Commodities as Collateral

Transcription

1 Commodtes as Collateral Ke Tang Haoxang Zhu November 16, 2013 Prelmnary, Comments Welcome Abstract Ths paper proposes a theory of usng commodty as collateral for fnancng, a practce that becomes ncreasngly mportant n recent years. In the presence of fundng frctons, fnancal nvestors mport commodtes and pledge them as collateral to capture the credt rsk premum, measured by the nterest rate dfferental between unsecured and secured loans. The demand for collateral commodtes ncreases the concurrent commodty spot prces, total nventory, convenence yelds, and the spread of commodtes futures rsk prema between the mportng and exportng countres. These effects are stronger f the nterest rate spread between mportng and exportng countres s larger. The predctons of our model are dstnct from those of the theory of storage. We test our model predctons n the copper market and fnd that the evdence supports our theory. Keywords: commodty, collateral, fnancalzaton, theory of storage JEL Codes: G12, Q02 Hanqng Advanced Insttute of Economcs and Fnance and School of Fnance, Renmn Unversty of Chna; ketang@ruc.edu.cn. MIT Sloan School of Management; zhuh@mt.edu.

2 Commodtes as Collateral November 16, 2013 Abstract Ths paper proposes a theory of usng commodty as collateral for fnancng, a practce that becomes ncreasngly mportant n recent years. In the presence of fundng frctons, fnancal nvestors mport commodtes and pledge them as collateral to capture the credt rsk premum, measured by the nterest rate dfferental between unsecured and secured loans. The demand for collateral commodtes ncreases the concurrent commodty spot prces, total nventory, convenence yelds, and the spread of commodtes futures rsk prema between the mportng and exportng countres. These effects are stronger f the nterest rate spread between mportng and exportng countres s larger. The predctons of our model are dstnct from those of the theory of storage. We test our model predctons n the copper market and fnd that the evdence supports our theory.

3 1 Introducton In ths paper we propose a theory and report evdence of commodtes as collateral for fnancng. Ths research s motvated by recent anecdotes, substantated later by our theoretcal and emprcal analyss, that producton commodtes such as copper are ncreasngly used as collateral to obtan fnancng, most notably n Chna. Because of the sze of Chna s economy and ts role as the leadng mporter of commodtes and raw materals, the demands for commodty as collateral have mportant mplcatons for the underlyng commodtes markets. The collateral use of commodtes also serves as a concrete llustraton of how fnancal frctons spll over to the real economy. A typcal trade that mports commodty as collateral works as follows (see Secton 2 for detals). A domestc Chnese frm n the mport-export busness apples for US dollar credt from a bank to buy commodtes n the nternatonal market. These commodtes are shpped to Chna and pledged as collateral to rase CNY (the Chnese currency) fundng on a secured bass, wth a harcut. The CNY cash s subsequently nvested, on an unsecured bass, n assets, loans or projects that generate a hgher expected return. At the maturty of the US dollar loan, all legs of the trades are unwound, and the commodty s sold n the spot market. As long as the return on the unsecured loan n CNY s suffcently hgh, ths trade s proftable. In other words, commodty s used as the means to capture the rsk premum n Chna s credt market. 1 We emphasze that the above sequence of trades rely on a couple of key frctons n Chna. Frst, captal flows are restrcted. The credt spreads between CNY loans n Chna and USD loans n developed countres cannot be closed out by pure fnancal transactons that nvolve no real mports. Second, large Chnese lenders such as state-owned banks may be unwllng or dscouraged to lend to small but hgh-return frms on an unsecured bass. We present a smple two-perod model that formalzes the causes and effects of fnancng usng commodty as collateral. In our model, a representatve commodty mportng country (e.g. Chna) buys commodtes from a representatve exportng country. Credt rsk prema, modeled as the spread between secured and unsecured loans, dffer between two countres. But ths gap cannot be elmnated by fnancal nvestors because of heavy restrctons of captal flows n and out of the mportng country. That s, the two fnancal markets are effectvely segmented, an extreme form of captal mmoblty (see Duffe (2010) and Duffe and Strulovc (2012)). Although fnancal captal flows are lmted, 1 In May 2013 the Chnese government announced a new regulaton on mportng frms, whch makes commodty fnancng more standardzed. Yuan, Layton and Curre (2013), Burton and Wong (2012), and Fayen Wong (2013) provde addtonal descrptons on copper fnancng. Although publc data on the true scale of copper fnancng s unavalable, The Economc Observer (2012) estmates that 90% of copper stored n the Wagaoqao tarff-free zone n Shangha are for copper fnancng, wth the total amount more than 500,000 tonnes. 1

4 trades of producton assets such as commodtes are not. To take advantage of the gap n rsk prema, fnancal nvestors n the mportng country engage the followng sequence of trades at perod 0. They borrow at the unsecured nterest rate abroad and mport commodtes; these commodtes are then pledged n the domestc market to get secured, low-nterest loans, whch are subsequently nvested n hgh-nterest assets. At the maturty of the loans, perod 1, all borrowng and lendng are unwound, and the collateral commodty s sold to fundamental consumers. The hgher s the unsecured nterest rate n the mportng country, the more proftable t s to mport commodty as fnancng collateral. The fnancal nvestor can further use the futures market n the mportng country to lmt the rsk exposure n the spot market. Our model predcts that the demand for commodty as collateral leads to hgher concurrent commodty prces. The larger s the nterest rate spread between Chna and US (as commodtes are traded n US dollars), the hgher s the demand for collateral commodty, and the hgher are the commodty spot prces n both the exportng country and the mportng country. A hgher spot prce crowds out some real demand for commodtes, but the overall effect s a hgher nventory level n Chna and globally. Snce owners of nventores need to hedge these postons, our model also predcts that the futures term structure n Chna exhbts more backwardaton,.e., a hgher convenence yeld. In other words, the convenence yeld n commodtes futures curve contans a component that s ncreasng n the fundng advantage of usng commodty as collateral. The rsk prema for nvestng n commodtes futures then behave more lke the rsk prema n credt markets; they ncreases n Chna but decrease n the exportng country. The predctons from our model are dstnct from the classcal theory of storage (see Workng 1939, Telser 1949, Brennan 1958, and Gorton, Hayash and Rouwenhorst 2012, among others). For example, whle the theory of storage predcts a negatve relaton between convenence yeld and nventory, our model predcts that the collateral demands for commodty cause a hgher total nventory and a hgher convenence yeld smultaneously, a postve relaton. Moreover, collateral demands result n a hgh total nventory and a hgh commodty prce smultaneously. Ths s agan opposte to the predcton from the theory of storage that a hgher nventory ndcates the abundance of commodty and hence a lower prce. Commodty fnancng also has mportant mplcatons for the relaton between macro-economc factors and commodty prces. Frankel (1986, 2008) fnd evdence that a hgher nterest rate (.e., a tghter monetary polcy) n the US and other countres tends to reduce commodty prces by lowerng the real demand and ncreasng the nventory costs. By contrast, our model shows that hgher nterest rates n commodty-mportng countres can ncrease commodty prces by makng them more attractve as collateral for fnancng. We test the model predctons n copper markets, usng weekly data from October

5 to October 2013 n the Shangha Futures Exchange (SHFE) and London Metal Exchange (LME). The evdence supports our theory. We focus on copper because t s an mportant ndustral commodty and has a hgh value-to-bulk rato, makng t a desrable collateral commodty. The man ndependent varable we use s the dfference between Shangha Interbank Offerng Rate (Shbor) and London Interbank Offerng Rate (Lbor), whch s a proxy for the attractveness of mportng copper for collateral purposes. In all tests we control for the effects predcted by the theory of storage and focus on the ncremental effects of collateral commodty. Frst, we fnd that a hgher Shbor-Lbor spread makes the convenence yeld-nventory relaton sgnfcantly less negatve (or more postve) n the SHFE. Second, a hgher Shbor- Lbor spread s assocated wth a sgnfcantly hgher nventory n the SHFE. In these two tests, the results usng LME data are not statstcally sgnfcant, suggestng that LME nventores are unlkely to be pledged as collateral for fnancng. Thrd, a hgher Shbor- Lbor spread predcts a sgnfcantly lower futures rsk premum n the LME, whereas the correspondng tests usng SHFE data have the predcted sgn but are not statstcally sgnfcant. Overall, these results support our theory and complement the theory of storage. Ths paper relates to an emergng strand of lterature on the fnancalzaton of commodtes, e.g. Masters (2008), Tang and Xong (2012), Cheng, Krlenko and Xong (2012), Basak and Pavlova (2013) and Henderson, Pearson and Wang (2013), among others. In these papers, fnancally motvated tradng on commodty futures markets, other dervatves and structured products lnked to commodtes tends to move underlyng commodty prces. Whle these exstng papers focus predomnantly on commodty dervatves, an essental element of our theory and evdence s the movement of the physcal commodtes. The nteracton between fnancng and the physcal market s a unque feature of our theory. Moreover, we provde a mcrofoundaton for how commodty prces can be nfluenced by nterest rates, monetary polcy and relatve fundng constrants between mportng and exportng countres of commodtes n a drecton dstnct from the theory of storage and exstng lterature on the fnancalzaton of commodtes. Our results also llustrate concretely how frctons n the fnancal markets can spll over to the prces of producton assets and hence the real economy through the collateral channel. The rest of ths paper s organzed as follows. Secton 2 dscusses the concrete procedures of usng commodty as collateral to obtan fnancng. Secton 3 presents the model and Secton 4 solves for the equlbrum. Secton 5 analyzes the effects of usng commodtes as collateral. Secton 6 tests the model predctons n copper markets. Secton 7 concludes. 3

6 2 Commodtes as Collateral n Practce In ths secton we dscuss the nsttutonal procedures of obtanng fnancng usng commodty as collateral. We focus on copper n our dscusson because t s one of the man commodtes used n ths process. The mportng country s Chna, and the exportng country s UK. We take UK n ths dscusson because the UK-based London Metals Exchange s a major tradng center for copper and keeps large copper nventores n ts warehouses. A typcal copper fnancng deal conssts of a few steps. Frst, a Chnese mportng frm sgns contract to buy copper from the UK and obtans credt. In many cases, the traded copper s stored n bonded warehouse n London, Shangha, or other locatons. The mportng frm then uses the purchase contract to apply for a letter of credt from a domestc or foregn bank. Typcally granted n dollars at the US dollar nterest rate, the letter of credt guarantees that the seller wll be pad by the bank. In order to secure the credt, the mportng frm needs to pay a margn, whch s about 20% to 30% of the notonal amount. The maturty of the letter of credt vares and are often between 3 to 6 months. For example, f the letter of credt s granted for 6 months, the mportng frm needs to pay the credt back after 6 months n USD and pay the USD nterest rates. Second, the mporter sells futures n Shangha futures exchange to hedge the prce rsk of holdng copper. The mporter then shps copper to bonded warehouses n Shangha and obtan the warehouse recept. Note that at ths stage the copper does not have to enter Chna customs, and the mporter does not have to pay the assocated taxes yet. The warehouse recept s subsequently provded to a domestc bank as collateral to obtan a CNY loan. (If the bank accepts warehouse recepts elsewhere n the world, then mportng s not necessary.) A typcal loan harcut s 30%, that s, the amount of the CNY loan s 70% of the market value of the copper. Even after adjustng for the harcut, the nterest on the collateralzed CNY loan s sgnfcantly lower than the expected return n other asset markets n Chna, such as short-term lendng to small busnesses. Effectvely, the mporter uses the copper collateral to capture the spread between the collateralzed and uncollateralzed CNY fundng rates n Chna. Thrd, after sx months, the copper mporter receves the hgh uncollateralzed return from ts CNY nvestments and then sells the copper n bonded warehouse n Shangha. The mporter also clears ts futures poston. The proceeds of copper sale and nvestment returns n ts CNY nvestment are used to pay for the domestc bank loan n CNY (wth relatvely low CNY nterest rates) and the foregn bank for the letter of credt (wth relatvely low USD nterest rate). Ths completes a typcal Chnese copper fnancng deal. There are some varatons of the above procedure. For nstance, at the maturty of the CNY loan, the mportng frm may re-sell the copper n Shangha bonded warehouse to 4

7 a UK frm, agan outsde Chnese customs, and subsequently repeat the copper fnancng procedure. Ths way, subsequent mportng of copper does not nvolve the physcal shpment of the metal because the nventores stay n Shangha. Thus, each tone of mported copper can be used to obtan fnancng multple tmes. An alternatve arrangement, commonly referred to as cash for copper, nvolves the mmedate sale of the mported copper to the domestc Chnese markets. The proceeds of the sale n CNY s then nvested to obtan hgher returns than the USD nterest rates. A man dfference of ths procedure s that the copper has to enter customs and ncur the assocated taxes, and repeatng ths fnancng arrangement nvolves mportng addtonal copper, nstead of recyclng exstng copper n Shangha warehouses. For addtonal dscusson of the nsttutonal arrangements of commodty fnancng, see Yuan, Layton and Curre (2013). 3 A Model of Commodty as Collateral In ths secton we present a model of commodty as collateral. There are two perods, t {0, 1}. There s an exportng country of commodty and an mportng country of commodty. The exportng country has a commodty suppler and a speculator. The mportng country has a commodty suppler, a fundamental user of commodty for producton, and a fnancal nvestor who mports commodty for collateral. For smplcty, we assume that the commodty s prced n dollars n both countres, so currency returns are not part of our model. (Alternatvely, the mportng frm completely hedges ts FX exposure.) We use the superscrpt e to denote quanttes and prces n the exportng country, and use the superscrpt to denote quanttes and prces n the mportng country. Table 1 lsts the exogenous and endogenous varables we use n ths model. 3.1 Suppler n the Exportng Country We drectly model the net supply n the exportng country. Our model n the exportng country s largely adopted from Acharya, Lochstoer and Ramadora (2012). Let It e and G e t be the aggregate commodty nventory and producton, respectvely. Let δ (0, 1) be the cost of storage; that s, the producer can store I unts of the commodty at t 1 and receve () I unts at t. We also assume that the producton schedule (G e 0, Ge 1 ) are fxed ex ante and common knowledge. The nventory I0 e, however, s a choce varable of the producer. Gven the choce of nventory I0 e, the commodty sales n perod 0 and 5

8 Table 1: Key model varables. Varables n the top block are exogenous; varables n the bottom block are endogenous. Varable Explanaton r, R The secured and unsecured nterest rate n country, respectvely δ Storage cost of commodty h Shppng cost of commodty G e t Commodty producton of the exportng country at tme t k t, l The fundamental demander s margnal proft of usng Dt unt of commodty s k t St ldt, where k t N(µ k, σk ) a, b Commodty supply n the mportng country s a + bst γp, e γs e Rsk averson coeffcents of commodty producer and fnancal speculator n exportng country γd, γ c Rsk averson coeffcents of fundamental commodty demander and fnancal player n mportng country St Spot commodty prce n perod t n country F Futures prce n country, traded at t = 0 and delvered at t = 1 It e Commodty nventory n the exportng country at tme t Dt,f, D t,d Fundamental demand at tme t of foregn and domestc commodty, respectvely C0 Collateral commodty demand at tme 0, all mported λ Lagrange multpler assocated wth constrant I0 e 0 η Lagrange multpler assocated wth constrant D0,f 0 h e p, h e s Postons of futures contracts of commodty producer and fnancal speculator n exportng country at tme 0 h d, h c Postons of futures contracts of fundamental commodty demander and fnancal player n mportng country at tme 0 perod 1 are, respectvely, Q e 0 = G e 0 I0, e (1) Q e 1 = G e 1 + ()I0. e (2) In addton to sellng the commodty n the spot market, the commodty suppler shorts h e p futures contracts at the prce of F e to hedge ts nventory and producton. Therefore, the termnal wealth of the producer s Wp e = S0(G e e 0 I0)(1 e + r e ) + S1(G e e 1 + ()I0) e h e p(s1 e F e ). (3) We emphasze that the commodty spot prce n perod 1, S1 e, s a random varable. As we dscuss shortly, S1 e s determned by the stochastc demand of the mportng country 6

9 n perod 1. We denote by σ e S the volatlty (standard devaton) of Se 1. The commodty producer has a mean-varance utlty of the form E[W e p ] γe p 2 V ar[w e p ]. (4) Substtutng n the expresson of W e p, we see that the producer solves the optmalty problem max {I0 e,he p} Se 0 (G e 0 I0) e (1 + r e ) + E [ S1((1 e δ) I0 e + G e 1) h e p (S1 e F e ) ] γe p 2 V ar [ S e 1(() I e 0 + G e 1) h e p (S e 1 F e ) ] (5) subject to :I e 0 0. We denote by λ 0 the Lagrange multpler assocated wth the nventory constrant I e 0 0. Takng the frst-order condton wth respect to the nventory Ie 0 poston h e p, we get I0 e = E [Se 1 ] () Se 0 (1 + re ) + λ ( ) γp e σ e 2 S () 2 and futures + he p G e 1 (), (6) h e p = I0 e () + G e 1 E [Se 1 F e ] ( ) γp e σ e 2, (7) S If I0 e > 0, λ = 0. If Ie 0 = 0, λ > 0. The endogenous λ affects the convenence yeld of holdng the commodty. 3.2 Speculator n the Exportng Country The speculators only trade futures n the exportng country, and ther futures poston s denoted by h e s. They have mean-varance utlty and solve the followng optmzaton problem The soluton s max E [h e h e s (S1 e F e )] γe s s 2 V ar [he s (S1 e F e )]. (8) h e s = E [Se 1 F e ] ( ) γs e σ e 2. (9) S 3.3 Market Clearng n the Exportng Country From (6) and (7) we obtan S e 0 F e S e 0 = λ S0 e () re + δ. (10) 7

10 Thus, the futures prce n the exportng country s By the futures market clearance, h e p = h e s, we have F e = Se 0 (1 + re ) λ, (11) E [S e 1 F e ] = γe sγ e p γs e + γp e (σs) e 2 [I0 e () + G e 1]. (12) Snce F e s solved, the above equaton has two unknowns, E[S e 1 ] and Ie 0. These two varables cannot be determned by varables n the exportng country alone; rather, we need the demand from the mportng country, whch we turn to now. 3.4 Suppler n the Importng Country Snce the commodty supply n the mportng country s not our man focus, we smply assume that the commodty supply n the mportng country s gven by Q t = a + bst, where a < 0 and b > 0 are commonly known constants. 3.5 Fundamental Demander n the Importng Country We model the fundamental demander n the mportng country as a consumer who uses commodty as nput to produce fnal goods. At tme t, the fundamental demander has a lnearly decreasng margnal proft per unt of commodty nput, k t St ldt, (13) where k t s a stochastc varable, l s a constant, and D t s the amount of commodty nput used at tme t. At tme 0, k 0 s commonly known, but k 1 s unobservable and s normally dstrbuted N ( µ k, σ k). Ths stochastc k1 can be nterpreted as the fundamental shock to the economy of the mportng country, only realzed at tme 1. All players n our model have symmetrc nformaton and the same probablty dstrbuton about k 1. The fundamental demander has three endogenous choces at tme 0: the amount of commodtes to mport, D 0,f, the amount of commodtes to buy n domestc market, D 0,d, and the amount of futures contracts to trade n the local market, h d. The shpment of one unt of commodty across the two countres ncurs the cost h > 0. For smplcty, we assume that shpment s nstantaneous, that s, commodty purchased n the exportng country at tme t can be used n the mportng country at tme t as well. The fundamental demander s termnal wealth n perod 1 that s derved from hs 8

11 producton and tradng actvty n perod 0 s Wd,0 = [ D 0,f k0 (S0 e + h) l ( D0,f + ( 0,d)] D 1 + r ) + D0,d [ k0 S0 l ( D0,f + )] ( D 0,d 1 + r ) + h ( d S 1 F ), (14) where the frst and second terms are the demander s total profts of usng foregn and domestc commodty supples, respectvely. solves 2 The fundamental demander has the mean-varance preference wth parameter γ d and max {D0,d,D 0,f,h d} E[W d,0 ] γ d 2 V ar[w d,0 ], (15) Subject to :D 0,f 0. (16) The soluton to the above problem s D 0,f = k 0 (S e 0 + h) D 0,d + η (17) D0,d = k 0 S0 D0,f (18) h d = E [ S1 F ] ( ) σ 2 (19) S γ s where η s the Lagrange multpler assocated wth the constrant (16). If D0,f = 0,.e. the fundamental demander only buys commodty locally, then η > 0. If D0,f > 0, then η = 0. Smlarly, we can solve the fundamental demander s problem at tme 1. We denote by D1,f and D 1,d the demand of foregn and domestc commodty, respectvely. The termnal wealth of the fundamental demander s W d,1 = D 1,f [ k1 (S1 e + h) l(d1,f + D 1,d )] + D1,d [ k1 S1 l(d1,f + D 1,d )]. (20) Snce the fundamental shock k 1 s realzed and becomes common knowledge at tme 1, the fundamental demander solves max {D1,d,D 1,f} W d,1. (21) 2 Because k 1 s an exogenous varable, the mean and varance of the fundamental demander s perod-1 wealth that comes from hs perod-1 producton actvty are not affected by hs perod-0 strategy, (D 0,f, D 0,d, h d ). Thus, we can solve hs optmal strateges perod by perod. 9

12 The soluton s D1,d = k 1 S1 D1,f, (22) D1,f = k 1 (S1 e + h) D1,d. (23) 3.6 Fnancal Demander n the Importng Country The fnancal demander n the mportng country mports commodty not for producton, but to use t as collateral to get secured fnancng at rate r and lend unsecured at rate R > r. (Wthout loss of generalty, the nterest rates R and r are after adjustng for the harcut mposed on the loan.) In other words, the commodty s mported as a means to capture the unsecured-secured spread, or rsk premum, of R r. We assume that the fnancal demander must frst borrow unsecured n the exportng country at the rate R e to pay for the costs of commodty and shppng. Snce borrowng and lendng take one perod, ths trade must be completed at tme 0. The expected tme-1 proft of mportng one unt of collateral commodty at tme 0 s S 0(R r ) + () E[S 1] (S e 0 + h) (1 + R e ). (24) The three terms capture, respectvely, the expected proft of borrowng S 0 at rate r and lendng at rate R, the proceeds from sellng the remanng () commodty at tme 1, and the payment of the unsecured loan at rate R e. We later specfy explctly condtons under whch the expected proft of mportng commodty as collateral s postve. denote by C0 the amount of commodty mported for collateral purposes at tme 0. We also observe that these collateral commodtes must be mported for ths trade to be proftable. If the fnancal demander were to use domestc commodty, he must frst pay the unsecured rate R, defeatng the purpose of lendng at R. The fnancal demander also uses futures contract to hedge hs poston. We denote by h c hs futures poston at tme 0. The fnancal demander s termnal wealth at tme 1 s Wf = [ C 0 S 0 (R r ) + () S1 (S0 e + h) (1 + R e ) ] h c(s1 F ). (25) We Suppose that the fnancal demander has a mean-varance utlty functon wth parameter γ c. At tme 0, he solves the optmzaton problem max {C0,h c} E[W f ] γ c 2 V ar[w f ]. (26) where the varance comes from the uncertanty about S 1. We denote the volatlty of S 0 10

13 by σ S. Solvng for the optmal C 0 and h c, we get C0 = S 0 (R r ) + () E [ S1 ] (S e 0 + h) (1 + R e ) ( ) γc σ 2 + h c S (), (27) h c = E [ S1 F ] ( ) σ 2 + C0 (). (28) S γ c 3.7 Market Clearng n the Importng Country From (17) and (18), we get S 0 = S e 0 + h η. (29) Recall that η s the Lagrange multpler assocated wth D 0,f 0; η > 0 whenever D 0,f = 0. Thus, f commodty mports are not for fundamental producton but for collateral, the commodty prce n the mportng country s lower than that n the exportng country after adjustng for shppng costs. From (22) and (23) we get S 1 = S e 1 + h. By the market-clearng condton of the futures market, h d = h c, we have ( γ C0 = d + γc ) [ E S 1 F ] γd γ c () ( ) σs 2. (30) For parameters consdered n ths paper, C0 0. Cases n whch C 0 = 0 are dentcal to the benchmark case wthout the collateral use of commodtes. From (27) and (28), we can solve the futures prce n the mportng country, F = (Se 0 + h) (1 + Re ) = 1 + r (R R e ) ( S 0 R r ) S0 + (1 + Re ) η. (31) 4 Equlbrum In ths secton we characterze the equlbrum prces and quanttes. We frst consder the equlbrum n whch demanders of collateral commodtes partcpate n the market. Then, we consder the equlbrum wthout collateral demand of commodtes. 4.1 Equlbrum wth Demand of Collateral Commodty Puttng the market-clearng condtons from the prevous secton, we have the followng 11

14 proposton. Proposton 1. Suppose that collateral demanders of commodtes partcpate n the market. In equlbrum, the spot prces (S0 e, Se 1, S 0, S 1 ), the nventory Ie 0 n the exportng country, and the fundamental demands (D0,d, D 1,d ) are gven by the soluton to the followng system of equatons: where a + bs 0 = D 0,d, (32) G e 0 I0 e = D0,f + C 0 [ k0 (S0 e = + h) ] ( γ D0,d + η + d + γc ) [ E S 1 F ] γd γ c () ( ) σs 2, (33) E [S1 e F e γsγ e p e ] = γs e + γp e (σs) e 2 [I0 e () + G e 1], (34) ( γ D1,d = a + bs1 + d + γc ) [ E S 1 F ] ( ) σ 2, (35) S I e 0 () + G e 1 = D 1,f γ d γ c = k 1 (S1 e + h) D1,d, (36) S1 = S1 e + h, (37) S 0 = S e 0 + h η, (38) The two Lagrange multplers (λ, η) satsfy: F e = Se 0 (1 + re ) λ, (39) F = (Se 0 + h)(1 + Re ) S0 (R r ). (40) and f I e 0 = 0, λ > 0, f I e 0 > 0, λ = 0; f D0,f = 0, η = D 0,d k 0 (S0 e + h) f D0,f > 0, η = 0. > 0, 12

15 The soluton of spot prces and nventores are: S 0 = [ (1 δ)(k0 2al) + mq + n (q h + zh) [G e 0 () + Ge 1 ] ] + n 1 δ λ (om + zn) η, (41) v + ( + w) m + ( + z) n S e 0 = S 0 h + η. (42) S 1 = 1 2bl + 1 [k 1 2al + () (k 0 2al) (() G e 0 + G e 1)] () S 0, (43) S e 1 = S 1 h, I e 0 = 1 [ n (q h + zh) ( + z) ns 0 G e 1 2nlzη + (44) nλ ], (45) where the constants (n, m, q, o, z, v, w) are defned n the Appendx A.1. The equlbrum demands (C0, D 0,d, D 1,d, D 0,f, D 1,f ) are calculated from (32) (36). Whle the exact solutons to the four cases are provded n Appendx A.1, we now outlne the man soluton steps. Snce each of the two Lagrange multplers λ and η can be bndng or non-bndng, there are four possble cases. Case 1 (λ = 0 and η = 0,.e., I e 0 > 0 and D 0,f > 0). In ths case, the demand of collateral commodty does not lead to stockout or zero mport by fundamental demanders. Snce nether constrant s bndng, the equlbrum prces and nventory are smply gven by (41) (45) after substtutng n λ = η = 0. There are seven unknowns and seven lnear equatons, from whch we obtan a unque soluton. Case 2 (λ = 0 and η > 0,.e., I e 0 > 0 and D 0,f = 0). In ths case, the collateral demand leads to zero mport by fundamental demanders. Intutvely, collateral demand drves up the commodty prce n the exportng country; f ths prce s above the spot prce n the mportng country after adjustng for shppng cost, the fundamental commodty demand n the mportng country s met entrely by local commodty supply. In ths case, the fundamental demanders mport nothng, and D 0,f s gven by Thus, D0,f = 0 mples that S D0,f = k 0 2al (2bl + 1) S0. (46) 0 = k 0 2al 2bl + 1. (47) Therefore, gven λ = 0, from (41) we can explctly obtan η. After gettng S0 and η, we can easly solve all other varables. Case 3 (λ > 0 and η = 0,.e., I e 0 = 0 and D 0,f > 0). In ths case, the collateral demand leads to zero nventory n exportng country. Ths can be the case f too much collateral demand drves up the prce n the exportng country 13

16 such that the commodty suppler does not keep any nventory. Snce I0 e = 0 and η = 0, combnng (41) and (45), one can get S 0 = (1 δ)(k 0 a) G e 0 () + mq. (48) v + ( + w)m Thus, combnng (41) and (48), one can solve for λ. After gettng S 0 varables can be easly solved. Case 4 (λ > 0 and η > 0,.e., I e 0 = 0 and D 0,f = 0). and λ, all other In ths case, too much collateral demand drves up the prce n the exportng country and produces two effects. Frst, the commodty producer has a stockout. Second, the fundamental commodty demand n the mportng country s met entrely by the cheaper local commodty supply (after adjustng for shppng cost). Ths corresponds to I e 0 = 0 and D0,f = 0. As shown n Case 2, D 0,f = 0 mples that S 0 = k 0 2al 2bl+1. Therefore, we have S 0 = I e 0 = [ (1 δ)(k0 2al) + mq + n (q h + zh) [G e 0 () + Ge 1 ] ] + n 1 δ λ (om + zn) η = k 0 2al v + ( + w) m + ( + z) n 2bl + 1, [ 1 n (q h + zh) ( + z) ns0 G e 1 2nlzη + nλ ] = 0. We can solve λ and η from the above two equatons. Then, t s easy to further solve all other varables n the equlbrum. 4.2 The Benchmark Equlbrum wthout Collateral Commodty In the benchmark case, we exogenously shut down commodty use as collateral,.e., forcng C 0 = 0. Snce the exportng country s a net suppler n tme 0 and tme 1, ts commodty must eventually be absorbed by fundamental demander n the mportng country. By the same reasonng as before, we have the followng proposton. Proposton 2. Suppose that collateral demanders of commodtes do not partcpate n the market. In equlbrum, the spot prces (S0 e, Se 1, S 0, S 1 ) and the nventory Ie 0 n the 14

17 exportng country are gven by the soluton to the followng system of equatons: where G e 0 I e 0 = k 0 S 0 I e 0 () + G e 1 = γe s + γ e p γ e sγ e p a bs 0, (49) E [S e 1 F e ] ( σ e S ) 2, (50) I0 e () + G e 1 = k 1 S1 ( a + bs 1), (51) S1 = S1 e + h, (52) S 0 = S e 0 + h η, (53) F e = Se 0 (1 + re ) λ. (54) The explct equlbrum soluton s provded n Appendx A.2. For smplcty, we restrct attenton to cases such that λ = 0 and η = 0. In ths case, the equlbrum S0 s gven by S 0 (1 δ)(k 0 2al) + n (q h + zh) [G e 0 () + Ge 1 ] v + ( + z) n, (55) where the constants (n, q, z, v) are defned n Appendx A, and where we use the bar notaton to emphasze the equlbrum varable n the benchmark case wthout collateral demand. The other spot prces, nventory, and quanttes can be calculated from the equatons gven n Proposton 2. In equlbrum, the restrcton of λ = η = 0 bols down to a + bs 0 < k 0 S 0 < G e 0 + a + bs 0. (56) That s, at the equlbrum prce S 0, the fundamental demand n the mportng country at tme 0 s larger than the domestc supply but s less than the total global supply (domestc plus foregn). Hence, the exportng country stll keeps postve nventory and fundamental demander mports postve amount of commodty. Evaluatng the above nequalty at the equlbrum S 0, we arrve at the followng techncal condton: Techncal Condton 1. k 0 a G e b < (1 δ)(k 0 2al) + n (q h + zh) [G e 0 () + Ge 1 ] < k 0 a v + ( + z) n 1 + b, (57) where the constants (n, q, z, v, w) are defned n Appendx A. For the collateral channel to be nontrval, we also restrct attenton to stuatons n 15

18 whch, once demanders of collateral commodty partcpate n the market, they trade a postve amount. Ths amounts to the condton that S 0(R r ) + ()E[S 1] (S e 0 + h)(1 + R e ) > 0. (58) Evaluatng the above nequalty at the equlbrum prces, we get the followng techncal condton: Techncal Condton 2. (1 δ)(k 0 2al) + n (q h + zh) [G e 0 () + Ge 1 ] < v + ( + z) n where the constants (n, q, z, v, w) are defned n Appendx A. q + w, (59) Techncal Condton 2 mples that a postve quantty of commodty s mported for collateral purposes n equlbrum. Because the fnancal nvestors engagng n ths trade are rsk-averse, the expected margnal proft of mportng commodty as collateral must be postve n equlbrum. That s, we have S 0 (R r ) + ()E[S 1] (S e 0 + h)(1 + R e ) > 0. (60) Evaluatng the above equaton at the equlbrum prces gven n Proposton 1, we have the followng corollary. Corollary 1. In the equlbrum of Proposton 1, S0 < q (1+Re ) 1 δ η + w. (61) 5 The Effects of Demand for Collateral Commodty and Comparatve Statcs In ths secton we analyze the effect of collateral commodty on the spot and futures prces, convenence yeld, nventory, rsk premum, and real demand of commodtes. We also study how the unsecured nterest rate R n the mportng country affects all these equlbrum varables. 5.1 Prces Proposton 3 (Effect on Prces). The collateral demand for commodtes: 1. Increases the spot prces at tme 0, S0 e and S 0. 16

19 2. Reduces the spot prces at tme 1, S e 1 and S 1, for a fxed fundamental shock k Makes S0 (Se 0 + h) < 0 f and only f η > 0. That s, the spot prce dfferental can become smaller than the shppng cost f fundamental commodty demand s entrely met by local supply (.e. D 0,f = 0). 4. Reduces the futures prce F n the mportng country. The ntuton for Proposton 3 s smple. Snce there s an extra collateral demand for commodtes at tme 0, t wll ncrease the spot prce at tme 0. However, because these extra collateral commodtes are sold at tme 1, they reduce the tme-1 spot prces S 1 as well as the tme-0 futures prce F n the mportng country. In partcular, f η > 0, the tme-0 spot prce n the exportng country becomes hgher than the local spot prce n the mportng country plus shppng cost. Thus, t s not optmal for the fundamental demanders to mport commodty; nstead, they buy commodtes exclusvely n the local markets. The commodty supply n the exportng country s absorbed entrely by the demand for commodty collateral. The top half of Fgure 1 plots the spot prces n the exportng country and mportng country as functons of the uncollateralzed rate R. As shown n Proposton 3, the collateral demand ncreases the spot prces n both countres. Wthout collateral demand, the spot prces are nvarant to R ; wth collateral demand, spot prces are weakly ncreasng n R. The knks are where the constrants start to bnd. In ths example, f R goes above 0.87, the spot prce n the exportng country S0 e becomes so large that fundamental demand s entrely met by domestc supply (Case 4 of Proposton 1). Consequently, S e 0 ncreases sharply n R, but S 0 s nvarant to R. The prce dfferental, S 0 Se 0, becomes smaller than the shppng cost h. The bottom half of Fgure 1 plots the spot prces n the two countres as functons of the fundamental actvty k 0. Naturally, a hgher fundamental actvty k 0 mples a hgher spot prce. Interestngly, for small k 0 (lower than about 34 n ths example), the prce n the exportng country s relatvely nsenstve to k 0. Ths s because collateral demand for commodty makes up for the shortfall of fundamental demand. If real actvty s suffcently low, commodty mports are entrely used as collateral, resultng n the spread S0 Se 0 beng smaller than the shppng cost h, whch corresponds to Case 2 of Proposton 1. Once k 0 becomes large enough, fundamental demand s partly met by mport, and spot prces n two countres ncrease n k 0 n tandem, whch corresponds to Case 1 of Proposton Convenence Yeld The convenence yeld of a commodty s the beneft of holdng ths commodty versus holdng a futures for forward contract. Economcally, the convenence yeld can come 17

20 Fgure 1: Spot prces as functons of R and k 0, wth and wthout collateral demand. Model parameters: r e = 0.01, r = 0.01; R e = 0.05, δ = 0.05, h = 3, G e 0 = 20, G e 1 = 21, k 0 = 45, u k = 80, σ k = 2, l = 0.1, a = 5, b = 10, γ e p = 2, γ e s = 2, γ d = 5, and γ c = Prces n the exportng country Collateral case Benchmark case Prces n the mportng country S 0 e S R Prces n the exportng country 16 Collateral case Benchmark case R Prces n the mportng country S 0 e 10 S k 0 k 0 from the real opton of startng producton anytme, especally f the commodty s scarce and cannot be bought quckly n the spot market. The convenence yeld can be mathematcally defned as the carry cost of the commodty less the spot-futures spread. In our model, the convenence yeld n country s obtaned as y = F S r. (62) From (11), one can see that the convenence yeld y e n the exportng country s y e = S e 0 λ (63) (). Note that the convenence yeld arses because stock-out can happen (I e 0 = 0) n the 18

21 exportng country, whch corresponds to a postve λ. Ths s consstent wth the theory of storage, n whch the convenence yeld arses because of the possblty of a stockout (see, for example, Deaton and Leraque (1992, 1996) and Routledge, Sepp and Spatt (2000)). From (31), wth collateral demand, the convenence yeld n the mportng country s y = R R e (1 + Re ) () S0 η. (64) Note that ths convenence yeld s not necessary non-negatve. The convenence yeld n the mportng country s lnearly related to the spread between the uncollateralzed nterest rates n the two countres. Ths convenence yeld arses because of collateral demand for fnancng, not real demand of producton. To dstngush t from the theory of storage, we call y the convenence yeld of collateral. Proposton 4 (Effect on Convenence Yeld). The collateral demand for commodtes: 1. Increases the convenence yeld n the mportng country and makes the futures curve more backwardaton (.e. reducng F /S 0 ). 2. Increases the convenence yeld n the exportng country and makes the futures curve more backwardaton (.e. reducng F e /S0 e ) f and only f λ > 0 (.e. a stockout). The results n Proposton 4 come from the convenence yeld of collateral. Intutvely, commodty as collateral offers the holders the beneft of earnng the spread between collateralzed and uncollateralzed nterest rates. As mentoned before, f the uncollateralzed nterest rates are the same n both countres, R = R e, the collateral demander wll not mport and hence the convenence yeld of collateral s zero. If, however, R > R e, collateral demander wll beneft from mportng commodty as collateral. In partcular, η > 0 arses f collateral demand drves up the spot prce n the exportng countres so much that the local spot prce n the mportng countres s lower than the foregn spot prces plus shppng cost, as shown n (29). Ths hgh spot prce n the mportng country corresponds to a hgher convenence yeld. By contrast, the convenence yeld n the mportng country wthout collateral demand s gven by y = q S 0 + () r, (65) whch s not a functon of R e or R. (The equlbrum S 0 does not depend on R e or R.) Proposton 4 reveals that n equlbrum, the collateral demand for commodty ncreases the convenence yeld relatve to the benchmark case,.e., y > y. The effect on convenence yeld s closely related to the effect on the shape of the spot-futures spread (F /S0 ). Snce the collateral holders n the mportng country need to 19

22 short futures contracts to hedge ther rsk, the hedgng actvty decreases the futures prce n the mportng country. As shown n Proposton 3, the collateral demand ncreases the spot prce. Thus, the futures curve exhbts more backwardaton. In the mean tme, the collateral demand of commodtes makes stockout more lkely the exportng country, leadng to a postve convenence yeld. Thus, the futures curve n the exportng country also exhbts more backwardaton. Fgure 2: Convenence yelds as functons of R and k 0, wth and wthout collateral demand. Model parameters are those of Fgure Conv. yeld n the exportng country Collateral case Benchmark case Conv. yeld n the mportng country 0.15 y e y R Conv. yeld n the exportng country 0.1 Collateral case Benchmark case R Conv. yeld n the mportng country y e y k 0 k 0 The top half of Fgure 2 plots the convenence yelds n the two countres as functons of R. The collateral demand of commodty weakly ncreases the convenence yelds n both countres; so does an ncrease n R. A postve convenence yeld n the exportng country obtans only f a stockout occurs, and the speed at whch y e ncreases n R s hgher once the mportng constrant D0,f 0 starts to bnd. In the mportng country, the convenence yeld ncreases almost one-for-one wth R (see (64)). Ths ncrease contnues untl R becomes so hgh that fundamental demand s entrely met by local supply, at 20

23 whch pont the convenence yeld becomes flat n R. The bottom half of Fgure 2 plots the convenence yelds as functons of k 0. As expected, a hgher k 0 mples a hgher demand at t = 0 and a hgher convenence yeld. 5.3 Inventores Proposton 5 (Effect on Inventores). The collateral demand for commodtes: 1. Reduces the nventory I e 0 n the exportng country. 2. Increases the total nventory I e 0 + C 0 n both countres. The extra collateral demand ncreases the spot prces n exportng country and hence makes the producers less lkely to hold nventory. It s also possble that the collateral demand can cause a stockout n the exportng country. Moreover, because the collateral commodty C0 s also a form of nventory, the exstence of collateral demand ncreases the total nventory. The top half of Fgure 3 plots the nventory n the exportng country and n both countres as functons of R. As we show n Proposton 5, the collateral demand for commodty reduces nventory n the exportng country I e 0 but ncreases the total nventory Ie 0 + C 0. Once R goes above 0.25, a stockout happens n the exportng country (Case 3). Any further ncrease n R leads to a hgher collateral demand and crowds out fundamental demand n the mportng country, leadng to a hgher total nventory. Once R goes above 0.87, there emerges a complete separaton: all commodtes n the exportng country are mported for collateral purposes, and all local commodty supply n the mportng country are used for fundamental producton. Thus, total nventory becomes nvarant to R. The bottom half of Fgure 3 plots the nventores as functons of k 0. As expected, a hgher real demand at t = 0 leads to a lower nventory n the exportng country as well as a lower total nventory. For small k 0, nventores declne slowly n k 0 (for I e 0 ) or stays constant (for I0 e+c 0 ) because collateral demand makes up for low real demands. For ntermedary levels of k 0, nventores are much more senstve to k 0 because both collateral demand and fundamental demand consume mported commodty. Once k 0 s hgher than about 45, a stockout occurs n the exportng country, and total nventores declnes slowly n k 0 agan. 5.4 Commodty Futures Rsk Premum Proposton 6 (Effect on Commodty Futures Rsk Premum). The collateral demand for commodtes: 1. Reduces the futures rsk premum n the exportng country, E[S1 e F e ]. 21

24 Fgure 3: Inventores as functons of R and k 0, wth and wthout collateral demand. Model parameters are those of Fgure 1. 3 Inventory n the exportng country 25 Total nventory n both countres I 0 e Collateral case Benchmark case I 0 e +C R Inventory n the exportng country 16 Collateral case 14 Benchmark case R Total nventory n both countres I 0 e 8 6 I 0 e +C k 0 k 0 2. Increases the futures rsk premum n the mportng country, E[S 1 F ]. Equaton (12) shows that the futures rsk premum s proportonal to [I e 0 () + Ge 1 ], whch can be consdered as the total amount the producers need to hedge. The theory of normal backwardaton as n Keynes (1923), Hrshlefer (1990) and Bessembnder (1992) argues that hedgers need to offer rsk premums n order to solct speculators to offset ther trades; thus the rsk premum relates postvely to the amount producers hedge. Ths s consstent wth our model. The extra collateral demands reduces the amount of nventory I0 e, whch n turn reduces the rsk premums n the exportng country. Smlarly, n the mportng country, the amount to hedge s just the amount of collateral n tme 1, () C0. Therefore, consstent wth the theory of normal backwardaton, our model shows that the rsk premum n the mportng country s proportonal to () C0. In the benchmark case, E[S1 F ] = 0; hence the rsk premum s zero. In the collateral case, however, the rsk premums are postve because C0 > 0. Thus, the futures rsk premums 22

25 n the mportng country ncrease wth the collateral demand for commodty. Fgure 4: Commodty futures rsk premum as functons of R and k 0, wth and wthout collateral demand. Model parameters are those of Fgure 1. Rsk premum n the exportng country Rsk premum n the mportng country E[S 1 e F e ] Collateral case Benchmark case F ] E[S R Rsk premum n the exportng country 16 Collateral case 15 Benchmark case R Rsk premum n the mportng country E[S 1 e F e ] F ] E[S k 0 k 0 Fgure 4 plots the futures rsk premum n two countres as functons of R and k 0. Collateral demand of commodty reduces the futures rsk premum n the exportng country but ncreases that n the mportng country. Comparng Fgure 4 wth Fgure 3, we observe that the shapes of the rsk premum resemble those of the nventores. In the exportng country, ths pattern s natural because the hgher s the futures rsk premum, the more expensve t s for the producers to hedge, and the larger s the nventory held by producers. In the mportng country, the futures rsk premum s postve because collateral commodty must be sold at t = 1. The larger s the mport for collateral purposes, the hgher s the sellng pressure at t = 1, and the hgher s the futures rsk premum. 23

26 5.5 Real Demand of Commodty Proposton 7 (Effect on Real Demand). For a fxed fundamental shock k 1, the collateral demand for commodtes: 1. Reduces the real demand of commodtes n the mportng country at tme Increases the real demand of commodtes n the mportng country at tme Reduces the sum of tme-0 and tme-1 real demands of commodtes n the mportng country. As the collateral demand ncreases the spot prce at tme 0 n the mportng country, t also decreases the real demand of the commodty. As the collateral commodtes are sold at tme 0, the spot prce at tme 1 decreases n the mportng country and the real demand ncreases. But because a larger amount of commodtes s stored between the two perods, a larger deadweght loss s ncurred. The total real demand s reduced by the demand for collateral commodty. Ths real effect of usng commodty as collateral complements to that of Kyotak and Moore (1997). In ther model, producton assets such as land and machneres can also be pledged as collateral. They show that a small, temporary negatve shock to frms net worth can be amplfed to large, persstent shock to the prces of assets and frms nvestments and producton. Our model s complementary n that the producton asset, commodty, s a traded asset, and frms not nvolved n the real producton can also mport commodty to generate fnancal returns. In our model, f the producton functons of the real sector s nvarant to the nterest rate, as we mplctly assume n Proposton 7, more fnancal demand for commodty crowds out the real demand by ncreasng commodty spot prces and by ncreasng the deadweght loss of commodty storage. If, however, producton constrant can be relaxed by mportng commodtes as collateral, we may reasonably expect the collateral demand for commodty to ncrease total output at the cost of amplfcaton and fraglty, as n Kyotak and Moore (1997). The latter effect s not n our current analyss because we expect t to be smlar to that modeled by Kyotak and Moore (1997). The welfare mplcatons of usng commodtes as collateral s therefore ambguous. 5.6 Comparatve Statcs wth respect to Interest Rate R By the same ntuton as the effects of collateral commodty, we can derve the effect of rasng nterest rates R n the mportng country. Proposton 8. Holdng other parameters fxed, n Case 1 of the equlbrum of Proposton 1, as the uncollateralzed nterest rate R ncreases n the mportng country: 24

27 1. The spot prces n mportng and exportng countres at tme 0, S 0 and Se 0, ncrease. n the exportng country de- 2. The collateral demand C0 ncreases, the nventory Ie 0 creases, and the total nventory ncreases. 3. The convenence yeld n the mportng country y ncrease. 4. The rsk premum n the exportng country E [S1 e F e ] decreases, and the rsk premum n the mportng country E [ S1 F ] ncreases. Moreover, holdng other parameters fxed: 5 If an ncrease n R causes the constrant I0 e n the exportng country, y e, ncreases. 0 to bnd, then the convenence yeld 6 If an ncrease n R causes the constrant D0,f S0 Se 0, decreases. 0 to bnd, then the spot prce spread, Parts 1 4 of Proposton 8 are wrtten for Case 1 of Proposton 1 for smplcty, but the same qualtatve results hold for other three cases. The only caveat s that f R s suffcently hgh, certan endogenous varables may become flat n R. For nstance, f η > 0, S0 and Ie 0 + C 0 are nvarant to R. Proposton 8 complements exstng theory and evdence on the relaton between nterest rate and (real) commodty prces. For example, Frankel (1986, 2008) show that hgh nterest rates reduce the prce of storable commodtes by ncreasng the ncentve for commodty extracton today rather than tomorrow, by decreasng frms desre to carry nventores, and by encouragng speculators to shft out of commodty contracts and nto Treasury blls. He fnds a sgnfcant and negatve coeffcent of real commodty prce on the real US nterest rate, representng global monetary polcy, as well as on the real nterest dfferental between the foregn economy and the US, representng local varatons n monetary polcy. The foregn countres used n Frankel s analyss nclude Australa, Brazl, Canada, Chle, Mexco, New Zealand, Swtzerland and UK. The frst sx countres are major exportng countres of commodtes, whereas the last two are mportant commodty tradng centers that hold large nventores. Frankel s results, as well as the explanaton based on costs of commodty extracton and nventory, apply well n these countres. Complementary to Frankel s work, our result focuses on the collateral channel, whch apples to countres that mport commodtes and exhbt slow-movng captal across the natonal border. For these countres, most notably Chna, a hgher nterest rate wll ncrease the demand of collateral and hence ncrease the global prce of the commodty, holdng all else equal. The net effect depends on whch demand, producton or collateral, reacts faster and n larger magntude to a hgher nterest rate n mportng countres. 25