledits abundance. Unless otherwise stated, we assume profits andlosses are reinvested, so that the wealth of each strategy variesaccording to its cumulative performance.A trading strategy is defined by its trading signal φ ( t ),which can depend on the price p ( t ) and other variables, suchas dividends and past prices. We modify φ by a tanh functionto ensure that the excess demand is bounded and differentiable.A strategy’s excess demand for the stock is E ( t ) = W ( t ) λ ∗ p ( t ) (cid:16) tanh ( c · φ ( t )) + 12 (cid:17) − S ( t − , [2]where S ( t −
1) is the number of shares of the stock held atthe previous time step. The parameter c > φ , and isstrategy specific. When the signal of the strategy is zero,the agent is indifferent between the stock and the bond andsplits its portfolio equally between the two (hence the factorof 1 / λ ( t ) of a strategy at any given time is p ( t ) S ( t ) K ( t ) = λ ∗ | tanh ( c · φ ( t )) + | . This equality holds whenthe market clears. B. Investment Strategies.
We study three typical tradingstrategies, which we call value investors, trend followers andnoise traders. We make a representative agent hypothesis,treating each strategy as though it were only used by a singlefund; however, these should be thought of as representing allinvestors using these strategies. We now describe each of themin turn.
Value Investors observe the dividend process, use a modelto derive the value of the stock, and seek to hold more of thestock when it is undervalued and hold more of the bond whenit is overvalued. The parameters of their model are estimatedbased on the historical dividends. However their model isinaccurate in that it contains estimation errors and it does nottake the autocorrelation of the dividend process into account(i.e. they assume ω = 0).Following Gordon and Shapiro (12), the fundamental value V ( t ) of a stock is V ( t ) = Z ∞ t +1 D ( t + 1) e ˆ gτ e − kτ dτ = D ( t + 1) k − ˆ g , k > ˆ g. [3]The parameter k is the discount rate, also called called the required rate of return . It is the sum of the risk-free rateand a risk-premium investors expect for the additional risksassociated with the stock. We follow (13) and use a fixeddiscount rate k = 2% based on the average rate of returnimplied by historical data.We define the trading signal for the value investor as thedifference in log prices between the estimated fundamentalvalue V ( t ) and the market price. φ VI ( t ) = log V ( t ) − log p ( t ) [4]This strategy will enter into a long position when the proposedprice is lower than the value estimated by the investor, andit will enter into a short position when the proposed price ishigher than the estimated fundamental value. The use of thebase two logarithm means that the value investor employs allof its assets when the stock is trading at half the perceivedvalue (14). Trend Followers expect that historical trends in returnscontinue into the short term future. Several variants exist inthe literature, including the archetypal trend follower that weuse here (15–18). There is evidence to suggest that trend-basedinvestment strategies are profitable over long time horizons,and reference (19) argues that investors earn a premium forthe liquidity risk associated with stocks with high momentum(momentum trading is a synonym for trend following).The trend strategy we use extrapolates the trend in pricebetween θ and θ time steps in the past as follows φ TF ( t ) = log p ( t − θ ) − log p ( t − θ ) , θ < θ . [5]We choose θ = 1 and θ = 2 and keep them fixed. Thischoice of parameters allows the trend follower to exploit theautocorrelation that the dividends impart to prices. The trendfollowers’ demand is a decreasing function of price. Trendfollowers will make profits if there is positive auto-correlationin the stock’s returns, e.g. due to the dividend process. Noise Traders represent non-professional investors whodo not track the market closely. Their transactions are mostly et al. or liquidity, but they are also somewhat aware of value, sothat they are slightly more likely to buy when the market isundervalued and slightly more likely to sell when the market isovervalued. The signal function of our noise traders containsthe product of the value estimate V ( t ) (which we assume is thesame as for the value investors) and a stochastic component X ( t ), φ NT ( t ) = log X ( t ) V ( t ) − log p ( t ) . [6]The noise process X ( t ) is an Ornstein-Uhlenbeck process,which has the form dX ( t ) = ρ ( µ − X ( t )) dt + γdW ( t ) [7]This process reverts to the long term mean µ = 1 with reversionrate ρ = 1 − × √ .
5, meaning the noise has a half life of 6years, in accordance with the values estimated by Bouchaud(20). W ( t ) is a Wiener process and γ = 12% is a volatilityparameter, which is twice the volatility of the dividend process.The parameters of the model are summarized in Table 5.We have chosen them for an appropriate compromise betweenrealism and conceptual interest, e.g. so that each strategy hasa region in the wealth landscape where it is profitable. Results
C. Density Dependence.
An ecology is density dependent ifthe characteristics of the ecology depend on the populationsof the species, as is typically the case. Similarly, a marketecology is density dependent if its characteristics depend onthe wealth invested in each strategy. The toy market ecologythat we study here is strongly density dependent.When the core ideas in this paper were originally introducedin reference (3), prices were formed using a market impactfunction, which translates the aggregate trade imbalance atany time into a shift in prices. This can be viewed as alocal linearization of market clearing. The use of a marketimpact function suppresses density dependence and neglectsnonlinearities that are important for understanding marketecologies.In contrast, using market clearing we see strong densitydependence. This is evident in Figure 2, which shows whichstrategy makes the highest profits as a function of the relativesize of each of the three strategies. To control the size ofeach strategy we turn off reinvestment, and instead replenishthe wealth of each strategy at each step as needed to holdit constant. We then systematically vary the wealth vector W = ( W NT , W VI , W TF ). We somewhat arbitrarily let W NT + W VI + W TF = 3 × , but we plot the relative wealth (as ifthe wealths sum to one). The results shown are averages overmany long runs; to avoid transients we exclude the first 252time steps, corresponding to one trading year.Roughly speaking, the profitability of the dominant strategydivides the wealth landscape into four distinct regions. Trendfollowers dominate at the bottom of the diagram, where theirwealth is small. Value investors dominate on the left side ofthe diagram, where their wealth is small, and noise tradersdominate on the right side of the diagram, where their wealthis small. There is an intersection point near the center wherethe returns of all three strategies are the same, correspondingto an efficient equilibrium. In addition, there is a complicatedregion at the top of the diagram, where no single strategydominates. The turbulent behavior in this region comes about Noise TraderValue Investor ≤
0% 2 4 6 8 10return rate (annual)Trend Follower ≤ % % % % %
38% 33% 28% 23% 18% 13% 8%
Trend Follower wealth -2.5%0%2 . R e t u r n
19% 24% 29% 34% 39% 44% 49%
Value Investor wealth . R e t u r n V o l a t ili t y Trend Follower
Value Investor BC Fig. 2.
The profitability of the dominant strategy in the wealthlandscape.
Panel A is a ternary plot which displays the returns achieved by thestrategy with the largest return. The axes correspond to the relative wealth investedin each strategy. The top corner is pure trend followers, the left corner pure noisetraders, and the right corner pure value investors. The color indicates the strategywith the highest returns at a given wealth vector W . The regions colored in redcorrespond to the noise traders, blue regions to value investors, and green to thetrend followers. The intensity of the color indicates the size of the average return. Theupper box of panel B shows the average returns to value investors (blue) and trendfollowers (green) while holding the noise trader wealth at its equilibrium value of .The lower box shows the volatility in the returns of each strategy. The horizontal axisis the relative wealth of the trend follower (top) and value investor (bottom). because the wealth invested by trend followers is large and theprice dynamics are unstable.A quantitative snapshot of the average returns and volatilityis given in Figure 2B, where we hold the size of the noisetraders constant at its equilibrium level of 42% and vary thewealth of the value investors and trend followers. The averagereturn to both trend followers and value investors increasesmonotonically as their wealth decreases. The volatility ofthe returns of both strategies, in contrast, is a monotonicfunction of the wealth of the trend followers – higher trendfollower wealth implies higher volatility. Although this is notshown here, the average return of the value investors increases Scholl et al.
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September 22, 2020 | vol. XXX | no. XX || vol. XXX | no. XX | trongly with the wealth of the noise traders; in contrast, theaverage return of the trend followers is insensitive to it. D. Adaptation.
We now investigate the dynamics of the ecology.To understand how the wealth of the strategies evolves throughtime, we allow reinvestment and plot trajectories correspondingto the average return from each point W . This is done byaveraging over many different runs. The result is shown inFigure 3A. Most of the wealth trajectories in the diagramevolve toward the efficient equilibrium in the center, wherethere is a fixed point where the wealths of the strategies nolonger change. At the equilibrium the returns to the threestrategies are all equal to π = 2 . S . In the idealcase of a stationary market and I.I.D. normally distributedreturns, the time required to detect excess performance ∆ S with a statistical significance of s standard deviations isapproximately τ = ( s/ ∆ S ) . To take an example, a buy andhold of the S&P index has a Sharpe ratio of roughly S = 0 .
5. Itthus requires roughly 400 years to confirm the performance ofa strategy that outperforms the index by 20% at two standarddeviations of statistical significance. Furthermore, as shownin reference (21), because the rate of approach to marketefficiency slows down as it is approached, it follows a powerlaw of the form t − α , where 0 ≤ α ≤
1. For large times this ismuch slower than an exponential.To better understand the long-term evolution, we samplethe space of initial wealth uniformly, simulate the ecologicaldynamics under reinvestment, and record the final wealthafter 200 years, as shown in Figure 3C. While the fixed pointequilibrium is contained in the region where the ecology ismost likely to be found, it is not in the center of this region,and the deviations in the relative wealth of the strategies fromthe equilibrium are substantial, often more than 20%.The autocorrelation of price returns is an indicator ofmarket efficiency. Efficient price returns should have anautocorrelation that is reasonably close to zero (close enoughthat it is not possible to make statistically significant excess profits). In Figure 3D we plot the autocorrelation of returnsacross the wealth landscape. There is a striking whiteband across the center of the simplex, corresponding tozero autocorrelation. This happens when trend followersinvest about 40% of the total wealth, thereby eliminatingthe autocorrelation coming from the dividend process.
E. Community matrix.
The community matrix is a tool usedin ecology to describe the pairwise effects of the population ofspecies j on the population growth rate of species i (22, 23).As originally pointed out by Farmer (3), who called it thegain matrix, analogous quantity is also useful for interpretingthe behavior of market ecologies. Assuming differentiability,let ∆ W i ( t ) = dW i /dt be the profits per unit time, so that π i ( t ) = ∆ W i ( t ) /W ( t ) is the return to strategy i , and let therelative wealth w i ( t ) = W i /W T , where W T is the total wealth.The analogue of the community matrix for market ecologies is G ij = ∂ ∆ W i ∂W j = ∂π i ∂w j . [8]This has units of one over time. The wealth W i ( t ) investedin strategy i replaces the population size of a species. Thesecond equation makes explicit the sense in which the termsin the community matrix are like elasticities in economics, i.e.they measure the response of the returns to relative changes inwealth. The possible pairwise interactions between strategiescan be classified according to the sign of G ij . If both G ij and G ji are negative, then strategies i and j are competitive; if G ij is positive and G ji is negative, then there is a predator-preyinteraction, with i the predator and j the prey; and if both G ij and G ji are positive, then there is a mutualistic interaction(24).Because we do not have a differentiable model for our toymarket ecology, we compute the community matrix numericallyusing finite differences (see Materials and Methods). Thecommunity matrix is strongly density dependent. If wecompute the community matrix near the equilibrium point inthe center of the simplex, we get the result shown in Table 1.The diagonal entries are all negative, indicating that thestrategies are competitive with themselves. This means thattheir average returns diminish as the strategy gets larger,causing what is called crowding in financial markets. Wealready observed this in Figure 2. Interestingly, however, thesize of the diagonal terms varies considerably, from − .
89 fornoise traders to − . et al. ig. 3. Profit dynamics as a function of wealth. A shows how wealth evolves on average through time under reinvestment. The intensity of the color denotesthe rate of change. B shows sample trajectories for a few different initial values of the wealth vector, making it clear that the trajectories are extremely noisy due to statisticaluncertainty, so that the deterministic dynamics of panel A is a poor approximation. The visualization displays three different initial wealth vectors, each color-coded. Themarker + indicates the initial wealth. The trajectories with the same color follows the system for T = 200 years and color saturation increases with time. Starting from uniformlydistributed initial conditions, C displays a density map of the asymptotic wealth distribution after 200 years. The system is initialized at random with a uniformly distributedwealth vector and then allowed to freely evolve for 200 years. The darkness is proportional to density. The black dot is the equilibrium point from Panel A. Panel D displays theautocorrelation in the realized prices.Scholl et al. PNAS |
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September 22, 2020 | vol. XXX | no. XX || vol. XXX | no. XX | able 1. Estimated community matrix near the equilibrium at W =( NT = 0 . , V I = 0 . , T F = 0 . . G ij NT VI TFNT -0.89% 0.89% 0.82%VI 26.6% -10.6% 22.4%TF 11.1% 15.2% -19.3%
Table 2.
Estimated community matrix near W = ( NT =0 . , V I = 0 . , T F = 0 . . G ij NT VI TFNT -0.46% 0.40% 0.36%VI 8.94% -0.77% -1.89%TF 6.81% 6.87% -9.65% pairwise community relations. As before, all of the terms inthe row corresponding to the noise traders are small, indicatingthat the noise traders are not strongly affected by otherstrategies, and that they compete only weakly with themselves.This should not be surprising – the noise traders’ strategyis mostly random, and is less influenced by prices than theother two strategies. Value investors, who have the majority ofthe wealth in this case, still strongly benefit from an increasein the wealth of noise traders (though less so than at theequilibrium). However, there is now a negative term in thesecond row, corresponding to the effect of trend followers. Incontrast, from the third row we see that trend followers benefitfrom an increase in the wealth of both noise traders and valueinvestors, implying that trend followers now prey on valueinvestors. Other variations in community relationships can befound at different points in the wealth landscape, illustratingdensity dependence.The Lotka-Volterra equations, which describe how thepopulations in an idealized predator-prey system evolvethrough time, are perhaps the most famous equations inpopulation biology. Their surprising result is that at someparameter values they have solutions that oscillate indefinitely.Using the assumption of no density dependence, Farmerderived Lotka-Volterra equations for market ecologies (18).Our results here indicate that the density dependence in thissystem is so strong that simple Lotka-Volterra equations area poor approximation, at least for this system. The existenceof oscillating solutions in financial markets remains an openquestion.
F. Food Webs and Trophic Level.
The food web providesan important conceptual framework for understanding theinteractions between species. If lions eat zebras and zebras eatgrass, then the population of lions is strongly affected by thedensity of grass, and similarly the density of grass dependson the population of lions, even though lions have no directinteractions with grass. The trophic level of a species is bydefinition one level higher than what it eats, so in this idealizedsystem grass has trophic level one, zebras have trophic leveltwo and lions trophic level three.The existence of animals with more complicated diets, suchas omnivores and detritivores, means that real food webs arenever this simple. If we let A ( ij ) be the share of species j inthe diet of species i , then the trophic level T i of species i can be computed by the relation T i = 1 + X j A ij T j . [9]The resulting trophic levels are typically not integers, but theystill provide a useful way to think about the role that a givenspecies plays in the ecology.We can also compute trophic levels for the strategies in amarket ecology. We define the analogous quantity A ij as thefraction of the returns of strategy i that can be attributed tothe presence of strategy j . We do this by simply comparingthe returns of strategy i at wealth W to those when strategy j is removed, i.e. when W j = 0 but all the other wealths remainthe same. In mathematical terms, A ij = max [0 , π i ( W , . . . , W j , . . . , W N ) − π i ( W , . . . , , . . . , W N )] . [10]The maximum is taken so that A ij is never negative. Forcomputing the trophic levels we only care about the strategiesthat i benefits from, not those that cause it losses.Equations (9) and (10) allow us to compute trophic levels foreach of the strategies. At the equilibrium point, for example,the trophic levels are (1, 2, 3). In order to better understandthe density dependence, we compute trophic levels at eachpoint in the wealth landscape. For three strategies there are3! = 6 possible orderings of the trophic levels. We display theordering of the trophic levels across the wealth landscape inFigure 4. Fig. 4.
A survey of the trophic levels across the wealth landscape.
We color the diagram according to the ordering of the trophic levels of the threestrategies (see legend). Red, for example, denotes the dominant zone where thenoise traders have the lowest trophic level and trend followers have the highest trophiclevel (with value investors in the middle). In the grey region there are cycles wherethe trophic levels become undefined. The black dots correspond to samples of thewealth vector after 200 years, as shown in Figure 3(c). The system spends most of itstime in the grey and red zones.
The computation of trophic levels is complicated by thefact that for some wealth vectors there are cycles in the et al. ood web. For example, for W = (0 . , . , . G. How ecological dynamics cause market malfunction.
Thewealth dynamics of the market ecology help explain whythe market malfunctions and illuminate the origins of excessvolatility and mispricing , i.e. deviations of prices fromfundamental values. Volatility and mispricings are bothfunctions of time – there are eras where they are large and eraswhere they are small. Volatility tends to vary intermittently,with periods of low volatility punctuated by bursts of highvolatility – this behavior is called clustered volatility. Thestandard explanation for clustered volatility is fluctuatingagent populations (15, 25, 26). Our analysis reinforces thisexplanation, but gives more insight into its causes. Clusteredvolatility can also be caused by leverage (27). While we observethat clustered volatility increases with increasing leverage, wehave not investigated this in detail here.Figure 5A presents the variation of the volatility across thewealth landscape. The landscape can roughly be divided intotwo regions. On the lower right there is a flat low volatility“plain" occupying most of the landscape. On the upper leftthere is a high volatility region, with a sharp boundary betweenthe two. As we will now show, excursions into the highvolatility region cause clustered volatility. A similar storyholds for mispricing.Figure 5A shows a sample trajectory that begins at theefficient equilibrium and spans 200 years. The statisticalfluctuations in the performance of the three strategies acts asnoise, causing large excursions away from equilibrium. Thetrajectory mostly remains on the volatility plain, but thereare several epochs where it ventures into the high volatility region causing bursts of high volatility.The wealth dynamics have strong explanatory power forboth mispricing and volatility. This is illustrated in Table 3,where we perform regressions of the strategies’ wealth againstvolatility using daily values for the time series shown inFigure 5A. For volatility R = 0 .
79 and for mispricing R = 0 .
33. In both cases the value investor wealth andthe trend follower wealth have large coefficients (in absolutevalue) and the fit is overwhelmingly statistically significant.The noise trader is also highly statistically significant butthe coefficients and the t-statistics are more than an order of
0% 20% 40% 60% 80% ≥ realized volatility (annual) V o l a t ili t y ( a nnu a l ) B predictor (OLS)volatility (1 year rolling)0.00.51.01.52.0 P r i c i n g e rr o r C predictor (OLS) mispricing | V ( t ) / p ( t ) 1|0 10000 20000 30000 40000 50000time (days)20%40%60% W e a l t h f r a c t i o n D Value Investor wealthTrend Follower wealth
Fig. 5.
How fluctuations in the ecology cause market malfunc-tions.
Panel A gives a color map of the price volatility over the wealth landscape;the volatility is low and constant throughout the lower right part of the diagram, wherethe system spends most of its time, but there is a high volatility region runningacross the upper left. A sample trajectory spanning 200 years beginning at theefficient equilibrium is shown in black. The noise caused by statistical fluctuationsin performance causes large deviations from equilibrium and excursions into thehigh volatility region. Panel B shows the volatility of this trajectory as a function oftime, plotted against the predicted volatility from equation (11). Panel C shows theactual mispricing plotted against the predicted mispricing from equation (12). Panel D shows the wealth of the value investors and trend followers.Scholl et al. PNAS |
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September 22, 2020 | vol. XXX | no. XX || vol. XXX | no. XX | able 3. Multivariate regressions with volatility and mispricing asdependent variables and the funds’ wealth as independent variables. volatility R = 0 . observations: 50,397independent variable coefficient tnoise trader 2.4 10value investor -68 -249trend follower 107 169 mispricing R = 0 . observations: 50,397independent variable coefficient tnoise trader -0.15 -18value investor -1.02 -107trend follower 1.5 69 magnitude smaller. In Figure 5B and C we compare a timeseries of the predicted volatility and predicted mispricing,ˆ ν = − w vi + 107 w tf + 2 . w nt , [11]ˆ m = − . w vi + 1 . w tf − . w nt , [12]against the actual values. The series match very well. Notethat in both cases the coefficient for trend followers is positive,indicating that they drive instabilities, and the coefficientfor value investors is negative, indicating their stabilizinginfluence.Nonetheless, due to their effect on the population of valueinvestors, the net effect of the trend followers on marketmalfunctions is not obvious. In Figure 5D we plot the wealthof value investors and trend followers. The strong mutualismpredicted by the community matrix is clearly evident from thefact that the wealth of trend followers and value investors risesand falls together. However, their dynamics are quite different– there are several precipitous drops in the value investors’wealth, whereas the trend followers tend to take more graduallosses. As predicted, the highest volatility episodes happenwhen the value investors’ wealth drops sharply while the trendfollowers’ wealth is high. Discussion
Our analysis here demonstrates how understanding fluctu-ations of the wealth of the strategies in the ecology canhelp us predict market malfunctions such as mispricings andendogenously generated clustered volatility. The toy modelthat we study here is simple and highly stylized, but itillustrates how one can import ideas from ecology to betterunderstand financial markets. Our analysis of this modelillustrates several properties of market ecologies that wehypothesize are likely to be true in more general settings.This model gives important insights into how deviationsfrom market efficiency occur and how they affect prices.While the market may be close to efficiency in the sensethat the excess returns to any given strategy are small, therecan nonetheless be substantial deviations in the wealth ofdifferent strategies, that can cause excess volatility and marketinstability.Market ecology is a complement rather than a substitutefor the theory of market efficiency. There are circumstances,such as pricing options, where market efficiency is a usefulhypothesis. Market ecology, in contrast, provides insight intohow and why markets deviate from efficiency, and what theconsequences of this are. It can be used to explain the time dependence in the returns of trading strategies, and in somecases it can be used to explain market malfunctions. One ofour main innovations here is to demonstrate how to computethe community matrix and the trophic web, which provideinsight into the interactions of strategies.There are so far only a few examples of empirical studiesof market ecologies (28, 29). This is because such a studyrequires counterparty identifiers on transactions in order toknow who traded with whom. Trying to study a marketecology without such data is like trying to study a biologicalecology in which one can observe that an animal ate anotheranimal without any information about the types of animalsinvolved. Unfortunately, for markets such data is difficult formost researchers to obtain.Regulators potentially have access to the balance sheetsof all market participants, which can allow them to track theecology of the markets they regulate in detail. Ideas suchas those presented here could provide valuable insight intowhen markets are in danger of failure, and make it possibleto construct models for the ecological effect of innovations,e.g. the introduction of new types of assets such as mortgage-backed securities.One of our most striking results is that the approach toefficiency is highly uncertain and exceedingly slow. As alreadypointed out, this should be obvious from a straightforwardstatistical analysis, but it is not widely appreciated. Ourresults demonstrate this dramatically and they indicate that,even in the long-term, we should expect large deviations fromefficiency.There are many possible extensions to this work. Anobvious follow up is to explore a larger space of strategies, or tolet new strategies evolve in an open-ended way through time.Does the process of strategy innovation tend to stabilize ordestabilize markets? Another follow up is to construct a modelthat is empirically validated against data with counterpartyidentifiers. Our analysis here provides concepts and methodsthat could be used to interpret the behavior of real worldexamples.
Materials and Methods
1. Accounting and Balance Sheets.
The funds in our model use a stylized balance sheet that is presentedin table 4. External investors endow the fund with a certain amountof equity capital E , in the forms of cash C in dollars and a numberof trading securities S . When S >
0, the fund holds this amount ofsecurities, and when
S <
0, it has borrowed this amount from othermarket participants to create a short position. In order to guaranteethat the short-selling fund can return the borrowed securities tothe lender at a later time, the fund sets aside a margin amount M equal to the current market value of the borrowings, in the form ofcash. Fund managers may decide to borrow cash L up to a certainmultiple of fund equity. For simplicity, only one interest rate appliesto cash holdings, loans, and margin. This interest rate is the sameas the interest rate obtained from holding the risk-free bond.Wealth is calculated as: W ( t ) = C ( t ) + S ( t ) p ( t ) − L ( t ) [13]The margin entry M ( t ) on the balance sheet does not occur in thisequation, as the margin account covers the negative part of S ( t ) p ( t )by holding its market value in cash. The funds can use leverage ,meaning using borrowed funds to purchase additional risky assets.Leverage is a tool commonly used by fund managers, with a cursory et al. able 4. This table details the balance sheet items used by all funds.All securities use the most recent market value in valuation. Assets LiabilitiesEquity cash C capital K Debt margin M loans L trading securities S + borrowed securities S − Table 5. This table provides a listing of the model parameters andtheir values. parameter value description g . dividend growth rate k . cost of equity σ p dividend growth volatility ω . dividend autocorrelation coefficient ρ − · p noise trader mean reversion rate σ NT p noise trader volatility λ ∗ NT , λ ∗ VI , λ ∗ TF c NT , c VI , c TF
5, 10, 4 signal scale look at public regulatory filings of U.S. institutional fund managersshowing leverage ratios between 1 and 10 ∗ .The investment mandate defines the fund managers’ leverageconstraints, which may be set by the external creditors, who areproviding the fund with the needed loans, or internally – as a formof trading risk management. Given the leverage constraint λ ∗ ∈ R + ,we can compute the maximum and minimum demand, in terms ofthe number of assets. Because we can have short positions, this setof portfolios is more general than the budget set as it also allows fornegative amounts of the stock. The wealth of the fund develops as: W ( t + 1) = W ( t ) + r [ C ( t ) − L ( t )]+ [ p ( t + 1) − p ( t ) + D ( t )] S ( t ) [14]The leverage constraint is an integral part of the excess demandfunction. A fund can only violate its leverage constraint when theproportion of risky assets changes faster than the amount of equitycapital. This can happen due to losses, or in rare cases when themarket fails to clear completely. In those cases, the fund has theopportunity to reduce its risky position during the period via theinclusion of the leverage limit in the excess demand function. Therestill may be losses that exceed the fund’s equity, making the networth of the fund less than zero, and we require that all funds meetthe solvency condition W ( t ) >
0. The simulation ends when oneor more funds are insolvent. The model parameters, particularlythe leverage constraint λ ∗ , influence the observed dynamics in themodel. Table 5 lists the parameters used for the analysis in thispaper.
2. Market Clearing.
Prices are set by a price setter who chooses prices such that demandand supply match as close as possible. The excess demand of agent a for property i is defined in equation 2. The market excess demandcurve for one particular investment i is the aggregate of the excessdemand of all agents. As in the classical Walrasian setting, theprice setter seeks to match demand and supply, so that aggregateexcess demand is zero for each investment, by finding a root of themarket excess demand curve.However, if no solution is found through the root-finding process,we must fall back to a heuristic that seeks for the best solution thatonly partially clears the market. We interpret the goodness of asolution as the extent to which the solution minimizes demand andsupply mismatch . We here use the square of excess demand, and this ∗ The Electronic Data Gathering, Analysis, and Retrieval system (EDGAR) way the root-finding problem is transformed into the correspondingminimization problem:minimize p X a ∈A E a ( p ) ! subject to finite p.
3. Model and Software.
The simulation in this paper builds on the
Economic SimulationLibrary , an open-source library for agent-based modeling which isaccessible at https://github.com/INET-Complexity/ESL . ACKNOWLEDGMENTS.
We thank Klaus Schenk-Hoppé, RobertMacKay, Michael Wooldridge, and Rama Cont for enlighteningdiscussion during the development of this work. We acknowledgefunding by the J.P. Morgan AI Faculty Awards, Baillie-Gifford andthe Rebuilding Macroeconomics program, funded by the Economicand Social Research Council.
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