Extrapolative Expectations and the Second-Hand Market for Ships

Transcription

1 Extrapolatve Expectatons and the Second-Hand Market for Shps IOANNIS C. MOUTZOURIS and NIKOS K. NOMIKOS Cass Busness School, Cty Unversty London December 2016 Abstract. Ths artcle nvestgates the jont behavour of vessel prces, net earnngs, and second-hand actvty n the dry bulk shppng ndustry. We develop and estmate emprcally a behavoural asset prcng model wth mcroeconomc foundatons that can account for some dstnct characterstcs of the market. Namely, among other features, our partal equlbrum model reproduces the actual volatlty of vessel prces, the average tradng actvty n the market, and the postve correlaton between net earnngs and second-hand transactons. In order to explan the formaton of vessel prces, we depart from the ratonal expectatons benchmark of the model, ncorporatng extrapolatve expectatons on the part of nvestors. In contrast to the majorty of fnancal markets behavoural models, however, n our envronment agents extrapolate fundamentals, not past returns. Ths form of extrapolaton s consstent wth the nature of the ndustry. Accordngly, we ntroduce two types of nvestors who hold heterogeneous belefs about the cash flow process. Formal estmaton of the model ndcates that a heterogeneous belefs envronment, where both agent types extrapolate fundamentals, whle smultaneously under(over)estmate ther compettors future demand responses, can explan the postve relaton between net earnngs, prces and second-hand vessel transactons. To the best of our knowledge, the second-hand market for vessels had never been examned from the perspectve of a structural, behavoural economc model n the shppng lterature before. Keywords: Asset Prcng, Vessel Valuaton, Based Belefs, Cash Flow Extrapolaton, Heterogeneous-agent, Tradng Actvty I. Introducton As t s well-establshed n the equty markets lterature, the majorty of ratonal expectatons models fal to explan numerous emprcal regulatons related to asset prces. Among others, a promnent example s the excess volatlty puzzle (Leroy and Porter, 1981), accordng to whch actual asset prces exhbt sgnfcantly hgher volatlty compared to the ratonal model-mpled ones. Addtonal stylsed facts are the postve correlaton between tradng volume and asset prces (Barbers et al, 2015b) and the strong postve relaton between the aggregate dvdend yeld and future returns n the post-wwii U.S. equty markets (Campbell and Shller, 1988a; Fama and French, 1988b; Cochrane, 2011). For the purpose of explanng these fndngs, researchers n the last decade have developed heterogeneous belefs economc models that ncorporate behavoural bases, manly termed as heurstcs (Barbers et al, 2015a).

2 2 I. C. MOUTZOURIS AND N. K. NOMIKOS Heterogeneous belefs models, however, have not been wdely appled for the modellng of other asset classes, and n partcular vessels. Shppng s a very mportant sector of the world economy, snce 90% of the world trade s transported by sea and shppng s justfably consdered as a leadng ndcator of world economc actvty (Kllan, 2009). The fact that vessels are assets wth fnte lves whose value deprecates over tme provdes dfferent challenges n the econometrc modellng of the market, compared to the case of an nfntely lved fnancal asset. Hence, from both a theoretcal and practcal perspectve, t s mportant to understand the prcng and tradng dynamcs of ths asset class. In ths artcle, we develop a heterogeneous belefs model that can explan numerous emprcal fndngs related to the sale and purchase market for second-hand vessels. The emprcal estmaton focuses on the dry bulk segment of the shppng ndustry, snce t consttutes by far the largest sector n terms of both cargo carryng capacty and quantty transported (Alzadeh and Nomkos, 2010). Furthermore, nvestgatng the dry bulk shppng market, as opposed to the tanker and contaner ones, provdes us wth the opportunty to employ a sgnfcantly larger dataset. In the context of ths artcle, we concentrate on the Handysze sector, however, our model s predctons have been tested and accordngly, can be extended to the entre dry bulk shppng ndustry. The proposed partal equlbrum framework explans the observed prce behavour of second-hand vessels and n partcular ther excess volatlty. Most mportantly, our model provdes a plausble economc nterpretaton for specfc features correspondng to the tradng actvty of vessels. We reproduce and justfy the stylsed fact that tradng actvty s postvely related to both market condtons and absolute changes n net earnngs between two consecutve perods. In our sample, the two correlaton coeffcents are equal to 0.53 and 0.65, respectvely. In other words, nvestors appear to trade more aggressvely durng prosperous market condtons, but also when net earnngs have sgnfcantly changed compared to the prevous perod. Interestngly, formal estmaton of the model shows that the postve correlaton between net earnngs and tradng actvty s accompaned by low average volume of transactons n the market. In addton, therefore, our model mplctly captures the fact that second-hand markets are rather llqud; durng the perod , the average annual tradng actvty was roughly 5.8% of the correspondng fleet sze. Fnally, the proposed framework accounts also for the emprcal fndngs that hgh net earnngs-prce ratos negatvely forecast future net earnngs growth, and that the bulk of the rato s volatlty s attrbuted to expected cash flow varaton, not tme-varyng expected returns (Nomkos and Moutzours, 2015). Our dscrete tme envronment conssts of two agent types, conservatves and extrapolators, the relatve populaton fractons of whch reman constant over tme (Barbers et al, 2015a). In the model, annual shppng net earnngs are the sole state varable, observed at each perod by the entre nvestor populaton. Accordngly, smlar to Barbers et al (2105b), when valung the asset at each

3 THE SECOND-HAND MARKET FOR SHIPS 3 perod, agents maxmse recursvely a constant absolute rsk averson (CARA) utlty functon, whch s defned over next perod s wealth. In addton, for the purpose of beng consstent wth the nature of the ndustry, both agents face short-sale constrants. Whle both types value vessels based on the evoluton of fundamentals, they are charactersed by bounded ratonalty whch stems from two facts. Frst, market partcpants form extrapolatve expectatons regardng the cash flow process; the conservatve at a lesser degree compared to the extrapolator. From a psychologcal perspectve, the extrapolaton of fundamentals can be the result of several heurstc-drven bases. The most frequently ncorporated bas s the one known as representatveness heurstc, accordng to whch, ndvduals beleve that small samples are representatve of the entre populaton (Tversky and Kahneman, 1974). Smlarly, nvestors may suffer from the avalablty heurstc whch causes subjects to overwegh readly avalable nformaton (Shefrn, 2000). Alternatvely, market agents may fall nto the ths tme s dfferent belef (Renhart and Rogoff, 2009). In our model, there s no need to specfy the exact form of psychologcal bas, snce the presence of ether of these heurstcs wll lead shppng agents to extrapolate current market condtons (Fuster et al, 2011). Second, each agent s nvestment strategy s ndependent of the other s. In partcular, both agents assume that, n all future perods, the other type wll mantan hs per-capta fracton of the rsky asset supply (Barbers et al, 2015b). From a psychologcal pont of vew, ths msbelef can be drven by a bas known as competton neglect (Camerer and Lovallo, 1999, Kahneman, 2011). Followng Glaeser (2013) and Greenwood and Hanson (2015), ths bas may be the result of bounded ratonalty whch leads agents to form forecasts about compettors reactons ncorporatng a smplfed economc framework nstead of a more elaborate model of the market. In our case, each agent under(over)estmates the future demand responses of the other type. Fnally, at each perod, the equlbrum prce of the vessel s defned through a market clearng condton. Havng expressed the equlbrum vessel prce and second-hand actvty as functons of the state varable, we can then estmate the parameters of nterest that allow us to capture the prevously analysed stylsed facts. In partcular, we are nterested n the populaton fracton and the perceved net earnngs persstence correspondng to each agent type. A frst-order effect of the proposed framework s that n the presence of extrapolatve expectatons n the market, vessel prces become more senstve to the prevalng cash flow. As a result, the extrapolatve model-generated prce devates from the asset s fundamental value whenever the correspondng cash flow varable devates from ts steady state. Ths fact mples an mmedate overor under-valuaton of the vessel, whch n turn generates excess prce volatlty. In lne wth the lterature, we defne as fundamental asset value the one generated by the ratonal benchmark of the model, or equvalently, the prce of the asset n a counterfactual economy where all agents form

4 4 I. C. MOUTZOURIS AND N. K. NOMIKOS ratonal expectatons about the net earnng process. However, one should not confuse the noton of over-or under-valuaton n our context wth the use of the term n the emprcal asset prcng lterature (Cochrane, 2011). In the latter, the term overvaluaton means that the asset s prce s hgh compared to the correspondng cash flow varable. Interestngly enough, n shppng, when the asset s overvalued compared to ts fundamental prce, t s n general undervalued compared to the correspondng cash flow varable (Papapostolou et al, 2014, and Nomkos and Moutzours, 2015). Ths stylsed fact s n lne wth our model s predctons. Whle a homogeneous-agent settng wth extrapolatve expectatons could capture the observed prce behavour, t would not be suffcent to justfy the second-hand market transactons. Hence, tradng actvty n our framework s the consequence of heterogeneous valuatons of the asset by market partcpants. Whle there can be alternatve explanatons for tradng actvty (e.g. lmts to arbtrage, portfolo dversfcaton polces, other nformaton frctons) or excess prce volatlty, the motvaton provded has the advantage of smultaneously explanng n a suffcent manner numerous emprcal regulartes. Furthermore, the economc nterpretaton of the model and the respectve results are plausble and n lne wth the nature of the shppng ndustry. To the best of our knowledge, ths s the frst tme n the shppng lterature that a structural model ncorporates the coexstence of heterogeneous belefs agents for the purpose of explanng the jont behavour of observed vessel prces and second-hand vessel transactons. Regardng the exstng shppng lterature, Beenstock (1985) and Beenstock and Vergotts (1989) construct and estmate a ratonal expectatons general equlbrum model relatng vessel prces to cash flows, new buldng and scrappng actvty, and demand for shppng servces. The homogeneous-agent settng, however, does not allow for the explanaton of the second-hand market actvty. Furthermore, as the authors argue, ther model s a smplfed verson of realty snce t does not capture the possblty of extrapolatve expectatons on the part of nvestors. Greenwood and Hanson (2015) ncorporate ths behavoural bas by developng an elaborate mcroeconomc model that reproduces several features of the ndustry. In ther homogeneous-agent framework, agents extrapolate current demand condtons whle smultaneously neglect ther compettors supply responses. The combned effect of those behavoural bases s overnvestment and overvaluaton of the asset durng prosperous market condtons and vce versa. The behavoural mechansm proposed here s smlar to that of Greenwood and Hanson (2015) but also allows for the degree of extrapolaton to depend on the agent type. Consequently, the heterogeneous nature of our model allow us to smultaneously capture the observed volatlty of prces and the relaton between net earnngs and second-hand actvty n the market.

5 THE SECOND-HAND MARKET FOR SHIPS 5 Kalouptsd (2014) examnes the mplcatons of demand uncertanty and constructon lags on shp prces, new buldng and scrappng actvty, and market partcpants surplus. Usng a ratonal expectatons model, she shows that the requred tme to buld and the level and volatlty of vessel prces are postvely related. Smlar to the prevous artcles, however, the assumpton of a unque agent type excludes the modellng of the sale and purchase market. Fnally, whle Nomkos and Moutzours (2015) address the queston regardng the volatlty of valuaton ratos as a whole, they do not examne ether the underlyng mechansm behnd the volatlty of asset prces per se or the relaton between prces, earnngs and second-hand actvty. Our paper looks at the man features of heterogeneous-agent models but also ntroduces mportant modfcatons whch are requred n order to capture stylzed features of the shppng markets. In partcular, recent artcles, manly n equty but also n commodty markets (Ellen and Zwnkels, 2010), have attempted to explan emprcal asset prcng fndngs usng heterogeneous belefs models n whch a fracton of the populaton forms based expectatons about future returns. Barbers et al (2015a) develop an extrapolatve captal asset prcng model (X-CAPM) that explans the volatlty of the aggregate stock market. Furthermore, Barbers et al (2015b) ncorporate a heterogeneous-belef extrapolatve model of returns n order to analyse the formaton of asset bubbles n equty markets. Some key features of ther envronment and model s soluton are very closely related to the one presented n ths artcle. However, n contrast to Barbers et al, n our model there s cash flow and not return extrapolaton, the correspondng asset s affected by economc deprecaton due to wear and tear, and most mportantly, all agents are extrapolators, however, at a dfferent degree. The latter feature allows us to capture also severe undervaluaton phenomena and moreover to explan the postve correlaton between market condtons and tradng actvty. As a byproduct, our model also reproduces the relatvely low lqudty of the shppng markets. The remander of ths artcle s organsed as follows. Secton II ntroduces the envronment of our economy and the soluton of the theoretcal model. Secton III presents the dataset employed along wth the emprcal estmaton of the model and a formal analyss of the produced results. In addton, t provdes an economc nterpretaton of the results. Secton IV examnes several alternatve hypotheses regardng the nvestor populaton composton. Secton V concludes. II. Envronment and Model Soluton Consder a dscrete-tme envronment where the passage of tme s denoted by t. The economy conssts of two asset classes: the frst one s rsk-free whle the second one s rsky. The rsk-free asset can be thought of as an nfntely lved fnancal nstrument n perfectly elastc supply, earnng an exogenously determned constant rate of return equal to R f. The rsky asset class conssts of otherwse

6 6 I. C. MOUTZOURIS AND N. K. NOMIKOS dentcal vessels whch are further categorsed accordng to ther age. Importantly, all age classes have fxed per capta supply over tme, equal to Q. In what follows, we restrct our attenton to the modellng of the market for 5-year old vessels. As we llustrate n the followng, n order to analyse the tradng actvty correspondng to ths age class, we need to examne also the demand and prce mechansm related to the 6-year old vessel. Notce that exactly the same prncples apply for the valuaton of the other age classes. Followng market practce, we assume that a newly-bult vessel has an average economy lfe of 25 years. At the end of ths perod, the asset s scrapped; hence, t exts from the economy. Accordngly, settng the tme-step of the model, Δt, equal to one year, mples that a 5-year old asset has T = 20 perods of remanng economc actvty. We focus on 5-year old secondhand vessels, nstead of new buldngs as one of the man objectves of our framework s the modellng of tradng actvty n the sale-and-purchase market for shps. An nherent characterstc of the shppng ndustry s that forthcomng perod s net earnngs are F t - measurable (Nomkos and Moutzours, 2015). Assumng no default on the part of the charterer, 1 the shp owner at tme t knows precsely hs net earnngs for the perod t t + 1, defned as Π t. In equty markets termnology the asset s sad to be tradng cum dvdend. Therefore, the owner of the 5- year old vessel at tme t s enttled to an exogenously determned stream of annual net earnngs, {Π n } t+t t. In the context of our theoretcal model, net earnngs are the sole state varable, the evoluton of whch s assumed to be followng a mean-revertng process Π t+1 = (1 ρ 0 )Π + ρ 0 Π t + ε t+1, (1) where Π s the long-term mean, ρ 0 [0,1), and ε t+1 N(0, σ 2 ε ),.. d. over tme. Notably, though, 2 n contrast to Π, parameters ρ 0 and σ ε are not publc nformaton. The economy conssts of two nvestor types, : 2 conservatves and extrapolators, denoted by c and e, respectvely. We normalze the nvestor populaton related to each asset age-class to a unt measure, 3 and we further assume that the fractons of conservatves, μ c, and extrapolators, μ e, are fxed both across all age classes and through each specfc asset s lfe. In what follows, we set μ c = μ; hence, μ e = 1 μ. The assumpton that each unt measure of nvestors s related to one and only 1 In realty, the agreed tme-charter rates are usually receved every 15 days (sometmes also n advance). As a result, the probablty of default on the part of the charterer s reduced. Moreover, an extensve network of compettve brokers and the fact that shp owners normally lease ther vessels to solvent charterers assure transparency and low probablty of default. In contrast, a tme-charter lease wth a less credtworthy charterer wll ncur hgher rates n order to compensate the owner for the hgher probablty of default on the part of the charterer. Fnally, addtonal contractual agreements ncluded n the charter party, ensure that the owner wll receve the full tme-charter rate agreed. 2 For llustratonal purposes, from the pont of vew of agent, the other agent s denoted by. 3 Ths assumpton s n lne wth both the characterstcs of the ndustry, and the exstng lterature (Kalouptsd, 2014), where shp ownng companes are assumed to be operatng on a one frm-one shp bass (Stopford, 2009).

7 THE SECOND-HAND MARKET FOR SHIPS 7 asset age-class s equvalent to assumng that each unt mass of nvestors has a fnte lfe equal to the correspondng lfe of the vessel. 4 We mpose ths assumpton n order not to overcomplcate mathematcally the model; n partcular, t permts us to treat each age-class n solaton. Accordngly, we can derve analytcal theoretcal predctons and closed-form results for many quanttes of nterest. The dfference between the two agent types les n the alternatve ways n whch they form expectatons about future cash flows. Specfcally, compared to extrapolators, conservatves percepton s closer (n prncple, t mght be even dentcal) to equaton 1. Therefore, snce conservatves mght also form extrapolatve expectatons, the terms conservatve and extrapolator are used n a comparatve manner. From a psychologcal perspectve, as analysed n the Introducton, both types suffer from a heurstc-drven bas, whch n turn leads to the extrapolaton of current cash flows. In order to capture mathematcally ths behavoural bas, we assume that n agent s mnd, net earnngs related to the valuaton of the 5-year old vessel evolve accordng to Π t+1 = (1 ρ )Π + ρ Π t + ε t+1, (2) n whch ρ 0 ρ c < ρ e < 1, and ε t+1 N(0, θ 5 σ ε 2 ),.. d. over tme, where 0 < θ 5 e < θ 5 c. The strctly postve parameter θ 5 adjusts the (true) varance of the cash flow shock accordng to agent s perspectve, whle the subscrpt denotes the current age-class of the vessel beng valued. The conservatve agent parameters, μ, ρ c, and θ 5 c characterse completely the nformaton structure of our model. When μ = 1, ρ c = ρ 0, and θ 5 c = 1, all agents have perfect nformaton about the economy. We defne ths case as the benchmark ratonal economy of our model, and we term ths agent type as fundamentalst, f; hence, ρ f = ρ 0 and θ f 5 = 1. When μ = 1, ρ c ρ 0, and θ c 5 1 or μ = 0, all agents have mperfect nformaton about the economy. However, n all cases above, there s no nformaton asymmetry among agents; hence, there s no tradng actvty n the market. Fnally, when μ (0,1), that s, when heterogeneous nvestors coexst n the market, nformaton s both mperfect and asymmetrc (Wang, 1993). As a result, tradng actvty s generated n the economy. The tmelne of the model s as follows. At each pont t, Π t s realsed and observed by all market partcpants. Furthermore, the 25-year old age class s scrapped and replaced by newly bult vessels (.e. the 0-year old age class). Accordngly, both agent types determne ther tme t demands for each 4 Essentally, we have a model of overlappng generatons; however, there s no nteracton among dfferent generatons.

8 8 I. C. MOUTZOURIS AND N. K. NOMIKOS age class asset wth the am of maxmzng a constant absolute rsk-averson (CARA) utlty functon, defned over next perod s wealth. For the 5-year old vessel, ths corresponds to max N 5,t Ε t [ e α w t+1 ], (3) where α and N 5,t are nvestor s coeffcent of absolute rsk-averson and tme t per-capta demand for the 5-year old vessel, respectvely. Agent s next perod s wealth, w t+1, s gven by w t+1 = (w t N 5,t P 5,t )(1 + R f ) + N 5,t (Π t + P 6,t+1 ), (4) n whch, P 5,t and P 6,t+1 are the prces of the 5- and 6-year old vessel at t and t + 1, correspondngly. 5 In what follows, we normalse the rate of return of the rsk-free asset to zero (Wang, 1993, and Barbers et al, 2015b). 6 Therefore, nvestor s objectve becomes max N 5,t Ε t [ e α (w t +N 5,t (Π t +P 6,t+1 P 5,t )) ]. (5) Accordngly, the tme t cum dvdend prce of the 5-year old vessel s endogenously determned, through the market clearng condton μ N c 5,t + (1 μ) N e 5,t = Q, (6) where Q s the fxed per capta supply of the rsky asset. The assumpton of fxed per captal supply over the vessel s lfe can be justfed by the fact that we are nterested n the modellng of a real asset wth economc deprecaton. Therefore, the supply of the age-specfc asset cannot ncrease over tme. Furthermore, snce scrappng very rarely occurs before the 20 th year of a vessel s lfe, 7 we assume that the supply of the rsky asset cannot be reduced ether. The mmedate effect of ths assumpton s that our model does not endogense the scrappng decson. Fnally, tradng actvty correspondng to tme t takes place n the market. In shppng, ths actvty refers to the sale and purchase market for second-hand vessels. Notce that, snce ths s a dscrete tme model, we mpose the assumpton that tradng occurs nstantaneously at each pont t. In analogy 5 In prncple, at each t, each agent could nvest a fracton of hs wealth n every age-class of the rsky asset. However, n order not to overcomplcate the analyss and obtan closed-form solutons for the demand functons, we assume that, at each t, a new unt mass of nvestors solely nterested n 5-year old vessels enters n the ndustry. Accordngly, at t + 1 that nvestor populaton wll be solely nterested n the 6-year old asset class, whle a new unt mass related to the 5-year old class wll enter n the market (Fgure 1). 6 It s straghtforward to ncorporate the rsk-free parameter n the analyss, and moreover, to adjust for a tmevaryng rsk-free return. We expect, however, the results to be qualtatvely the same. 7 In practce, the supply of the fleet may be reduced due to accdents and losses as well as conversons of vessels to other uses; these n general consttute an nsgnfcant proporton of the fleet and thus are not consdered here.

9 THE SECOND-HAND MARKET FOR SHIPS 9 to heterogeneous belefs models for equty markets (Barbers et al, 2015b), tradng actvty s estmated through V t 1 t V t = μ N 6,t N 5,t 1, (7) where N 6,t s agent s tme t per-capta demand for the 6-year old vessel. Intutvely, we defne as tradng actvty the agent-specfc change n demand for the rsky asset between ponts t 1 and t, multpled by the respectve populaton fracton. Snce supply s fxed, the quantty sold by one agent s equal to the quantty acqured by the other. Notce that n the equty markets lterature, where researchers are nterested n the behavour of the same, nfntely lved ntangble asset over tme, there s no need for the age subscrpt n (7). Snce, however, vessels are real assets wth lmted economc lves, ther values are substantally affected by economc deprecaton. In partcular, at each pont n tme, a 6-year old vessel s less valuable than an dentcal 5-year one. Therefore, we need to estmate the demand functons for both the 5 and 6-year old vessels, at each tme t. Fgure 1 summarses the tmelne of the model. Fgure 1: Tmelne of the Model. t 1 t ۓ ە ۓ ە Π t 1 s realsed. 5-year populaton determnes N 5,t 1 and P 5,t 1. At t 2, ths group had determned N 4,t 2 and P 4,t year populaton determnes N 6,t 1 and P 6,t 1. At t 2, ths group had determned N 5,t 2 and P 5,t 2. Tradng actvty for the 6-year old vessel: V t 1 = μ N 6,t 1 N 5,t 2 Π t s realsed. 5-year populaton determnes N 5,t and P 5,t. At t 1, ths group had determned N 5,t 1 and P 5,t year populaton determnes N 6,t 1 and P 6,t 1. At t 1, ths group had determned N 5,t 1 and P 5,t 1. Tradng actvty for the 6-year old vessel: V t = μ N 6,t N 5,t 1 Appendx A shows that the tme t per-capta demand of agent for the 5-year old vessel s where N 5,t = 1 ρ 21 1 ρ (Π t Π ) + 21Π X 5 σ ε 2 Q P 5,t Y 5 σ ε 2, (8a)

10 10 I. C. MOUTZOURIS AND N. K. NOMIKOS Xۓ 20 5 = [ (1 ρ ) 2 (1 ρ 20 )(1 + 2ρ ρ 20 ) (1 + ρ )(1 ρ ) 3 ] α θ 5 Y 5 = ( 1 ρ (8b) ) α θ ە 1 ρ 5 Furthermore, n order to be consstent wth the nature of the ndustry, we mpose short-sale constrants for each nvestor type. Hence, followng Barbers et al (2015b), equaton 8a becomes 21 1 ρ 1 ρ (Π t Π ) + 21Π X 5 σ 2 ε Q P 5,t N 5,t = max { Y, 0}. (9) 2 5 σ ε Equaton 9, along wth the market clearng condton 6, determne the equlbrum 5-year old vessel prce at each t. Note that n order to derve the agent-specfc demand functons, we have assumed that apart from the extrapolaton of fundamentals, both types of agent suffer from an addtonal form of bounded ratonalty. Namely, agent, nstead of takng nto account the strategy of agent, that s, tryng to forecast the evoluton of s demand, makes the smplfyng assumpton that n all future perods wll just hold hs per-capta fracton of the rsky asset supply constant at μ Q. (Barbers et al, 2015b). Equvalently, assumes that s future demand s ndependent of the correspondng future net earnngs varable. Therefore, we can argue that each agent s optmsaton problem s not a functon of agent s strategy. From a psychologcal perspectve, ths msbelef can be drven by a bas known as competton neglect (Camerer and Lovallo, 1999, Kahneman, 2011). Followng Glaeser (2013) and Greenwood and Hanson (2015), ths bas may be the result of bounded ratonalty n whch agents form forecasts about compettors reactons by ncorporatng a smplfed economc framework nstead of a more elaborate dynamc supply and demand model of the market. In our case, each agent under(over)estmates the future demand responses of the other type. The assumed form of competton neglect mples that each agent type expects the prce of the rsky asset to revert to ts far value (far accordng to ther belefs) wthn one perod. As a result, agent trades more aggressvely aganst any msprcng compared to the case they were explctly forecastng agent s future demand responses. From an economc perspectve, the quantty 1 ρ 21 (Π t Π ) + 21Π X 5 σ 2 ε Q n the numerator of 1 ρ (8a) s the expected ncome for nvestor from holdng the vessel for 1 perod; namely, from the 5 th to the 6 th year of ts economc lfe. Hence, the numerator corresponds to the expected one-perod net ncome for nvestor. Accordngly, the numerator s scaled by the product Y 5 σ 2 ε, whch conssts of

11 THE SECOND-HAND MARKET FOR SHIPS 11 nvestor s rsk averson and the perceved aggregate effect of the condtonal one-perod varances of the cash flow shock (.e. the rsk agent s bearng for the 1-year nvestment accordng to hs percepton). Snce extrapolators have more ncorrect belefs about the net earnngs process compared to conservatves, one would expect that ther expectatons about future returns wll be more naccurate compared to the ones of conservatves. Whle ths belef s n general vald, the assumpton of competton neglect complcates further the ssue. As we llustrate n Secton III, the dscrepancy between agent-specfc returns expectatons and realsed returns depends on both the agent s belefs about the net earnngs process and the relatve populaton fractons. Namely, the fact that each agent neglects the strategy of the other mples that conservatves do not explctly attempt to explot the more ncorrect belefs of extrapolators. In contrast, as mentoned above, conservatves valuaton and n turn ther nvestment strategy are based on the msbelef that the prce of the vessel wll revert to ts far value (far accordng to ther belefs) wthn one perod. If ths were not the case, then ndeed, rrespectve of the relatve populaton fractons, conservatves would always have sgnfcantly more precse expectatons than extrapolators. Secton III llustrates that for a model parametersaton that reproduces suffcently well the emprcal results, conservatves always form substantally more accurate forecasts of future returns compared to extrapolators. As a result, the nvestment strategy of the former s notceably less rsky than the latter s, as measured by the one perod change n wealth. Consequently, t mght be the case that n the long horzon, extrapolators wealth has become severely reduced compared to that of conservatves, and n turn, they are not able to support an nvestment n the rsky asset. Notce, however, that due to the exponental utlty assumpton, the demand functon s ndependent of the respectve wealth level. Ths property of the exponental utlty functon allows us to abstract from the survval on prces effect (Barbers et al, 2015a). Hence, t permts us to focus solely on the prcng and tradng mplcatons of the heterogeneous-agent economy. Nevertheless, from an economc perspectve, even f extrapolators are not able to nvest due to lmted wealth, t s not unrealstc to assume that they wll be mmedately replaced by a new fracton of extrapolator nvestors wth exactly the same characterstcs (Barbers et al, 2015a). In shppng, ths cohort could correspond to dversfed nvestors wth substantal cash avalablty (e.g. prvate equty frms), but lttle or no pror experence of the ndustry. Equatons 6 and 8a suggest that f the market conssted of only one agent type, the equlbrum prce of the 5-year old vessel would be

12 12 I. C. MOUTZOURIS AND N. K. NOMIKOS P 5,t = 1 ρ 21 (Π 1 ρ t Π ) + 21Π [X 5 + Y 5 ]σ 2 ε Q. (10) Incorporatng the uncondtonal volatlty operator on both sdes of equaton 10, we obtan σ(p 5,t ) = 1 ρ 21 σ(π 1 ρ t ). (11) We observe that n ths smplfed case, vessel prce volatlty depends on the volatlty of the net earnngs varable and on the value of ρ. Snce 1 ρ 21 s a strctly ncreasng functon of ρ n the nterval 1 ρ [ρ 0, 1), the hgher the perceved persstence of net earnngs, that s, the hgher the degree of extrapolaton, the hgher the volatlty of generated prces. Furthermore, as we dscuss n the emprcal estmaton of the model, the uncondtonal mean of the net earnngs varable s set equal to ts longterm mean; that s, Ε[Π t ] = Π. Hence, takng uncondtonal expectatons on both sdes of equaton 10 yelds Ε[P 5,t ] = 21Π [X 5 + Y 5 ]σ 2 ε Q. (12) In what follows, t wll be useful from both a theoretcal and emprcal perspectve to examne the benchmark ratonal economy of our model. Recall that n ths scenaro, denoted by f, the market conssts solely of agents who know precsely the actual stochastc process that governs the evoluton of net earnngs. Accordngly, the equlbrum prce and the uncondtonal volatlty, and mean of 5-year old vessel prces are obtaned from equatons 10, 11, and 12, respectvely, for = f. Fnally, as equatons 8a and 8b ndcate, fundamentalsts percepton of the rsk they are bearng s gven by the product ( 1 ρ 0 20 ) 2 σ 2 1 ρ ε. In ths benchmark case, ths percepton s correct. In the presence of 0 extrapolators, though, t s just an approxmaton snce future asset prces wll also depend on extrapolators future demand responses and not just on the rskness of cash flows. In a smlar manner, f the market conssts solely of one extrapolator type, denoted by e; that s, f all market partcpants form expectatons about net earnngs based on (2), the equlbrum prce and the uncondtonal volatlty, and mean of vessel prces are obtaned from (10), (11), and (12), respectvely, after substtutng e for. The most nterestng scenaro, however, s the one where heterogeneous-belef agents coexst n the market. In the context of our model, second-hand actvty s the result of heterogeneous estmaton of the asset s worth. Hence, whle a homogeneous extrapolatve envronment can account for the actual volatlty of vessel prces, t cannot explan the

13 THE SECOND-HAND MARKET FOR SHIPS 13 observed second-hand transactons. 8 As mentoned above, we defne the two agent types as conservatves, c, and extrapolators, e. Notce that a specal case of the heterogeneous envronment s when conservatves form expectatons based on equaton 1, that s, when they are fundamentalsts. Proposton: Equlbrum prce for 5-year old vessels. In the envronment presented above, a marketclearng prce for the 5-year old vessel, P 5,t, always exsts. 9 The equlbrum prce of the vessel depends on the prevalng market condtons. We denote the net earnngs thresholds at whch extrapolators and conservatves related to the 5-year old vessel class ext the market by Π 5 e and Π 5 c, respectvely. Frst, when (X e 5 X c 5 Y 5 c Π e μ 5 = Π )σ ε 2 Q (X e 5 X c 5 + Y 5 e 1 μ + 1 ρ 21 e 1 ρ 1 ρ c 21 < Π t < Π )σ ε 2 Q + 1 ρ 21 e e 1 ρ c 1 ρ 1 ρ c 21 = Π c 5, e 1 ρ c (13a) both agents are present n the market, and the market clearng prce, denoted by P c+e 5,t, s equal to e 1 ρ μy c c 1 ρ 5 + (1 μ)y e P c+e 1 ρ 5 5,t = 21Π + c 1 ρ e μy e c 5 + (1 μ)y 5 (Π t Π ) (13b) μy 5 e X c 5 + (1 μ)y c 5 X e 5 + Y c e 5 Y 5 μy e c σ (1 μ)y ε Q. 5 Second, n the case where (X e 5 X c 5 Y 5 c μ Π t Π )σ ε 2 Q + 1 ρ 21 e 1 ρ c 21 = Π e 5, 1 ρ e 1 ρ c (14a) c extrapolators ext the market, and the clearng prce, P 5,t, s gven by c P 5,t = 21Π + 1 ρ c 21 (Π 1 ρ t Π ) [X c 5 + Y 5 c μ ] σ ε 2 Q. c (14b) Thrd, n the scenaro where 8 We exclude lmts to arbtrage, ndvdual lqudty concerns (Kalouptsd, 2014), and nvestors dversfcaton polces from beng potental causes of second-hand transactons. 9 Ths Proposton s smlar to Proposton 1 n Barbers et al (2015b).

14 14 I. C. MOUTZOURIS AND N. K. NOMIKOS Y 5 e (X e 5 X c 5 + Π c 1 μ 5 = Π )σ ε 2 Q + 1 ρ 21 e 1 ρ 1 ρ c 21 Π t, (15a) e 1 ρ c e conservatves ext the market, and the equlbrum prce, P 5,t, s gven by e P 5,t = 21Π + 1 ρ m 21 (Π 1 ρ t Π ) [X e 5 + Y 5 m 1 μ ] σ ε 2 Q. (15b) e The ntuton behnd the equlbrum prces descrbed through equatons 13b, 14b and 15b s the followng one. As the frst term of each equaton ndcates, the prce of the vessel heavly depends on the long-term mean of the cash flow varable, multpled by the total number of payments to be receved untl the end of the asset s economc lfe. The second term corresponds to the effect of the product of the perceved persstence of the net earnngs varable tmes ts current devaton from the long-term mean. Essentally, ths term s responsble for the man bulk of over(under)valuaton n the prce of the rsky asset. Evdently, as we move from equaton 14b to 13b to 15b, the degree of the perceved aggregate extrapolaton ncreases, whch n turn mples an ncrease n the coeffcent of the devaton term. Furthermore, n analogy to the standard present value formula wth constant requred returns (Shller, 1981), the last term of the equlbrum prce equatons corresponds to the aggregate dscountng (.e. the perceved market rsk n equlbrum), by whch future cash flows are reduced n order for nvestors to be compensated for the rsk they bear (Wang, 1993). In a smlar manner, we derve the demand functons for 6-year old vessels. In addton, we assume that agents become more suspcous or equvalently, more rsk averse, as the specfc asset s age grows. Ths suspcon stems from the fact that they realse that the evoluton of net earnngs does not evolve precsely n the way they expected n the prevous perod. As a result, agents ndrectly respond by ncreasng the perceved rsk assocated wth ths partcular nvestment. In order not to overcomplcate thngs, we model the update n agents belefs n a smple and straghtforward manner. Namely, we assume that agent, at tme t, ncreases the value of the perceved cash flow shock varance correspondng to the valuaton of the 6-year old vessel, θ 6 σ 2 ε, compared to the one ncorporated for the valuaton of the 5-year old asset, at t 1. Hence, for a gven t, nvestors related to dfferent vessel-age classes have dfferent belefs about the varance of the error term. Of course, n the specal case where conservatves are fundamentalsts, the agent knows the precse stochastc process; hence, no varance update occurs between perods t 1 and t. Alternatvely, we could have assumed that agent becomes more rsk averse, whch would mply an ncrease of the CARA from perod t to t + 1. Both methods, however, yeld exactly the same results. Fnally, notce that ths

15 THE SECOND-HAND MARKET FOR SHIPS 15 assumpton s not mportant ether for the theoretcal or the emprcal predctons of our model. We mpose t, however, n order for the steady state equlbrum of our economy to be well-defned from a mathematcal perspectve. Even f we do not mpose ths assumpton, the steady state equlbrum restrctons wll hold approxmately and our results wll be essentally the same. Therefore, accordng to agent, net earnngs related to the valuaton of the 6-year old vessel evolve through Π t+1 = (1 ρ )Π + ρ Π t + ε t+1, n whch ρ 0 ρ c < ρ e < 1, and ε t+1 N(0, θ 6 σ ε 2 ),.. d. over tme, where 0 < θ 6 e < θ 6 c. Despte ther ncreased suspcon, however, agents reman rratonal, snce they stll do not form unbased forecasts of ether the cash flow process or ther compettors demand responses. Fnally, followng precsely the same procedure as for the 5-year old asset, agent s tme t demand for the 6-year old vessel gven by 20 1 ρ (Π 1 ρ t Π ) + 20Π X 6 σ 2 ε Q P 6,t N 6,t = max { Y, 0}, 2 (16a) 6 σ ε where Xۓ 19 6 = [ (1 ρ ) 2 (1 ρ 19 )(1 + 2ρ ρ 19 ) (1 + ρ )(1 ρ ) 3 ] α θ 6 Y 6 = ( 1 ρ (16b) ) α θ ە 1 ρ 6 Snce agents adjust upwardly the perceved rskness of the nvestment, θ 6 s hgher compared to θ 5. For our parameter values, ths n turn mples that Y 6 σ ε 2 > Y 5 σ ε 2. Therefore, as mentoned above, the expected one-perod net ncome related to the 6-year old nvestment s scaled by a hgher quantty, compared to the respectve 5-year old one. Corollary 1: Market clearng prce for 6-year old vessels. Extendng the arguments llustrated n the Proposton, t s straghtforward to show that also n the case of the 6-year old vessel, a market- clearng prce, P 6,t always exsts. Frst, n the case where both agents are present n the market, that s, when (X e 6 X c 6 Y 6 c Π e μ 6 = Π )σ ε 2 Q (X e 6 X c 6 + Y 6 e 1 μ + 1 ρ 20 e 1 ρ 1 ρ c 20 < Π t < Π )σ ε 2 Q + 1 ρ 20 e e 1 ρ c 1 ρ 1 ρ c 20 = Π c 6, e 1 ρ c (17a)

16 16 I. C. MOUTZOURIS AND N. K. NOMIKOS the prce s gven by 20 e 1 ρ μy c c 1 ρ 6 + (1 μ)y e P c+e 1 ρ 6 6,t = 20Π + c 1 ρ e μy e c 6 + (1 μ)y 6 20 (Π t Π ) μy 6 e X c 6 + (1 μ)y c 6 X e 6 + Y c e 6 Y 6 μy e c σ (1 μ)y ε Q. 6 (17b) Second, when only conservatves hold the vessel, that s, when (X e 6 X c 6 Y 6 c μ Π t Π )σ ε 2 Q + 1 ρ 20 e 1 ρ 1 ρ c 20 = Π e 6, e 1 ρ c (18a) the prce s gven by c P 6,t = 20Π + 1 ρ c 20 (Π 1 ρ t Π ) [X c 6 + Y 6 c μ ] σ ε 2 Q. c (18b) Thrd, n the scenaro where only extrapolators hold the rsky asset; namely, when Y 6 e (X e 6 X c 6 + Π c 1 μ 6 = Π )σ ε 2 Q + 1 ρ 20 e 1 ρ c 20 Π t, (19a) 1 ρ e 1 ρ c the prce equals e P 6,t = 20Π + 1 ρ e 20 (Π 1 ρ t Π ) [X e 6 + Y 6 e 1 μ ] σ ε 2 Q. (19b) e Notce that, as mentoned above, due to the exponental utlty assumpton, the demand functons and the equlbrum asset prces, for each age-class, are unaffected by the relatve levels of wealth. Corollary 2: Steady state equlbrum. We defne the steady state of our economy as the one n whch the net earnngs varable s equal to ts long-term mean, Π. As equaton 1 ndcates, the economy reaches ths state after a sequence of zero cash flow shocks. In the steady state the prce of the rsky asset, for each age-class, s equal to ts respectve fundamental value. Furthermore, under a

17 THE SECOND-HAND MARKET FOR SHIPS 17 specfc condton analysed below, both fundamental agents and extrapolators are present n the market, and each type holds the rsky asset n analogy to hs fracton of the total populaton. Accordngly, the steady state equlbrum prce of the 5-year old vessel, P, 5 s gven by P 5 = 21Π [X 5 + Y 5 ]σ 2 ε Q, (20a) under the restrcton X 5 c + Y 5 c = X 5 e + Y 5 e = X 5 f + Y 5 f. (20b) In a smlar manner, the steady state equlbrum prce of the 6-year old vessel s P 6 = 20Π [X 6 + Y 6 ]σ 2 ε Q, (21a) under the restrcton X 6 c + Y 6 c = X 6 e + Y 6 e = X 6 f + Y 6 f. 10 (21b) Therefore, f n two consecutves dates the net earnngs varable s equal to ts long-term mean, the change n the prce of the asset s gven by P 6 P 5 = Π [X 6 + Y 6 (X 5 + Y 5 )]σ 2 ε Q. (22) Notce that the rght hand sde of (22) s negatve and corresponds to the one-year economc deprecaton n the value of the vessel. Fnally, n ths scenaro, there s no actvty n the second-hand market, snce the change n share demand of each agent s equal to zero. The equlbrum condtons above mply that our model parameters are nested. Ths nterrelatonshp can be llustrated n a smple manner through the followng system of equatons α = 21Π P 5 [ 20 + (1 ρ 20 ) 2 (1 ρ ) 2 (1 ρ 20 )(1 + 2ρ ρ 20, ) (23a) (1 + ρ )(1 ρ ) 3 ] θ 5 σ 2 ε Q and 10 The consequence of the update mechansm analysed above s that restrcton 21b holds wth exact equalty. Otherwse, the restrcton holds as an approxmate equalty.

18 18 I. C. MOUTZOURIS AND N. K. NOMIKOS α = 20Π P 6 [ 19 + (1 ρ 19 ) 2 (1 ρ ) 2 (1 ρ 19 )(1 + 2ρ ρ 19 ) (1 + ρ )(1 ρ ) 3 ] θ 6 σ 2 ε Q. (23b) The mplcatons of ths fact are analysed n the emprcal estmaton of the model. Corollary 3: Devaton from the fundamental value. Whenever the value of the net earnngs varable devates from ts long-term mean, the model generated prce of the 5-year old vessel devates from the fundamental value as well. In the followng, we denote the degree of devaton by D t ; namely, a postve (negatve) value of D t corresponds to over (under) valuaton of the asset. In order to derve the equatons that quantfy the degree of devaton from the fundamental value, we have to dstngush between three cases. Frst, n the case where both agents are present n the market, we can estmate the devaton, denoted by D t c+e, by subtractng the fundamental prce from the one ndcated by equaton 13b. Accordngly, we obtan D t c+e = (1 μ) ( 1 ρ e 21 1 ρ e 1 ρ c 21 1 ρ c ) Y c μy 5 e + (1 μ)y 5 c (Π t Π ). (24) Snce the fracton s always postve, the sgn of the prce devaton solely depends on the sgn of the net earnngs devaton. Therefore, durng prosperous (deteroratng) market condtons, the rsky asset s over (under) prced. Second, when only conservatves exst n the market, the devaton, D t c, s estmated by subtractng P t f from equaton 14b. Ths yelds D c t = ( 1 ρ c 21 1 ρ 21 f ) (Π 1 ρ c 1 ρ t Π ) [X c 5 + Y c 5 f μ X f 5 Y f 5 ] σ 2 ε Q, (25) whch s always negatve, snce Π t Π e 5 < Π and by condton 20b the second term s always negatve. Therefore, when only conservatve agents are present n the market, the vessel s undervalued. Notce that snce extrapolators consttute a fracton of the nvestor populaton, even f conservatve agents hold ratonal belefs, that s ρ c = ρ f, the vessel s stll undervalued. Namely, the rght hand sde of 25 s always negatve, as mpled by the fact that μ < 1 along wth the equlbrum restrcton 20b. Thrd, n the scenaro n whch only extrapolators are present, the dscrepancy, D e t, s calculated by subtractng the fundamental prce from equaton 15b. Ths yelds

19 THE SECOND-HAND MARKET FOR SHIPS 19 D e t = ( 1 ρ e 21 1 ρ 21 f ) (Π 1 ρ e 1 ρ t Π ) μy e 5 f 1 μ σ ε 2 Q. (26) The frst term of the rght hand sde of 26 s always postve n ths scenaro, snce Π t Π c 5 > Π. Therefore, devaton of vessel prces s a strctly ncreasng functon of net earnngs dscrepancy n ths nterval. Furthermore, whle the second term s negatve, t s straghtforward to llustrate that n the correspondng nterval (.e. Π t Π c 5 ), ts absolute value s always smaller than the frst term. e Therefore, D t s always postve, and when market condtons sgnfcantly mprove, the degree of overvaluaton becomes severe. Corollary 4: Senstvty of ext ponts to the fracton of conservatves. As condtons 14a and 15a suggest, the agent-specfc ext ponts dffer due to the quanttes Y 5 c and Y e 5. The mplcaton of ths μ 1 μ fact s that whenever Y c 5 μ Y e 5 (1 μ), there s no symmetry around Π between the two ponts. As a result, the postve and negatve shock cases are not mrror mages of each other. Takng the frst partal dervatve of the extrapolators 5-year ext pont wth respect to the fracton of conservatves yelds μ = 1 μ 2 Y c 5 σ 2 ε Q 1 ρ 21 e 1 ρ c 21, (27) 1 ρ e 1 ρ c Π 5 e whch for ρ c < ρ e s strctly postve. As a result, extrapolators ext pont ncreases wth the relatve proporton of conservatve nvestors n the market. Equvalently, the hgher the fracton of conservatves, the more prone extrapolators are to ext from the market durng deteroratng condtons. Smlarly, the frst partal dervatve of conservatves ext pont wth respect to ther relatve fracton s equal to μ = 1 (1 μ) 2 Y e 5 σ 2 ε Q 1 ρ 21 e 1 ρ c 21, (28) 1 ρ e 1 ρ c Π 5 c whch for ρ c < ρ e s strctly postve. Consequently, conservatves ext pont ncreases wth ther relatve proporton n the market. Hence, the hgher ther fracton, the less prone they are to ext the market durng prosperous condtons. The same prncples apply for the 6-year old vessel valuaton. Hence, the asymmetry ncreases as μ devates from the mdpont 0.5.

20 20 I. C. MOUTZOURIS AND N. K. NOMIKOS Furthermore, as we demonstrate graphcally n the next secton, each agent s ext pont s a strctly decreasng functon of ther own persstence, and a strctly ncreasng functon of agent s perceved persstence. Corollary 5: Tradng volume and net earnngs. To begn wth, ncorporatng equatons 9 and 16a n (7) results n 20 1 ρ V t = μ 1 ρ (Π t Π ) + 20Π X 6 σ 2 ε Q P 6,t max { Y, 0} 2 6 σ ε 21 1 ρ 1 ρ (Π t 1 Π ) + 21Π X 5 σ 2 ε Q P 5,t 1 max { Y, 0}. 2 5 σ ε (29) Due to the short-sale constrants, the agent-specfc demand functons are not strctly monotonc wth respect to the net earnngs varable n the entre Π t doman; namely, strct monotoncty dsappears whenever the constrants are bndng. Accordngly, n order to examne the tradng actvty varable, we have to dstngush between several cases. In the frst scenaro, both agents are present n the market for two consecutve dates. Equvalently, conservatve agents demands for 5 and 6-year old vessels are postve. Incorporatng the equlbrum prces from (13b) and (17b) n equaton 29, and applyng straghtforward algebra, we obtan V t = μ A 6 Π t A 5 Π t 1 + (A 6 A 5 )Π, (30a) where A 6 Π t A 5 Π t 1 + (A 6 A 5 )Π = N 6,t N 5,t 1, (30b) s agent s change n demand for the asset between perods t 1 and t. Te agent-specfc constants are gven by and ۓ (1 μ) ( 1 ρ c 21 1 ρ 21 e ) A c 1 ρ 5 = c 1 ρ e [μy e 5 + (1 μ)y c 2 < 0 5 ]σ ε (1 μ) ( 1 ρ c 20 1 ρ 20, (30c) e ) A c 1 ρ 6 = c 1 ρ e ە [μy e 6 + (1 μ)y c 6 ]σ2 < 0 ε

21 THE SECOND-HAND MARKET FOR SHIPS 21 ۓ μ ( 1 ρ e 21 A e 1 ρ 1 ρ c 21 5 = e 1 ρ ) c [μy e 5 + (1 μ)y c 2 > 0 5 ]σ ε μ ( 1 ρ e 20 A e 1 ρ 1 ρ 20, (30d) c 6 = e 1 ρ ) c ە [μy e 6 + (1 μ)y c 6 ]σ2 > 0 ε Snce tradng volume n the market s the same rrespectve of the agent type, n the followng we examne ths varable from the conservatve agent s perspectve. Accordngly, equaton 30d becomes V t = μ A 6 c Π t A 5 c Π t 1 + (A 6 c A 5 c )Π. (30e) The second scenaro s when both agents are present at tme t 1, but conservatves ext at t. In ths case, tradng actvty s V t = μ A 5 c (Π t 1 Π ) + Q. (31) In the thrd scenaro, conservatves are not present n the market at tme t 1, but both agent types are actve at t. Accordngly, equaton 29 becomes V t = μ A 6 c (Π t Π ) + Q. (32) The fourth scenaro refers to the case where both agents are present n the market at tme t 1, but extrapolators ext at t. In ths case, tradng actvty s gven by V t = μ A 5 c (Π t 1 Π ) (1 μ) Q. (33) μ The ffth scenaro s when only conservatves are present n the market at tme t 1, but both types at t. Therefore, equaton 29 becomes V t = μ A 6 c (Π t Π ) (1 μ) Q. (34) μ In the sxth (seventh) scenaro, only agents of type are present n the market at tme t 1, and only of type at t. Namely, (39) smplfes to V t = Q. (35)

22 22 I. C. MOUTZOURIS AND N. K. NOMIKOS Furthermore, f n two consecutve dates agents are out of the market, there s no tradng actvty. Fnally, f μ = 0, or equvalently, ρ c = ρ e, the market clearng condton along wth equatons 9 and 16a suggest that there are no second-hand transactons n the economy. As t becomes evdent from Corollary 5, short-sale constrants have a major mplcaton for the relatonshp between tradng actvty and net earnngs. In order to smplfy ths pont, let s defne tradng actvty as n the equty markets lterature; namely, we set N 6,t = N 5,t Equvalently, we substtute A 5 c for A 6 c n equatons 30e, 32, and 34, above. Therefore, n the absence of short-sale constrants, tradng actvty would always be equal to μ A 5 c Π t Π t 1. As a result, corr( Π t Π t 1, V t ) = corr( Π t Π t 1, μ A 5 c Π t Π t 1 ) = 1. In other words, f there are no constrants, absolute net earnngs changes are perfectly correlated wth tradng actvty. Due to the exstence of short-sale constrants, however, the two varables are sgnfcantly less correlated. Furthermore, Corollary 4 demonstrates that both ext ponts ncrease (decrease) wth the fracton of conservatves (extrapolators) and the perceved persstence on behalf of extrapolators (conservatves). Accordngly, the hgher the values of the ext ponts, the more agents coexst durng prosperous market condtons and the less they nteract durng a downfall. Taken together, these two observatons suggest that a hgh value of μ, along wth a sgnfcant spread between ρ c, and ρ e wll smultaneously result n postve correlaton between current net earnngs and tradng actvty and less than perfect correlaton between absolute net earnngs changes and tradng actvty. These theoretcal predctons are confrmed n the emprcal estmaton of the model. III. Emprcal Estmaton of the Model n the Dry Bulk Shppng Industry In ths secton, we dscuss the dataset employed and the constructon of the varables of nterest. Accordngly, we evaluate emprcally the theoretcal predctons of our model by performng a large number of smulatons. In order to provde a deeper ntuton of the results, we mplement mpulse response and senstvty analyses. Fnally, we dscuss our fndngs from an economc and ndustral perspectve. III.A. Data on Net Earnngs, Prces, and Tradng Actvty The dataset employed conssts of annual observatons on second-hand vessel prces, 1-year tme-