经融学essay/report/assignment代写-Estimating Tipping Points in Feedback-Driven Financial Networks



Estimating Tipping Points in Feedback-Driven

Financial Networks

Zvonko Kostanjcar , Member, IEEE , Stjepan Begusi c, Harry Eugene Stanley, and Boris Podobnik

Abstract Much research has been conducted arguing that tip- ping points at which complex systems experience phase transitions are difficult to identify. To test the existence of tipping points in financial markets, based on the alternating offer strategic model we propose a network of bargaining agents who mutually either co- operate or compete, where the feedback mechanism between trad- ing and price dynamics is driven by an external hidden variable R that quantifies the degree of market overpricing. Due to the feed- back mechanism, R fluctuates and oscillates over time, and thus periods when the market is underpriced and overpriced occur re- peatedly. As the market becomes overpriced, bubbles are created that ultimately burst as the market reaches a crash tipping point Rc. The market starts recovering from the crash as a recovery tip- ping point Rr is reached. The probability that the index will drop in the next year exhibits a strong hysteresis behavior very much alike critical transitions in other complex systems. The probability distribution function of R has a bimodal shape characteristic of small systems near the tipping point. By examining the S&P index we illustrate the applicability of the model and demonstrate that the financial data exhibit tipping points that agree with the model. We report a cointegration between the returns of the S&P 500 index and its intrinsic value.

Index Terms Complex systems, cooperative game theory, feedback, networks, market crash, tipping point.



LTHOUGH a lot of network science research has focused on how network collapse occurs when certain internal parameters approach their tipping points [1], there is a broad class of real-world complex networks in which the dynamics are driven by hidden external variables [2][4] in the form of feedback mechanisms [5], [6]. The tipping points at which these networks collapse have not yet been adequately understood [7][13].

Manuscript received September 28, 2015; revised March 06, 2016 and June 24, 2016; accepted July 15, 2016. Date of publication July 19, 2016; date of current version August 12, 2016. This work was supported in part by the Croatian Science Foundation under the project 5349. The work of B. Podobnik was supported in part by the University of Rijeka. The work of H. E. Stanley was supported by National Science Foundation under Grants CMMI 1125290 and PHY 1505000. The guest editor coordinating the review of this manuscript and approving it for publication was Dr. Danilo Mandic. Z. Kostanjcar and S. Begu si c are with the Faculty of Electrical Engineer- ing and Computing, University of Zagreb, Zagreb 10 000, Croatia (e-mail: zvonko.kostanjcar@fer.hr; Stjepan.Begusic@hotmail.com). H. E. Stanley is with the Center for Polymer Studies and Department of Physics, Boston University, Boston, MA 02215 USA (e-mail: hes@bu.edu). B. Podobnik is with the Faculty of Civil Engineering, University of Rijeka, Rijeka 51 000, Croatia, also with the Center for Polymer Studies and Department of Physics, Boston University, Boston, MA 02215 USA, and also with the Zagreb School of Economics and Management, Zagreb 10 000, Croatia (e-mail: bp@phy.hr). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JSTSP.2016.

In a buyer/seller financial network the trading dynamics are
strongly affected by trader perceptions that the market is over-
priced or underpriced. In a landlord/renter network a hidden
variable is the ratio between the average apartment price and
the average renter income. In a university/student network a
hidden variable is the ratio between the average tuition and the
average family income. In each of these examples, increasing
the ratio increases the probability of network collapse, and our
goal is to quantify the ratio that causes networks to collapse.
Here we focus on financial trading markets where the trading
decisions made by the market players are influenced by their
expectations about the future. With the passing of time they
learn whether their expectations about future market behavior
were accurate. Overly-optimistic predictions in particular are
very time-limited and are often followed by adjustments in the
market price so abrupt that they cause the market to collapse.
Extensive research developments exist on the topic of col-
lapses in real-world networks such as ecological, social and
economic systems [1], [2]. Scheffer et al. [7] report evidence of
hysteretic behavior in ecological systems around their tipping
points, and later suggest generic early-warning signals, such
as critical slowing down, that precede catastrophic shifts [10],
[14]. The core concepts in microscopic modelling of ecological
systems [11] include mechanisms of cascading failures, herd
behavior and contagion [15], [16], and are crucial to a number
of other real-world networked systems [8].
Ideas of herd behavior and contagion and their application in
agent-based models are in the foundations of explaining critical
events in financial markets [17][19]. Lux [20] explains the
emergence of bubbles as an infection process among traders
which leads to equilibrium prices deviating from fundamental
values. Eventually, Lux and Marchesi [21], [22] challenge the
efficient market hypothesis and present a model of interacting
agents switching between fundamentalist and noisy trader
strategies, based on the fundamental price as the input. Although
their fundamental prices have Gaussian relative changes, the
interaction of agents generates power-law scaling properties and
temporal dependence in volatility. The ContBouchaud model
[23] accounts for heavy tails in stock market returns through
agent interaction which induces herd behavior. They illustrate
the fact that a market model without such agent interaction
would give rise to normally distributed aggregate fluctuations,
whereas agent interaction accounts for known stylized facts on
market returns.
Sornette [24] identifies risk-driven and price-driven ap-
proaches to modelling financial bubbles and crashesin the
risk-driven model the crash hazard comes from herding which
drives the bubble price, whereas in the price-driven model
the price itself creates the crash hazard through feedback
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mechanisms. Abreu and Brunnermeier [25] present a model where market bubbles emerge, persist and burst due to synchro- nization of agent strategies. They argue that rational arbitrageurs are aware that the market will eventually collapse, but before it happens they want to ride the bubble and generate high returns. They indicate that a bubble bursts when the fraction of speculative traders leaving the market exceeds some threshold, which is the tipping point in their model. Recently, studies applying concepts of herding and imitation in both market crashes as well as rebounds report evidence of similar mechanics in these phenomena [26][28]. Furthermore, analytical studies of agent-based market models provide new opportunities for parametrization directly from market data, making such models applicable in a number of risk manage- ment scenarios [29]. A particularly important problem is the analysis of market drawdowns (so called Dragon-kings or Black swans) [24], [30], which novel studies attempt to ac- count for [31] since understanding them is of special value in risk management [32]. According to Johnston and Djuri c [33], with the advances of signal processing, we can afford to study risk management with high complexity models based on computational meth- ods that include nonlinearities and many hidden unknowns. So, from a signal processing perspective, we are presented with the challenge of designing an appropriate model, as un- conditional approaches will have an inherent misspecification. Schweitzer et al. [34] identify network structures, heterogenous agents and systemic feedbacks as some of the most important challenges in modelling economic networks. In addition, a sig- nificant issue in explaining critical phenomena is the lack of empirical datathere are few occurrences of such phenom- ena (bubbles and crashes) for which historic data are avail- able, which makes identifying systemic risk factors difficult but nevertheless crucial. Here we introduce a combined approach, employing opinion clustering, competition/cooperation and feedback mechanisms, with the fundamental price (identified as the intrinsic value in our model) as the input. We propose a network of bargain- ing agents [35][39] along with the competition and coopera- tion mechanisms between them [40][43]. This paper presents a computational model which, with an intrinsic value input, generates market pricesunlike generic computational models (such as neural networks), the proposed model is built specif- ically to illustrate market dynamics. We demonstrate how the degree of market overpricing through a feedback mechanism be- tween trading and price dynamics induces further market over- pricing, market bubbles, and ultimately market collapse [25], [44][47]. Beyond analyzing the crash tipping points, we also inspect the points at which the network model rebounds and the markets start recovering. We find that a well-known US finan- cial index, the S&P 500 index, exhibits a hysteresis behavior, as well as market-collapse tipping points and recovery tipping points that confirm the predictions of our network model. We report that the model, with only the change in the S&P 500 intrinsic value as input, consistently generates prices which ex- hibit appropriate heavy-tailed drawdown and drawup distribu- tions. Furthermore, we justify the applicability of the model by

Fig. 1. Competition, cooperation, and feedback mechanism between trading
process and price dynamics in a coupled network model. Inflow of information
determines intrinsic price, a genuine, fundamental information about an asset.
The ratio between market price and intrinsic price affects the trading dynamics
which in turn change the trading pattern.
demonstrating the possibility of deriving early-warning indica-
tors and drawing predictions of future market performances for
the S&P 500 index.
A. Initial Networks
Because most human activity is limited by the finite avail-
ability of resources, individuals are compelled to bargain over
the division of those resources [35], [43]. Bargaining has been
at the core of trade from the earliest recorded human history
when, prior to the introduction of currency, goods and services
were bartered. Today bargaining is ubiquitous and ranges from
haggling for food items in certain cultures to negotiations be-
tween large international business firms [3], [47]. The bottom
line in every market is the outcome of the bargaining process,
e.g., market price. Although standard axiomatic bargaining the-
ory idealizes the bargaining problem by assuming that indi-
viduals are highly rational as they negotiate their desires for
various resources, panic and irrational behavior [44] do occur
in real-world complex systems including political networks and
financial markets. This irrational behavior strongly affects the
bargaining process, affects entire complex network systems, and
causes bubbles and crashes. Irrespective of market trading rules,
players use strategic reasoning and the information available to
them when proposing initial bargaining prices [29], [35], [43],
[48][51]. For example, someone selling an apartment needs to
know the prices being listed by other sellers of similar apart-
ments in the same neighborhood before they can list an initial
bargaining price for their own apartment. The seller takes into
account that this initial price will probably be challenged by
potential buyers and that bargaining in competition with other
apartment sellers in the neighbor will ensue.
We propose a coupled network model composed of equally
sized demand (buyer) and supply (seller) networks, where agents
represented as network nodes, cooperate with agents on the
opposite side of the market [43], but compete with agents on
the same side of the market, as depicted in Fig. 1. Both demand


Fig. 2. Above (left and right): Clustered chain networks of competing agents forN=10, withp=0(chain) andp=1(fully connected graph). Below: Clustered chain network of competing agents forN=50andp=0. 3 .Thefirst two are the trivial cases of the network model, and the last case is representative of the networks in our model, since the values ofpare non-trivial and vary as defined in equation (3).

(buyer) and supply (seller) networks are initialized with a single node, and new nodes together with their prices are added one by one until each network hasNnodes. At any given time step and for each demand and supply network the new node is added as follows.

  1. With a constant probabilityp, a new nodeviis added to the already existing network structure:
  2. nodevirandomly links a nodevj, where eachj [1,i1]has equal probability 1 /(i1)and node viis then connected to each neighbor ofvj,
  3. nodevi sets its market (traded) priceSi as the arithmetic mean of all its neighbors prices
Sk (1)
wherekruns over all ofvis neighbors and the size
ofvis neighborhood isni=nj+1.
2) With a probability 1 pa new nodeviconnects to nodee
in the current network, which has the highest/lowest price
Sein the buyer/seller network, and sets its market price
Sias a percentage increase/decreasefromSe
Si=Se(1 + ). (2)

For the limit casep=0, all the nodes form a chain. In the opposite limit casep=1, each new node connects to a node from the existing cluster and all its neighbors, forming a com- plete graph. These trivial cases together with the casep=0. 3 are shown in Fig. 2.

B. Bargaining Process

Once both demand and supply networks are generated, the trading between buyers and sellers is initiated. Trading and price dynamics are driven by the inflow of information about an as- set, which determines the intrinsic priceSI. In contrast to this

relatively stable intrinsic price, the dynamics of which will be
explained below, the market price is volatile. To generate bub-
bles and crashes and the feedback mechanism between the trad-
ing and price processes, we use network parameters that are
time-dependent, not constant, i.e.,
S(t) (3)
whereS(t)is the last traded price at timet.Hereis a
parameter controlling overall network clustering. We define this
ratioR=SSI((tt))between the market priceSand and the intrinsic
valueSIas the Market-to-Intrinsic ratio.
Note that Eq. (3) implies that when the market priceS(t)
greatly exceeds the intrinsic valueSI(t),ps 1 , the competi-
tion between supply agents increases, their strategies begin to
converge, and their trading prices become increasingly similar
(generating the current market price). As a consequence, sup-
ply agents become increasingly insecure about their bargaining
position. In addition,ps 1 impliespd 0 , which intensi-
fies the market situation because the demand agents now have
more strategic options and face less competition. This combina-
tion of increasing confidence levels in the demand network and
decreasing confidence levels in the supply network increases
the probability that there will be an abrupt price crash. This
is in agreement with Scheinkman and Xiong who report that
overconfidence generates disagreements among agents regard-
ing asset fundamentals which then causes a significant bubble
component in asset prices [52].
Demand and supply agents bargain with each other to reach
agreements and execute trades. Here we present a bargaining
framework based on the alternating offer strategic model pro-
posed by Rubinstein [36], [43]. Agents are randomly selected,
alternating between the demand and supply networks, to make
moves. A move is either accepting the best current price offered
by the other side or proposing a new price. Each time a trade
is executed, the trading agents are removed from their networks
and a new one is added in both demand and supply network,
thus keeping the number of agentsNconstant (since the net-
work dynamics determine which and how many agents will be
able to trade, this is not a critical assumption).
Agents joining the network follow the proposed network al-
gorithm but are not allowed to connect to the first cluster (the
neighborhood of the first node). This is due to the fact that these
traders have just bought/sold the asset and believe the price is
going to rise/fall, and thus would not go back into their former
position immediately. After each trade, there is a probability
that the last node from each network will reconnect into the
existing network structure following the proposed algorithm,
giving the more fundamental long-term investors the possi-
bility of changing their position. The utility gained by agent
ifrom trading at priceSis quantified by a monotonic utility
functionui(S)(see Appendix A). Before making a move, agent
ievaluates the possible outcome at two steps={ 0 , 1 }using
a random utility functionUi(S, )
Ui(S,0) =ui(S) (4)


Ui(S,1) =
ui(S), with probabilityi
0 , with probability 1 i

wherei[0,1]quantifies agent confidence, the probability calculated by agentithat his/her offer (priceS) will be accepted in the next step at=1. The expected utility of immediately accepting offerS(at=0) is deterministic and equal to the agents utility function of priceS, as noted in Eq. (4). The expected utility of proposing a new priceScan be calculated from Eq. (5) as

E[Ui(S,1)] =ui(S)i. (6)

Agentiaccepts the current offer at=0if the expected utility at=0is larger than the expected utility at=1. Agent i evaluates his/her confidence level i based on his/her position in the network, taking into account any avail- able information about trends in the intrinsic price value. De- mand and supply agents evaluate their confidence based on the probabilities

n( 0 d)
n( 0 s)

wheren(is)andn(jd)are the neighborhood sizes of supply agent

iand demand agentj,n( 0 s), andn( 0 d)is the neighborhood size of the supply and demand agents (the best offers), andrI(t)is the proportional change in the intrinsic value. Hereis a parameter controlling how much a change in the intrinsic value impacts the market price. When the intrinsic value is approximately constant then the agents evaluate their confidence based on their position in the network only. For supply agentinote that the

more competing supply agentsn(is)rely on similar information and have synchronized trading strategies, and the smaller the

number of demand agentsn (d) 0 , the lower the value of

(s) i .More details regarding utility functionsui(S), new proposed prices S, and agent confidenceiare given in Appendix A. Note that the competition-cooperation mechanism in trading dynamics in which agents within the same group compete differs from the mechanisms that drive the evolution of cooperation in which agents within the same group cooperate.


In our modelling we were led by the conjecture that market bubbles are caused by complex cooperation-competition bar- gaining dynamics and that market collapse occurs when the ra- tio between market price and intrinsic priceclose to the Tobin qratio and its expansions [53], [54]reaches a critical (tip- ping) point. To calculate this ratio, which quantifies the degree of market overpricing, we must first estimate the intrinsic value of the market. Our model for determining the intrinsic value is based on the widely used free cash flow model (FCFM) [55], [56], in which the stock of a company is worth the sum of all of

Fig. 3. Price dynamics. (a) The S&P 500 prices, earnings, and cash flows,
monthly recorded. (b) The S&P 500 index, and the intrinsic value of the S&P
500 index. Both charts are adjusted for inflation. (c) Different time series of
model prices together with the intrinsic value of the S&P 500 index from
January 1920 to March 2015, and the (market) S&P 500 price index.
its discounted future free cash flows (FCFs) [57]
k=1(1 +WACCt+k)
and the discount rate is the future weighted average cost of
capital (WACC). Using information about past realizations of
FCF and WACC and Eq. (8) we determine the intrinsic valueSI
at some past timetT.TheSIvalue is approximate because
for thetTtime period we know only those FCF and WACC
values fromtTup to timetand thus the FCF and WACC
values for times longer thantmust be estimated. Thus the further
back we go (tT), the better will be the past price estimate


Fig. 4. Early-warning indicators through the 19902015 period. (a) The scaled variance of model price as an early-warning indicator. As the market is getting overpriced, the variance of fluctuations is increasing. (b) Average clustering coefficients of the supply and demand networks demonstrate the dynamics of supply and demand agent opinion clustering.

attT. Because our goal is to estimate the current intrinsic value, we use pastSIvalues we estimate the current intrinsic value and assume that the growth in cash flows is constant and exponential. For a detailed discussion, see Appendix B. Our model model has two stages, a volatile growth phase that lasts Tyears followed by a stable steady state growth phase. We first examine the steady state growth phase where the ultimate question to be answered is: what are the dynamics that control cash flow? To this end, Fig. 3(a) shows the time series of Shillers monthly recorded S&P price, S&P earnings, and S&P cash flows, which represent the behavior of the total US economy for the last 95 years [58]. Note that although the US economy has radically changed over the last century, with traditional in- dustry being replaced by advanced technology, both earning levels and cash flows have on average increased exponentially and the exponential growth rates are approximately constant. This result indicates that the earning dynamics in the US econ- omy during the last two decades have been similar to those during the first two decades of the 20th century. According to Abreu and Brunnermeier [25] bubbles are sometimes caused when less sophisticated, overly optimistic traders believe that some new technological innovation will guarentee permanently higher growth rates. Fig. 3(b) also shows the S&P 500 index

Fig. 5. The Market-to-Intrinsic ratio for the S&P 500 index. (a) Historical
values, compared with the scaled Shillers P/E (green), the latter obtained by
dividing with 20 for clarity reasons. (b) Histogram of the ratio (in blue) resembles
a bimodal functional form, characteristic for nonlinear phenomena of small
systems near the tipping point. The red vertical line indicates todays value of
the ratio, and the fitted Gaussian mixture model, scaled for visibility, with two
components at 1 =0. 8 and 2 =1. 3 is shown in green. The^2 statistical
test of the null hypothesis that the Gaussian mixture accounts for the data gives
a^2 statistic of 65. 17 , and the limit for 95% certainty is 81. 38  therefore, the
null hypothesis stands at a significance level of 0. 05.
SMover the last 95 years together with the intrinsic value of
the S&P 500 index,SI, obtained using Shillers S&P 500 data
from January 1920 until March 2015 [58].
Recalling that in our model trader confidence [44] is quan-
tified by the ratio between between market price and intrinsic
value, Fig. 3(c) shows several realizations of the model mar-
ket price. These outputs represent all model outputs for this
intrinsic price input and no other realizations were discarded.
When the market becomes unsustainably overpriced a specu-
lative bubble is created [44] which in turn makes the agents
increasingly uncertain about current and future trading prices.
This accumulated collective uncertainty among agents we quan-
tify as the variance of model price, and Fig. 4(a) shows that the
variance is at its highest levels just before a market crash when
market price greatly exceeds intrinsic value. We also show the
average clustering coefficients (calculated as the mean of local
clustering coefficients of all vertices, as proposed by [59]) for
the supply and demand networks, shown in Fig. 4(b), which
demonstrate opinion clustering and global confidence levels in
the networks. Moreover, the clustering coefficients cross paths


as the market became underpriced in the 2008. crisis. The re- sults reported in Fig. 4(a)(b) are in agreement with a complex system phenomenon called critical slowing down [11] charac- terized by the increase in the variance as the system approaches criticality, which is, in our case, market collapse. To test how the market and intrinsic prices are related in the long run, Fig. 5(a) shows the ratio between the market S&P 500 index and the intrinsic value of the S&P 500 index. Although the intrinsic value of the S&P 500 is not calculated by fitting the market index, the average value of the ratio is close to one (1.04), which is a result that contrasts with the results of a theoretical model that assumes rational agents who do not know the beliefs of other agents and the market price is larger than the intrinsic value [45]. Fig. 5(a) further validates our intrinsic price since we find that the peaks of the market-to-intrinsic ratio follow the peaks of the Shiller P/E indicator widely used to estimate the degree of market overpricing. These correspond to the timings of known major financial crashes and crises in the last century, marked in the figure. This fact affirms that ourSIrepresents a reasonable estimation of the intrinsic value of the S&P 500 index. Fig. 5(a) further reveals that the periods when the market is underpriced and overpriced occur repeatedly, which agrees, when speaking of companies, with the suggestion that expectations for a company should not be too high or too low [60]. If a company promises investors blue-sky expected outcomes that are not realized, not only will the share price drop when the market realizes that the company cannot deliver, but it may take years for the company to regain credibility. To paraphrase Abraham Lincoln, by overestimating the market you can fool some of the people all of the time, but all of the people only some of the time. The final punishment of the market comes in the ensuing period when the market is underpriced. We confirm the intriguing possibility that two distinct under- priced and overpriced modes are present in the financial complex system. Fig. 5(b) shows the probability of the ratio between mar- ket price and intrinsic value for the S&P 500 index. We apply a statistical mixture model to reveal the presence of a substructure in the ratio and show that the probability distribution function of the ratio fits the Gaussian mixture model, which resembles the bimodal shape characteristic for small nonlinear systems near a tipping point [61]. Note that we can locate this substructure of the financial system with the underpriced and overpriced modes because we have a model that can estimate the intrinsic price. Fig. 5(a) suggests that over a long period of time the mar- ket price and the intrinsic value should follow each other. We quantify this assumption by investigating the long-term rela- tionship [62] between the S&P 500 index,SM(t), and the in- trinsic value of the S&P 500 index,SI(t), i.e. the S&P 500 Market-to-Intrinsic ratio. Motivated by the finding of Camp- bell and Shiller that dividends and the present discounted value of expected future dividends cointegrate [62], we next employ the EngleGranger cointegration test [63], [64] and report the cointegration relationship

log(SM(t))log(SI(t)) = 0 (9)

between two series at a 5% confidence level (see Table II). The test is based on a PhillipsPerronZtunit root test of

Test statistics (intrinsic value) Test statistics (index price)
0. 2645 0. 387
First differences
 20. 063  919. 714
Test Statistic Critical values Significance levels
 16. 642  20. 5032 1 %
 14. 034 5 %
 11. 213 10 %
regression residuals, with 8 lags included in the NeweyWest
estimator of the long-run variance (the lag parameter was set to
8 in accordance with the StockWatson method [64] 0. 75 N
in whichNis the number of observations). Details about the
test are provided in Appendix B and in Table II. Intrinsic value
and index price may deviate from each other in the short run,
but in the long run the intrinsic value catches up to the market
realizations. As a consequence of the cointegration, the longer
the market is overvalued, the longer we may expect the market
to stay in the undervalued mode.
We hypothesize that financial crashes occur when a sin-
gle parameterthe ratioRbetween market price and intrinsic
pricereaches a crash tipping pointRc, and that the recovery
begins as it reaches a recovery tipping pointRr. This extends
the idea of herding-induced equilibrium prices deviating from
fundamental values by Lux and Marchesi [21], and includes
the interpretation by Abreu and Brunnermeier [25] that bub-
bles burst when agent synchronization reaches a tipping point.
Moreover, we include crash as well as recovery tipping points
in our analysis. The tipping points are not deterministic, but
rather stochastic variables. In models in which a nodes activity
is dependent upon the activity of neighboring nodes, quantified
by thresholds as in the Watts model [15], we define thresh-
olds to be fixed numbers and the result are well-defined critical
points [27], [28]. However, if we assume that the thresholds are
stochastic variables, the critical points will also be stochastic
variables. Here we propose a procedure for estimating tipping
pointsRcandRrin both the network model and U.S. financial
market data. We first apply a variation on Hodrick-Prescott fil-
tering proposed in [65] (referred to as 1 trend filtering) on the
log-prices. The method minimizes the sum of squared errors of


the estimated trend ( 2 norm) and penalizes trend variations by a sum of absolute values ( 1 regularization [66])

z= argmin
Sz^22 +Dz 1

wherez=(z(1),…,z(n))Rnis the estimated trend vec- tor, S=(S(1),…,S(n))Rn are either network model prices or S&P 500 prices,is the smoothing parameter and DR(n2)nis the second-order difference matrix: 1 21 1 21 .. . 1 21 1 21

. (11)

The resulting trend estimatezis a piecewise linear approxima- tion of the original dataS, with sparse slope changes. According to Kim et al. [65], the changes of the estimated trend can be in- terpreted as abrupt changes or events in the underlying dynamics of the time serieshere we identify them as the systems tipping points. To estimate the tipping points we define the set of time points which correspond to the changes of the estimated trend from positive to negative as the crash tipping timestc.Like- wise, points at which the estimated trend changes from negative to positive are defined as the recovery tipping timestr

tc={t:z(t1) 0 ,z(t+1)< 0 } (12)
tr={t:z(t1) 0 ,z(t+1)> 0 }. (13)

The values of R we consider for the estimation of the crash tipping points and recovery tipping points are therefore {R(t):ttc}for crash tipping points and{R(t):ttr}for recovery tipping points. Fig. 6(a) demonstrates the estimated financial trends and the identified crash and recovery tipping moments in the S&P 500 log-prices, with the smoothing parameter=10chosen to iden- tify major financial trends (and consequentlycrashes) in the considered period (19202015). Fig. 6(b) shows the values of the Market-to-Intrinsic ratio of the S&P 500 with highlighted tipping points identified from the log-prices. The tipping points vary between different realizationsas one would expect for tipping points in the social sciences, and we report the mean values of the identified crash and recovery tipping points, to- gether with their respective standard deviations. The S&P 500 market crashes as it reaches the crash tipping point 1. 64 0. 41 , and rebounds as it reaches the subsequent recovery tipping point

  1. 72 0. 32. Our network model outputs (none of the model outputs were discarded) follow the exact same pattern with the values of its tipping points being 1. 66 0. 22 , and 0. 85 0. 28. Evidently, the network model generates trajectories which not only exhibit crashes and rebounds at appropriate moments in time [as demonstrated earlier in Fig. 3(c)], but also at corre- sponding values of the variableR. To further inspect the properties of the market and the net- work model, we take on the approach proposed by Sornette [24] and look into the distribution of drawdowns. A drawdown is defined as a persistent decrease in the price over consecutive
Fig. 6. (a) S&P 500 log-prices from 1920 to 2015, with 1 trend estimates
and the identified tipping points. (b) S&P 500 Market-to-Intrinsic ratio and the
estimated tipping points.
daysand is thus the cumulative loss from the last maximum
to the next minimum of the price [24]. Similarly, a drawup is a
persistent increase in the price over consecutive days and is the
cumulative gain from the last minimum to the next maximum
of the price. Johansen and Sornette [24], [30] find that the draw-
down distribution of the Dow Jones Industrial Average index
prices exhibits heavy tails, which is not the case when the same
price series is shuffled. This means that drawdowns reveal the
subtle time-dependences, and that major financial crashes are in
fact outliers in the standard assumption of independent succes-
sive price variations [24]. Our network model accounts for these
intricate properties and in Fig. 7 we demonstrate the correspon-
dence between the distribution of drawdowns and drawups of
the S&P 500 index and the model for all the generated model
outputs. The heavy tails in the model-generated prices are due
to the emergence of herding modelled by opinion clusters in our
model, which can explain the occurrences of so-called black
swans (large drawdowns of huge impact)in other words, ma-
jor market crashes.
In addition, we introduce a procedure to investigate the sys-
temic behavior around tipping points of the network model
and U.S. financial market data. In order to facilitate this, we
must first identify the model output and the parameter space
within which we search for the tipping points. For the model we


Fig. 7. Normalized natural logarithm of the cumulative distribution of draw- downs and of the complementary cumulative distribution of drawups for the S&P 500 index and all of the model generated output prices.

Fig. 8. Hysteresis and critical points of financial networks. (a) Model: Proba- bility that the price will drop during[t, t+Nt],whereN=12andt= month, versus the Market-to-Intrinsic price ratio,R, shows a hysteresis behavior. Approaching a tipping point, the probability of price decline exhibits an abrupt change. (b) S&P 500: Probability that the price will drop during[t, t+Nt] versusR.

define the network output at timetas the fraction of future times within the time interval[t+Nt]which has a lower price than the current, wheretis the time step andNis an integer. We generate a large number of simulations for the last two major financial crises (Dot-com bubble in 2000 and the financial crisis of 2008), and at anytfor each price time seriesSi,twe record the Market-to-Intrinsic price ratioRi,tand the probabilityIi,t,

Fig. 9. Forecasting power. (a) Different time series of model price (in blue),
with the the S&P 500 price index (in green) for the estimated intrinsic values (in
red). (b) Cumulative distribution of potential lossL,P(L<x)(in red), with
cumulative distribution of potential gainG,P(G>x)(in blue) in period from
April 2015 to December 2016.
which indicates what fraction of the future prices are lower
than the current (i.e.S(t+kt)<S(t), 1 kN). Com-
bining all pairs (Ri,t,Ii,t), Fig. 8(a) shows that the probability
of a future price decline versus the ratioRexhibits a hysteresis
behavior. The hysteresis is revealed by analyzing the network
model when it moves from its underpriced to its overpriced
phase and vice versa.
Fig. 8(a) shows that the hysteresis obtained for the network
model agrees with the hysteresis shown Fig. 8(b) for the U.S.
financial market, represented by the S&P 500. The hysteresis
shown in Fig. 8(a) was obtained using a large number of numer-
ical simulations, and the hysteresis in Fig. 8(b) was obtained by
analyzing the S&P 500 during the 2000 dot-com crash and the
2007 recession. At anytfor the S&P 500precisely the price
time seriesSi,twe record the ratioRtof the S&P 500 and
the probabilityIi,tindicating what fraction of the future prices
are lower than the currentSi,tduringt+Nt. Collecting all
of the pairs (Rt,Ii,t), we calculate for a givenRthe probability
that the price will decline during the time interval[t+Nt].
We find that the tipping point at which the S&P 500 crashes
strongly resembles the tipping point we obtain by our model
shown in Fig. 8(a).


Because theories with predictive power are highly valued in science and in finance in particular [67], [68], we calculate fu- ture intrinsic values of the S&P 500 by assuming that market earnings and WACC in the future will follow the historical trends. Specifically we use autoregressive models to make predictions of earnings and WACC trends. We further generate a large number of the network realiza- tions, resulting in modeled prices shown in Fig. 9(a), and then we count the modeled prices according to their relative change with respect to the maximum or minimum value (which ever occurs earlier) in period from April 2015 to December 2016. Fig. 9(a) shows the cumulative frequencies for the cases with the negative change, i.e. loss (in red) and cumulative frequen- cies with the positive change, i.e. gain (in blue). Fig. 9(b) reveals that the probability of S&P 500 declining from its peak for more than 10% is approximately 80%. The same Figure further re- veals that there is only 20% chance that the S&P 500 index will grow more than 10% from its minimal value in the same time period. This calculations assumes that the earning trends and WACC obtained from the historical data will continue to hold, at least in the near future, and Fig. 3 shows that this assumption is reasonably correct.


In this paper we introduce a model based on a network of bargaining agents who mutually either compete or cooperate. This model assumes that agents compete when they use sim- ilar strategies. Due to the feedback mechanism, driven by the Market-to-Intrinsic ratioR, the network model produces mar- ket prices which oscillate around the input (intrinsic values) and exhibit bubbles and crashes. Moreover, due to the stochastic na- ture of the model, the same input (intrinsic values) can generate different outputs (market prices), which we can use to analyze tipping points, drawdowns and trader confidence. To test the model, we calculated the intrinsic values of the S&P 500 market index, using the FCFM. The estimated intrinsic values were shown to be cointegrated with the market prices in the last 95 year period (19202015), and the estimated Market- to-Intrinsic ratioRfor the S&P 500 market oscillates around 1, on the long run. These results affirm our modelling approach and support the use of the estimated intrinsic values as the input to our network model. The results show that crashes occur in the network model when the ratioRbetween the model generated market prices and the input intrinsic values reaches a crash tipping point, and the recovery begins as the ratio reaches a recovery tipping point. This is in line with the results from the S&P 500 market index, which is shown to exhibit this exact bahavior at similar tipping points. Moreover, we demonstrate that the network model pro- duces heavy-tailed drawdowns, as seen in real-world markets the comparison with S&P 500 drawdowns further confirms the value of the results. In addition, based on a range of estimates for the future intrinsic values, we generate a number of predictions for the future behavior or the market prices and demonstrate the forecasting power of the model by calculating the the cumu-

lative distribution of potential gain and loss for the following
period (until December 2016).
These findings show that the proposed modelling approach
provides a novel perception about the mechanisms behind
tipping points in marketboth crash and recovery. With fur-
ther insight into agent trading behavior and the effects of dif-
ferent parameter settings, future research will be able to apply
our model to the analysis of other important market phenomena
(such as bid-ask spread dynamics).
Assume two agents (players) are involved in the sale of
a book. Player 1 is selling the book and values it at $60, and
player 2 wants to buy the book and values it at $30. The two
players are the only agents involved in the transaction, both
are rational, and both want the transaction to occur (neither
will withdraw). Because there are no other agents who might
be competitors and influence the transaction, the two players
rationally meet halfway and agree on a price of $45. We can
make the situation more complex by adding the assumption
that there is a probability 1 =0. 8 that either player will
quit the bargaining process before an agreement is reached. In
this case each players anxiety that the other will back out of the
transaction gives them greater motivation to reach an agreement.
As a result, two equilibrium prices, $35 and $55, emerge. The
seller is assured of $35 without risk but must weigh that against
getting $55 with a confidence of=0. 8. The seller will accept
any price above $35 but will not offer a price below $55. These
described bargaining scenarios are rudimentary, and for a more
detailed discussion see Nash [35], Rubinstein [36] and a number
of other contributors[43], [49], [50].
The agent utility functions are linear functions of the offered
price that monotonically increase for supply agents and mono-
tonically decrease for buyer agents, i.e.,
i S
whereS(inits) andS(initd)are the initial bargaining prices of the
supply and demand agents (the best supply and demand offers),
respectively. These are reset when a trade occurs and are fixed
between trades.
Agents evaluate the utility of acceptingS, the offer in the
current step, using
Ui(s)(S,0) =u(is)(S),Uj(d)(S,0) =u(jd)(S). (15)
When evaluating whether to accept an offer in the next bar-
gaining step, an agent must assess the probability that another
trading agent will intervene and steal the trading opportunity.
For all supply and demand agents, the probability that this
breakdown scenario (b) will occur isu(is)(b)=u(jd)(b)=0.


The utility of accepting offerSin the next bargaining step is given by

E[Ui(s)(S,1)] = u(is)(S)(is)+u(is)(b)
1 (is)
E[Uj(d)(S,1)] = u(jd)(S)(jd)+u(jd)(b)
1 (jd)
=u(jd)(S)(jd). (16)

Here(is)and(jd)are the evaluated probabilities (the agent confidences) of the supply agentiand demand agentj,re- spectively, poised to trade (or not trade) in the next step. The agents use their neighborhoods to evaluate their confidence fol- lowing the expression (7). Since external information affects one side positively and the other negatively, supply and demand agents interpret this information reciprocally. In effect, the in- trinsic value returnrI(t)tilts the overall confidence evaluations of agents and renders one side more confident and the other less, and is calculated as

. (17)

The network parameters used in this study are:=0. 2 , =0. 3 and=90. These values were empirically chosen to produce reasonable model outputs. We use a version of the monotonic concession protocol to simulate agent price movement during the bargaining process [48]. Agents from the supply and demand sides alternate in making their moves. A move can either be accepting the current offer from the other side or proposing a new price. If the offer is accepted and a trade in completed the trading agents are removed from their networks and the agreed-upon price becomes the trading price. An agent accepts the current offer if the expected advantage of accepting it is greater than or equal to the expected advantage of risking that a more attractive price will be agreed upon in the next step. This condition can be expressed using equations (15) and (16) for supply and demand agents

S 0 (d), 0
, 1
S( 0 s), 0
, 1
If the condition is not met, the agent proposes a new price,



. The multiplicative concession steps

i(s)for supply and demand agents are defined

1 (is)
1 (jd)

where(s) 0 and(d) 0 are fixed concession step constants for the supply and demand networks, respectively. This means that the magnitude of concession steps made by agents will be determined by the level of agent confidence. Less confident agents offer a larger concession step. After a trade occurs and the trading agents are removed from their networks, a new agent is added to each network, which preserves the orders of both

networks. The new nodes are added according to the protocol
described above, with the probabilitypdefined in Eq. (3). The
parameter values used arepc=0. 2 ,(d)=(s)=0. 005 .The
supply and demand networks each have 500 agents and each are
initialized with a single node with the initial price equal to the
initial intrinsic value. The network steps are(s)=0. 01 and
(d)= 0. 01 for supply and demand networks, respectively.
The FCFM is one of the most respected methods for deter-
mining the intrinsic value of a company. The model assumes
that a companys stock is equal to the sum of all of its future
FCFs, discounted back to their present value using the future
WACC as the discount rate [see Eq. (8)]. FCF is the cash flow
generated by the core operations of the business after deducting
investments in new capital, and WACC is the rate of return that
investors expect to earn from investing in the company. Because
investors do not know future values of FCF and WACC, they
use estimates based on their own experience, knowledge, and
available information. To avoid false precision errors, investors
often split their forecast into two periods, (i) a detailed five-
year forecast that develops complete balance sheets and income
statements with as many links to real variables as possible, and
(ii) a simplified forecast for the remaining years. The stock is
then traded at a price
S(0) =
k=1(1 +WACC
k=1(1 +WACC
whereFCFjEis the estimated FCF at the stepj,WACCEk
the estimated WACC at time stepk, andCVTEthe estimated
continuing value at time stepT[60]. Because the correct FCF
and WACC are known for the previous five yearstime period
Tan estimation of a companys intrinsic value at the beginning
of that five-year time period,SI(T), made today will be more
accurate than one made five years ago
k=T+1(1 +WACCk)
k=T+1(1 +WACCk)
k=1(1 +WACC
Although estimates of FCF and WACC must extend to the
indeterminate future, because of the discount effect these esti-
mates will have a decreasing impact on the price. For the sake
of simplicity, we assume (i) that the future FCF will grow at a
constant growth rateg, and (ii) that the future WACC will be
constant. Therefore (21) becomes
k=T+1(1 +WACCk)
k=T+1(1 +WACCk)(WACCg)


whereFCF 1 is the expected FCF one time step in the future (e.g., one year from now), and WACCthe expected long-term WACC. The goal is to estimate the current intrinsic value of the company now, not its value five years ago. Because over a long period of time the growth rate of the company earnings will be similar to its stock prices, we can estimate the current intrinsic value of the company using

SI(0) = (1 +g)TSI(T) (23)

wheregis the constant growth rate of earnings. Fig. 3(a) shows that this assumption is reasonable, and Fig. 3(b) shows the in- trinsic values with and without the forward correction from Eq. (23). Based on the ratio between these two time series, we define the 90% confidence interval, also shown in Fig. 3(b). Because data for FCF and WACC are usually unavailable, when we implement and test the model we replace these vari- ables with quantities supplied in financial databases. FCF is, in the first approximation,

FCF = ENI (24)

where E is firm earnings and NI net investmentsthe increase in invested capital from one year to the next, i.e., the portion of earnings the firm reinvests. The investment rate (IR), the portion of earnings invested back into the business, is

IR =
. (25)

The return on invested capital (ROIC) is usually defined as the return a company earns on each dollar invested in the business [60]. This is approximately

Invested Capital
. (26)

Growth (g) is usually defined as the rate at which the companys earnings grows each year [60]. Note that FCF=E( 1 IR) and from [60]:g= ROICIR. Finally,

. (27)

In FCFM the discount rate is the WACC. In its simplest form, the WACC is the market-based weighted average of the after-tax cost of debt and cost of equity,

ke (28)

where D/(D + E) is the debt-to-value target level, E/(D + E) the equity-to-value target level,kdthe cost of debt,Tmthe marginal tax rate, andkethe cost of equity [60]. We use data on the S&P 500 from January 1900 to March 2015 supplied by Robert Shiller when we implement our model. These include data on real (inflation adjusted) earnings, real prices, real long-term interest rates, and S&P500 historical av- erages (inflation adjusted). The median debt-to-equity ratio for the S&P500 is 19.7%, the median return on invested capital 7%, and the median growth rate of earnings g=1.7%. We use the long-term interest rate as a proxy for the after-tax cost of debt, and the S&P 500 earning yield as a proxy for the cost of equity.

Because the time period isT=5years, at each time step the
intrinsic value for the five previous years is determined, and
(23) is employed to determine the intrinsic value at the current
time step.
We next use the Engle-Granger cointegration test to investi-
gate the long-term relationship [62] between S&P500 intrinsic
values and S&P500 index prices. Prior to testing for cointegra-
tion, we ensure that both series have the same order of inte-
gration. To determine the order of integration, we employ the
Phillips-PerronZt(PP) unit root test of the null hypothesis that
indicates whether the variable has a unit root against a station-
ary alternative. Table I shows the results of PP unit root tests
for both levels and the first differences of log-price series. The
results imply that both series contain a unit root in levels and
thus should be first differentiated to achieve stationarity. We
conclude that both series are integrated at the first order,I(1),
at a 5% confidence level. To determine whether there is cointe-
gration between the S&P500 intrinsic log-value and the index
log-price, we employ the EngleGranger test [63], [64], which
is based on a PP unit root test of regression residuals, with
8 lags included in the NeweyWest estimator of the long-run
variance (the lag parameter was set to 8 in accordance with the
StockWatson method [64] 0. 75 N

(^13) , whereNis the number of observations). Table II shows that the cointegration between the two series is at a 5% confidence level. Note that intrinsic value and index price can deviate slightly from each other in the short term, but that market forces, government policies, and investor behavior bring them back to a equilibrium in which market realizations and our market expectations converge. Because theories with predictive power are highly valued in science we calculate future intrinsic values of the S&P 500 by assuming that market earnings and WACC in the future will follow the historical trends. Specifically we use AR(5) model with constant term to make prediction of earnings trend and AR(2) model with constant term to make prediction of WACC trend. Historical earnings and WACC are used for estimation of the models and Akaike information criterion is used for lag order selection in both cases. The fitted models are used in simulation of many possible estimates of the future intrin- sic values. We then use these estimates to predict the future market performance.

The authors would like to thank J. Cvitanic for useful
suggestions, and the reviewers for helpful comments.
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Zvonko Kostanjcar received the Dipl.Eng. de- gree in electrical and computer engineering and in financial mathematics and the Ph.D. degree, in 2010, both from the University of Zagreb, Zagreb, Croatia. He is currently an Assistant Professor with the University of Zagreb, also the President of the IEEE Signal Processing Society, Croatia Chapter. His research interests include development of the stochas- tic models, statistical methods and machine learning algorithms for data analysis, risk measurement, and asset allocation. He received the Roberto Giannini Teaching Award from the Faculty of Electrical Engineering and Computing, University of Zagreb.

Stjepan Begusic  received the B.S. and M.S. de-
grees in information and communication technology
with specialization in information processing from
the University of Zagreb, Zagreb, Croatia, in 2012
and 2014, respectively, where he is currently work-
ing toward the Ph.D. degree in the Department of
Electronic Systems and Information Processing, Fac-
ulty of Electrical Engineering and Computing. His
research interests include signal processing and com-
putational methods for finance, risk analysis, and
asset allocation.
Harry Eugene Stanley received the Ph.D. degree
from Harvard University, Cambridge, MA, USA, in
1967 (T.A. Kaplan & Nobelist J.H. Van Vleck). In
1979, he became a University Professor, and in 2011,
a William Fairfield Warren Distinguished Professor
with Boston University. He is a Professor of physics,
chemistry, biomedical engineering, and physiology.
He is also the Director of the Center for Polymer
Studies. He is a Member of the National Academy
of Sciences. His research interests include statistical
physics, complex systems, and economics.
Boris Podobnik received the B.S.E., M.S., and Ph.D.
degrees in physics from the University of Zagreb, Za-
greb, Crotia. In 2014, he became a Professor with
the University of Rijeka. He is also a Professor
with Zagreb School of Economics and Management.
His research interests include theories regarding data
analysis, complex systems, and economics.


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