Country Portfolios in Open Economy Macro Models *

Transcription

1 Federal Reserve Bank of Dallas Globalzaton and Monetary Polcy Insttute Workng Paper No. 9 Country Portfolos n Open Economy Macro Models * Mchael B. Devereux Unversty of Brtsh Columba CEPR Globalzaton and Monetary Polcy Insttute, Federal Reserve Bank of Dallas Alan Sutherland Unversty of St Andrews CEPR Aprl 2008 Abstract Ths paper develops a smple approxmaton method for computng equlbrum portfolos n dynamc general equlbrum open economy macro models. The method s wdely applcable, smple to mplement, and gves analytcal solutons for equlbrum portfolo postons n any combnaton or types of asset. It can be used n models wth any number of assets, whether markets are complete or ncomplete, and can be appled to stochastc dynamc general equlbrum models of any dmenson, so long as the model s amenable to a soluton usng standard approxmaton methods. We frst llustrate the approach usng a smple two-asset endowment economy model, and then show how the results extend to the case of any number of assets and general economc structure. JEL codes: E52, E58, F41 * Mchael B. Devereux, Department of Economcs, Unversty of Brtsh Columba, East Mall, Vancouver, B.C. Canada V6T 1Z1. devm@nterchange.ubc.ca. Alan Sutherland, School of Economcs and Fnance, Unversty of St Andrews, St Andrews, Ffe, KY16 9AL, UK. ajs10@st-and.ac.uk. We are grateful to Phlp Lane, Klaus Adam, Perpaolo Bengno, Ganluca Bengno, Berthold Herrendorf, Fabrzo Perr, Robert Kollmann, Gancarlo Corsett, Morten Ravn, Martn Evans and Vktora Hnatkovska for comments on an earler draft of ths paper. Ths research s supported by the ESRC World Economy and Fnance Programme, award number Devereux also thanks SSHRC, the Bank of Canada, and the Royal Bank of Canada for fnancal support. The vews n ths paper are those of the authors and do not necessarly reflect the vews of the Federal Reserve Bank of Dallas or the Federal Reserve System.

2 1 Introducton Ths paper develops a smple and tractable approach to computng equlbrum nancal asset portfolos n open economy dynamc stochastc general equlbrum (DSGE) models. To a large extent, exstng open economy macroeconomc models gnore portfolo composton, analyzng nancal lnkages between countres n terms of net foregn assets, wth no dstncton made between assets and labltes. But recent research has hghlghted the presence of large cross-country gross asset and lablty postons, and consderable heterogenety among countres n portfolo composton among d erent classes of assets. Lane and Mles-Ferrett (2001, 2006) show that these gross portfolo holdngs have grown rapdly, partcularly n the last decade. Ther measures show that even large countres such as the UK hold gross assets and labltes that are multples of GDP. The growth n nternatonal nancal portfolos rases a number of mportant questons for open economy macroeconomcs. What are the determnants of the sze and composton of gross portfolo postons? Can standard theores account for the observed structure of portfolo holdngs? Moreover, the large sze of gross postons makes t lkely that the portfolo composton tself a ects macroeconomc outcomes. Wth gross postons as large as GDP, unantcpated changes n exchange rates or asset prces can generate valuaton e ects that are the same order of magntude as annual current accounts 1. Ths rases questons about how portfolo composton may a ect the nternatonal busness cycle and nternatonal transmsson of shocks. Fnally, by generatng sgn cant wealth re-dstrbutons n response to uctuatons n exchange rates and asset prces, nternatonal portfolo composton may have sgn cant mplcatons for economc polcy. How should monetary and scal polces be desgned n an envronment of endogenous portfolo choce? Whle these questons are obvously of nterest to open economy macroeconomsts and polcymakers, current theoretcal models and soluton methods cannot answer them n any very systematc way. Ths s because the standard approaches to solvng general equlbrum models make t d cult to ncorporate portfolo choce. The usual method of analyss n DSGE models s to take a lnear approxmaton around a non-stochastc steady 1 Lane and Mles-Ferrett (2001) emphasze the quanttatve mportance of valuaton e ects on external assets and labltes. See also subsequent work by Ghron et al. (2005), Gournchas and Rey (2005), and Tlle (2003, 2004). 1

3 state. But optmal portfolos are not unquely de ned n a non-stochastc steady state, so there s no natural pont around whch to approxmate. Moreover, portfolos are also not de ned n a rst-order approxmaton to a DSGE model, snce such an approxmaton sats es certanty equvalence, so all assets become perfect substtutes. As a result, the analyss of portfolo choce n DSGE models appears to be ntractable n all but the most restrcted of cases. 2 In ths paper we develop and present an approxmaton method whch overcomes these problems. Our method can be appled to any standard open economy model wth any number of assets, any number of state varables, and complete or ncomplete markets, so long as the model s amenable to soluton by the usual approxmaton methods. We nd a general formula for asset holdngs whch can be very easly ncorporated nto the standard soluton approach for DSGE models. The technque s smple to mplement and can be used to derve ether analytcal results (for su cently small models) or numercal results for larger models. A key feature of our approach s to recognze that, at the level of approxmaton usually followed n open economy macroeconomcs, one only requres a soluton for the steady-state portfolo holdngs. The steady state portfolo s de ned as the constant (or zero-order ) term n a Taylor seres approxmaton of the true equlbrum portfolo functon. Hgher-order aspects of portfolo behavour are not relevant for rst-order accurate macro dynamcs. Equvalently, tme varaton n portfolos s rrelevant for all questons regardng rst-order responses of macroeconomc varables lke consumpton, output, real exchange rates, etc. n a DSGE model. Therefore, the soluton we derve exhausts all the macroeconomc mplcatons of portfolo choce at ths level of approxmaton. How do we obtan the zero-order component of the equlbrum portfolo? We do so 2 If there are enough nancal assets to allow perfect rsk sharng (so that nternatonal nancal markets are e ectvely complete) then the problem becomes somewhat easer. In ths case, t s possble to dentfy an equlbrum macroeconomc allocaton ndependent of nancal structure, and then, gven ths allocaton, one can derve the mpled portfolos whch support the equlbrum. Engel and Matsumoto (2005) and Kollmann (2006) represent examples of such an approach. However, when markets are ncomplete (n the sense that there are not su cent assets to allow perfect rsk sharng) optmal portfolos and macroeconomc equlbrum must be derved smultaneously. Ths makes the problem consderably more d cult. Heathcote and Perr (2004) provde one example of an ncomplete markets model n whch t s possble to derve explct expressons for equlbrum portfolos. Ther model s, however, only tractable for a spec c menu of assets and for spec c functonal forms for preferences and technology. 2

4 usng a combnaton of a second-order approxmaton of the portfolo selecton condton wth a rst-order approxmaton to the remanng parts of the model. Of course, these two approxmatons wll be nterdependent; the endogenous portfolo weghts wll depend on the varance-covarance matrx of excess returns produced by the general equlbrum model, but that n turn wll depend on the portfolo postons themselves. We show that ths smultaneous system can be solved to gve a smple closed-form analytcal soluton for the equlbrum portfolo. Whle our soluton procedure s novel, the mathematcal foundatons of the soluton we derve are already establshed n the lterature, n partcular n the work of Samuelson (1970), and n d erent form by Judd (1998) and Judd and Guu (2001). Samuelson shows how a mean-varance approxmaton of a portfolo selecton problem s su cent to dentfy the optmal portfolo n a near-non-stochastc world. In a related paper, Judd and Guu show how the same equlbrum can be dent ed by usng a combnaton of a Bfurcaton theorem and the Implct Functon Theorem. Our soluton approach reles on rst-order and second-order approxmatons of the model, rather than the Implct Functon and Bfurcaton Theorems, but the underlyng theory descrbed by Judd and Guu (2001) s applcable to our equlbrum soluton. In partcular, the steady-state portfolo derved usng our technque corresponds to a bfurcaton pont n the set of nonstochastc equlbra. The man contrbuton of ths paper s to show how ths soluton can easly be derved n standard DSGE models. We note n addton, that there s nothng about the approxmaton method that restrcts ts use to open economy models. It can be appled to any heterogeneous agent DSGE model, whether n a closed or open economy context. 3 As we have already stated, the steady-state portfolo s all that s needed n order to analyze the rst-order propertes of a general equlbrum model. But for many purposes, t may be useful to analyze the dynamcs of portfolo holdngs themselves. In addton, n order to do welfare analyss, t s usually necessary to analyze a second-order approxmaton of a model. At the level of second-order approxmaton, tme varaton n portfolos becomes relevant for macroeconomc dynamcs. But these features can be obtaned by an extenson of our method to hgher-order approxmatons of the model. In 3 Samuelson (1970) and Judd and Guu (2001) dd not develop ther results n open economy (or general equlbrum) contexts. 3

5 partcular, the state-contngent, or rst-order aspects of the equlbrum portfolo, can be obtaned by combnng a thrd-order approxmaton of the portfolo selecton equatons, wth a second-order approxmaton to the rest of the model. The current paper focuses on the dervaton of steady-state portfolos because ths represents a dstnct and valuable rst-step n the analyss of portfolo choce n open-economy DSGE models. We do, however, dscuss bre y the extenson of the method to hgher orders. In a companon paper, Devereux and Sutherland (2007), we show how hgher-order solutons to portfolos also have an analytcal representaton. In the related lterature a number of approaches have been developed for analysng portfolo choce n ncomplete-markets general equlbrum models. In a recent paper, Tlle and Van Wncoop (2007) show how the zero and hgher-order components of portfolo behavour n an open economy model can be obtaned numercally va an teratve algorthm. Ther approach delvers a numercal soluton for steady-state portfolos n manner analogous to the analytcal solutons derved n ths paper. Judd et al (2002) develop a numercal algorthm based on splne collocaton and Evans and Hnatkovska (2005) present a numercal approach that reles on a combnaton of perturbaton and contnuoustme approxmaton technques. 4 The methods developed by Judd et al and Evans and Hnatkovska are very complex compared to our approach and they represent a sgn cant departure from standard DSGE soluton methods. Devereux and Sato (2005) use a contnuous tme framework whch allows some analytcal solutons to be derved, but ther approach can not handle general nternatonal macroeconomc models wth dmnshngreturns technology or stcky nomnal goods prces. Ths paper proceeds as follows. The next secton sets out a two-asset portfolo choce problem wthn a smple two-country endowment model and shows how our method can be appled n ths context. Secton 3 develops a more general n-asset portfolo problem wthn a generc two country DSGE model and shows how the method can be generalsed to accommodate a wde class of models. Secton 4 bre y outlnes how the method can be extended to derve a soluton for the rst-order component of the equlbrum portfolo. Secton 5 concludes the paper. 4 Evans and Hnatkovska (2005) develop an approach smlar to that of Campbell and Vcera (2005), who present a comprehensve analyss of optmal portfolo allocaton for a sngle agent. 4

6 2 Example: A Smple Two-Asset Endowment Model 2.1 The Model We rst llustrate how the soluton procedure works n a smple two-country example wth only two nternatonally traded assets, where agents consume an dentcal consumpton good, and ncome takes the form of a exogenous endowment of the consumpton good. Agents n the home country have a utlty functon of the form U t = E t 1 X =t t u(c ) (1) where C s consumpton and u(c ) = (C 1 )=(1 ). The budget constrant for home agents s gven by 1;t + 2;t = 1;t 1 r 1;t + 2;t 1 r 2;t + Y t C t (2) where Y s the endowment receved by home agents, 1;t 1 and 2;t 1 are the real holdngs of the two assets (purchased at the end of perod t 1 for holdng nto perod t) and r 1;t and r 2;t are gross real returns: It s assumed that the vector of avalable assets s exogenous and prede ned. The stochastc process determnng endowments and the nature of the assets and the propertes of ther returns are spec ed below. De ne W t = 1;t + 2;t to be the total net clams of home agents on the foregn country at the end of perod t (.e. the net foregn assets of home agents). The budget constrant can then be re-wrtten as W t = 1;t 1 r x;t + r 2;t W t 1 + Y t C t (3) where r x;t = r 1;t r 2;t Here asset 2 s used as a numerare and r x;t measures the "excess return" on asset 1. At the end of each perod agents select the portfolo of assets to hold nto the followng perod. Thus, for nstance, at the end of perod t home agents select 1;t to hold nto perod t+1. The rst-order condton for the choce of 1;t can be wrtten n the followng form E t [u 0 (C t+1 )r 1;t+1 ] = E t [u 0 (C t+1 )r 2;t+1 ] (4) 5

7 Foregn agents face a smlar portfolo allocaton problem wth a budget constrant gven by Wt = 1;t 1r x;t + r 2;t Wt 1 + Yt Ct (5) where an astersk ndcates foregn varables. In equlbrum t follows that W t = W t : Foregn agents have preferences smlar to (1) so the rst-order condton for foregn agents choce of 1;t s E t u 0 (C t+1)r 1;t+1 = Et u 0 (C t+1)r 2;t+1 Assets are assumed to be n zero net supply, so market clearng n asset markets mples 1;t 1 + 1;t 1 = 0; 2;t 1 + 2;t 1 = 0 To smplfy notaton, n what follows we wll drop the subscrpt from 1;t and smply refer to t : It should be understood, therefore, that 1;t = 1;t 1 = t, 2;t = W t t and 2;t = W t + t : Endowments are the sum of two components, so that (6) Y t = Y K;t + Y L;t ; Y t = Y K;t + Y L;t (7) where Y K;t and YK;t represent captal ncome and Y L;t and YL;t labour ncome. The endowments are determned by the followng smple stochastc processes log Y K;t = log Y K + " K;t ; log Y K;t = log Y K + " K;t; log Y L;t = log Y L + " L;t log Y L;t = log Y L + " L;t where " K;t ; " L;t ; " K;t and " L;t are zero-mean..d. shocks whch are symmetrcally dstrbuted over the nterval [ ; ] wth V ar[" K ] = V ar[" K ] = 2 K ; V ar[" L] = V ar[" L ] = 2 L. We assume Cov[" K ; " K ] = Cov[" L; " L ] = 0 and Cov[" K; " L ] = Cov[" K ; " L ] = KL: The two assets are assumed to be one-perod equty clams on the home and foregn captal ncome. 5 The real payo to a unt of the home equty n perod t s de ned to be 5 Notce that we are assumng that, by default, all captal n a country s owned by the resdents of that country. Ths allows us to treat equty clams to captal ncome as nsde assets,.e. assets n zero net supply. Ths s purely an accountng conventon. Our soluton method works equally n the alternatve approach, where captal s not ncluded n the de nton of Y and Y and equty s treated as an outsde asset whch s n postve net supply. The present approach makes our dervatons easer however. 6

8 Y K;t and the real prce of a unt of home equty s denoted Z E;t rate of return on home equty s 1. Thus the gross real Lkewse the gross real return on foregn equty s r 1;t = Y K;t =Z E;t 1 (8) r 2;t = Y K;t=Z E;t 1 (9) where ZE;t 1 s the prce of the foregn equty. The rst-order condtons for home and foregn consumpton are C t = E t C t+1r 2;t+1 ; C t = E t C t+1 r 2;t+1 (10) Fnally, equlbrum consumpton plans must satsfy the resource constrant C t + C t = Y t + Y t (11) 2.2 Zero-order and rst-order components Despte the extreme smplcty of ths model, t s only n specal cases that an exact soluton can be found, e.g. when there s no labour ncome (n whch case trade n equtes supports the perfect rsk-sharng equlbrum). 6 The model s also not amenable to standard rst-order approxmaton technques, so standard lnearsaton approaches to DSGE models can not provde even an approxmate soluton to the general case. Our method, nevertheless, does yeld an approxmate soluton to the general case. Before descrbng the method, t s useful to show why standard soluton technques do not work for ths model, and to demonstrate how our method o ers a way around the problems. Frst, we de ne some terms relatng to the true and approxmate portfolo solutons. Notce that agents make ther portfolo decsons at the end of each perod and are free to re-arrange ther portfolos each perod. In a recursve equlbrum, therefore, the equlbrum asset allocaton wll be some functon of the state of the system n each perod - 6 If there s no labour ncome then equtes can be used to trade all ncome rsk. It s easy to show that the equlbrum portfolo s for home and foregn agents to hold portfolos equally splt between home and foregn equty. Ths mples perfect consumpton rsk sharng. Ths s a useful benchmark for comparson wth the soluton yelded by our method. 7

9 whch s summarsed by the state varables. We therefore postulate that the true portfolo (.e. the equlbrum portfolo n the non-approxmated model) s a functon of state varables. In the model de ned above there s only one state varable, W - so we postulate t = (W t ). 7 Now consder a rst-order Taylor-seres expanson of (W t ) around the pont W = W (W t ) ' ( W ) + 0 ( W )(W t W ) Ths approxmaton contans two terms: ( W ); whch s the zero-order component (.e. at the pont of approxmaton) and 0 ( W )(W t W ); whch s the rst-order component (assumng (W t W ) s evaluated up to rst-order accuracy). Notce that, by de nton, the zero-order component of s non-tme varyng. The approxmate dynamcs of the portfolo are captured by the rst-order component. When analysng a DSGE model up to rst-order accuracy the standard soluton approach s to use the non-stochastc steady-state of the model as the approxmaton pont, (.e. the zero-order component of each varable) and to use a rst-order approxmaton of the model s equatons to solve for the rst-order component of each varable. Nether of these steps can be used n the above model. It s very smple to see why. In the non-stochastc equlbrum equatons (4) and (6) mply r 1;t+1 = r 2;t+1.e. both assets pay the same rate of return. Ths mples that, for gven W, all portfolo allocatons pay the same return, so any value for s consstent wth equlbrum. Thus the non-stochastc steady state does not te down a unque portfolo allocaton. A smlar problem arses n a rst-order approxmaton of the model. Frst-order approxmaton of equatons (4) and (6) mply E t [r 1;t+1 ] = E t [r 2;t+1 ].e. both assets have the same expected rate of return. Agan, any value of s consstent wth equlbrum. 7 Optmal portfolo allocaton wll of course depend on the propertes of asset returns generated by the model. In equlbrum, however, the stochastc propertes of asset returns wll also be a functon of state varables, so the mpact of asset returns on portfolo allocaton s mplct n the functon (W t ): 8

10 So nether the non-stochastc steady state nor a rst-order approxmaton of the model provde enough equatons to te down the zero or rst-order components of. The basc problem s easy to understand n economc terms. Assets n ths model are only dstngushable n terms of ther rsk characterstcs and nether the non-stochastc steady state nor a rst-order approxmaton capture the d erent rsk characterstcs of assets. In the case of the non-stochastc steady state there s, by de nton, no rsk, whle n a rst-order approxmaton there s certanty equvalence. Ths statement of the problem mmedately suggests a soluton. It s clear that the rsk characterstcs of assets only show up n the second-moments of model varables, and t s only by consderng hgher-order approxmatons of the model that the e ects of second-moments can be captured. Ths fundamental nsght has exsted n the lterature for many years. It was rst formalsed by Samuelson (1970), who establshed that, n order to derve the zero-order component of the portfolo, t s necessary to approxmate the portfolo problem up to the second order. Our soluton approach follows ths prncple. We show that a second-order approxmaton of the portfolo optmalty condtons provdes a condton whch makes t possble to te down the zero-order component of. The second-order approxmaton captures the mpact of the portfolo on the correlaton between portfolo returns and the margnal utlty of consumpton. It therefore captures d erences between assets n ther ablty to hedge consumpton rsk and thus tes down an optmal portfolo allocaton. In ths paper we show n detal how to use second-order approxmatons of the portfolo optmalty condtons to solve for the zero-order component of : 8 Havng establshed ths startng pont, t s relatvely straghtforward to extend the procedure to hgher-order components on : Samuelson (1970) n fact states a general prncple that, n order to derve the N th-order component of the portfolo, t s necessary to approxmate the portfolo problem up to order N + 2. In secton 4 we bre y outlne how, by followng ths prncple, the soluton for the rst-order component of can be derved from thrd-order approxmatons of the portfolo optmalty condtons. The full detals of the soluton procedure for the rst-order component are gven n a companon paper, Devereux and Sutherland (2007). 8 Note that Samuelson approached the problem by approxmatng the agent s utlty functon, whle we take approxmatons of agents rst-order condtons. It s possble to show that the two approaches produce dentcal results. 9

11 Whle Samuelson (1970) was the rst to show how solutons for the zero and hgherorder components of the portfolo may be derved, more recently Judd and Guu (2001) have demonstrated an alternatve soluton approach whch sheds further lght on the nature of the zero-order portfolo. They show how the problem of portfolo ndetermnacy n the non-stochastc steady state can be overcome by usng a Bfurcaton theorem n conjuncton wth the Implct Functon Theorem. Ther approach shows that the zero-order portfolo s a bfurcaton pont n the set of non-stochastc equlbra. Lke Samuelson (1970), our soluton approach reles on second-order approxmatons of the model to dentfy the zero-order component, but the underlyng theory descrbed by Judd and Guu (2001) s also applcable to our equlbrum soluton. In partcular, the zero-order portfolo derved usng our technque corresponds to the soluton that emerges from the Judd and Guu approach. Our soluton can therefore be ratonalsed n the same way,.e. t s a bfurcaton pont n the set of non-stochastc equlbra. 9 The general underlyng prncples of the soluton we derve are thus well establshed. The man contrbuton of ths paper s to provde a soluton approach whch can easly be appled to DSGE models. 10 We now demonstrate ths by solvng for the zero-order component of n the smple two-asset endowment model descrbed above. 2.3 Solvng for the zero-order portfolo In what follows, a bar over a varable ndcates ts value at the approxmaton pont (.e. the zero-order component) and a hat ndcates the log-devaton from the approxmaton pont (except n the case of ^; ^W and ^rx ; whch are de ned below). Notce that the non-stochastc steady state, whle falng to te down ; stll provdes solutons for output, consumpton and rates of return. We therefore use the non-stochastc steady state of the 9 As already explaned, n a non-stochastc world all portfolo allocatons are equvalent and can be regarded as vald equlbra. A stochastc world on the other hand (assumng ndependent asset returns and sutable regularty condtons on preferences) has a unque equlbrum portfolo allocaton. If one consders the lmt of a sequence of stochastc worlds, wth dmnshng nose, the equlbrum portfolo tends towards a lmt whch correspond to one of the many equlbra n the non-stochastc world. Ths lmtng portfolo s the bfurcaton pont descrbed by Judd and Guu (2001),.e. t s the pont n the set of non-stochastc equlbra whch ntersects wth the sequence of stochastc equlbra. 10 Note that both Samuelson (1970) and Judd and Guu (2001) demonstrate ther results usng statc partal equlbrum models of portfolo allocaton. 10

12 model as the approxmaton pont for all varables except. In partcular we use the symmetrc non-stochastc steady state, where W = 0: It follows from equatons (4) and (6) that r 1 = r 2 = 1= and thus r x = 0: Equatons (3) and (5) therefore mply that Y = Y = C = C : Snce W = 0, t also follows that 2 = 1 = 2 = 1 = : As argued above, solvng for the zero-order component of requres a second-order expanson of the portfolo problem. So we start by takng a second-order approxmaton of the home-country portfolo rst-order condton, (4), to yeld E t ^r x;t (^r2 1;t+1 ^r 2 2;t+1) ^C t+1^r x;t+1 = O 3 (12) where ^r x;t+1 = ^r 1;t+1 ^r 2;t+1 and O ( 3 ) s a resdual whch contans all terms of order hgher than two. Applyng a smlar procedure to the foregn rst-order condton, (6), yelds E t ^r x;t (^r2 1;t+1 ^r 2 2;t+1) ^C t+1^r x;t+1 = O 3 (13) These expresson can now be combned to show that, n equlbrum, the followng equatons must hold E t h ( ^C t+1 ^C t+1 )^r x;t+1 = 0 + O 3 (14) and E t [^r x;t+1 ] = 1 2 E 2 t ^r 1;t+1 ^r 2;t E t h( ^C t+1 + ^C t+1)^r x;t+1 + O 3 (15) These two equatons express the portfolo optmalty condtons n a form whch s partcularly convenent for dervng equlbrum portfolo holdngs and excess returns. Equaton (14) provdes an equaton whch must be sats ed by equlbrum portfolo holdngs. And equaton (15) shows the correspondng set of equlbrum expected excess returns. We wll now show that equaton (14) provdes a su cent condton to te down the zero-order component of : In order to do ths we rst state two mportant propertes of the approxmated model. Property 1 In order to evaluate the left hand sde of equaton (14) t s su cent to derve expressons for the rst-order accurate behavour of consumpton and excess returns. Ths s because the only terms that appear n equaton (14) are products, and second-order accurate solutons for products can be obtaned from rst-order accurate solutons for ndvdual varables. 11

13 Property 2 The only aspect of the portfolo decson that a ects the rst-order accurate behavour of consumpton and excess returns s,.e. the zero-order component of the. The rst-order component,.e. the devaton of from the approxmaton pont, does not a ect the rst-order behavour of consumpton and excess returns. To see why ths s true notce that portfolo decsons only enter the model va the portfolo excess return,.e. va the term 1;t constrants. A rst-order expanson of ths term s ^r x;t + r x^ 1;t only ^r x;t remans. 1 r x;t n the budget 1 : But r x = 0 so It s now straghtforward to show that equaton (14) provdes a condton whch tes down : Property 2 tells us that t s possble to evaluate the rst-order behavour of ( ^C t+1 ^C h t+1 ) and ^r x;t+1 condtonal on a gven value of : Property 1 tells us that E t ( ^C t+1 ^C t+1 )^r x;t+1 can therefore also be evaluated condtonal on a gven value of : h Equaton (14) tells us that a soluton for s one whch mples E t ( ^C t+1 ^C t+1 )^r x;t+1 = 0: In order to derve ths soluton for t s rst necessary to solve for the rst-order accurate behavour of ( ^C t+1 ^C t+1 ) and ^r x;t+1 condtonal on a gven value of : The rst-order accurate behavour of ^r x;t+1 s partcularly smple n ths model. Frst-order approxmatons of (8) and (9) mply ^r x;t+1 = ^Y K;t+1 ^Y K;t+1 ( ^Z E;t ^Z E;t ) + O 2 where O ( 2 ) s a resdual whch contans all terms of order hgher than one, so E t [^r x;t+1 ] = E t [ ^Y K;t+1 ] E t [ ^Y K;t+1] ( ^Z E;t ^Z E;t ) + O 2 Notce that (15) mples that, up to a rst-order approxmaton, E t [^r x;t+1 ] = 0 so ( ^Z E;t ^Z E;t ) = E t [ ^Y K;t+1 ] E t [ ^Y K;t+1] + O 2 and thus, snce Y K and Y K are..d:, ^r x;t+1 s gven by 11 ^r x;t+1 = ^Y K;t+1 ^Y K;t+1 + O 2 (16) 11 Notce from ths dervaton that, n ths model, r x s completely ndependent from : Ths makes the applcaton of our soluton process partcularly smple. In the next secton we wll show that our method can easly be appled to more general models where may have a drect or ndrect mpact on r x. 12

14 The rst-order accurate soluton for ( ^C t+1 ^C t+1 ) s also straghtforward to derve. A rst-order approxmaton of the home and foregn budget constrants mples ^W t+1 = 1 ^W t + ^Y t+1 ^Ct+1 + ~^r x;t+1 + O 2 (17) ^W t+1 = 1 ^W t + ^Y t+1 ^C t+1 ~^r x;t+1 + O 2 (18) where ^W t = (W t W )= C and ~ = =( Y ). Combnng (17) and (18) wth (16) and an approprate transversalty condton mples 1X =0 E t+1 ( ^C t+1+ ^C t+1+ ) = 2 ^W t + ( ^Y t+1 ^Y t+1 ) +2~( ^Y K;t+1 ^Y K;t+1 ) + O 2 (19) where use has been made of he fact that E t+1 [ ^Y t+1+ ] = E t+1 [ ^Y t+1+] = E t+1 [ ^Y K;t+1+ ] = E t+1 [ ^Y K;t+1+ ] = 0 for all > 0. The rst-order condtons for consumpton, equatons (10), mply E t+1 [ ^C t+1+ ^C t+1+ ] = ^C t+1 ^C t+1 + O 2 for all > 0 (20) so ( ^C t+1 ^C t+1 ) s gven by ^C t+1 ^C t+1 = 2(1 ) ^W t + (1 )( ^Y t+1 ^Y t+1 ) +2(1 )~( ^Y K;t+1 ^Y K;t+1 ) + O 2 (21) Equatons (16) and (21) show the rst-order accurate behavour of ( ^C t+1 ^C t+1 ) and ^r x;t+1 condtonal on a gven value of : Combnng these expresson yelds h E t ( ^C t+1 ^C t+1 )^r x;t+1 = h (1 )E t ( ^Y t+1 ^Y t+1 ) + 2~( ^Y K;t+1 ^Y K;t+1 ) ( ^Y K;t+1 ^Y K;t+1 ) + O 3 (22) It follows from (14) and (22) that the soluton for ~ s or ~ = 1 E t [( ^Y t+1 ^Y t+1 )( ^Y K;t+1 ^Y K;t+1 )] 2 E t [( ^Y + O () (23) K;t+1 ^Y K;t+1 ) 2 ] ~ = + (1 ) KL= 2 K 2 + O () (24) 13

15 where = Y K =( Y K + Y L ) = Y K = Y. Notce that the resdual n ths expresson s a rst-order term. The soluton for s then gven by = ~ Y : To provde an economc nterpretaton of our soluton t s helpful to re-express (24) n terms of the proporton of home equty held by home resdents. The total value of home equty s Y K, so the proporton held by home resdents s gven by Y K + Y = 1 + (1 ) KL= 2 K K 2 (25) The most obvous benchmark aganst whch to compare (25) s the case where there s no labour ncome,.e. where = 1 and 2 L = 0: In ths case there s a known exact soluton to the model where home and foregn agents hold a balanced portfolo of home and foregn equtes. It s easy to see from (25) that our soluton yelds exactly ths outcome..e. home agents hold exactly half of home equty (and by mplcaton half of foregn equty). It s also easy to check from (21) that the equlbrum portfolo yelds full consumpton rsk sharng. More generally, n cases where ths s labour ncome rsk,.e. 0 < < 1 and 2 L > 0; there s no exact soluton to the model, but our zero-order soluton provdes an approxmate soluton. Equaton (25) shows that f KL = 0 (.e. labour and captal ncome are uncorrelated) agents contnue to hold a balanced portfolo of home and foregn equty, but equaton (21) shows that full consumpton rsk sharng s not acheved n ths case. The equlbrum portfolo devates from an equal balance of home and foregn equty when there s some correlaton between captal and labour ncome. For nstance, when there s a negatve correlaton,.e. KL < 0; there wll be home bas n equty holdngs (.e. home agents wll hold more then half of home equty and foregn agents wll hold more than half of foregn equty). 12 Before showng how the soluton procedure can be appled to a more general model, we use (24) to address a number of potentally puzzlng ssues. Frst, notce that despte the presence of tme subscrpts, all the terms n (23), ncludng the condtonal secondmoments, are constant. So our soluton for s non-tme-varyng (whch s consstent wth our de nton of the zero-order component). At rst sght t may seem contradctory that portfolo allocatons are non-tme varyng whle net wealth, n the form of ^W t, s tme varyng. But ths s to confuse orders of approxmaton. s the zero-order component of the portfolo, and should be compared to the zero-order component of net wealth, W, 12 Conversely, when KL > 0, we have a bas aganst home assets, as n Baxter and Jermann (1997). 14

16 whch, lke ; s non-tme varyng. ^Wt on the other hand, s the rst-order component of net wealth, and ths should be compared to the rst-order component of portfolos, ^ t. Both ^W t and ^ t are tme varyng. But notce, by Property 2 t s possble to solve for the dynamcs of ^W wthout havng to know the behavour of ^. As explaned above, havng solved for t s possble to solve for ^ t by analysng a thrd-order approxmaton of the portfolo problem. Ths s dscussed below n Secton 4. A more general mplcaton of Property 2, whch s worth emphassng, s that t s not necessary to solve for the rst-order behavour of ^ n order to solve for the rstorder behavour of other varables n the model. It s therefore possble to analyse the mplcatons of the above model for the rst-order behavour of all varables other than wthout havng to solve for ^: The logc presented above mples that the zero-order component of the portfolo, ; s analogous to the zero-order component of the other varables n the model. At rst sght ths may also seem contradctory, snce the zero-order components of other varables are derved from the non-stochastc steady state, whle our soluton for s derved from an explctly stochastc analyss. The way to resolve ths apparent contradcton s to nterpret as the equlbrum for portfolo holdngs n a world wth an arbtrarly small amount of stochastc nose,.e. the equlbrum n a near-non-stochastc world. If one consders the lmt of a sequence of stochastc worlds, wth dmnshng nose, the equlbrum portfolo tends towards a lmt whch correspond to one of the many portfolo equlbra n the non-stochastc world. Ths lmtng portfolo s a bfurcaton pont descrbed by Judd and Guu (2001),.e. t s the pont n the set of non-stochastc equlbra whch ntersects wth the sequence of stochastc equlbra. Our soluton for corresponds to the portfolo allocaton at ths bfurcaton pont. 13 Fnally, we note a techncal ssue that arses regardng the pont of approxmaton 13 Suppose that the covarance matrx of the nnovatons s gven by = 0 where > 0 s a scalar and 0 s a vald covarance matrx. Notce that the soluton for ~ gven n (24) s ndependent of : So the value of ~ gven by (24) (and therefore the value of ) s equvalent to the value that would arse n the case of an arbtrarly small, but non-zero, value of -.e. the value of ~ that would arse n a world whch s arbtrarly close to a non-stochastc world. Furthermore, notce that as tends to zero (whch s equvalent to tendng to zero) the sze of the resdual n (24) tends to zero. So, as the amount of nose tends to zero, the value of ~ becomes arbtrarly close to the true value of portfolo holdngs n the non-approxmated model. Our soluton for can therefore be thought of as the true portfolo equlbrum n a world whch s arbtrarly close to the non-stochastc equlbrum. 15

17 of W t. In the example gven above, there s a unt root n the dynamcs of net foregn assets at the level of rst-order approxmaton. Ths means that we would not be able to compute uncondtonal second moments from the model. But ths has no bearng on the portfolo soluton. Equlbrum portfolos depend only on condtonal second moments, whch are well de ned. The unt root property could easly be elmnated usng any of the approaches dscussed n Schmtt-Grohe and Urbe (2003), and t should be clear from the above presentaton that our approach works equally well n ths case. We chose to use the model here however, because t gves very smple and ntutve expressons for optmal portfolos. 3 Generalsng to an n-asset Model 3.1 The Model We now show how the soluton method can be extended to a much more general model wth many assets. The model we now descrbe s general enough to encompass the range of structures that are wdely used n the recent open economy macro lterature. However, only those parts of the model drectly necessary for understandng the portfolo selecton problem need to be explctly descrbed. Other components of the model, such as the labour supply decsons of households and the producton and prcng decsons of rms, are not drectly relevant to the portfolo allocaton problem, so these parts of the model are suppressed. The soluton approach s consstent wth a wde range of spec catons for labour supply, prcng and producton. Thus, the non-portfolo parts of the model may be charactersed by endogenous or exogenous employment, stcky or exble prces and wages, local currency prcng or producer currency prcng, perfect competton or mperfect competton, etc. We contnue to assume that the world conssts of two countres. The home country s assumed to produce a good (or a bundle of goods) wth aggregate quantty denoted Y H (whch can be endogenous) and aggregate prce P H. Smlarly the foregn country produces quantty Y F of a (potentally d erentated) foregn good (or bundle of goods) at prce PF. In what follows foregn currency prces are denoted wth an astersk. 16

18 Agents n the home country now have a utlty functon of the form U t = E t 1 X =t t [u(c ) + v(:)] (26) where C s a bundle of the home and foregn goods and u(:) s a twce contnuously d erentable perod utlty functon. The functon v(:) captures those parts of the preference functon whch are not relevant for the portfolo problem. 14 The aggregate consumer prce ndex for home agents s denoted P. There are n assets and a vector of n returns (for holdngs of assets from perod t 1 to t) gven by h rt 0 = r 1;t r 2;t ::: r n;t Asset payo s and asset prces are measured n terms of the aggregate consumpton good of the home economy (.e. n unts of C). Returns are de ned to be the sum of the payo of the asset and captal gans relatve to the asset prce. As before, t s assumed that the vector of avalable assets s exogenous and prede ned. The budget constrant for home agents s gven by X ;t = X r ;t ;t 1 + Y t C t (27) where [ 1;t 1 ; 2;t 1 ::: n;t 1 ] are the holdngs of the n assets purchased at the end of perod t 1 for holdng nto perod t. Y s the total dsposable ncome of home agents expressed n terms of the home consumpton good. Thus, Y may be gven by Y H P H =P + T where T s a scal transfer (or tax f negatve). 15 Usng the followng de nton of net wealth (net foregn assets) W t = X ;t (28) 14 For these other aspects of the preference functon to be rrelevant for portfolo selecton t s necessary to assume utlty s addtvely separable n u(c) and v(:): Extensons to cases of non-addtve separablty (e.g. habt persstence n consumpton) are straghtforward, as wll become more clear below. Usng (26) allows us to llustrate the method wth mnmal notaton. 15 Wthout changng any of the results below, we could augment Y to allow for convex adjustment costs n W arsng from havng net foregn assets away from ther long term mean W. Ths would ensure a statonary dstrbuton for W. Thus, the model developed n ths secton does not necessarly dsplay the unt root property for W. 17

19 the budget constrant may be re-wrtten n the followng form W t = 0 t 1r x;t + r n;t W t 1 + Y t C t (29) where h 0 t 1 = 1;t 1 2;t 1 ::: n 1;t 1 and h rx;t 0 = (r 1;t r n;t ) (r 2;t r n;t ) ::: (r n 1;t r n;t ) h = r x;1;t r x;2;t ::: r x;n 1;t Here the nth asset s used as a numerare and r x;t measures the "excess returns" on the other n 1 assets. There are n 1 rst-order condtons for the choce of the elements of t whch can be wrtten n the followng form E t [u 0 (C t+1 )r 1;t+1 ] = E t [u 0 (C t+1 )r n;t+1 ] E t [u 0 (C t+1 )r 2;t+1 ] = E t [u 0 (C t+1 )r n;t+1 ] : E t [u 0 (C t+1 )r n 1;t+1 ] = E t [u 0 (C t+1 )r n;t+1 ] (30) Foregn-country agents face a smlar portfolo allocaton problem wth a budget constrant gven by 1 Wt = 1 0 t Q t Q 1r x;t + r n;t Wt 1 + Y t Ct (31) t where Q t = P t S t =P t s the real exchange rate. The real exchange rate enters ths budget constrant because Y and C are measured n terms of the foregn aggregate consumpton good (whch may d er from the home consumpton good) whle asset holdngs and rates of return are de ned n terms of the home consumpton good. Foregn agents are assumed to have preferences smlar to (26) so the rst-order condtons for foregn-country agents choce of t are E t Q 1 t+1u 0 (Ct+1)r 1;t+1 = Et Q 1 t+1u 0 (Ct+1)r n;t+1 E t Q 1 t+1u 0 (Ct+1)r 2;t+1 = Et Q 1 t+1u 0 (Ct+1)r n;t+1 : E t Q 1 t+1u 0 (Ct+1)r n 1;t+1 = Et Q 1 t+1u 0 (Ct+1)r n;t+1 The two sets of rst-order condtons, (30) and (32), and the market clearng condton t = t, provde 3(n 1) equatons whch determne the elements of t ; t and E t [r x;t+1 ]: 18 (32)

20 Clearly, n any partcular general equlbrum model, there wll be a set of rst-order condtons relatng to ntertemporal choce of consumpton, labour supply, etc., for the home and foregn consumers, and a set of rst-order condtons for prce settng and factor demands for home and foregn producers. Taken as a whole, and combned wth an approprate set of equlbrum condtons for goods and factor markets, ths full set of equatons wll de ne the general equlbrum of the model. As already explaned, the detals of these non-portfolo parts of the model are not necessary for the exposton of the soluton method, so they are not shown explctly. In what follows these omtted equatons are smply referred to as the "non-portfolo equatons" of the model. The non-portfolo equatons of the model wll normally nclude some exogenous forcng varables. In the typcal macroeconomc model these take the form of AR1 processes whch are drven by zero-mean..d. nnovatons. We assume that there are m such dsturbances, summarsed n a vector, x, whch s determned by the followng process x t = Nx t 1 + " t (33) where " s a vector of zero-mean..d. nnovatons wth covarance matrx : It s assumed that the nnovatons are symmetrcally dstrbuted over the nterval [ ; ]: Solvng for the zero-order portfolo Agan we use the symmetrc non-stochastc steady state of the model as the approxmaton pont for non-portfolo varables. Thus W = 0; Y = Y = C = C and r 1 = r 2 ::: = r n = 1=: Note agan that ths mples r x = 0: As before we proceed by takng second-order approxmatons of the home and foregn portfolo rst-order condtons. For the home country ths yelds E t h(^r 1;t+1 ^r n;t+1 ) (^r2 1;t+1 ^r n;t+1) 2 ^C t+1 (^r 1;t+1 ^r n;t+1 ) = O ( 3 ) E t h(^r 2;t+1 ^r n;t+1 ) (^r2 2;t+1 ^r n;t+1) 2 ^C t+1 (^r 2;t+1 ^r n;t+1 ) = O ( 3 ) (34) : E t h(^r n 1;t+1 ^r n;t+1 ) (^r2 n 1;t+1 ^r n;t+1) 2 ^C t+1 (^r n 1;t+1 ^r n;t+1 ) = O ( 3 ) 16 Clearly there must be a lnk between and : The value of places an upper bound on the dagonal elements of : So an experment whch nvolves consderng the e ects of reducng mplctly nvolves reducng the magntude of the elements of : 19

21 where u 00 ( C) C=u 0 ( C) (.e. the coe cent of relatve rsk averson). Re-wrtng (34) n vector form yelds E t ^r x;t ^r2 x;t+1 ^C t+1^r x;t+1 = O 3 (35) where h ^r x;t+1 0 ^r 1;t+1 ^r n;t+1 ^r 2;t+1 ^r n;t+1 ::: ^r n 1;t+1 ^r n;t+1 and h ^r x;t+1 20 ^r 2 1;t+1 ^r 2 n;t+1 ^r 2 2;t+1 ^r 2 n;t+1 ::: ^r 2 n 1;t+1 ^r 2 n;t+1 Applyng a smlar procedure to the foregn rst-order condtons yelds E t ^r x;t ^r2 x;t+1 ^C t+1^r x;t+1 + ^Q t+1 = 0 + O 3 (36) The home and foregn optmalty condtons, (35) and (36), can be combned to show that, n equlbrum, the followng condtons must hold E t h ( ^C t+1 ^C t+1 ^Qt+1 =)^r x;t+1 = 0 + O 3 (37) and E [^r x ] = 1 2 E ^r x E t h( ^C t+1 + ^C t+1 + ^Q t+1 =)^r x;t+1 + O 3 (38) These equatons are equvalent to (14) and (15) n the example from before. There we showed that equaton (14) provded a su cent condton to te down the zero-order component of the portfolo allocaton. We now show that equaton (37) provdes a su cent condton to te down the zero-order component of the portfolo n the general model. Propertes 1 and 2 played a central role n dervng the soluton to the example above. These propertes also hold for the general model, and reman central n the dervaton of the soluton. Clearly, Property 1 apples n the general model. The left hand sde of equaton (37) conssts entrely of products of varables and can thus be evaluated to second-order accuracy usng rst-order accurate expressons for ^C ^C ^Q= and ^rx : Lkewse, Property 2 holds n the general model. Agan, the portfolo allocaton enters only va the excess portfolo return, 0 r x. And, just as n the smple model, r x = 0; so the rst-order approxmaton of the excess portfolo return s ^r x : Thus only the zero-order component of enters the rst-order approxmated model. 20

22 The general outlne of the soluton strategy s the same as that descrbed for the smple model. Frst we solve for the rst-order accurate behavour of ^C ^C ^Q= and ^rx n terms of. Then we solve for the that ensures (37) s sats ed. But now thngs are somewhat more complcated because the behavour of ^C ^C ^Q= and ^r x s determned by a potentally complex set of rst-order dynamc equatons. Indeed, at rst sght, the general model may seem too complex to be solved explctly, and t may appear that a numercal approach s necessary to solve for the. We show, however, that t s possble to derve a closed-form analytcal soluton for n the general model. In fact, we derve a formula for whch s applcable to any model wth the same general features as the one descrbed above. To see why t s possble to obtan a closed-form soluton, t s necessary to state a further mportant property of the approxmated model. Property 3 To a rst-order approxmaton, the portfolo excess return, ^r x;t+1, s a zero mean..d. random varable. Ths follows from equaton (38), whch shows that the equlbrum expected excess return contans only second-order terms. So, up to a rst order approxmaton, E t 1 [^r x;t+1 ] s zero,.e. there s no predctable element n ^r x;t+1 : The rst-order approxmaton of the portfolo excess return, ^r x;t+1, s therefore a lnear functon of the..d. nnovatons, " t+1 ; and must therefore tself be an..d. random varable. Property 3 greatly smpl es the soluton process because t mples that a ects the rst-order behavour of the economy n a very smple way. In partcular, does not a ect the egenvalues of the rst-order system. Thus, n any gven perod (e.g. perod t) the dynamc propertes of the expected path of the economy from perod t + 1 onwards are ndependent of. The perod t behavour of the economy s a ected by only through ts e ect on the sze and sgn of..d. nnovatons to wealth arsng from the portfolo excess return, ^r x;t. The only remanng potental complcaton s that ^r x;t may tself depend on perod t nnovatons to wealth (and therefore ). Ths complcaton s, however, easly overcome by breakng the soluton process for ^C ^C ^Q= and ^rx nto two stages. In the rst stage we treat the portfolo excess return, ^r x, as an exogenous..d. random varable, and solve the rst-order model to yeld an expresson for ^r x n terms of exogenous nnovatons to wealth. In the second stage we use ths expresson to solve out for the behavour of 21

23 and t+1 ^r xt+1 = R 1 t+1 + R 2 " t+1 + O 2 (42) ^C ^C ^Q= and ^rx n terms of " (.e. the true exogenous nnovatons of the model). Ths provdes the expressons requred to evaluate (37) and thus to solve for. 17 We now apply ths procedure to the general model. Frst note that the rst-order approxmaton of the home budget constrant s gven by ^W t = 1 ^W t 1 + ^Y t ^Ct + ~ 0^r xt + O 2 where ^W t = (W t W )= Y and ~ = =(Y ). The soluton procedure wll be descrbed n terms of dervng a soluton for ~: The correspondng soluton for s obvously gven by = ~ Y : We now rewrte the budget constrant n the form ^W t = 1 ^W t 1 + ^Y t ^Ct + t + O 2 (39) where ~ 0^r xt has been replaced by t. We temporarly treat as an exogenous..d. varable. The rst-order approxmaton of the model can now be summarsed n a matrx equaton of the form " A 1 s t+1 E t [c t+1 ] # " # st = A 2 + A 3 x t + B t + O 2 (40) c t where s s the vector of predetermned varables, c s the vector of jump varables, x s de ned n (33) and B s a column vector wth unty n the row correspondng to (39) and zero n all other rows. The state-space soluton to (40) can be derved usng any standard soluton method for lnear ratonal expectatons models. It can be wrtten as follows s t+1 = F 1 x t + F 2 s t + F 3 t + O ( 2 ) c t = P 1 x t + P 2 s t + P 3 t + O ( 2 ) (41) By extractng the approprate rows from (41) t s possble to wrte the followng expresson for the rst-order accurate relatonshp between excess returns, ^r xt+1 ; and " t+1 where the matrces R 1 and R 2 are formed from the approprate rows of (41). Equaton (42) shows how rst-order accurate realsed excess returns depend on exogenous..d. shocks, 17 Notce from equaton (15) that, n the example, r x does not depend on, so ths two-step process for was not necessary. 22

24 " t+1 and t In partcular, t shows how ^r xt+1 depends on..d. shocks to wealth. Ths completes the rst stage n solvng for the rst-order behavour of ^C ^C ^Q= and ^rx : Now we mpose the condton that, rather than beng exogenous, the nnovatons to wealth, t+1, are endogenously determned by excess portfolo returns va the relatonshp t+1 = ~ 0^r xt+1 (43) where the vector of portfolo allocatons, ~; s yet to be determned. Ths equaton, together wth (42), can be solved to yeld expressons for t+1 and ^r xt+1 n terms of the exogenous nnovatons as follows where t+1 = ~ H" t+1 (44) ^r xt+1 = ~ R" t+1 + O 2 (45) ~H = ~0 R 2 1 ~ 0 R 1 ; ~ R = R1 ~ H + R2 (46) Equaton (45), whch shows how realsed excess returns depend on the exogenous..d. nnovatons of the model, provdes one of the relatonshps necessary to evaluate the lefthand sde of (37). The other relatonshp requred s the lnk between ^C t+1 ^C t+1 ^Qt+1 = and the vector of exogenous nnovatons, " t+1. Ths relatonshp can derved n a smlar way to (45). Frst extract the approprate rows from (41) to yeld the followng " # xt ^C t+1 ^C t+1 ^Qt+1 = = D 1 t+1 + D 2 " t+1 + D 3 + O 2 (47) where the matrces D 1 ; D 2 and D 3 are formed from the approprate rows of (41). After substtutng for t+1 usng (44) ths mples ^C t+1 ^C t+1 ^Qt+1 = = ~ D" t+1 + D 3 " xt s t+1 s t+1 # + O 2 (48) where ~D = D 1 ~ H + D2 (49) Equatons (45) and (48) are the equvalents of (16) and (21) n the example. They show the rst-order accurate behavour of ^r xt+1 and ^C t+1 ^C t+1 ^Qt+1 = and they can 18 Notce that, as follows from Property 3, ^r xt+1 does not depend on the values of the state varables contaned n x t or s t. 23