Dynamic Portfolio Optimization with a Defaultable Security and Regime-Switching

Transcription

1 Dynamic Porfolio Opimizaion wih a Defaulable Securiy and Regime-Swiching Agosino Capponi José E. Figueroa-López Absrac We consider a porfolio opimizaion problem in a defaulable marke wih finiely-many economical regimes, where he invesor can dynamically allocae her wealh among a defaulable bond, a sock, and a money marke accoun. The marke coefficiens are assumed o depend on he marke regime in place, which is modeled by a finie sae coninuous ime Markov process. By separaing he uiliy maximizaion problem ino a pre-defaul and pos-defaul componen, we deduce wo coupled Hamilon-Jacobi-Bellman equaions for he pos and pre-defaul opimal value funcions, and show a novel verificaion heorem for heir soluions. We obain explici consrucions of value funcions and invesmen sraegies for invesors wih logarihmic and Consan Relaive Risk Aversion CRRA) uiliies, and provide a precise characerizaion of he direcionaliy of he bond invesmen sraegies in erms of corporae reurns, forward raes, and expeced recovery a defaul. We illusrae he dependence of he opimal sraegies on ime, losses given defaul, and risk aversion level of he invesor hrough a deailed economic and numerical analysis. AMS 2 subjec classificaions: 93E2, 6J2. Keywords and phrases: Dynamic Porfolio Opimizaion, Credi Risk, Regime-Swiching Models, Uiliy Maximizaion, Hamilon-Jacobi-Bellman Equaions. 1 Inroducion Coninuous ime porfolio opimizaion problems are among he mos widely sudied problems in he field of mahemaical finance. Since he seminal work of Meron 1969), who explored opimal sochasic conrol echniques o provide a closed form soluion o he problem, a large volume of research has been done o exend Meron s paradigm o oher frameworks and porfolio opimizaion problems see, e.g., Karazas e al. 1996), Karazas and Shreve 1998), and Fleming and Pang 24)). Mos of he models proposed in he lieraure rely on he assumpion ha he uncerainy in he asse price dynamics is governed by a coninuous process, which is ypically chosen o be a Brownian moion. In recen years, here has been an increasing ineres in he use of regime-swiching models o capure he macroeconomic regimes affecing he behavior of he marke. More specifically, he price of he securiy evolves wih a differen dynamics, ypically idenified by he drif and he diffusion coefficien, depending on he macroeconomic regime in place. Uiliy maximizaion problems under regime-swiching have been invesigaed in Soomayor and Cadenillas 29), who considered he infinie horizon problem of maximizing he expeced uiliy from consumpion and erminal wealh in a marke consising of muliple socks and a money marke accoun, where boh shor rae and sock diffusion parameers evolve according o Markov-Chain modulaed dynamics. Siu 211) exended he marke o accommodae inflaion-linked bonds and solved he opimal porfolio selecion problem under a hidden regime-swiching model. In he imporan conex of risk managemen, Ellio and Siu 21) formulaed he risk minimizaion problem as a sochasic differenial game in a regime-swiching framework, and provided a verificaion heorem for he resuling School of Indusrial Engineering, Purdue Universiy, Wes Lafayee, IN, 4797, USA capponi@purdue.edu). Deparmen of Saisics, Purdue Universiy, Wes Lafayee, IN, 4797, USA figueroa@purdue.edu). 1

2 Hamilon-Jacobi-Bellman HJB) equaion. Wihin a similar framework, Ellio and Siu 211) considered he opimal invesmen problem of an insurer when he model uncerainy is modeled by a hidden Markov chain. Zhang e al. 21) solved he porfolio selecion problem afer compleing he coninuous-ime Markovian regime-swiching model wih jump securiies. Zariphopoulou 1992) considered an infinie horizon invesmen-consumpion model where he agen can consume and disribue her wealh across a risk-free bond and a sock, while Nagai and Runggaldier 28) considered a finie horizon porfolio opimizaion problem for a risk averse invesor wih power uiliy, assuming ha he underlying Markov chain is hidden. Korn and Kraf 21) relaxed he assumpion of consan ineres rae and derived expressions for he opimal percenage of wealh invesed in he money marke accoun and sock, under he assumpion of a diffusive shor rae process wih deerminisic drif and consan volailiy. Mos of he research done on coninuous ime porfolio opimizaion has concenraed on markes consising of a risk-free asse and of securiies which only bear marke risk. These models do no ake ino accoun securiies carrying defaul risk, such as corporae bonds, even hough he laer represen a significan porion of he marke, comparable o he oal capializaion of all publicly raded companies in he Unied Saes. In recen years, porfolio opimizaion problems have sared o incorporae defaulable securiies, bu assuming ha he risky facors are modeled by coninuous processes and more specifically by Brownian Iô processes. Bielecki and Jang 26) derived opimal finie horizon invesmen sraegies for an invesor wih Consan Relaive Risk Aversion CRRA) uiliy funcion, who opimally allocaes her wealh among a defaulable bond, risk-free accoun, and sock, assuming consan ineres rae, drif, volailiy, and defaul inensiy. Bo e al. 21) considered an infinie horizon porfolio opimizaion problem, where an invesor wih logarihmic uiliy can choose a consumpion rae, and inves her wealh across a defaulable perpeual bond, a sock, and a money marke. They assume ha boh he hisorical inensiy and he defaul premium process depend on a common Brownian facor. Unlike Bielecki and Jang 26), where he dynamics of he defaulable bond price process was derived from he arbirage-free bond prices, Bo e al. 21) posulaed he dynamics of he defaulable bond prices parially based on heurisic argumens. Lakner and Liang 28) analyzed he opimal invesmen sraegy in a marke consising of a defaulable corporae) bond and a money marke accoun under a coninuous ime model, where bond prices can jump, and employ dualiy heory o obain he opimal sraegy. Callegaro e al. 21) considered a marke model consising of several defaulable asses, which evolve according o discree dynamics depending on parially observed exogenous facor processes. Jiao and Pham 21) combined dualiy and dynamic programming o opimize he uiliy of an invesor wih CRRA uiliy funcion, in a marke consising of a riskless bond and a sock subjec o counerpary risk. Bielecki e al. 28) developed a variaional inequaliy approach o pricing and hedging of a defaulable game opion under a Markov modulaed defaul inensiy framework. In his paper, we consider for he firs ime finie horizon dynamic porfolio opimizaion problems in defaulable markes wih regime-swiching dynamics. We provide a general framework and explici resuls on opimal value funcions and invesmen sraegies in a marke consising of a money marke accoun, a sock, and a defaulable bond. Similarly o Soomayor and Cadenillas 29), we allow he shor rae, drif, and volailiy of he risky sock o be all regime dependen. For he defaulable bond, we follow he reduced form approach o credi risk, where he global marke informaion, including defaul, is modeled by he progressive enlargemen of a reference filraion represening he defaul-free informaion, and he defaul ime is a oally inaccessible sopping ime wih respec o he enlarged filraion, bu no wih respec o he reference filraion. We also make he defaul inensiies and loss given defaul raes o be all regime dependen. The use of regime-swiching models for pricing defaulable bonds has proven o be very flexible when fiing he empirical credi spreads curve of corporae bonds as illusraed in, e.g., Jarrow e al. 1997), where he underlying Markov chain models credi raings. Our main conribuions are discussed nex. Using he resuls on he dynamics of defaulable bond prices obained in Capponi, Figueroa-López, and Nisen 212b), we firs deduce he Hamilon-Jacobi-Bellman HJB) equaion of he dynamical opimizaion problem. The HJB equaion enables us o separae he uiliy maximizaion problem ino predefaul and pos-defaul dynamical opimizaion subproblems, for which novel verificaion heorems are obained. We show ha he regime dependen pre-defaul opimal value funcion and bond invesmen sraegy may be obained as he 2

3 soluion of a coupled sysem of nonlinear parial differenial equaions saisfied by he pre-defaul value funcion) and nonlinear equaions saisfied by he bond invesmen sraegy), each corresponding o a differen regime. Moreover, we obain he ineresing feaure ha he pre-defaul opimal value funcion and bond invesmen sraegy depend on he corresponding regime dependen pos-defaul value funcion. Thirdly, we develop an explici numerical and economic analysis of value funcions and invesmen sraegies for he case of a logarihmic and CRRA invesor facing boh defaul and regime-swiching risk. For he logarihmic invesor, we demonsrae ha he compuaion of he opimal pre-defaul and pos-defaul value funcions amouns o solving a sysem of ordinary linear differenial equaions, while he opimal bond sraegy may be recovered as he unique soluion of a decoupled sysem of equaions, one for each regime. For he CRRA invesor, we show ha he opimal bond invesmen sraegy and pre-defaul value funcion can be uniquely recovered as he soluion of a coupled sysem composed of ordinary differenial equaions and nonlinear equaions. Under mild assumpions, we provide condiions guaraneeing local exisence and uniqueness of he soluion of he coupled sysem and show numerically, via a fixed poin algorihm, ha global convergence is ypically achieved. Ineresingly, in a differen conex of liquidiy risk, where invesors can only rade in socks a Poisson random imes, Pham and Tankov 29) also found ha he opimal conrol problem leads o solving a coupled sysem of inegro-parial differenial equaions. We also provide necessary and sufficien condiions under which he logarihmic and CRRA invesor go long or shor in he defaulable securiy, and show ha hese depend on he inerplay beween corporae bond reurns, insananeous forward rae of he defaulable bond, and expeced recovery he precise saemen is given in Secion 6.2). The res of he paper is organized as follows. Secion 2 inroduces he marke model. Secion 3 formulaes he dynamic opimizaion problem. Secion 4 gives and proves he wo verificaion heorems associaed o he pos-defaul and pre-defaul case. Secion 5 specializes he heorems given herein o he case of invesors wih logarihmic and CRRA uiliies. Secion 6 characerizes he direcionaliy of he bond invesmen sraegy in erms of meaningful economic quaniies, and numerically illusraes how i behaves as a funcion of ime, risk aversion level of he invesor, and loss experienced a defaul, under a meaningful realisic economic scenario. Secion 7 summarizes he main conclusions of he paper. The proofs of he main heorems and necessary lemmas are deferred o he appendix. 2 The Model Assume Ω, G, G, P) is a complee filered probabiliy space, where P is he real world probabiliy measure also called hisorical probabiliy), G := G ) is an enlarged filraion given by G := F H he filraions F and H will be inroduced laer). Le {W } be a sandard Brownian moion on Ω, G, F, P), where F := F ) is a suiable filraion saisfying he usual hypoheses of compleeness and righ coninuiy. We also assume ha he saes of he economy are modeled by a coninuous-ime Markov process {X } defined on Ω, G, F, P) wih a finie sae space {x 1, x 2,..., x N }. Wihou loss of generaliy, we can idenify he sae space of {X } o be a finie se of uni vecors {e 1, e 2,..., e N }, where e i =,..., 1,...) R N and denoes he ranspose. We also assume ha {X } and {W } are independen. The following semi-maringale represenaion is well-known cf. Ellio e al. 1994)): X = X + A s)x s ds + M P ), 1) where M P ) = M1 P ),..., MN P )) is a R N -valued maringale process under P and A) = [a i,j ) i,j=1,...,n is he so-called infiniesimal generaor of he Markov process. Specifically, denoing p i,j, s) := PX s = e j X = e i ), for s, and δ i,j = 1 i=j, we have ha p i,j, + h) δ i,j a i,j ) = lim ; 2) h h cf. Bielecki and Rukowski 21). In paricular, a i,i ) := a i,j). For fuure references, we also inroduce he process C := i1 {X=e i}. 3) i=1 3

4 We consider a fricionless financial marke consising of hree insrumens: a risk-free bank accoun, a defaulable bond, and a sock. The dynamics of each of he following insrumens will depend on he underlying saes of he economy as follows: Risk-free bank accoun. The insananeous marke ineres rae a ime is r := r, X ) := r, X, where, denoes he sandard inner produc in R N and r := r 1, r 2,..., r N ) is a vecor of posiive consans. This means ha, depending on he sae of he economy, he ineres rae r will be differen; i.e., if X = e i hen r = r i. Then, he price process of he risk-free asse associaed wih risk-free bank accoun follows he dynamics db = r B d. 4) Sock price. We assume ha he sock appreciaion rae {µ } and he volailiy {σ } of he sock also depend on he economy regime in place X in he following way: µ := µ, X ) := µ, X, σ := σ, X ) := σ, X, 5) where µ := µ 1, µ 2,..., µ N ) and σ := σ 1, σ 2,..., σ N ) are vecors denoing, respecively, he differen values ha he drif and volailiy can ake depending on he differen economic regimes. Hence, we assume ha he sock price process follows he dynamics ds = µ S d + σ S dw, S = s. 6) Risky Bond price. Unlike he previous wo securiies, whose dynamics have been wrien under he hisorical measure, he bond prices are defined under a suiably chosen risk-neural measure Q and he hisorical dynamics of he process i.e. dynamics under he acual probabiliy measure P) will have o be inferred from he risk-neural dynamics. Before defining he bond price, we need o inroduce a defaul process. Le τ be a nonnegaive random variable, defined on Ω, G, P), represening he defaul ime of he counerpary selling he bond. Le H = σhu) : u ) be he filraion generaed by he defaul process H) := 1 τ, afer compleion and regularizaion on he righ, and also le G := G ) be he filraion G := F H. We use he canonical consrucion of he defaul ime τ in erms of a given hazard process {h }, which will also be assumed o be driven by he Markov process X. Specifically, hroughou he paper we assume ha h := h, X, where h := h 1, h 2,..., h N ) are posiive consans. For fuure reference, we now give he deails of he consrucion of he random ime τ. We assume he exisence of an exponenial random variable Θ defined on he probabiliy space Ω, G, P), independen of he process X ). We define τ by seing { τ := inf R + : } h u du Θ. 7) I can be proven ha h ) is he F, G)-hazard rae of τ see Bielecki and Rukowski 21), Secion 6.5 for deails). Tha is, h ) is such ha ξ P := H) 1 Hu ))h u du 8) is a G-maringale under P, where Hu ) = lim s u Hs) = 1 τ<u. Inuiively, Eq. 8) says ha he single jump process needs o be compensaed for defaul, prior o he occurrence of he even. An imporan consequence of he previous consrucion is he following so-called H hypohesis. Le us fix > and F = s F s. For any u R +, we have Pτ u F ) = 1 e u hsds. Therefore, for any u, Pτ u F ) = E P [Pτ u F ) F = 1 e u hsds = Pτ u F ). Plugging u = inside he above expression, we obain Pτ F ) = Pτ F ). 9) 4

5 I was proven in Bremaud and Yor 1978) ha Eq. 9) is equivalen o saying ha any F-square inegrable maringale is also a G-square inegrable maringale. The laer propery is also referred o as he H hypohesis or he maringale invariance principle wih respec o G, and we will make use of his propery laer on. For furher deails abou his propery in his conex he reader is referred o Secions and in Bielecki and Rukowski 21). The final ingredien in he bond pricing formula is he recovery process z ), i.e., an F-adaped righ-coninuous wih lef-limis process o be fully specified below. In erms of z ), he ime- price of he risky bond wih mauriy T is given by [ T p, T ) := E Q e u rsds z u dhu) + e T rsds 1 HT )) G, 1) where Q is he equivalen risk-neural measure used in pricing. Furhermore, we adop a pricing measure Q such ha, under Q, W is sill a sandard Wiener process and X is a coninuous-ime Markov process independen of W ) wih possibly differen generaor A Q ) := [a Q i,j ) i,j=1,2,...,n. The exisence of he measure Q in he previous paragraph follows from he heory of change of measures for denumerable Markov processes see, e.g., Secion 11.2 in Bielecki and Rukowski 21)). Concreely, for i j and some bounded measurable funcions κ i,j : R + 1, ), define and for i = j, define a Q i,j ) := a i,j)1 + κ i,j )), 11) a Q i,i ) := N k=1,k i We also fix κ i,i ) = for i = 1,..., N. Now, consider he processes where M i,j := H i,j H i := 1 {X=e i}, and H i,j := <u a Q i,k ). a i,j u)h i udu, 12) 1 {Xu =e i}1 {Xu=e j}, i j). 13) The process M i,j ) is known o be an F-maringale for any i j see Lemma in Bielecki and Rukowski 21)) and, since he H-hypohesis holds in our defaul framework, his is also a G-maringale. Then, by virue of Proposiion in Bielecki and Rukowski 21), he probabiliy measure Q on G = G ) wih Radon-Nikodym densiy {η } given by η = 1 +, i,j=1 η u κ i,j u)dm i,j u, 14) is such ha X is a Markov process under Q wih generaor [a Q i,j ) i,j=1,2,...,n. In paricular, noe ha X = X + where M Q is a R N -valued maringale under Q, and also, M Q ) = M P ) + A Q s) X s ds + M Q ), 15) As) A Q s) )X s ds. 16) Wihou loss of generaliy, Q can be aken o be such ha W is sill a Wiener process independen of X under Q. We emphasize ha he disribuion of he hazard rae process h = h, X under he risk-neural measure differs from he one under he hisorical measure. Therefore, our framework allows modeling he defaul risk premium, defined as he raio beween risk-neural and hisorical inensiies, hrough he change of measure of he underlying Markov chain. 5

6 We now proceed o obain he bond price dynamics under he hisorical probabiliy measure. Eq. 1) may be rewrien as [ T [ p, T ) = 1 τ> E Q e u rs+hs)ds zu)h u du F + 1 τ> E Q e T rs+hs)ds F, 17) which follows from Eq. 8), along wih applicaion of he following classical ideniy [ E [1 Q τ>s Y G = 1 τ> E Q e s hudu Y F, where s and Y is a F s -measurable random variable see Bielecki and Rukowski 21), Corollary 5.1.1, for is proof). We assume he recovery-of-marke value assumpion, i.e. z := 1 L )p, T ), where L is F-predicable. As wih he oher facors in our model, we shall assume ha L is of he form L := L, X, where L := L 1,..., L N ) [, 1 N. Under he recovery-of-marke value assumpion, i follows using a resul in Duffie and Singleon 1999) see also Proposiion in Bielecki and Rukowski 21)) ha [ p, T ) = 1 τ> E Q e T rs+hsls)ds F. 18) The following resul from Capponi, Figueroa-López, and Nisen 212b) will play a key role in obaining he bond price dynamics see Secion 3 herein for is proof): Lemma 2.1. Suppose ha, for any i j, he funcion a Q i,j defined in 11) is coninuously differeniable in, T ) and such ha < inf s [,T aq i,j s) sup a Q da Q i,j i,j s) < & sup s) <. 19) s [,T s,t ) ds Then, he ime- bond price under he i h -regime given by he formula is differeniable for any, T ). ψ i ) = E Q [ e T rs+hsls)ds X = e i, 2) The following resul, saed as a lemma in his paper, gives he dynamics of he defaulable bond price process under he hisorical measure P. I can be obained as a special case of he semi-maringale represenaion formulas for vulnerable claims provided in Capponi, Figueroa-López, and Nisen 212b), by seing he erminal payoff of he claim equal o one. Lemma 2.2. Under he condiions of Lemma 2.1, he pre-defaul dynamics of he bond price p, T ) under he hisorical measure P is given by { ψ), dm P ) } dp, T ) = p, T ) [r + h L 1) + D) d + ψ), X dξ P, 21) where M P )) is he N-dimensional F, P)-maringale defined in 1), ξ P ) is he G, P)-maringale defined in 8), ψ) = ψ 1 ),..., ψ N )), and D) := D 1 ),..., D N )), X wih D i ) := a i,j ) a Q i,j ))ψ j) ψ j=1 i ) = ) a i,j ) a Q i,j )) ψj ) ψ i ) 1. 22) 6

7 3 Opimal Porfolio Problem We consider an invesor who wans o maximize her wealh a ime R T by dynamically allocaing her financial wealh ino he risk-free bank accoun, he risky asse, and he defaulable bond defined in Secion 2. The invesor does no have inermediae consumpion nor capial income o suppor her purchase of financial asses. Le us denoe by ν B he number of shares of he risk-free asse B ha he invesor buys ν B > ) or sells ν B < ) a ime. Similarly, ν S and ν P denoe he invesor s porfolio posiions in he sock and risky bond a ime, respecively. The process ν B, ν S, ν P ) is called a porfolio process. We denoe V ν) he wealh of he porfolio process ν = ν B, ν S, ν P ) a ime, i.e. V ν) = ν B B + ν S S + ν P p, T ). As usual, we require he processes ν B, ν S, and ν P o be F-predicable. We also assume he following self-financing condiion: dv = ν B db + ν S ds + ν P dp, T ). Given an iniial sae configuraion x, z, v) E := {e 1, e 2,..., e N } {, 1}, ), we define he class of admissible sraegies A := Av, i, z) o be a se of self-financing) porfolio processes ν such ha V ν) for all when X = x, H = z, and V = v. Le π := π B, π S, π P ) be defined as π B := νb B V ν), πs := νs S V ν), πp = νp p, T ), 23) V ν) if V ν) >, while π B = π P = π S =, when V ν) =. The vecor π := π B, π S, π P ), called a rading sraegy, represens he corresponding fracions of wealh invesed in each asse a ime. Noe ha if π is admissible, hen he dynamics of he resuling wealh process V π can be wrien as dv π = V π {π B db B + π S ds S + π P } dp, T ) p,, T ) under he convenion ha / =. This convenion is needed o deal wih he case when defaul has occurred > τ), so ha p, T )= and we fix π P =. Using he dynamics derived in Proposiion 2.2 and ha π B + π P + π S = 1, we have he following dynamics of he wealh process [ {r dv π = V π + π S µ r ) + π P [h L 1) + D) } ψ), dm d + π S σ dw + π P P ) π P dξ P, 24) ψ), X under he hisorical probabiliy P. 3.1 The uiliy maximizaion problem For an iniial value x, z, v) E and an admissible sraegy π = π B, π S, π P ) Ax, z, v), le us define he objecive funcional o be J R x, z, v; π) := E [UV P R π ) X = x, H = z, V = v ; 25) i.e. we are assuming ha he invesor sars wih v dollars is iniial wealh), ha he iniial defaul sae is z z = means ha no defaul has occurred ye), and he iniial value for he underlying sae of he economy is x. The consrain V = v is also called he budge consrain. As usual, we assume ha he uiliy funcion U : [, ) R { } is sricly increasing and concave. Our goal is o maximize he objecive funcional Jx, z, v; π) for a suiable class of admissible sraegies π := π B, π S, π P ). Furhermore, we shall focus on feedback or Markov sraegies of he form π = π B, V C, H ), π S, V C, H ), π P, V C, H )), 7

8 for some funcions πi B, πp i, πs i : [, ) [, ) {, 1} R such ha πi B, v, z) + πs i, v, z) + πp i, v, z) = 1. Above, we had used he process C ) defined in 3). As usual, we consider insead he following dynamical opimizaion problem: [ ϕ R, v, i, z) := sup E P UV π,,v R ) V = v, X = e i, H) = z, 26) π A v,i,z) for each v, i, z), ) {1, 2,..., N} {, 1}, where dv π,,v s = V π,,v s V π,,v = v. [ {rs + π S s µ s r s ) + π P s 1 Hs ))[h s L s 1) + Ds) } ds + π S s σ s dw s + π P s 1 Hs )) ψs), dm P s) ψs), X s π P s 1 Hs ))dξ P s, s [, R, 27) The class of processes A v, i, z) is defined as follows: Definiion 3.1. Throughou, A v, i, z) denoes a suiable class of F-predicable locally bounded feedback rading sraegies π s := πs S, πs P ) := πc S π,,v s s, V s, Hs )), π P π,,v C s s, V s, Hs ))), s [, R, > for any s [, R when X = e i and H) = z. Throughou his paper, a rading sraegy saisfying hese condiions is simply said o be -admissible wih respec o he iniial condiions V = v, X = e i, and H = z). such ha 27) admis a unique srong soluion {V π,,v s } s [,R and V π,,v s Remark 3.1. As i will be discussed below see Eqs. 32), 33), and 83)), he jump V s := V s V s of he process 27) a ime s is given by V s = V s N i=1 ) πi P s, V s, Hs )) ψ js) ψ i s) Hs i,j πs P Hs) ψ i s). 28) Since for {V s } s o be sricly posiive, i is necessary and sufficien ha V s > V s for any s > a.s. cf. Jacod and Shiryaev, 23, Theorem 4.61)), we conclude ha in order for π P o be admissible, i is necessary ha, ) ψ i s) M i := max < πi P s, v, z) < 1, 29) :ψ is)<ψ js) ψ j s) ψ i s) for any s, v >, z {, 1}, and i = 1,..., N, where we se M i := if ψ i s) ψ j s) for all j i. 4 Verificaion Theorems As i is usually he case, we sar by deriving he HJB formulaion of he value funcion 26) via heurisic argumens. We hen verify ha he soluion of he proposed HJB equaion when i exiss and saisfies oher regulariy condiions) is indeed opimal. Such a resul is called he verificaion heorem of he opimizaion problem. Le us assume for now ha ϕ R, v, i, z) is C 1 in and C 2 in v for each i and z. Then, using Iô s rule along he lines of Appendix A, we have ha ϕ R, V π, C, H)) = ϕ R r, V π r, C r, H r ) + r Lϕ R s, V π s, C s, H s )ds + M M r, where C ) is he Markov process defined in 3), L is he infiniesimal generaor of, V, C, H ) given in Eq. 86), and M ) is he maringale given by Eq. 87). Nex, if r < < R, by virue of he dynamic programming principle, we expec ha ϕ R r, Vr π, C r, H r ) = max E [ ϕ R, V π, C, H)) G r. 3) π 8

9 [ Therefore, we obain E r LϕR s, Vs π, C s, H s )ds G r, wih he inequaliy becoming an equaliy if π = π, where π denoes he opimum. Now, evaluaing he derivaive wih respec o, a = r, we deduce he following HJB equaion: max π LϕR r, v, i, z) =, 31) wih boundary condiion ϕ R T, v, i, z) = Uv). In order o furher specify 31), le us firs noe ha he dynamics 27) can be wrien in he form dv s = α Cs ds + ϑ Cs dw s + β dm P Cs,j j s) γ dξp Cs s, < s < R), 32) wih coefficiens j=1 β i,j, v, z) = vπ P i, v, z)1 z) ψ j) ψ i ), γ i, v, z) = vπ P i, v, z)1 z), α i, v, z) = v [ r i + π S i, v, z)µ i r i ) + π P i, v, z)1 z)h i L i 1) + D i )) 33) ϑ i, v, z) = π S i, v, z)σ i v, where D i ) is defined as in 22). Using he expression for he generaor in Eq. 86), he noaion ϕ i,z, v) := ϕ R, v, i, z), and he relaionship π B = 1 π S π P, and dropping he dependence of he sraegies from he riple, v, z) o lighen noaion, 31) can be wrien as follows for each i = 1,..., N: = ϕ i,z ϕ i,z + vr i v + z a i,j ) [ϕ j,z, v) ϕ i,z, v) { πi S µ i r i )v ϕ } i,z v + πs i ) 2 σ2 i 2 v2 2 ϕ i,z v 2 + max π S i + 1 z) max π P i + a i,j ) { πi P θ i )v ϕ i,z v [ [ϕ j,z, v 1 + πi P + h [ i ϕi,1, v1 πi P )) ϕ i,z, v) )) } ψj ) ψ i ) 1 ϕ i,z, v), 34) where θ i ) := h i L i ) a Q i,j ) ψj ) ψ i ) 1. 35) We can consider wo separae cases and ϕ R, v, i) = ϕ i,, v) = ϕ R, v, i, ), pre-defaul case) 36) ϕ R, v, i) = ϕ i,1, v) = ϕ R, v, i, 1), pos-defaul case). 37) Secion 4.1 proves a verificaion heorem for he pos-defaul case, while Secion 4.2 proves a verificaion heorem for he pre-defaul case. 4.1 Pos-defaul case In he pos-defaul case, we have ha p, T ) =, for each τ < T. Consequenly, π P = for τ < T and, since π B = 1 π S π P, we can ake π = π S as he unique conrol. Below, η i := µi ri σ i denoes he Sharpe raio of he risky asse under he i h sae of economy and C 1,2 denoes he class of funcions ϖ : [, R R + {1,..., N} R + such ha ϖ,, i) C 1,2, R) R + ) C[, R R + ), ϖ v s, v, i), ϖ vv s, v, i), for each i = 1,..., N. We have he following verificaion resul, whose proof is repored in Appendix B: 9

10 Theorem 4.1. Suppose ha here exiss a funcion w C 1,2 ha solves he nonlinear Dirichle problem w s, v, i) η2 i 2 w 2 vs, v, i) w vv s, v, i) + r ivw v s, v, i) + a i,j s) ws, v, j) ws, v, i)) =, 38) for any s, R) and i = 1,..., N, wih erminal condiion wr, v, i) = Uv). We assume addiionally ha w saisfies i) ws, v, i) Ds) + Es)v, ii) w v s, v, i) w vv s, v, i) Gs)1 + v), 39) for some locally bounded funcions D, E, G : R + R +. Then, he following saemens hold rue: 1) w, v, i) coincides wih he opimal value funcion ϕ R, v, i) = ϕ R, v, i, 1) in 26), when A v, i, 1) is consrained o he class of -admissible feedback conrols π S s = π Cs s, V s ) such ha π i, ) C[, R R + ) for each i = 1,..., N and vπ i s, v) Gs)1 + v), 4) for a locally bounded funcion G. If he soluion w is non-negaive, hen condiion 4) is no needed. 2) The opimal feedback conrol {π S s } s [,R), denoed by π S s, can be wrien as π S s = π Cs s, V s ) wih 4.2 Pre-defaul case π i s, v) = η i σ i w v s, v, i) vw vv s, v, i). 41) In he pre-defaul case z = ), we ake π S and π P as our conrols. We hen have he following verificaion resul, proven in Appendix B: Theorem 4.2. Suppose ha he condiions of Theorem 4.1 are saisfied and, in paricular, le w C 1,2 be he soluion of 38). Assume ha w C 1,2 and p i = p i s, v), i = 1,..., N, solve simulaneously he following sysem of equaions: θ i s) w v s, v, i) h i ϕ R s, v1 p v i), i) + ) [ ) ) ψj s) a i,j s) ψ i s) 1 ψj s) w v s, v 1 + p i ψ i s) 1, j =, 42) w s, v, i) η2 i w 2 { vs, v, i) 2 w vv s, v, i) + r iv w v s, v, i) + p i θ i )v w v s, v, i) + h i [ws, v1 p i ), i) ws, v, i) + a i,j ) [ w s, v 1 + p i ψj s) ψ i s) 1 )), j ) ws, v, i) } =, 43) for < s < R, wih erminal condiion wr, v, i) = Uv). We also assume ha p i s, v) saisfies 29) and 4) uniformly in v and i) and w saisfies 39). Then, he following saemens hold rue: 1) w, v, i) coincides wih he opimal value funcion ϕ R, v, i) = ϕ R, v, i, ) in 26), when A v, i, ) is consrained o he class of -admissible feedback conrols πs S, πs P ) = π S s, V C s s, Hs )), π P s, V C s s, Hs ))) such ha π S i,, z), π P i,, z) C[, R R + ), for each i = 1,..., N, π S saisfies 4) for a locally bounded funcion G, and π P saisfies 29) and 4) uniformly in v, i, z). If he soluion w is non-negaive, hen hese bound condiions are no needed. 2) The opimal feedback conrols are given by π s S := π S s, V C s, Hs)) and π s P s := π P, V C, Hs)) wih s π i S s, v, z) = η i w v s, v, i) σ i v w vv s, v, i) 1 z) η i w v s, v, i) z, 44) σ i vw vv s, v, i) π P i s, v, z) = p i s, v)1 z). 45) 1

11 5 Consrucion of Explici Soluions In his secion, we specialize he framework developed above o defaulable regime-swiching markes wih logarihmic and CRRA invesors. Secion 5.1 analyzes a logarihmic invesor, while secion 5.2 considers a CRRA invesor. 5.1 Logarihmic invesor We consider an invesor wih uiliy funcion given by Uv) = logv). We will show ha he coupled sysem yielding he opimal pre-defaul value funcion and bond invesmen sraegy decouples, hereby faciliaing he consrucion of explici soluions. We sar by giving a lemma, which will be used laer o characerize he opimal pre-defaul value funcions, as well as he long/shor direcionaliy of he bond invesmen sraegy. Lemma 5.1. The sysem of equaions h i θ i s) + ψ j s) ψ i s) a i,j s) =, 46) 1 p i ψ i s) + p i ψ j s) ψ i s)) for i = 1,..., N, admis a unique real soluion p i s) in he inerval M i, 1), where M i [, ) is defined as in 29). Moreover, if for each i, j = 1,..., N, a i,j and a Q i,j are coninuous funcions, hen ps, i) is a coninuous funcion of s. Proof. For fixed i and s, consider he funcion fp i, s, i) := θ i s) h i + ψ j s) ψ i s) a i,j s) 1 p i ψ i s) + p i ψ j s) ψ i s)). We firs observe ha fp i, i, s) is a coninuous funcion of p i in he inerval M i, 1). Indeed, we can wrie he above summaion as a i,j s) a i,j s) ψ is) :ψ js)>ψ is) ψ + p + ψ is) js) ψ is) i :ψ js)<ψ is) ψ + p, js) ψ is) i and since 1 < ψis) ψ when ψ ψ js) ψ is) js) < ψ i s), we have p i + is) ψ < for p js) ψ is) i M i, 1). Moreover, he previous decomposiion also shows for each fixed s, fp i, s, i) is sricly decreasing in p i from M i, 1) ono, ). This implies he exisence of a unique p i s) such ha fp i s), s, i) =, for any s >. In ligh of Kumagai 198) implici heorem, we will also have ha p i s) is coninuous if we prove ha fp i, s, i) is coninuous in p i, s). The laer propery follows because, by assumpion, a i,j and a Q i,j are coninuous, which implies direcly he coninuiy of he funcions θ i. The coninuiy of he funcions ψ j will follow from a similar argumen o ha of Lemma 2.1. The following resul characerizes he opimal pre/pos pos defaul value funcions. Appendix C.1. The proof is repored in Proposiion 5.2. Assume ha he a Q i,j s and a i,j s are coninuous in [, T. Then, he following saemens hold: 1) The opimal pos-defaul value funcion is given by ϕ R, v, i) = logv) + K, i), where R, and K) = [K, 1), K, 2),... K, N) is he unique posiive soluion of he linear sysem of firs order differenial equaions where = [,..., R N and K ) = F )K) + b), KR) =, 47) [F ) i,j := a i,j ), [b) i := ) r i + η2 i, i, j = 1,..., N. 48) 2 11

12 2) The opimal percenage of wealh invesed in sock is given by π S ) = [ π 1 S ), π 2 S ),..., π N S ), where π j S ) = µ j r j σj 2, R. 3) The opimal percenage of wealh invesed in he defaulable bond is π P j ) = p j)1 τ>, while he opimal predefaul value funcion is ϕ R, v, i) = logv) + J, i), where J) = J, 1), J, 2),... J, N)) is he unique posiive soluion of he linear sysem of firs order differenial equaions J, i) = G)J) + d), JR) =, 49) wih [G) i,j = a i,j ), i j), [G) i,i = h i a i,i ), [ [d) i = r i + η2 i 2 + p i)θ i ) + h i log1 p i )) + K, i)) + )) ψj ) a i,j ) log 1 + p i ) ψ i ) 1, 5) and p) = [p 1 ), p 2 ),..., p N ) is he unique coninuous soluion of he nonlinear sysem of equaions 46). We noe ha he difference beween he pre-defaul and pos-defaul opimal value funcion lies in he ime and regime dependen componen. Moreover, he opimal proporion of wealh invesed in socks is consan in every economic regime, and independen on ime and curren level of wealh, similarly o he findings in Soomayor and Cadenillas 29) and Bo e al. 21), where infinie-ime horizon problems are considered. We also find ha he opimal proporion of wealh allocaed o he defaulable bond depends on ime and regime, bu no on he curren level of wealh. Bo e al. 21) find ha he opimal allocaion only depends on ime hrough he defaul risk premium. We also have he following corollary. Corollary 5.3. Assume he generaor marix A o be ime-invarian or homogenous i.e. a i,j ) a i,j, for all ). Then, K) = R e s )F b s)ds, and J) = where F and b are given in Eq. 48) and G and d in Eq. 5). R e s )G d s)ds, Proof. I follows direcly from applicaion of Lemma C.1, par 3), given he equaions for K and J given in Eq. 47) and 49), respecively. 5.2 CRRA invesor In his secion, we consider a CRRA invesor wih uiliy Uv) = vγ γ, wih < γ < 1. In conras o a logarihmic invesor, we will see ha he sysem characerizing he opimal bond sraegy and pre-defaul value funcion does no decouple. Neverheless, we provide condiions for he exisence of soluions. We sar giving he expressions for he pos-defaul value funcion and sock invesmen sraegy, which similarly o he logarihmic case can be compued explicily. Proposiion 5.4. Assume ha he a Q i,j s and a i,j s are coninuous in [, T. Then, he following saemens hold: 12

13 i) The opimal pos-defaul value funcion is given by ϕ R, v, i) = v γ K, i), R), where K) = [K, 1), K, 2),... K, N) is he unique posiive soluion of he linear sysem of firs order differenial equaions K ) = F )K), KR) = 1 1, 51) γ wih 1 = [1,..., 1 R N and [F ) i,j = { ) γr i η2 i γ 2 γ 1 + a i,i), if i = j, a i,j ). if j i. 52) ii) The opimal percenage of wealh invesed in sock a ime in a pos defaul scenario is given by π S j ) = µ j r j σ 2 j 1 1 γ. 53) Proof. i) I can be checked by direc subsiuion ha ϕ R, v, i) = K, i)vγ solves he Dirichle problem 38), wih erminal condiion Uv) = v γ /γ, if and only if he funcions K, i), i = 1,..., N, R, saisfy he sysem of ODE s given by Eq. 51). Using he subsiuion s = R, we have ha he soluion Ks) of he iniial value problem given by K s s) = F R s) Ks) 1 s R), K, i) =, i = 1,..., N), 54) γ is such ha K) = KR ). Using Lemma C.1, par 1), we have ha he unique soluion of sysem 54) can be wrien as Ks) = φ F R s, R)γ 1 1. Therefore, using ha K) = KR ), we obain ha K) = φ F, R)γ 1 1 = φ F R, )γ 1 1. As for all i j, and for all, we have [F ) i,j, hen [ F s) i,jds. Therefore, using Lemma C.1, par 2), we obain ha φ F R, ) has all nonnegaive enries, and consequenly K, i) for all R and i = 1,..., N. Hence, ϕ R, v, i) C 1,2 due o facs ha K, i) and v γ is concave and increasing in v. Under he choice D) =, E) = max i=1,...,n K, i), and G) = γ 1) 1, he funcion ϕ R, v, i) saisfies he condiions in 39). Therefore, applying Theorem 4.1, we can conclude ha, for each i = 1,..., N, ϕ R, v, i) is he opimal pos-defaul value funcion. ii) Plugging he expression for ϕ R, v, i) inside Eq. 41), we obain immediaely Eq. 53). We also have he following corollary. Corollary 5.5. Assume he rae marix F defined in 52) o be ime invarian. Then, we have ha 1) The pos-defaul value funcion is given by K) = e R)F 1 γ 1. 55) 2) For each i {1,..., N}, K, i) is a decreasing funcion of. Proof. 1) I follows direcly from Lemma C.1, par 3), using he expression Ks) = φ F R s, R). 2) I is enough o prove ha he ime derivaive vecor K ) consiss of all negaive enries. From Eq. 55), we obain ha K ) = F e R)F 1 γ 1. Using he well known fac ha if wo marices A and B commue, hen Ae B = e B A, we ge K ) = e R)F F 1 γ 1. We noice ha F 1 γ 1 is a vecor whose enries are negaive and given by [F 1γ 1 i = γr i + η2 i 2 13 γ γ 1

14 Since R)F consiss of posiive off-diagonal enries, from lemma C.1, par 2), we have ha e R)F nonnegaive enries, and consequenly K, i) for all i and, hus compleing he proof. has all We now consider he pre-defaul case. The following resul gives sufficien condiions for he exisence of he pre-defaul value funcion, provided ha a cerain non-linear sysem of ODE s is well-posed. Proposiion 5.6. Assume ha he a Q i,j s and a i,j s are coninuous in [, T and le K) = [K, 1), K, 2),... K, N) be he unique posiive soluion of he linear sysem of firs order differenial equaions 51). Suppose solve simulaneously he sysem of equaions: J) = [J, 1), J, 2),... J, N), and p) = [p 1 ),..., p N ), J ) = G, p))j) + d, p)), JR) = 1 1, 56) γ θ i )J, i) h i K, i)1 p i )) γ 1 + ) )) γ 1 ψj ) a i,j )J, j) ψ i ) 1 ψj ) 1 + p i ) ψ i ) 1 =, 57) where G : R + R N R N N and d : R + R N R N 1 are given by )) γ ψj ) [G, p) i,j = a i,j ) 1 + p i ψ i ) 1, i j), ) [G, p) i,i = η2 i γ 2 γ 1 + r iγ + p i γθ i ) h i + a i,i ), Then, he opimal pre-defaul value funcion is given by [d, p) i = h i 1 p i ) γ K, i), p = [p 1,..., p N. 58) ϕ R, v, i) = v γ J, i). 59) The opimal percenage of wealh invesed in sock in he pre-defaul scenario is given by π S j ) = µ j r j σ 2 j 1 1 γ, while he opimal percenage of wealh invesed in bond is π P j ) = 1 τ>p j ). The proof of he previous proposiion follows immediaely by plugging he funcion ϕ R, v, i) in Eq. 59) inside he coupled sysem given by Eq. 42) and Eq. 43). The opimal sock sraegy follows immediaely from Theorem 4.2, iem 2), using Eq. 59). Therefore, he opimal demand in sock is myopic and independen from he value funcions and from he defaul even, while he opimal defaulable bond sraegy is non-myopic and dependen on he relaion beween hisorical and risk neural regime-swiching probabiliies. Noe ha he sysem 56)-57) can be formulaed as a non-linear sysem of differenial equaions on R + R N + of he form: J ) = Ĝ, J))J) + d, J)), JR) = 1 1, 6) γ where Ĝ : R + R N + R N N and d : R + R N + R N N are defined for J = [J 1,..., J N R N + and as Ĝ, J) = G, p, J)), and d, J) = d, p, J)), wih G and d given as in Proposiion 5.6, and p, J) := [p 1, J),..., p N, J) defined implicily by he sysem of equaions θ i )J i h i K, i)1 p i, J)) γ 1 + ) )) γ 1 ψj ) a i,j )J j ψ i ) 1 ψj ) 1 + p i, J) ψ i ) 1 =. 61) The following Lemma shows ha indeed p, J) is well-defined for, J) R + R N +. 14

15 Lemma 5.7. Assume J R N +. The sysem 61) admis a unique real soluion p i, J) in he inerval M i ), 1), where M i ) is defined as in 29). Moreover, if for each i, j = 1,..., N, a i,j and a Q i,j are differeniable funcions of, hen, J) p, J) is differeniable a each, J) R +, ) N. Proof. For fixed i and s, consider he funcions f i, p, J) = θ i )J i h i K, i)1 p i ) γ 1 + ) )) γ 1 ψj ) a i,j )J j ψ i ) 1 ψj ) 1 + p i ψ i ) 1, 62) where J = [J 1,..., J N and p = [p 1,..., p N. We observe ha f i, p, J) is a coninuous funcion of p i in he inerval M i ), 1). Moreover, we know by Proposiion 5.4 ha K, i) and, by assumpion J j, hus implying ha p i f i, p, J) is sricly decreasing in p i M i, 1). We consider wo cases: M i, ) and M i =. In he firs case, i is easy o check ha lim p M + f i, p, J) = and lim p 1 f i, p, J) =. Therefore, applying he i Inermediae Value Theorem, here exiss unique p, J) = [p 1, J),..., p N, J) such ha f i, p, J), J) =, for i = 1,..., N. The case M i = means ha ψ j )/ψ i ) 1 for all j i. Then, we have lim p f i, p, J) = θ i )J i. By he definiion 35), θ i > as h i, L i > ) and, hence, Inermediae Value Theorem implies again he exisence of a unique p, J) = [p 1, J),..., p N, J). The differeniabiliy of p, J) follows direcly from he implici funcion heorem. Nex, we prove ha he non-linear sysem 6) has a unique soluion in a local neighborhood {, J) R + R+ N : R < a, J i 1/γ < b, i = 1,..., N} for some a > and b >. For illusraion purposes, le us consider in deail he case N = 2. In ha case, he sysem 56-57) akes he form: )) γ ψ2 ) J, 1) = a 1,2 )J, 2) 1 + p 1, J)) ψ 1 ) 1 63) ξ 1 ) + γθ 1 )p 1, J))) J, 1) h 1 K, 1)1 p 1, J))) γ, )) γ ψ1 ) J, 2) = a 2,1 )J, 1) 1 + p 2, J)) ψ 2 ) 1 64) ξ 2 ) + γθ 2 )p 2, J))) J, 2) h 2 K, 2)1 p 2, J))) γ, JR, 1) = JR, 2) = 1 γ, 65) where ξ i ) := η2 i γ 2 γ 1 + r iγ h i + a i,i ) and he funcions p 1, J), p 2, J) : R + R 2 + R are defined implicily by he following equaions for any J := [J 1, J 2 : ) )) γ 1 = θ 1 )J 1 h 1 K, 1)1 p 1, J)) γ 1 ψ2 ) + a 1,2 )J 2 ψ 1 ) 1 ψ2 ) 1 + p 1, J) ψ 1 ) 1, 66) ) )) γ p 2, J). 67) = θ 2 )J 2 h 2 K, 2)1 p 2, J)) γ 1 + a 2,1 )J 1 ψ1 ) ψ 2 ) 1 ψ1 ) ψ 2 ) 1 Noe ha while he range of one of he funcions p i s is bounded, he oher funcion will be unbounded. For insance, if ψ 2 )/ψ 1 ) > 1, hen p 1, J) will ake values on he bounded domain ψ 2 )/ψ 1 ) 1) 1, 1), while p 2, J) will ake values on, 1). In urn, his fac makes he righ hand-side of equaion 64) poenially unbounded and also is he main reason why i is no possible o obain global exisence wihou furher resricions see Remark C.2 in Appendix C.2 for more informaion). The following resul shows he local exisence and uniqueness of he soluion he proof of Proposiion 5.8 is repored in Appendix C.2). Proposiion 5.8. Suppose ha a i,j ) and a Q i,j ) are differeniable funcions. Then, for any b > γ, here exiss an α := αb) > and a unique funcion J : R α, R [b 1, b N saisfying 63-64) wih erminal condiion JR, 1) = JR, 2) = γ 1. 15

16 Remark 5.9. Under he condiions of Proposiion 5.8, i is known see, e.g., Theorem in Chicone 26)) ha if R α, R + ᾱ) wih ᾱ, α, ) is he maximal inerval of exisence of he soluion of 63-65) and α <, hen eiher J), J, 1), or J, 2) as R α. Moreover, he soluion of 63-65) can be found by he sandard Picard s fixed-poin algorihm. Hence, for insance, one can show numerically wheher he soluion is well defined in he whole inerval [, R by analyzing wheher he numerical soluion blows up or converges o. Based on he above analysis, we deduce ha he pre-defaul scenario of he power invesor differs from he one of he logarihmic invesor. In he logarihmic case, he wo sysems decouple, and he pre-defaul value funcion may be obained in erms of an inegral of a marix exponenial for ime invarian generaors, see Corollary 5.3 for deails. 6 Economic Analysis In his secion, we provide a deailed economic analysis of he corporae bond invesmen sraegies and value funcions for he ype of invesors considered in Secion 5. The objecive is o invesigae how he inerplay beween he hisorical and risk-neural generaors of he Markov chain, ime o mauriy, defaul inensiy and loss parameers, affec he direcionaliy of he bond invesmen sraegy. Moreover, we illusrae how he risk aversion level γ of he power uiliy invesor affecs he bond invesmen sraegy, including he limiing case of he logarihmic invesor. We firs inroduce he necessary noaion and erminology in Secion 6.1. Secion 6.2 characerizes he direcionaliy of he bond invesmen sraegy for CRRA and logarihmic invesors in erms of corporae reurns, insananeous forward raes, and expeced recovery a defaul. We presen a comparaive saic analysis under a realisic simulaion scenario in Secion Noaion and erminology Throughou, R n m respecively, R n m + ) denoes he se of n m resp., posiive) real marices G. Given G R n m, [G i,j denoes is i, j) enry. Nex, we give some definiions, which will be used o characerize he opimal sraegies. Le us recall ha C is given by 3). We noice ha he pre-defaul regime condiioned bond price ψ i ), defined in 2) depends on he mauriy T. In he following, we will someimes use he noaion ψ i, T ) o emphasize his dependence. In all definiions o follow, we assume he macroeconomy o be in he i h regime a. Le A Υ ) = [a Υ i,j ) i,j=1,...,n be he infiniesimal generaor of he Markov process X ), under a given equivalen probabiliy measure Υ. Definiion 6.1. For any s <, we have he following erminology: The expeced corporae bond reurn per uni ime under he measure Υ, during he inerval [, s, is defined as E Υ i, s) := 1 [ ψcs s, T ) ψ i, T ) s EΥ ψ i, T ) X = e i. The expeced insananeous corporae bond reurn, under he measure Υ, is defined as E Υ i ) := lim s + EΥ i, s). 68) The insananeous forward rae of he defaulable bond a ime is defined as g i ) := log ψ i, T ) T. 69) T = Noe ha he above definiions are meaningful because he funcion ψ i ) is differeniable in ime, as i has been shown in Capponi, Figueroa-López, and Nisen 212b) Lemma 4.1 herein). We have he following useful resuls. 16

17 Lemma 6.1. The insananeous forward rae is given by [ g i ) = a Q ψ j, T ) ψ i, T ) i,j, 7) ψ i, T ) while he insananeous corporae bond reurn, under he equivalen measure Υ, is given by E Υ i ) = Proof. For he firs ideniy, a simple calculaion shows ha ψ i, T ) T = lim T = ψ i, T ) ψ i +, T ) = a Q i,j ψ j, T ) ψ i, T )). a Υ i,j) ψ j, T ) ψ i, T ). 71) ψ i, T ) p Q i,j, + ) = lim ψ j +, T ) ψ i +, T )) Using he previous equaion and he definiion of insananeous forward rae given in Eq. 69), we obain Eq. 7). For he second ideniy, le p Υ i,j, s) denoe he probabiliy ha he chain wih generaor AΥ ) ransis o regime j a ime s >, given ha i is in regime i a ime. Then, Ei Υ 1 ) = lim s + s EΥ = [ ψcs s, T ) ψ i, T ) ψ i, T ) X = e i = lim a Υ i,j) ψ js, T ) ψ i, T ). ψ i, T ) p Υ i,j s +, s) s ψ j s, T ) ψ i, T ) ψ i, T ) 6.2 Characerizaion of long/shor opimal invesmen sraegies In his secion, we provide condiions under which he logarihmic and CRRA invesor would go long or shor in he defaulable bond. Recall ha, under he i h economy regime, he opimal percenage of wealh invesed in he defaulable bond is given by π P i ) = 1 τ>p i ), where p i ) is idenified in he Lemma 5.1. We firs characerize he direcionaliy of he sraegy for a logarihmic invesor. Lemma 6.2. I holds ha p i ) > if and only if E P i ) + g i ) > h i 1 L i ). 72) Proof. Firs, we esablish ha p i ) > if and only if he following relaion holds. ) ψj ) ) ψ i ) 1 a i,j) a Q i,j ) > h i 1 L i ). 73) Using Lemma 5.1 and Eq. 35), we have ha he fracion of wealh invesed in he bond a ime saisfies he following equaion 1 h i a i,j ) = ψ p i ) + i) 1 p i ) h il i + a Q i,j ) 1. 74) ψ i) ψ j) ψ i) ψ j) ψ i) Noice ha, for each fixed, he lef hand side is a sricly decreasing funcion of p i ) from M i, 1) o, ). The righ hand side, insead, is a sricly increasing funcion of p i ) defined from M i, 1) o, ). Moreover, we know from lemma 5.1 ha here exiss a unique p i ) saisfying Eq. 74). Evaluaing boh lef and righ hand side a p i ) = 17

18 leads o he conclusion ha p i ) > if and only if Eq. 73) holds. From he definiion of expeced insananeous corporae bond reurn given in Eq. 68), compued under he hisorical measure Υ = P, and using Eq. 2), we obain ha E P i ) = a i,j) ψj) ψi) ψ i). Using his relaion and Eq. 7), we may rewrie Eq. 73) as in Eq. 72). We say ha he long condiion of he logarihmic invesor is saisfied when he relaionship 72) holds. The following corollary provides sufficien condiions for he logarihmic invesor o always go shor in he defaulable securiy. Corollary 6.3. For a logarihmic invesor, he following saemens hold: i) If N = 1, hen for each fixed, we have p) = 1 1 L 1 <. ii) For fixed, i, if a Q i,j ) = a i,j) for any j i, hen p i ) <. Proof. Boh in he case when N = 1 and in he case when a Q i,j ) = a i,j), we have ha E P i ) = g i). Therefore, he long condiion in 72) will be never saisfied. Moreover, in case when N = 1, we can see direcly from Eq. 46) ha p i ) = 1 1 L 1. This ends he proof. Corollary 6.3 show ha in he mono-regime scenario, or in he case when a Q i,j ) = a i,j), he corporae bond reurn ges reduced by he insananeous forward credi spreads g i, T ) by an amoun which makes i smaller han he expeced recovery a defaul. This leads he invesor o go always shor in he securiy, because he compensaion offered by he marke is no enough o compensae him for he credi risk incurred. Moreover, iem i) of Corollary 6.3 shows ha 1) in case of zero recovery on he defaulable bond L i = 1), he logarihmic invesor would no rade a all in he defaulable securiy, and 2) he amoun of bond unis shored is a decreasing funcion of he loss incurred a defaul. Nex, we characerize he direcionaliy of he sraegy for he power invesor. Le us define a measure P, equivalen o he hisorical measure P, via he generaor A P = [a P i,j of he Markov process given by J, j) a P i,j ) := a i,j ) J, i), j i), and a P i,i ) := k=1,k i a P i,j ), 75) where J, j) > is he ime componen of he opimal pre-defaul value funcion defined by Eq. 56), 57), and 58). Inuiively, he measure P is redisribuing he mass of he hisorical disribuion P owards hose regimes j wih higher values of he pre-defaul value funcion wih respec o regime i. We nex characerize he direcionaliy of he sraegy for he CRRA invesor, where p i ) is coupled wih he pre-defaul value funcions as indicaed in Proposiion 5.6. Lemma 6.4. Under he assumpions of Proposiion 5.6, we have ha p i ) > if and only if ) K, i) E P i ) + g i ) > h i J, i) L i. 76) Proof. Firs, we show ha p i ) > if and only if he following relaion holds. a i,j )J, j) ψ j) ψ i ) ψ i ) We may rewrie Eq. 57) from Proposiion 5.6 as ψ i) ψ j) ψ i) a i,j )J, j) 1 + p i ) ψj) ψi) ψ i) > h i K, i) L i J, i)) + h i ) 1 γ = 1 p i )) 1 γ K, i) h il i J, i) + a Q i,j )J, i)ψ j) ψ i ). 77) ψ i ) a Q i,j ψ i) ψ j) ψ i) )J, i), 78) where we have used he expression for θ i ) given in Eq. 35). I can be easily checked ha he lef hand side of Eq. 78) is a decreasing funcion of p i ) from M i, 1) o, ). The righ hand side, insead, is a sricly increasing funcion of p i ), defined from M i, 1) o, ). Since we are assuming ha here exiss a unique soluion p i ) o he 18