Less of a puzzle: a new look at the forward forex market

Transcription

1 Less of a puzzle: a new look a he forward forex marke Michael J. Moore a *, Maurice J. Roche b a School of Managemen and Economics, The Queen s Universiy of Belfas, Belfas BT7 NN, Norhern Ireland b Deparmen of Economics, Naional Universiy of Ireland, Maynooh, Co. Kildare, Ireland Absrac The wo-counry moneary model is exended o include a consumpion exernaliy wih habi persisence. The model is simulaed using he arificial economy mehodology. The puzzles in he forward marke are re-examined. The model is able o accoun for (a) he low volailiy of he forward discoun (b) he higher volailiy of expeced forward speculaive profi (c) he even higher volailiy of he spo reurn (d) he persisence in he forward discoun (e) he maringale behavior of spo exchange raes and (f) he negaive covariance beween he expeced spo reurn and expeced forward speculaive profi. I is unable o accoun for he forward marke bias because he volailiy of he expeced spo reurn is oo large relaive o he volailiy of he expeced forward speculaive profi. Keywords: Arificial Economy; Forward Foreign Exchange; Habi Persisence JEL classificaion: F3; F4; G *The address for correspondence is Michael J. Moore, School of Managemen and Economics, The Queen s Universiy of Belfas, Belfas BT7 NN, Norhern Ireland. Tel +44 (0) ; Fax +44 (0) ; m.moore@qub.ac.uk

2 . Inroducion Campbell and Cochrane (999) have proposed a preference specificaion in which here is boh an aggregae consumpion exernaliy and uiliy is ime-inseparable because of habipersisence. They apply his o he equiy premium puzzle and show ha heir model can replicae he relevan sylized facs for US daa. In his paper, we exend he Campbell and Cochrane (999) preferences o boh a moneary and an inernaional seing. The puzzles associaed wih he forward foreign exchange marke are revisied in he ligh of he new heoreical framework. Numerous wriers (a good example is Backus, Gregory and Telmer, 993) have shown ha he sandard deviaion of he forward discoun is smaller han ha of he expeced forward speculaive profi 3 and is an order of magniude less han ha of he spo reurn. We call his ordering of relaive sandard deviaions he volailiy puzzle. The sandard cash-inadvance for goods model is unable o replicae his sylized fac a levels of risk aversion ha are commonly used in he lieraure. Bekaer (996) has argued ha he sandard for udging a successful heory of he forward marke is is abiliy o replicae he volailiy puzzle a plausible absolue levels of volailiy. Our model is no only able o replicae hese relaive volailiies bu goes mos of he way in maching he absolue volailiies also. A second feaure of he daa is ha he forward discoun is highly persisen while he spo reurn is virually whie noise (see Bekaer, 996). The sandard model is usually able o explain one of hese bu no boh a he same ime (see Moore and Roche, 00). This is he persisence puzzle. Our model succeeds in geing close o he sylized facs for every case we examine. The bes-known deficiency in our undersanding of he forward marke is he unbiasedness puzzle: ha he forward discoun is a poor predicor of realized fuure spo

3 reurn. Indeed, he forward discoun ypically fails o even forecas he correc direcion of spo exchange rae changes. The mos challenging 4 ask of a good heory is o replicae his bias. Since Fama (984), i has been undersood ha wo condiions need o be me o replicae he bias. Firsly, here needs o be a negaive covariance beween expeced spo reurn and expeced forward speculaive profi. This is he condiion ha has received mos emphasis (see Alvarez, Akeson and Kehoe, 999) and our model succeeds in saisfying i. Fama s second condiion has received less aenion in he lieraure. I requires ha he volailiy of expeced forward speculaive profi mus exceed he volailiy of expeced spo reurns. In his respec our model falls shor. Ironically, is very success in explaining he volailiy of spo reurn is accouned for by an excessive increase in he volailiy of expeced spo reurn. I is his feaure of he model ha prevens i from meeing Fama s second condiion. The plan of he paper is as follows. Secion provides a brief inroducion o Campbell and Cochrane (999) preferences, develops he heoreical framework and provides inuiion for explaining our resuls. Secion 3 summarizes he resuls of simulaing an arificial economy. The conclusion is in Secion 4.. A heoreical framework. Background Campbell and Cochrane (999) have proposed a novel soluion o he equiy premium puzzle in closed economies. They inroduce a simple and racable modificaion o he uiliy funcion in he sandard model 5. An exernal habi is inroduced. In heir model he real riskfree rae is consan. This is achieved by ensuring ha precauionary savings effecs and ineremporal subsiuion effecs cancel each oher ou. This removes he risk-free rae

4 puzzle (see Weil, 989). Campbell and Cochrane (999) preferences overcome he irriaing problem wih exising habi persisence models ha marginal uiliies can someimes ake on negaive values. The habi aduss slowly in response o consumpion. This differs from he usual habi persisence specificaions ha model he habi as proporional o pas consumpion choices. Campbell and Cochrane (999) habi persisence akes he form of an aggregae consumpion exernaliy i.e. Keeping Up wih he Jones s effecs along he lines of Abel (990) and Duesenberry (949). Habi persisence preferences have already been proposed before as a soluion o he puzzles in he forward marke. For example, Bekaer (996) inroduces habi persisence ino a convenional Lucas (98) wo-counry model where equilibrium consumpion in each good equals half of he curren endowmen. He is able o mach he observed volailiies of he spo reurn, he forward discoun and expeced forward speculaive profi bu never a he same configuraion of parameers. The reason why we believe ha he Campbell and Cochrane (999) framework can help o solve he forward marke puzzles is as follows. The sandard cash in advance model is able o mach he sylized facs of he forward discoun relaively well. However, i is unable o accoun for he observed high volailiy in spo reurn. I is even more unsuccessful a explaining he volailiy of expeced forward speculaive profi. The rouble is ha exising aemps o improve performance wih he laer leads o grealy increased volailiy in he forward discoun. This is an inernaional analogue o he equiy premium and risk-free rae puzzles in closed economy sock markes. Campbell and Cochrane (999) preferences provide he opporuniy o anchor he volailiy of he forward premium because i forces real risk-free raes o display low volailiy. This is because he ineremporal marginal rae of subsiuion (IMRS) is more volaile han he sandard case, because of he habi specificaion, while he expeced IMRS is consan by consrucion. 3

5 . A cash in advance model wih habi persisence The basic srucure of he model is he well-known Lucas (98) wo-counry, wo-good, wo-money represenaive agen sory. The households in boh counries 6 have he same ineremporal uiliy funcion. They maximize he discouned expeced value of lifeime uiliy: ( C H ) ( C H ) UC (, C) = +, i=, ( γ ) ( γ ) β i i i i i i β = 0 = 0 γ γ () where β is he discoun facor, /γ is he ineremporal elasiciy of subsiuion, C is he consumpion of goods and services of counry by he household of counry i and H i i is he subsisence consumpion (or habi) of goods and services of counry by he household of counry i. We reparameerize he uiliy funcion in erms of raio of goods and services of counry by he household of counry i: X i, he surplus consumpion Ci Hi X i =, i =,, =, C i () When C = H, X = 0 : his is he wors possible sae. By conras, as i i i C i rises, he surplus consumpion raio converges on uniy. The agen in he goods marke faces he following cash-in-advance consrain: M P C, i =,, =, (3) i i where M is he amoun of money of counry held by he household of counry i for i ransacions in he goods marke a ime and P is he price of counry goods in erms of counry money. If ineres raes are posiive he cash-in-advance consrain will hold wih equaliy. A he end of period (or he beginning of period +), he domesic households holding of domesic currency: 4

6 M + + P C B FG (4) is made up of proceeds from he sale of he endowmen, he redempion of nominal discoun bonds, B, i and proceeds from forward conracs, where F is he one-period ahead forward foreign exchange rae expressed as he home price of foreign currency. NoeG > 0 consiues he number of long forward conracs for foreign currency. The domesic household's holding of foreign currency is: M + + B G (5) Analogously he foreign households holding of foreign currency is: M + + P C B G (6) and of domesic currency is: M + + B FG (7) whereg > 0 consiues a shor posiion in forward foreign exchange for he foreign counry. The only role for he governmen is o have a cenral bank ha engages in open marke operaions. In each period he cenral bank of each counry changes he money sock by issuing one-period discoun bonds. The bonds are redeemed a he end of period (or he beginning of period +). Equilibrium in he goods marke is given by: = + C C, =, C (8) Equilibrium in he money marke given by: M = M +, =, M (9) Equilibrium in he forward foreign exchange marke is given by: G = G (0) Each household maximizes Eq. () subec o Eqs. (3)-(0). Like Lucas (98) and Bekaer (996) we assume ha here is perfec inernaional risk pooling in equilibrium. Thus 5

7 in equilibrium consumpion of each good equals half of he curren endowmen. There are four firs-order efficiency condiions o each household s maximizaion problem. They are all well known in he inernaional asse pricing lieraure bu differ from he sandard model in he argumens of he marginal uiliy funcion. The firs wo are Fisher equaions and relae he ineremporal marginal rae of subsiuion for each good o is counry s nominal ineres rae and inflaion 7 : γ δu / δc + ( C+ X+ ) P + P+ ( + ) = E ( + ) β i E β i = 0 =, γ δu / δc ( C X ) P P () where i, =,, are he nominal raes of ineres for he home and foreign counry respecively. 8 The inuiion is he usual one ha if he individual decides o decrease M by one uni of currency his would decrease consumpion spending by one uni of currency. This money can be saved in he form of discoun bonds and used o buy goods in he nex period. The hird efficiency condiion is he purchasing power pariy relaionship. I equaes he real exchange rae wih he marginal rae of subsiuion beween domesic and foreign goods and can be wrien as: i SP δu/ δc ( C X ) / ( ) γ = = γ P δu δc C X () wheres is he nominal spo exchange rae, measured as he home price of foreign currency a ime. Noe ha, if X = X = in Eqs. () and (), he condiions in a sandard (no habi) model wih addilog preferences hold 9. The final efficiency condiion is covered ineres rae pariy and is common o mos models in inernaional finance: F S = + i + i (3) 6

8 We define exogenous consumpion and money growh as: C C + = ( + µ ), =, + (4) and M M + = ( + π ), =, + (5) respecively. When we calibrae he model in Secion 3 we will assume ha consumpion and money growh raes, µ and he sochasic processes denoed by π, follow vecor auoregressive processes wih innovaions o v and u respecively. We closely follow Campbell and Cochrane (999) by assuming ha he log of he surplus consumpion raios evolve as follows: ( ) x = ( φ) x + φx + λ ( x ) v, =, (6) + + whereφ <, is he habi persisence parameer and x is he seady sae value for he logarihm of he surplus consumpion raio for good. The funcion λ( ) describes he sensiiviy of he log surplus consumpion raio o endowmen innovaions. I depends non-linearly on he curren log surplus consumpion raio. The form of he sensiiviy funcion λ( ) is: x x ( x x ) λ( x ) for x x X = 0 for x > x = max max where xmax = x + = ( X ),, (7) X is he seady sae value of he surplus consumpion raio for good and is defined as: X = σ v γ φ (8) 7

9 We defineσ v as he sandard deviaion of he innovaion o he consumpion of he h good. The local curvaure of he uiliy funcion wih respec o good a he seady sae is: ( C ) i γ / ( ) = = i, =, (9) X U C δ U δ γ φ C i i δ / δ i σ v Noe ha his expression is posiively relaed o he coefficien of relaive risk aversion for he one good case 0. The main difference beween our specificaion and ha of Campbell and Cochrane (999) is ha we have wo goods in he model. We assume ha here is a separae habi in each good. There are hree advanages o specifying he habi along he lines of Eqs. (6)-(8). Two of hem are sensible echnical feaures. Firsly, he habi is predeermined a he seady sae. This means ha i akes ime for he consumpion exernaliy o affec an individual agens habi. The second advanage avoids a possible difficuly wih he firs. If he habi were always predeermined, a sufficienly low realizaion of consumpion would mean ha habi exceeded curren consumpion. The argumens of he uiliy funcion () would become negaive. Our habi specificaion prevens his by ensuring ha he habi moves nonnegaively wih consumpion everywhere. These wo feaures are illusraed in deail in Campbell and Cochrane (999). The final and mos imporan advanage of our specificaion is ha he real ineres rae is consan. This poin is illusraed in he nex sub-secion..3 A benchmark example Under some simplifying assumpions on he forcing processes, we can derive expressions for he forward discoun, spo reurn and expeced forward speculaive profi. This will provide some inuiion for our simulaed resuls. We assume ha consumpion growh is a 8

10 whie noise process wih a non-zero mean as in Campbell and Cochrane (999) and money growh follows a simple AR() process as in Chrisiano (99): µ µ σ (0) + = + v+, v+ ~ N(0, ), =, v and π ρ π ρπ σ () + = ( ) + + u+, u+ ~ N(0, ), =, u respecively. The uncondiional means of consumpion and money growh in counry are defined as µ and π respecively. The variances of shocks o consumpion and money growh in counry are defined as σ and σ respecively. We make he following simplifying v u assumpions: i) The uncondiional mean of consumpion growh may differ from he uncondiional mean of money growh bu he parameers are he same across counries. ii) The variance of shocks o consumpion growh may differ from he variance of shocks o money growh bu he parameers are he same across counries. iii) The covariances beween all shocks are zero. 3 iv) The firs-order auocorrelaion coefficien for money growh, ρ, is he same in boh counries. The assumpion on σ implies ha he seady sae surplus consumpion raio is he same for v boh goods: x = x = x. In wha follows we drop he superscrips on all seady sae parameers. As shown in Campbell and Cochrane (999) he real ineres rae, r, for counry is consan and is given by 4 : δu / δc + ln( + r) = r ln E β = ln( β) + γµ + γ( φ) δu / δc () 9

11 Thus boh domesic and foreign real ineres raes are consan. The reason for his cenral feaure of he habi specificaion is as follows. If he surplus consumpion raio is low, he marginal uiliy of consumpion is high. Consequenly, households wish o borrow pushing up he real ineres rae in he face of exogenous resource consrains i.e. he ineremporal subsiuion effec. In conras, when he surplus consumpion raio is low, uncerainy is high; consumers wish o increase precauionary saving and his pushes ineres raes down. These wo forces offse each oher. Similarly we can derive an expression for domesic and foreign nominal ineres raes. Using Eq. () he nominal ineres rae for counry is given by 5 : ( ( ) ) i ( ) = r + ρπ+ ρπ µ σu + γ( + λ( x )) σ v, =, (3) The firs erm is us he consan real ineres rae ha we have already presened in Eq. (). The second erm is he expeced rae of moneary growh minus he seady sae rae of real consumpion growh. This erm can be inerpreed as he expeced rae of inflaion. The final erm is he risk premium. Noe ha if λ ( ) = 0, his simplifies o he risk premium in he x sandard model. The habi specificaion adds an addiional risk premium ha has he following propery. When surplus consumpion is low, uncerainy (in he form of λ ( ) ) is high and he risk premium in he nominal ineres rae is higher. In shor, he lower he surplus consumpion raio, he more consumers have o be compensaed for aking on inflaion risk. Taking logarihms of he covered ineres pariy Eq. (3) and using Eq. (3), i is easy o obain he following expression for he forward discoun: x 0

12 = ( ) ( ) ( ) ( ) f = = ( ) + ( ) ( ) s i i ρπ π γσv λx λx or equivalenly X f s ρπ π γ φ x x γ (4) The second expression for he forward discoun in Eq. (4) is obained from a linearizaion of λ( ) in Eq. (7). 6 In he case of sandard addilog preferences, he forward discoun is x simply he erm in he firs se of square brackes in boh expressions in Eq. (4) i.e. ρ π π. This is he home and foreign expeced money growh differenial. Habis add ( ) real influences in he form of he raio beween he home and foreign expeced IMRS. When he home counry has a lower surplus consumpion raio han he foreign counry (i.e. uncerainy is higher in he home counry), he home currency is a a deeper forward discoun. Nex we derive an explici expression for spo reurn. Subsiuing he cash-in advance consrains from Eq. (3) ino Eq. () gives: S γ γ X C M = X C M (5) Thus spo reurn is given by: γ γ γ γ = S C C M M X X S C C M M X X (6) Using Eqs. (4)-() in Eq. (6) we derive an expression for he logarihm of he spo reurn: s s = ρπ ( π ) + ( u u ) ( + + ) γ( φ)( x x ) ( γ( + λ( x )) v ( γ( + λ( x )) v or equivalenly s+ s = ρπ ( π ) + ( u+ u+ ) + ( γ)( v+ v+ ) + γ( x+ x+ ) (7) The second expression in Eq. (7) is obained from he firs by noing ha Eq. (6) can be reparameerized as:

13 ( ) ( ) x = ( φ) x x + λ ( x ) v, =, (8) + + The conribuion of habis o spo reurn is sraighforward. Under sandard addilog preferences he spo reurn is he firs square-brackeed erm in he second expression in Eq. (7). I consiss of he difference beween home and foreign money growh as well as a erm ha is proporional o he difference beween home and foreign real innovaions. The remainder of he expression arises solely because of he habi specificaion. I inroduces differences in he changes in he growh of home and foreign surplus consumpion raios. I is clear ha one of he main sources of variaion in spo reurn is he raio of he IMRS of home and foreign goods. The volailiy of his is ypically very low leading o he conclusion by Flood and Rose (999) ha macroeconomics is incapable of explaining exchange rae volailiy. Our habi specificaion makes he IMRS dependen on he change in he surplus consumpion raios, which are volaile by consrucion. Consequenly, he spo reurn is much more variable under our habi specificaion han under he sandard addilog case. I is useful a his sage o inroduce he expeced spo reurn condiional on ime. Taking expecaions of Eq. (7) a ime gives: E( s+ s) = ρπ ( π ) γ( φ )( x x ) (9) Like he forward discoun in Eq. (4), he expeced spo reurn is influenced (in he same direcion) by he home and foreign counry s relaive uncerainy, as measured by he difference beween he home and foreign log surplus consumpion raios. The expeced forward profi is he expeced reurn from aking a shor posiion in forward foreign exchange i.e. f E s+. This can be derived as he difference beween he forward discoun and he expeced spo reurn: ( f s ) E ( s + s ) i.e. Eq. (4) minus Eq. (9).

14 ( f Es+ ) = γ( φ) ( x x ) X γ (30) I is well known (see, for example, Engel 999) ha he expeced forward profi consiss of wo elemens: a risk premium and a non-convexiy erm. However under he specific assumpions ha we have made abou he forcing processes in his example, he non-convexiy erm is zero. In addiion, he risk premium iself is zero under sandard addilog preferences wihou habis. 7 Consequenly, Eq. (30) is exclusively a risk premium ha is aribuable o habis. I is ime varying and is sign has wo deerminans. As in he cases of he forward discoun and expeced spo reurn, i depends on relaive uncerainy in he home and foreign counries, as measured by he difference beween he home and foreign surplus consumpion raios. However, i also depends on he local curvaure of he uiliy funcion a he seady sae funcion X γ and in paricular wheher i is greaer or less han uniy. I is sraighforward o provide an inerpreaion of he ime-varying risk premium of equaion (30) in erms of he properies of he pricing kernels ha price nominal asses in each counry. Backus, Foresi and Telmer (00) 8 show ha for log-normal pricing kernels such as ours, he risk premium is proporional o he difference beween he condiional variances of he log of he home and foreign pricing kernels. I is clear ha equaion (30) can be reparamaerised in hose erms. I also provides a new undersanding of he surplus consumpion raio in his model: i is proporional o he condiional variance of he pricing kernel. We defineσ as he uncondiional variance of x, =,. Given he definiion of X in Eq. x (8) i is easy o show ha he uncondiional variance of he forward discoun, spo reurn and expeced forward profi using Eqs. (4), (7) and (30) are 9 : ρσ var( f ) ( ) ρ u s = + γ φ σσ v x (3) 3

15 σ u γ var( s+ s) = + ( φ) γ σ + x σv ρ X (3) and X var( f E s ) = ( φ) γ σ γ + x (33) respecively. We now summarize he impac of he habi specificaion on relaive and absolue volailiies in he following proposiion 0 : Proposiion (Volailiy) Under he assumpions of (i) addilog uiliy for he sandard case, (ii) he forcing processes in Eqs. (0) and (), (iii) ha he mean surplus consumpion raio X < and (iv) he local γ curvaure of he uiliy funcion a he seady sae >, Campbell and Cochrane (999) X habis : (a) increase he absolue volailiies of (i) spo reurn, (ii) he forward discoun and (iii) expeced forward profi (b) increase he volailiy of spo reurn by (i) more han he increase in he volailiy of he forward discoun and (ii) more han he increase in he volailiy of expeced forward profi (c) increase he volailiy of expeced forward profi by more han he increase in he volailiy of he forward discoun. Proposiion shows ha he habi specificaion moves he hree volailiies in he desired absolue and relaive direcions. Wheher his is enough o mach he daa depends on he 4

16 relaive imporance of real and moneary shocks. The less volaile he moneary shocks, he greaer he imporance of habis. The nex proposiion relaes o he effec of he habi specificaion on he persisence of he spo reurn and he forward discoun. We measure persisence for a ime series y as ( y y ) ( y ) Cov Var, is firs order auocorrelaion coefficien. The firs order auocorrelaion coefficien of he forward discoun and he spo reurn are : ρσu ρ + φγ( φ) σσ ρ ρσ u + γ( φ) σ vσ x ρ v x (34) and σ u ρ + ( )( ) ( ) σ vγ φ γ γ φ σ x ρ X σ u γ + γ ( φ) σ + x σv ρ X (35) respecively. We are now ready o sae Proposiion 3 : Proposiion (Persisence) Under he assumpions of (i) addilog uiliy for he sandard case, (ii) he forcing processes in Eqs. (0) and (), (iii) φ > ρ and (iv) erms inσ v are small in relaion o erms in boh σ u and σ x, Campbell and Cochrane (999) habis: (a) increase he persisence of he forward discoun (b) decrease he persisence of he spo reurn. 5

17 Proposiion can be explained easily by reference o Eqs. (4) and (7). In he sandard addilog case, he only persisen elemen deermining boh spo reurn and he forward premium is expeced money growh differenials. Consequenly, hey boh share he firs order auocorrelaion coefficien of money growh, ρ. Under he habi specificaion, he forward discoun in Eq. (4) also depends on he log surplus consumpion raios. This is because he expeced IMRS depends on he level of he surplus consumpion raios. Since heir firs order auocorrelaion coefficien φ is close o uniy, he persisence of he forward discoun rises. By conras, under he habi specificaion, spo reurn in Eq. (7) depends on he IMRS iself. The IMRS is a funcion of he change in he log surplus consumpion raio, he firs-order auocorrelaion coefficien of which is negaive. Consequenly, he persisence of he spo reurn falls. Backus, Gregory and Telmer (993) and Bekaer (996) argue ha he forward discoun bias can be summarized by he inabiliy of he sandard model o explain he high volailiy of expeced forward speculaive profi. The inroducion of our habi specificaion increases his volailiy bu is i enough? Backus, Foresi and Telmer (995) remind us ha he slope coefficien, b, from a regression of he spo reurn on he forward discoun can be wrien as b cov( Es + s, f Es + ) + var( Es + s) = var( f s ) (36) Many sudies repor negaive esimaes of his coefficien and hus he bias puzzle. For his o happen he covariance erm mus be (i) negaive and (ii) larger in absolue value han he variance erm in he numeraor. Poin (i) is Fama s firs necessary condiion and poin (ii) is equivalen o Fama s second condiion: ha he variance of expeced forward profi exceeds he variance of he expeced spo reurn. Mos wriers concenrae on he firs of Fama s necessary condiions (See, for example, Akeson e al., 999). In Proposiion 3 below, we show ha he habi specificaion, unlike he sandard addilog model, indeed delivers his 6

18 resul. The reason for his is sraighforward. In he sandard model, he expeced forward profi is, in fac, a consan and herefore is uncorrelaed wih expeced spo reurns. Wih he habi specificaion, boh expeced spo reurn, Eq. (9) as well as he expeced forward profi, Eq. (30), depend on x x, he difference beween he home and foreign log surplus consumpion raios. I has already been suggesed ha his measures relaive uncerainy in he home and foreign counries. When uncerainy is higher in he home counry, a depreciaion is expeced. The opposie happens o forward profi. The reason for his is embedded in Proposiion (b) (i): spo reurns are more sensiive o habis han he forward discoun. Consequenly, f E ( s + ) is dominaed by movemens in E ( s + ), which eners he expeced forward profi wih he opposie sign o which i appears in E ( s + ) s. However, we also show ha he second of Fama s condiions is violaed by he habi specificaion. This violaion is so significan ha i compleely overshadows he success in obaining he negaive covariance o he exen ha he coefficien slope b becomes larger han uniy. We are now ready o sae Proposiion 3 4 : Proposiion 3 (unbiasedness) Under he assumpions of (i) addilog uiliy for he sandard case, (ii) he forcing processes in Eqs. (0) and (), (iii) he local curvaure of he uiliy funcion a he seady sae γ > 5 X, Campbell and Cochrane (999) habis imply ha: (a) he uncondiional covariance beween expeced spo reurn and expeced forward profi is negaive (b) he uncondiional variance of expeced spo reurn is greaer han he variance of expeced forward profi (c) he slope coefficien from a regression of he spo reurn on he forward discoun exceeds uniy. 7

19 The reason why his perverse resul occurs is relaed o our success in increasing he volailiy of he spo reurn: oo much of he increase is accouned for by he volailiy of he expeced spo reurn. We require he volailiy of spo reurn o exceed he volailiy of expeced forward profi bu his mus, in urn, exceed he volailiy of expeced spo reurn. All of he resuls in his sub-secion depend on he specific error processes in Eqs. (0) and (). In realiy, home and foreign money and consumpion growh do no follow AR() processes wih scalar variance covariance marices. In he nex secion, we calibrae he model o processes esimaed from he daa and re-evaluae he resuls. 3. Empirical and model evidence 3. Some sylized facs abou he forward exchange marke Hodrick (987), Backus, Gregory and Telmer (993) and Bekaer (996) presen he sylized facs for a number of bilaeral exchange raes. In order o remind he reader of hese facs we presen some quarerly saisics over he period 973:-000:4 for bilaeral exchange raes beween he US dollar, serling and he yen in Table. In he inroducion we discussed hree ses of well-known facs abou he forward exchange rae marke. The firs is ha he sandard deviaion of he forward discoun is smaller han ha of expeced forward speculaive profi 6 and is an order of magniude less han ha of he spo reurn. The second is ha while he spo reurn is no very persisen, he AR() coefficien of he forward discoun is high. The hird fac is ha he coefficien on he forward discoun from a regression for predicing spo reurn, b, is usually he wrong sign (negaive). 8

20 3. Calibraion and resuls from baseline parameerizaion There is no closed form soluion for he firs wo non-linear sochasic raional expecaions efficiency condiions, i.e. Eq. (). Thus we find an approximae soluion using a linear-quadraic mehod of undeermined coefficiens, see Chrisiano (99) 7. This yields opimal linearised rules for domesic and foreign bond prices (and hus nominal ineres raes) as funcions of he consumpion and money growh shocks and he log of surplus consumpion raio. Spo and forward exchange raes can easily be calculaed using Eqs. (3), () and (3). We calibrae he model discussed in Secion and compare he momens generaed from he model wih hose ypically found in quarerly daa. Two approached are adoped. Firs, we assume very simple AR() ime series processes for he exogenous consumpion and money growh shocks, as in Secion.3 above. In his case he baseline parameers are aken sraigh from exising lieraure. This allows us o show ha he predicions from he simulaed linearquadraic approximaion are indeed wha are heoreically derived in Secion.3 above. Second, we use daa on consumpion and money growh for he U.S., U.K. and Japan and esimae bivariae vecor auoregressions. We calibrae he model using a VAR from each bilaeral relaionship and compare he momens generaed from he model wih hose found wih each bilaeral exchange rae. We presen he baseline parameerizaion in Table. The discoun rae, β, is assumed o be (.03) -.5 which is based on an annual real rae of ineres of 3 percen; a value commonly used in he lieraure (see for example Chrisiano, 99). This parameer remains consan in he various experimens we simulae. The elasiciy of ineremporal subsiuion, /γ, and he AR() coefficien of he log of he surplus consumpion raio, φ, have maor effecs in he habi persisence model. In he baseline parameerizaion we se he elasiciy of 9

21 ineremporal subsiuion equal o 5 (γ=0.). The value of γ is low compared o Campbell and Cochrane (999): heir lowes value for his parameer is 0.7. However, even wih γ=0., γ i is worh remembering ha he local curvaure of he uiliy funcion is = 3.8. In X Campbell and Cochrane (999) his is The AR() coefficien of log surplus consumpion equals o 0.97, which is used in Campbell and Cochrane (999). Laer in his secion we perform sensiiviy analysis and examine how he resuls change when he elasiciy of ineremporal subsiuion varies beween 0 and 0.5 and when we change he AR() coefficien of he log of he surplus consumpion raio from 0.9 o The parameers of he exogenous endowmen and money growh rae AR() processes are also aken from he lieraure. Campbell and Cochrane (999) use U.S. real consumpion expendiure on non-durables and services o proxy for endowmens. They assume ha consumpion growh is a whie noise process wih a non-zero mean esimaed o be 0.44% per quarer. They esimae ha he sandard deviaion of shocks o consumpion growh is 0.56%. Chrisiano (99) esimaes he mean of U.S. base money growh o be.6% per quarer. He also esimaed he firs-order auocorrelaion coefficien o be 0.3 and he sandard deviaion of he shock o money growh o be 0.38%. We simulaed he sandard and habi models,000 imes generaing 0 observaions for each series. The resuls from he baseline parameerizaions for he momens of ineres are presened in Table 3. We presen he mean of he simulaed momen and is sandard error (in parenhesis). Compare he resuls in Table 3 o Proposiion. The laer claims ha he habi specificaion increases he hree volailiies, in absolue erms. A comparison of he columns habi preferences and sandard preferences shows ha his is indeed he case. In paricular, noe ha he simulaed volailiies have very low sampling sandard errors. Proposiion also claims ha habis increase spo reurn volailiy by more han he oher wo 0

22 and increases expeced forward profi volailiy by more han he volailiy of he forward discoun. This clearly occurs and he ranking of he hree volailiies in he habi model is consisen wih he sylised facs in Table. Proposiion claims ha Campbell and Cochrane (999) habis raise he persisence of he forward discoun and lower ha of he spo reurn. From Table 3, he resuls from he simulaions demonsrae ha his occurs. A comparison wih he sylised facs in Table shows ha boh he simulaed AR() coefficiens are wihin wo sandard errors of he esimaes. Finally, Proposiion 3 argues ha in conras o he sandard case where he covariance beween forward profi and expeced spo reurn is zero, he habi specificaion delivers a negaive covariance. The row headed Covariance in Table 3 shows he average simulaed value of his covariance and is sandard error in he sample. The row headed Negaive covariances in Table 3 shows he percenage ha is negaive. In he sandard case, abou one-half of he covariances are negaive. By conras, he habi preferences deliver negaive covariances more han 90% of he ime. Finally, as prediced by Proposiion 3, Campbell-Cochrane (999) preferences give rise o a perverse resul in he forward rae regression. The sandard model suggess ha his slope is uniy in conras o he negaive values ha we see in he daa (see Table ). Because of he fac ha he volailiy of expeced spo reurns is so high in he habi model, he average simulaed value of his slope is greaer han uniy as demonsraed by Proposiion 3. I is worhwhile showing he effec of varying γ, he inverse of he ineremporal elasiciy of subsiuion on he exchange rae volailiies. This is illusraed in Fig. where he elasiciy varies from 0 (γ = 0.) o 0.5 (γ = ). As he elasiciy of ineremporal subsiuion falls, he lack of ineremporal subsiuion opporuniies means ha i is more difficul o diffuse he effec of shocks over differen ime periods. Consequenly, all exchange rae volailiies rise. Consider he effec of varying he habi persisence parameer. This is illusraed in Fig.. The AR() coefficien of he log of he surplus consumpion raio, φ, is allowed o vary from 0.9

23 o As φ increases, he variabiliy of he change in he surplus consumpion raio declines. Consequenly, he variabiliy of he ineremporal marginal rae of subsiuion, and herefore he hree exchange rae volailiies decline as φ approaches uniy 8. Anoher way of hinking abou i is ha he seady sae surplus consumpion raio, X, rises as φ rises. As he seady sae surplus consumpion raio rises, he local curvaure of he uiliy funcion, X γ, falls and herefore exchange rae volailiies decline. 3.3 Calibraion and resuls from alernaive parameerizaions In his secion we use daa on consumpion and money growh for he US, UK and Japan o esimae parameers from bivariae vecor auoregressions. We use quarerly seasonally adused daa on real consumpion expendiure on non-durables and services and he nominal moneary base. The daa on consumpion and money are obained from he Federal Reserve Bank of S. Louis for he US and from Daasream for he UK and Japan. The sample period is 973:-000:. The uncondiional means of consumpion and money growh for he US, UK and Japan are presened in Table 4. We use hese in furher simulaions of boh he sandard and habi models. We esimaed hree versions of he following vecor auoregression: W = Φ + ΦW + U, U ~ MN(0, Σ), (37) where W = ( µ, µ, π, π ) andu = ( v, v, u, u ). The hree versions used daa for US-UK, US-Japan and UK-Japan consumpion and money growh wih one lag in each of he variables. We performed likelihood raio ess for various resricions on (37) for each counry pair and presen p-values in Table 5. We es he following hree hypoheses on he coefficiens in Φ : wheher real growh variables affeced nominal variables (and vice versa),

24 wheher domesic variables affeced foreign variables (and vice versa) and wheher Φ is diagonal. In he op half of Table 5 we allow Σ o be unresriced and in he boom half he covariance marix was resriced o conform o he resricions on Φ. The UK-US daa do no reec he resricion ha he slope coefficiens on money growh in he consumpion growh equaions are zero (and vice versa) and ha he money growh shocks are uncorrelaed wih consumpion growh shocks. By conras, he US-Japanese daa do no reec he resricion ha he slope coefficiens on foreign growh variables in he domesic growh equaions are zero (and vice versa) and ha foreign growh shocks are uncorrelaed wih domesic growh shocks. Finally, we were unable o find any valid resricions on he UK-Japanese vecor auoregression. The remaining parameers in he simulaions are given he same value as in he baseline parameerisaion. We use he esimaed Φ and Σ based on he resricions discussed in he above o simulae hree versions of he model 9. The simulaed resuls are presened in Table 6. The resuls are very similar o hose presened for he baseline parameerizaion even hough we use hree differen ses of parameers for he exogenous processes. We noe ha he volailiy of he forward discoun increases relaive o he baseline resuls. This is due o he fac ha he real ineres rae is no consan. 4. Conclusion This paper has proposed a modelling sraegy ha makes subsanial progress owards resolving many of he ousanding puzzles in respec of he forward foreign exchange marke. A model ha combines habi persisence and a consan real rae of ineres in a moneary framework is capable of explaining mos of he volailiy, persisence and forward bias puzzles. 3

25 However, Campbell and Cochrane (999) habis ulimaely fail o explain he forward bias puzzle because hey make he volailiy of expeced spo reurns oo high. The main consequence of his failure is ha he regression slope in he forward marke equaion is greaer han uniy raher han negaive as is empirically found. I is ineresing ha Bekaer (996) repors ha he was unable o generae a single simulaion ha gave rise o a negaive esimae of his slope coefficien. Since he was also modelling habi persisence i would be worh revisiing his work o deermine if he was also obaining our perverse resul. Backus, Foresi and Telmer (00) have already noed he possibiliy ha oherwise sensible models can be creaed where his can occur. Ineresingly, hey find i in a wo-currency version of Cox, Ingersoll and Ross (985). Alvarez, Akeson and Kehoe (999) claim o have resolved he forward bias anomaly. They inroduce rading fricions by relaxing he assumpion of complee markes and by allowing he exen of he rading fricion o be endogenously deermined. Their sraegy is as follows: asse markes and goods markes are segmened in he sense ha agens mus pay a fixed cos o ransfer money beween he wo markes. This segmenaion is endogenous because, in equilibrium, some agens chose o pay he fixed cos while some do no. They derive sufficien condiions for one of Fama s necessary condiions for he forward premium anomaly viz., ha he covariance beween expeced forward speculaive profis and expeced spo rae changes be negaive. They do no deermine he implicaion of heir model for Fama s second necessary condiion: ha he variance of expeced forward speculaive profis exceed he variance of expeced spo rae changes. This paper shows he imporance of his condiion. Overall, i is difficul o come o any definie conclusion abou heir model a his sage because hey do no simulae i. A few research direcions are immediaely suggesed. I would be ineresing o apply his approach o nominally denominaed asse prices in closed economies. There are many issues 4

26 unresolved in inernaional real business cycle heory, such as he behaviour of real exchange raes. I would be worhwhile o examine wheher his model cass any ligh on hem. A producion secor could be included in he models discussed in he paper. This is common in mos inernaional real business cycle models. Raher han using binding cash-in-advance consrains money could be inroduced ino he model in a manner ha does no impose he counerfacual uni consumpion velociy of money. 30 Finally, any fuure research mus focus explicily on his issue: how can he high uncondiional volailiy of spo reurns be explained wihou excessively high volailiy of expeced excess spo reurns condiional on he forward discoun? In oher words, high spo reurn volailiy mus come from high volailiy in spo reurn surprises. Acknowledgemens We are graeful o John Cochrane and Richard Friberg for heir commens. This paper also benefied from he conribuions of seminar paricipans a he Universiy of Srahclyde, he Universiies of California a boh Berkeley and Sana Cruz, Queen s Universiy, Belfas, he Naional Universiy of Ireland a Maynooh, Cork and Galway and he Dublin Economics Workshop. We would like o hank wo anonymous referees for helpful commens. All errors remain our own. References Abel, A., 990. Asse prices under habi formaion and caching up wih he Joneses. American Economic Review, Papers and Proceedings 80, Alvarez, F., Akeson, A., Kehoe, P., 999. Volaile exchange raes and he forward premium anomaly: a segmened asse marke view. Mimeo, Federal Reserve Bank of Minneapolis. 5

27 Backus, D.K., Foresi, S., Telmer, C.I., 995. Inerpreing he forward premium anomaly. Canadian Journal of Economics 8, Backus, D.K., Foresi, S., Telmer, C.I., 00. Affine erm srucure models and he forward premium anomaly. Journal of Finance 56, Backus, D.K., Gregory, A.W., Telmer, C.I., 993. Accouning for forward raes in markes for foreign currency. Journal of Finance 58, Bekaer, G., 996. The ime-variaion of risk and reurn in foreign exchange markes: a general equilibrium perspecive. Review of Financial Sudies 9, Campbell, J.Y., Cochrane, J.H., 999. By force of habi: a consumpion-based explanaion of aggregae sock marke behaviour. Journal of Poliical Economy 07, 05-5 Chrisiano, L.J., 99. Modelling he liquidiy effec of a money shock. Federal Reserve Bank of Minneapolis Quarerly Review (winer), Cox, J., Ingersoll, J., Ross, S., 985. A heory of he erm srucure of ineres raes. Economerica 53, Duesenberry, J.S., 949. Income, saving and he heory of consumer behaviour. Harvard Universiy Press, Cambridge, MA. Engel, C., 99. On he foreign exchange risk premium in a general equilibrium model. Journal of Inernaional Economics 3, Engel, C., 996. The forward discoun anomaly and he risk premium: a survey of recen evidence. Journal Of Empirical Finance 3, 3-9. Engel C., 999. On he foreign exchange risk premium in sicky price general equilibrium models, in Inernaional Finance and Financial Crises: Essays in Honor of Rober P. Flood, Peer Isard, Assaf Razin and Andrew Rose, eds. (IMF and Kluwer),

28 Fama E., 984. Forward and spo exchange raes. Journal of Moneary Economics 4, Flood, R.P., Rose, A.K., 999. Undersanding exchange rae volailiy wihou he conrivance of macroeconomics, Economic Journal, 09, Hodrick, R.J., 987. The empirical evidence on he efficiency of forward and fuures foreign exchange markes. Harwood Academic Publishers, Swizerland. Lungqvis, L., Uhlig, H., 000. Tax policy and aggregae demand managemen under caching up wih he Joneses. American Economic Review 90, Lucas, R.E., 98. Ineres raes and currency prices in a wo-counry world. Journal of Moneary Economics 0, Moore, M.J., 00. Covered Purchasing Power Pariy, Ex-Ane PPP and Risk Aversion. Journal of Business Finance and Accouning 4, Moore, M.J., Roche, M.J., 00. Liquidiy in he forward exchange marke. Journal of Empirical Finance 8, Obsfeld, M. and K. Rogoff, 998, Risk and Exchange Raes, NBER working paper no Weil, P, 989. The equiy premium puzzle and he risk-free rae puzzle, Journal of Moneary Economics 4,

29 Table Properies of forward and spo exchange raes US-UK US-Japan UK-Japan Volailiy Spo reurn 5.44% 6.% 6.% Expeced forward profi.75%.85%.58% Forward discoun 0.7% 0.7% 0.47% Persisence Spo reurn 0.09 (0.09) Forward discoun 0.74 (0.08) 0.3 (0.09) 0.80 (0.08) 0.9 (0.09) 0.65 (0.0) Unbiasedness Poin esimae of b -.43 (0.89) (0.76) -4.4 (0.93) The daa are quarerly and obained from Daasream. The sample period for all spo exchange raes is 973:-000:4. The sample period for he US-UK forward exchange raes is 976:-000:4. The sample period for he Japan-US forward exchange raes is 978:3-000:4. Volailiy is measured by he sandard deviaion. Persisence is measure by he firs-order auocorrelaion coefficien. The expeced forward profi is calculaed as he fied value from regressing realized profis on he lagged forward premium. The coefficien b is esimaed from regressing he spo reurn on he lagged forward premium. HAC sandard errors are in parenhesis. Table Baseline parameerizaion Discoun facor β Curvaure of he uiliy funcion γ 0.0 Persisence of he log surplus-consumpion raio φ 0.97 Consumpion growh Money growh Uncondiional mean 0.44%.60% AR() coefficien Sandard deviaion of shock 0.56% 0.38% 8

30 Table 3 Momens in he heoreical economy using baseline parameerizaion Sandard Preferences Habi Preferences Volailiy Spo reurn 0.84% (0.00) 9.85% (0.6) Expeced forward profi 0.07% (0.00).50% (0.04) Forward discoun 0.8% (0.000) 0.8% (0.04) Persisence Spo reurn 0.3 (0.003) Forward discoun 0.30 (0.003) (0.005) 0.77 (0.006) Unbiasedness Covariance (0.000) ( ) Negaive covariances 48.50% 93.50% Mean of b (0.00) (0.06) The resuls presened are he mean of,000 replicaions of 0 observaions using he baseline parameerizaion in Table. The sandard error of he mean is repored in parenhesis. Volailiy is measured by he sandard deviaion. Persisence is measure by he firs-order auocorrelaion coefficien. The expeced forward profi is calculaed as he fied value from regressing realized profis on he lagged forward premium. The coefficien b is esimaed from regressing he spo reurn on he lagged forward premium. Table 4 Uncondiional means of consumpion and money growh raes US UK Japan Consumpion growh 0.75% 0.58% 0.69% Money growh.89%.70%.0% Quarerly daa on real consumpion expendiure on non-durables and services and he money base are obained from he Federal Reserve Bank of S. Louis for he US and from Daasream for he UK and Japan. The sample period is 973:-000:. 9

31 Table 5 Probabiliy values for likelihood raio ess on a VAR() model of consumpion and money growh US-UK US-Japan UK-Japan Unresriced covariance marix VAR() wih no real-nominal coefficiens VAR() wih no cross-counry coefficiens Simple AR() processes Resriced covariance marix VAR() wih no real-nominal coefficiens VAR() wih no cross-counry coefficiens Simple AR() processes Quarerly daa on real consumpion expendiure on non-durables and services and he money base are obained from he Federal Reserve Bank of S. Louis for he US and from Daasream for he UK and Japan. The sample period is 973:-000:. VAR() wih no real-nominal coefficiens is a vecor auoregression where money growh does no effec consumpion growh and vice versa. VAR() wih no cross-counry coefficiens is a vecor auoregression where domesic growh variables do no effec foreign growh variables and vice versa. 30

32 Table 6 Momens in he heoreical economies Sandard preferences Volailiy Spo reurn.4% (0.004) Expeced forward profi 0.0% (0.00) Forward discoun 0.64% (0.00) Habi preferences US-UK US-Japan UK-Japan US-UK US-Japan UK-Japan 3.6% (0.007) 0.4% (0.006) 0.46% (0.000).93% (0.006) 0.% (0.005) 0.48% (0.000) 9.49% (0.00).09% (0.039).04% (0.09) 9.87% (0.05).3% (0.04).00% (0.0) 9.56% (0.0).% (0.04).00% (0.0) Persisence Spo reurn 0.36 (0.003) Forward discoun 0.47 (0.003) 0. (0.003) 0. (0.003) 0.08 (0.003) 0.04 (0.003) -0.0 (0.005) 0.66 (0.005) -0.0 (0.004) 0.58 (0.007) (0.00) 0.46 (0.009) Unbiasedness Covariance (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) Negaive covariances 4.70% 53.70% 54.0% 83.80% 88.50% 87.60% Mean of b 0.96 (0.006) 0.93 (0.0).00 (0.08).79 (0.03). (0.036).07 (0.035) The resuls presened are he mean of,000 replicaions of 0 observaions using parameers for he forcing processes described in Secion 3. The sandard error of he mean is repored in parenhesis. Volailiy is measured by he sandard deviaion. Persisence is measure by he firs-order auocorrelaion coefficien. The expeced forward profi is calculaed as he fied value from regressing realized profis on he lagged forward premium. The coefficien b is esimaed from regressing he spo reurn on he lagged forward premium. 3

33 Figure. The effecs of changing γ on he volailiies of exchange raes 45% Sandard Deviaion 40% 35% 30% Spo Reurn Forward Discoun Expeced Profi 5% 0% 5% 0% 5% 0% Gamma Figure : The effecs of changing φ on he volailiies of exchange raes 30% Sandard Deviaion 5% 0% Spo Reurn Forward Discoun Expeced Profi 5% 0% 5% 0% Phi 3

34 Endnoes See Abel (990) and Duesenberry (949). For discussions of forward marke puzzles, see Bekaer (996), Engel (996) and Hodrick (987). 3 Expeced forward speculaive profis are ofen referred o as he risk premium. This is, in general, incorrec because of non-convexiies: see Engel (999). 4 For a recen aemp o address he forward bias puzzle, see Obsfeld and Rogoff (998). For a horough criique of ha paper s conribuion o resolving he puzzle, see Engel (999). 5 The Campbell-Cochrane model, like he one used here, is an exchange economy. Lungqvis and Uhlig (000) have poined ou ha here are problems in expanding his framework o a producion economy. Consumpion bunching raher han consumpion smoohing becomes welfare opimal. However, his only arises if he habi is inernalised. 6 The superscrip denoes counry of origin and he subscrip denoes counry of use. Uppercase leers denoe variables in levels; lowercase leers denoe variables in log levels, including growh and ineres raes. Greek leers wihou ime subscrips denoe parameers. Bars over variables denoe seady saes. 7 From his poin he counry of use subscrip is suppressed for ease of noaion 8 As usual, he price of bonds is ( + i ) =,. 9 Throughou he remainder of he paper, he phrase sandard model or sandard addilog model is used o mean addilog preferences i.e. Eq. () wih H = 0 for i=, and =,. 0 Defining risk aversion in a muli-good model is no rivial (see Engel, 99 and Moore, 997). An ineremporal model has as many goods as ime periods. In addiion, our model has wo goods in each ime period. We evade his problem by only considering is value a he seady sae. An alernaive sraegy would be o specify a single habi in a baske of home and foreign goods. We explored his possibiliy bu were unable o develop he Campbell-Cochrane propery of consan real ineres raes. The main reason for his goes back o Lucas (98) who poined ou ha here is no sraighforward definiion of he real ineres rae in a muligood conex. In principle all of he assumpions are relaxed for he calibraions of Secion 3. 3 Engel (99) argues ha he empirical covariance beween real and nominal shocks is low or zero. 4 The proofs of expressions used hroughou he paper are derived in a echnical appendix ha is available from 5 For proof, see Secion B, equaion (B) in he echnical appendix. 6 For proof, see Secion A, equaion (A6) in he echnical appendix. 7 For proof, see Secion C in he echnical appendix. 8 See heir equaion (4). 9 For proof see Secion D in he echnical appendix. 0 For proof see Secion D in he echnical appendix. Assumpions (iii) and (iv) are no very resricive, as he mean surplus consumpion raio is ypically a small posiive fracion. For example, in Campbell and Cochrane (999), i is approximaely.08. For proof see Secion E in he echnical appendix. 3 For proof see Secion E in he echnical appendix. 4 For proof see Secion F in he echnical appendix. 5 Noe ha he condiion on he local curvaure of he uiliy funcion is less sringen han i 33