Parameter Uncertainty: The Missing Piece of the Liquidity Premium Puzzle?

Transcription

1 Parameer Uncerainy: The Missing Piece of he Liquidiy Premium Puzzle? Ferenc Horvah Tilburg Universiy November 14, 2016 Absrac I analyze a dynamic invesmen problem wih sochasic ransacion cos and parameer uncerainy. I solve he problem numerically and obain he opimal consumpion and invesmen policy and he leas-favorable ransacion cos process. Using reasonable parameer values, I confirm he liquidiy premium puzzle, i.e., he represenaive agen model (wihou robusness) produces a liquidiy premium which is by a magniude lower han he empirically observed value. I show ha my model wih robus invesors generaes an addiional liquidiy premium componen of 0.05%-0.10% (depending on he level of robusness) for he firs 1% proporional ransacion cos, and hus i provides a parial explanaion o he liquidiy premium puzzle. Addiionally, I provide a novel non-recursive represenaion of discree-ime robus dynamic asse allocaion problems wih ransacion cos, and I develop a numerical echnique o efficienly solve such invesmen problems. JEL classificaion: C61, G11, G12 Keywords: liquidiy, dynamic asse allocaion, robusness, uncerainy, ambiguiy, liquidiy premium puzzle Tilburg Universiy, Deparmen of Finance, P.O. Box 90153, 5000 LE Tilburg, he Neherlands. Tel.: address: f.horvah@ilburguniversiy.edu

2 1 Inroducion One of he many asse pricing puzzles in he finance lieraure is he liquidiy premium puzzle. Theoreical work finds ha ransacion cos only has a second-order effec on expeced reurns, while empirical research documens a firs-order effec. For example, he seminal work of Consaninides (1986) finds ha a 1% proporional ransacion cos on socks increases he expeced sock reurn only by 0.2%. On he oher hand, Amihud and Mendelson (1986) empirically find ha for a 1% proporional ransacion cos, he liquidiy premium is abou 2%, based on NYSE daa. In his paper, I approach he liquidiy premium puzzle from a new angle. A key feaure of dynamic asse allocaion problems wih ransacion cos is he se of assumpions abou he ransacion cos. In my model I no only allow he ransacion cos o be sochasic, bu I also relax he assumpion ha he invesor knows he precise probabiliy disribuion of fuure ransacion coss. She akes his uncerainy abou he ransacion cos process ino accoun when she solves a dynamic invesmen problem, and she makes robus decisions wih respec o consumpion and asse allocaion. A robus invesor is uncerain abou one or several parameers of he underlying model. Acknowledging his uncerainy, she makes invesmen decisions ha work well no only if he model ha she had in mind is correc, bu which work reasonably well even if her model urns ou o be misspecified. In his paper I use he minimax approach of Anderson, Hansen, and Sargen (2003). The invesor is uncerain abou he (fuure) ransacion coss in he underlying model, bu she has a base ransacion cos level in mind, which she considers o be he mos reasonable. Since she is uncerain abou he rue fuure ransacion coss, she considers oher alernaive ransacion cos levels as well. Bu insead of seing an explici consrain on he ransacion coss o be considered, I inroduce a penaly erm in he goal funcion. If he ransacion cos level under consideraion is very differen from he base ransacion cos level, he invesor is penalized hrough his funcion. Since she wans o make robus decisions, she prepares for he wors-case scenario, i.e., she uses he 1

3 alernaive ransacion cos level which resuls in he lowes value funcion (including he penaly erm). The invesor s olerance owards uncerainy is expressed by he uncerainyolerance parameer, which muliplies he penaly erm. I solve he robus dynamic opimizaion problem numerically, using parameer esimaes of he exising lieraure. I compare he opimal consumpion and asse allocaion decision of a robus invesor using several differen levels of uncerainy aversion o he decisions of an oherwise equivalen non-robus invesor, and I deermine he effecs of robusness on asse allocaion. Uncerainy-aversion affecs he invesor s decision abou asse allocaion hrough wo channels, and hese influence he invesor s decision in opposie direcions. On he one hand, a robus invesor is induced o buy more of he risky asse now, so ha she will no have o buy i laer a an unknown (and poenially high) ransacion cos. A he same ime, she is also induced o buy less risky asse now, o avoid having o sell hese asse in he fuure for consumpion purposes again, a an unknown and poenially high ransacion cos. The argumen is he same if he invesor wans o sell some of her risky asse, insead of buying. Besides he parameer values of he model, he curren value of he ransacion cos and he inheried invesmen raio joinly deermine which of he wo effecs will dominae. Afer showing he effecs of robusness on he opimal asse allocaion, I inroduce he definiions of liquidiy premium and is several componens, and I sudy he asse pricing implicaions of robusness in my model. Concreely, I show ha even a moderae level of robusness can generae a modes addiional liquidiy premium (0.05% for he firs 1% proporional ransacion cos), and his addiional liquidiy premium can be even higher if I assume higher levels of robusness. Thus, parameer uncerainy can provide a parial resoluion o he liquidiy premium puzzle. Besides hese economic conribuions, my paper also provides wo echnical conribuions. Firs, I provide a novel, non-recursive represenaion of a robus, discree-ime, dynamic asse allocaion problem. In is original form, his problem is of non-sandard recursive naure, and pracically impossible o solve numerically. My novel, non-recursive 2

4 represenaion of he problem no only gives more economic insigh ino he effecs of robusness on he invesor s decisions, bu i makes i possible o derive he Bellman equaion, which is already in sandard recursive form, and hus i is possible o solve he problem wihin a reasonable amoun of ime 1. As a second echnical conribuion, I provide a novel approach o effecienly solve discree-ime robus dynamic asse allocaion problems involving ransacion cos. Given he robus naure of he problem, one canno make use of he sandard opimizaion echniques which work well for maximizaion and minimizaion problems. Moreover, he value funcion is of class C 0, i.e., is firs parial derivaive is non-coninuous, due o he presence of a no-rade region, as documened in Consaninides (1986). To circumven hese difficulies, I develop a semi-analyical approach o solve he opimizaion problem. Firs I derive he firs-order condiions analyically, hen I solve he resuling non-linear equaion sysem numerically. Since hese equaions are he parial derivaives of he value funcion, hey conain non-coninuous funcions. These non-coninuiies preven me from making use of he Gaussian quadraure rule o evaluae numerically he arising double inegrals 2. So I deermine he no-rade boundaries explicily, and I calculae he double inegrals par-wise numerically, applying he Gaussian quadraure rule. My paper relaes o he lieraure on he liquidiy premium puzzle. The firs paper documening he puzzle is by Consaninides (1986). Consaninides repeas he calculaion including quasi-fixed ransacion cos 3, and he concludes ha is effec is sill of second order. Liu (2004) no only exends he model wih a fixed ransacion cos, bu he also proposes a differen definiion of he liquidiy premium: insead of maching he expeced uiliy, he maches he average holding of he risky asse. The generaed liquidiy premium is, however, lower han in Consaninides (1986). Vayanos (1998) proposes a general equilib- 1 Using a compuer wih 16 hreads and an Inel Xeon E v2 Processor, i akes approximaely 4 hours o solve a robus dynamic opimizaion problem wih an invesmen horizon of 10 years and annual discreizaion. 2 These double inegrals are a resul of having o evaluae a double expecaion due o he wo sources of risk: a sochasic sock reurn and a sochasic ransacion cos. 3 In he case of quasi-fixed ransacion cos, each rade incurs a ransacion cos ha is proporional o he invesor s curren wealh, regardless of he value of he rade. 3

5 rium model insead of he parial equilibrium model of Consaninides (1986), bu he finds ha general equilibrium models can generae even lower liquidiy premiums han parial equilibrium models. Buss, Uppal, and Vilkov (2014) propose a model wih an Epsein-Zin uiliy funcion insead of he CRRA uiliy funcion of Consaninides (1986), bu hey find ha here are no subsanial differences in he generaed liquidiy premiums. Jang, Koo, Liu, and Loewensein (2007) inroduce a sochasic invesmen opporuniy se ino he model, and hey find ha his more han doubles he generaed liquidiy premium o 0.45% for he firs 1% proporional ransacion cos from he original 0.20%. The model of Lynch and Tan (2011) feaures hree novel elemens, which can all generae addiional reasonable liquidiy premium, bu sill no enough o mach empirical facs. Their model feaures unhedgeable labor income, reurn predicabiliy, and counercyclical sochasic ransacion cos. Besides Lynch and Tan (2011) and he presen paper, o he bes of my knowledge only hree oher papers assume sochasic ransacion cos: Driessen and Xing (2015) focus on quanifying he liquidiy risk premium, while Garleanu and Pedersen (2013) and Glasserman and Xu (2013) analyze he effecs of sochasic ransacion cos on porfolio allocaion, bu due o he unconvenional uiliy funcion which hey use, hey canno derive conclusions regarding he level of he liquidiy premium. My paper also relaes o he lieraure on robus dynamic asse allocaion. The seminal paper inroducing he penaly approach ino he robus dynamic asse allocaion and asse pricing lieraure is Anderson, Hansen, and Sargen (2003). Then Maenhou (2004) applies his approach o analyze equilibrium sock prices. In a follow-up paper, Maenhou (2006) allows for a sochasic invesmen opporuniy se, and he finds ha robusness increases he relaive imporance of he ineremporal hedging demand, compared o he non-robus case. Oher papers analyzing differen aspecs of he effecs of robusness on dynamic asse allocaion include Branger, Larsen, and Munk (2013), Flor and Larsen (2014), and Munk and Rubsov (2014). This paper is organized as follows. Secion 2 inroduces my model, i.e., he financial marke and he robus dynamic asse allocaion problem. Secion 2 also provides he leas- 4

6 favorable ransacion cos, and he opimal consumpion and invesmen policy. Secion 3 inroduces he definiions of liquidiy premium and is componens, and i provides he model-implied liquidiy premiums. Secion 4 concludes. Appendix A provides he numerical procedure ha I used o solve he robus opimizaion problem. Appendix B conains he proofs of heorems and lemmas. 2 Robus Invesmen Problem The financial marke consiss of a money-marke accoun wih consan coninuously compounded risk-free rae r f and a risky securiy (a sock) wih coninuously compounded reurn r, which follows an i.i.d. normal disribuion wih mean µ r and variance σ 2 r. Buying and selling he money-marke accoun is free. On he oher hand, when he invesor buys or sells he sock, she encouners a ransacion cos, which is proporional o he raded dollar amoun. This ransacion cos is denoed by Φ, and i follows an i.i.d. log-normal disribuion wih parameers µ φ and σ 2 φ. I assume ha Φ and r are uncorrelaed 4. The order of decisions is shown in Figure 1. A each ime he invesor inheris wealh W. A proporion of his wealh, ˆπ is inheried in he risky asse. This proporion I call he inheried invesmen raio. The invesor firs learns abou he proporional ransacion cos Φ, hen she consumes C. I assume ha she finances her consumpion from he riskless asse. This assumpion is common in he lieraure, and i is in line wih economic inuiion, allowing one o hink abou he riskless asse as a money marke accoun, which can be direcly used o buy any goods for consumpion purposes 5. Afer consumpion, she decides abou her porfolio weigh π, and hen she rades in he securiies (and pays he ransacion 4 Empirically, ransacion coss are counercyclical, i.e., here is a negaive correlaion beween curren ransacion coss and expeced fuure reurns. Driessen and Xing (2015) show ha his resuls in a negaive liquidiy risk premium, however, is magniude is very small, approximaely 0.03%. Thus, assuming zero correlaion beween he curren ransacion cos and boh curren and fuure sock reurns in my model does no influence my resuls significanly. 5 In conras o his assumpion, Lynch and Tan (2011) assume ha he invesor finances her consumpion by coslessly liquidaing her risky and riskless asses in proporions ˆπ and 1 ˆπ. They jusify his by he fac ha equiies pay dividend, and his jusificaion is reasonable in an infinie-horizon seup. However, in he finie-horizon seup of my model he opimal consumpion raio becomes higher and higher as he invesor approaches he end of her invesmen horizon (and evenually becomes equal o one when = T ), making my assumpion abou financing consumpion from he riskless asse more realisic. 5

7 Figure 1. Order of he represenaive invesor s decisions A ime he invesor inheris wealh W. A proporion of his wealh, ˆπ is inheried in he risky asse. The invesor learns abou he proporional ransacion cos Φ, hen she consumes C. Afer consumpion, she decides abou her porfolio weigh π, and hen she rades in he securiies (and pays he ransacion cos) o obain his porfolio weigh. Then one ime period elapses, and a ime + 1 he invesor observes he reurn R +1, and he series of decisions sars over again. cos) o obain his porfolio weigh. The invesor s wealh process is hus W +1 = W + R f + π (R +1 R f )], (1) where R f = exp (r f ), R +1 = exp (r +1 ), and W + denoes he invesor s wealh a ime, afer she has consumed, and rebalanced her porfolio (and paid he ransacion cos). I.e., W + = W C Φ W + π W ˆπ. (2) Since W + appears on boh sides of equaion (2), i is convenien o express i in a form which does no conain W + on he righ-hand side. I.e., W + = W (1 + I Φ ˆπ ) C 1 + I Φ π, (3) 6

8 where I is an indicaor funcion, which is equal o 1 if he invesor is buying addiional risky asses when rebalancing her porfolio (i.e., if W + π > W ˆπ ), -1 if he invesor is selling par of her risky asses when rebalancing her porfolio (i.e., if W + π < W ˆπ ), and 0 if he invesor is no rading o rebalance her porfolio (i.e., if W + π = W ˆπ ). A ime + 1 he invesor learns abou he oucome of R +1, and her invesmen in he risky securiy as a fracion of her oal wealh becomes ˆπ +1 = π R +1 R f + π (R +1 R f ). (4) This ˆπ +1 I call he inheried invesmen raio a ime +1. Now I consider an invesor wih a finie invesmen horizon T. She derives uiliy from consumpion, and she has a CRRA uiliy funcion wih relaive risk aversion γ. Her goal is o maximize her expeced uiliy, bu she is uncerain abou he mean of he sochasic logransacion cos process, µ φ. She has a base parameer value in mind, which she considers o be he mos likely. This parameer value is denoed by µ B φ. Bu she is uncerain abou he rue value of µ φ, so she considers oher (alernaive) parameer values as well. These alernaive parameer values are denoed by µ U φ. I formalize he relaionship beween µb φ and µ U φ as µ U φ = µ B φ + u, (5) where u is a sochasic decision variable, jus as C and π are. The invesor wans o proec herself agains unfavorable oucomes, so she makes robus invesmen decisions. Now I formalize he invesor s robus opimizaion problem. Problem 1. Given W 0, ˆπ 0, and Φ 0, find an opimal riple C, π, u } 0, T 1] for he robus uiliy maximizaion problem V 0 (W 0, ˆπ 0, Φ 0 ) = inf u sup E U 0 C,π } T =0 exp ( δ) C1 γ 1 γ + Υ u 2 2 ], (6) 7

9 subjec o he budge consrains (1) and (2), o he erminal condiion C T = W T (1 ˆπ T Φ T ), and where E U 0 means ha he expecaion is calculaed assuming µu φ, condiional on all informaion available up o ime 0. To ensure homoheiciy of he opimal consumpion raio, he opimal invesmen policy, and he leas-favorable disorion 6, I scale he uncerainy olerance parameer Υ (following Maenhou (2004)) as Υ = (1 γ) V +1 (W +1, ˆπ +1, Φ +1 ). (7) θ Subsiuing he parameerizaion of he uncerainy olerance (7) ino he value funcion (6), he value funcion becomes V 0 (W 0, ˆπ 0, Φ 0 ) = inf u sup E U 0 C,π } T exp ( δ) C1 γ 1 γ =0 + (1 γ) u2 V +1 (W +1, ˆπ +1, Φ +1 ) ]. (8) This represenaion of he value funcion is recursive, i.e., he righ-hand side of (8) conains fuure values of he value funcion iself. The following heorem (which I prove in Appendix B) gives a represenaion of he value funcion which is no recursive. Theorem 1. The soluion o (8) wih iniial wealh x is given by V 0 = inf u sup E U 0 C,π } T exp ( δ) C1 γ 1 1 γ =0 subjec o he budge consrain (1) and (2), and o he erminal condiion C T = W T (1 ˆπ T Φ T ). s=0 ] } 1 + u2 s (1 γ), (9) In line wih Horvah, de Jong, and Werker (2016), equaion (9) shows ha inroducing robusness effecively increases he subjecive discoun rae δ. I.e., a robus invesor is effec- 6 Homoheiciy means ha he opimal consumpion raio, he opimal invesmen policy, and he leasfavorable disorion will all be independen of he wealh level. 8

10 ively more impaien han an oherwise equal non-robus invesor. However, his is no he only place where robusness plays a role in (9), bu i also affecs he expecaion operaor by changing he probabiliy densiy funcion of he ransacion cos Φ 1,..., T }. Acually, as i is shown in Meron (1969) and in Meron (1971), he change in he subjecive discoun facor affecs only he opimal consumpion policy, bu no he opimal invesmen policy. The effec of robusness on he opimal invesmen policy is due o he change from E B 0 o EU 0. As I menioned previously, he opimal consumpion raio, he opimal invesmen rule, and he leas-favorable disorion are homoheic, i.e., hey do no depend on he wealh level. This is formally saed in he following heorem, which I again prove in Appendix B. Theorem 2. Denoing he consumpion raio by c = C /W, he opimal soluion c, π, u } o he robus opimizaion problem (1) wih parameerizaion (7) is independen of he wealh level W, 0, 1,..., T }. Moreover, he value funcion can be expressed in he form V 0 (W 0, ˆπ 0, Φ 0 ) = W 1 γ 0 1 γ v 0 (ˆπ 0, Φ 0 ). (10) To solve he opimizaion problem, I apply he principle of dynamic programming (Bellman (1957)). As a firs sep of his approach, I formulae he one-period opimizaion problem a ime T 1 as v T 1 = inf sup u T 1 c T 1,π T 1 } exp ( δ (T 1)) c 1 γ T 1 ) ( + exp ( δt ) 1 + u2 T 1 (1 γ) E U T 1 (1 c T 1 + I T 1 Φ T 1ˆπ T 1 ) 1 γ (1 + I T 1 Φ T 1 π T 1 ) 1 γ (1 ˆπ T Φ T ) 1 γ (R f + π T 1 (R T R f )) 1 γ]}, (11) he deailed derivaion of which can be found in Appendix B, in he proof of Lemma 1. Afer obaining he opimal c T 1, π T 1, u T 1 } riple, I apply backward inducion o find he opimal c, π, u } riples for 0, 1,..., T 2}. To his end, I formulae he Bellman 9

11 equaion (he derivaion of which can be found in Appendix B, in he proof of Theorem 2). v (ˆπ, Φ ) = inf u sup c,π } E U exp ( δ) c 1 γ u2 (1 γ) (R f + π (R +1 R f )) 1 γ v +1 (ˆπ +1, Φ +1 ) ] ( 1 + I Φ ˆπ c ) 1 γ 1 + I Φ π ]}. (12) A closed form soluion o he robus uiliy maximizaion Problem 1 does no exis, herefore I solve he problem numerically. Sill, obaining he opimal consumpion and invesmen policies and he leas-favorable ransacion cos µ U φ parameer is compuaionally challenging for several reasons. Firs, he robus naure of he problem (i.e., he minimax seup) resuls in a saddle-poin soluion, which prevens one from making use of sandard numerical opimizaion echniques, ha are oherwise well suied for maximizaion and minimizaion problems. Second, he value funcion is of class C 0, since is firs parial derivaive wih respec o he invesmen raio, π, is no coninuous due o he presence of a no-rade region. This implies ha when I calculae he expeced value of he firs parial derivaive of he value funcion wih respec o π numerically, I canno direcly apply he Gaussian quadraure rule. Third, since my model conains wo sources of risk (R and Φ ), calculaing he firs parial derivaive of he value funcion wih respec o π involves he numerical approximaion of a definie double-inegral insead of he inegral of a funcion of only one variable. To circumven hese difficulies, I provide a semi-analyical approach: firs I derive he firs-order condiions on he Bellman equaion (12) in closed form 7, hen I solve he resuling non-linear equaion sysem numerically. To efficienly approximae he involved double inegrals numerically, I firs deermine he boundaries of he no-rade region, hen apply he Gaussian quadraure rule separaely on he sell, no-rade, and buy regions, in which he firs parial derivaive of he value funcion wih respec o he invesmen raio is coninuous. Using his approach, I can solve he robus opimizaion Problem 1 numerically 7 Since he firs-order condiion on he Bellman equaion (12) conains he parial derivaive of he value funcion a ime + 1 wih respec o π, I apply he Benvenise-Scheinkman Condiion (Envelope Theorem) o ransform his firs-order condiion ino a closed-form equaion. The deails can be found in Appendix A. 10

12 in a feasible ime. 2.1 Model parameerizaion Regarding he choice of he parameer values, I follow he lieraure o make my findings comparable o exising resuls. I assume ha he invesmen horizon is 9 years, and he discreizaion frequency is annual. The effecive annual risk-free rae is 3%. The gross sock reurns, R, are i.i.d. log-normally disribued wih parameers µ r = 8% and σ r = 20%. Following Consaninides (1986) and he esimaes of Lesmond, Ogden, and Trzcinka (1999), I assume ha he expeced fuure ransacion cos is 1%. Abou he sandard deviaion of he ransacion cos, I assume i o be 0.5%. In heir paper, Lynch and Tan (2011) use 0.76% sandard deviaion and 2% expeced value for he ransacion cos process. Since higher sandard deviaion of he ransacion cos produces higher negaive liquidiy risk premium, I use 0.50% sandard deviaion, which is higher han half of he value used by Lynch and Tan (2011) for 2% ransacion cos. This way I underesimae he oal liquidiy premium, which makes my resuls sronger 8. The represenaive invesor s relaive risk-aversion parameer is γ = 5, her subjecive discoun facor is δ = 5%, and I vary her uncerainy-aversion parameer θ beween 0 and 100, 0 corresponding o a non-robus invesor, and 100 corresponding o a highly robus invesor. Deermining a reasonable level of robusness is an imporan aspec of he model. Unforunaely, he uncerainy-aversion parameer θ does no have such a universal inerpreaion as he relaive risk-aversion parameer γ, he reasonable value of which is beween 1 and 5 according o he lieraure. The uncerainy-aversion parameer θ is always model specific. To sill give a general measure of uncerainy aversion, he dynamic asse allocaion lieraure provides wo approaches. A saisical approach is o make an addiional assumpion on he Deecion Error Probabiliy of he represenaive invesor, as in, e.g., Anderson, Hansen, and Sargen (2003). The oher approach relies more on economic inuiion: i 8 I model he logarihm of he ransacion cos, insead of he ransacion cos iself, o ensure ha he realized ransacion cos values are non-negaive. Alhough he log-normal disribuion heoreically allows he ransacion cos o ake values ha are above 100%, given my parameer choices he probabiliy of such an even is negligible. 11

13 suggess deriving he wors-case scenario (e.g., wors-case mean ransacion cos parameer in my model) explicily, and he difference beween he base parameer and he wors-case parameer gives an economic meaning o he uncerainy-aversion level of he paricular invesor. This approach is also followed by Maenhou (2004). In his paper, since calculaing he exac deecion-error probabiliies o he various uncerainy-aversion parameer values would require compuaionally inensive recursive calculaions, I use he second approach and provide he leas-favorable expeced fuure ransacion cos levels for for hree oherwise idenical invesors wih differen levels of uncerainy aversion: θ = 0 (no uncerainy aversion), θ = 50 (moderae uncerainy aversion), and θ = 100 (high uncerainy aversion). These are shown as a funcion of he curren ransacion cos in Figure 3, for hree differen inheried invesmen raios. As Consaninides (1986) showed, he opimal invesmen decision is deermined by a single sae variable, he inheried invesmen raio. If his raio is wihin given boundaries (deermined by he model parameers), he opimal decision is o no rade. If he raio is above he upper boundary, he invesor opimally sells a fracion of her risky asses, while if i is below he lower boundary, she buys addiional risky asses. The sell, no-rade, and buy regions in my calibraed model are shown in Figure 2. The solid line represens he boundaries of he no-rade region for he non-robus invesor. If her inheried invesmen raio is above he upper boundary, she has o sell par of her risky asses, while if i is below he lower boundary, she has o buy addiional risky asses. Allowing he invesor o be uncerainy-averse shrinks he no-rade region. If she is in he sell region, an uncerainy-averse invesor is supposed o sell more (so ha she will have o sell less laer a he anicipaed higher expeced ransacion cos). On he oher hand, if she is in he buy region, an uncerainy-averse invesor is supposed o buy more, so ha she will have o buy less in he fuure a he anicipaed higher expeced ransacion cos. Le me emphasize ha his conclusion abou he shrinkage of he no-rade boundaries depends on he level of he invesor s uncerainy aversion. An invesor wih a differen uncerainy-aversion parameer migh buys less (if she is in he buy region), so ha she 12

14 Figure 2. Sell, no-ransacion, and buy zones The invesmen horizon is 9 years, he discreizaion frequency is annual, he effecive annual risk-free rae is 3%, he gross sock reurns are IID and log-normally disribued wih parameers µ r = 8% and σ r = 20%. The proporional ransacion coss are IID and log-normally disribued wih parameers µ B φ = and σ φ = , which correspond o a base expeced ransacion cos of 1% and sandard deviaion of 0.50%. The relaive risk aversion is γ = 5, and he subjecive discoun facor is δ = 5%. has o sell less laer a he anicipaed higher expeced ransacion cos. This is so because in he case of his oher robus invesor he buy less, so ha you have o sell less laer moive dominaes, while in he case of he uncerainy-averse invesor shown in Figure 2 he dominan moive is he buy more, so ha you will have o buy less laer a he anicipaed higher expeced ransacion cos. The effec of uncerainy on he consumpion policy is quaniaively negligible, less han 0.04% on average. Moreover, he opimal consumpion raio is also very sable among he differen saes: i is around 11.85%. 2.2 Leas-favorable ransacion cos The leas-favorable fuure expeced ransacion cos is sae-dependen, i.e., i varies among differen combinaions of he curren values of he wo sae variables of my model: he 13

15 inheried invesmen raio and he curren ransacion cos. In he op panel of Figure 3 he inheried invesmen raio is se o ˆπ 0 = 0, i.e., he invesor inheris everyhing in he riskless asse. In he middle panel she inheris 31% (which is approximaely equal o he zero-curren-ransacion-cos opimal invesmen raio) of her wealh in he risky asse. In he boom panel she inheris everyhing in he risky asse. Regardless of he curren sae of he sysem, higher robusness always means a higher expeced fuure ransacion cos. This is inuiive: a robus invesor prepares for he wors-case scenario, hus she considers a higher expeced ransacion cos in he fuure. On he oher hand, i is no sraighforward how he leas-favorable expeced fuure ransacion cos changes if ceeris paribus we change one of he curren sae variables. If he invesor has inheried everyhing in he riskless asse (op panel of Figure 3), hen increasing he curren ransacion cos induces a higher leas-favorable expeced fuure ransacion cos. The inuiion behind his is ha a higher curren ransacion cos will resul in a lower opimal invesmen raio, i.e., he invesor will have less of her wealh invesed in he risky securiy. Since uncerainy abou he ransacion cos shows up via having o sell he risky securiy in he fuure o finance consumpion 9, having a lower opimal invesmen raio now means ha a bad oucome will hur he invesor less. Thus, given he same level of uncerainy aversion, o prepare for he wors-case scenario, she will consider a higher fuure expeced ransacion cos. If he invesor inheris par of her porfolio in he risky asse (middle panel of Figure 3), hen he effec of increasing he curren ransacion cos is he opposie: a higher ransacion cos induces a lower leas-favorable expeced fuure ransacion cos. The inuiion is ha a higher curren ransacion cos will resul in a higher opimal invesmen raio (since he inheried invesmen raio is above he no-ransacion-cos opimum), hus he invesor is exposed o more uncerainy. To compensae for his, she will consider a lower leas- 9 There is a second effec of a higher curren ransacion cos, which has he opposie direcion: since he invesor buys less risky asse now, she will have o buy more in he fuure, and if she considers a higher expeced fuure ransacion cos now, his addiional demand will decrease her value funcion now. This effec is, however, of second order, and quaniaively i is dominaed by he effec described in he main ex. 14

16 Figure 3. Leas-favorable expeced fuure ransacion coss, for differen inheried invesmen raios. The invesmen horizon is 9 years, he discreizaion frequency is annual, he effecive annual risk-free rae is 3%, he gross sock reurns are IID and log-normally disribued wih parameers µ r = 8% and σ r = 20%. The proporional ransacion coss are IID and lognormally disribued wih parameers µ B φ = and σ φ = , which correspond o a base expeced ransacion cos of 1% and sandard deviaion of 0.50%. The relaive risk aversion is γ = 5, and he subjecive discoun facor is δ = 5%. The inheried invesmen raios in he hree graphs are 0%, 31%, and 100%, respecively. 15

17 favorable ransacion cos. Regarding he comparaive saics changing he oher sae variable, he inheried invesmen raio, we can observe ha he inerceps in Figure 3 are he same. This means ha if he curren ransacion cos is zero, he leas-favorable expeced fuure ransacion cos is he same, regardless of he inheried invesmen raio. Moreover, if he inheried invesmen raio is he same as he zero-curren-ransacion-cos opimal invesmen raio (pos-consumpion), hen regardless of he curren ransacion cos, he invesor does no have o rade, and he leas-favorable ransacion cos will be consan a he same level as he inerceps in Figure 3. This is inuiive, since in his paricular case he curren ransacion cos does no have any effec on he probabiliies of he fuure saes of he sysem. If he inheried invesmen raio is beween zero and he zero-curren-ransacioncos opimal invesmen raio, hen he leas-favorable ransacion cos will be an increasing funcion of he curren ransacion cos, and he higher he inheried invesmen raio, he lower he slope of his funcion. The same is rue if he inheried invesmen raio is beween he zero-curren-ransacion-cos opimal invesmen raio and one: he leas-favorable expeced fuure ransacion cos is an increasing funcion of he curren ransacion cos, and he higher he inheried invesmen raio, he higher he slope of his funcion. 2.3 Opimal invesmen policy Similarly o he leas-favorable expeced fuure ransacion cos, he opimal invesmen raio is also sae-dependen, as i is shown in Figure 4. If he invesor inheris everyhing in he risk-free asse, and she is non-robus, hen she will buy he risky securiy o have an invesmen raio of 31.21%. This is he inercep of he solid line in he op panel of Figure 4. If he curren ransacion cos is higher, he non-robus invesor buys less of he risky asse, and a a curren ransacion cos of jus above 8% she will no rade a all, bu leave her enire wealh invesed in he riskless asse. If he invesor is uncerainy-averse, she will anicipae a higher expeced ransacion cos han a non-robus invesor (see Figure 3). In he case of such an uncerainy-averse 16

18 invesor, here are wo effecs working in he opposie direcions. On he one hand, he invesor wans o buy less risky asse han her non-robus counerpar, so ha laer she will have o sell less risky asse a he higher anicipaed expeced ransacion cos. Insead, she will hold more of her wealh now in he riskless asse, which laer she can use for consumpion purposes wihou having o pay ransacion cos o sell i. On he oher hand, she wans o buy more risky asse now a he known ransacion cos, so ha laer she will have o buy less risky asse a he higher anicipaed expeced ransacion cos. As we can see in he op panel of Figure 4, in he case of an uncerainy-averse invesor who inheried everyhing in he risk-free asse he firs effec is he dominan: she will buy less risky securiy now han her non-robus counerpar. If he invesor inheris everyhing in he risky asse (boom panel of Figure 4), she will have o sell par of her porfolio o achieve he opimal invesmen raio. If he invesor is non-robus, he zero-curren-ransacion-cos opimal invesmen raio for her is 31.21% he same is for he non-robus invesor who inheried everyhing in he risk-free asse. This is inuiive: if he curren ransacion cos is zero, hen he invesor can rebalance her porfolio for free, hence he inheried porfolio allocaion does no maer for her. Bu if he curren ransacion cos is non-zero, he opimal invesmen raio becomes higher i.e., she will sell less of her risky asse o save on he ransacion cos. And as he curren ransacion cos is higher and higher, she will sell less and less, unil she achieves he poin where she will no rade any more and raher keep all of her wealh in he risky asse. If his invesor is uncerainy-averse, hen again here will be wo effecs working in he opposie direcions. On he one hand, he robus invesor wans o sell more of he risky securiy so ha she will have o sell less laer a he higher anicipaed expeced ransacion cos. On he oher hand, she wans o sell less of he risky securiy o avoid having o buy addiional risky securiy in he fuure for rebalance purposes. In he boom panel of Figure 4, we can see ha regardless of he curren ransacion cos, he second effec will be he dominan one, i.e., a robus invesor will always sell more of he risky securiy now han a non-robus invesor. This is due o he fac ha her zero-curren- 17

19 Figure 4. Opimal invesmen raio, for differen inheried invesmen raios. The invesmen horizon is 9 years, he discreizaion frequency is annual, he effecive annual risk-free rae is 3%, he gross sock reurns are IID and log-normally disribued wih parameers µ r = 8% and σ r = 20%. The proporional ransacion coss are IID and log-normally disribued wih parameers µ B φ = and σ φ = , which correspond o a base expeced ransacion cos of 1% and sandard deviaion of 0.50%. The relaive risk aversion is γ = 5, and he subjecive discoun facor is δ = 5%. The inheried invesmen raios in he hree graphs are 0%, 31%, and 100%, respecively. 18

20 ransacion-cos opimal porfolio is very differen from her inheried porfolio. If hese wo porfolios were more similar in erms of asse allocaion o each oher as i is he case in he op panel of Figure 4, hen he firs effec migh be he dominan. Bu even now we can observe he fac ha we encounered in he case of he invesor who inheried everyhing in he risk-free asse ha as he invesor becomes more and more uncerainyaverse, he dominance of he second effec is less and less srong, especially if he curren ransacion cos is no oo high (less han 2%). In he middle panel of Figure 4 his is refleced in he fac ha below 2% curren ransacion cos he opimal invesmen raio of a moderaely uncerainy-averse invesor is acually higher, han an oherwise idenical, bu more uncerainy-averse invesor s. 3 Liquidiy Premium Afer showing he effecs of uncerainy abou fuure ransacion cos on asse allocaion in Secion 2, now I go one sep furher and I analyze he effecs of uncerainy abou fuure ransacion coss on asse pricing. To be more precise, I show ha my model wih a robus represenaive invesor generaes an addiional liquidiy premium of 0.05%-0.10% for he firs 1% proporional ransacion cos, depending on he level of uncerainy aversion of he invesor. To separae he effecs of he level of he ransacion cos, he volailiy of he ransacion cos, and he parameer uncerainy abou he ransacion cos process, I inroduce he definiion of he liquidiy-uncerainy premium, he liquidiy-risk premium, and he liquidiylevel premium. 10 Definiion 1. Le us consider a represenaive agen wih uncerainy-aversion parameer θ solving he robus opimizaion Problem 1, and obaining V 0. If we impose he resricion u = 0 0, 1,..., T 1}, he coninuously compounded expeced reurn on he risky securiy, µ r, has o be decreased o µ θ=0 r so ha he invesor achieves he same level of value 10 My liquidiy premium definiions are in line wih he majoriy of he lieraure, hough some researchers use alernaive definiions, e.g., Liu (2004). 19

21 V 0 as wihou his resricion. The difference µ r µ θ=0 r premium. is called he liquidiy-uncerainy Definiion 2. Le us consider he represenaive invesor in Definiion 1. If we impose he resricion u = 0 0, 1,..., T 1}, and we change he volailiy parameer of he ransacion cos process from σ φ o 0, he coninuously compounded expeced reurn on he risky securiy, µ r, has o be decreased o µ θ=0,σ φ=0 r level of value V 0 as originally. The difference µ θ=0 r µ θ=0,σ φ=0 r premium. so ha he invesor achieves he same is called he liquidiy-risk Definiion 3. Le us consider he represenaive invesor in Definiion 1. If we impose he resricion ha u = 0 0, 1,..., T 1}, we change he volailiy parameer of he ransacion cos process from σ φ o 0, and we also change he base mean parameer of he ransacion cos from µ B φ o µb φ so ha EB Φ +1 } = 0 0, 1,..., T 1}, he coninuously compounded expeced reurn on he risky securiy, µ r, has o be decreased o µ θ=0,σ φ=0,µ Φ =0 r so ha he invesor achieves he same level of value V 0 as originally. The difference µ θ=0,σ φ=0 r µ θ=0,σ φ=0,µ Φ =0 r is called he liquidiy-level premium. Using Definiions 1-3 and he parameer values described in Secion 2, my model generaes he liquidiy-level premium, liquidiy-risk premium, and liquidiy-uncerainy premium values described in Table 1, and also shown in Figure 5 and Figure 6. Table 1. Model implied liquidiy premiums Model implied liquidiy-level premiums, liquidiy-risk premiums, and liquidiy-uncerainy premiums for differen levels of uncerainy aversion. The invesmen horizon is T=10 years, he discreizaion is annual. The coninuously compounded expeced sock reurn is µ r = 8%, he volailiy of he reurn is σ r = 20%, he risk-free rae is 3%, he base ransacion cos mean parameer is µ B φ = 1%, he ransacion cos volailiy parameer is σ φ = 0.5%. The invesor s risk aversion parameer is γ = 5, and her subjecive discoun rae is δ = 5%. θ = 0 θ = 50 θ = 100 Liquidiy uncerainy premium 0.00% 0.05% 0.10% Liquidiy risk premium 0.03% 0.02% 0.04% Liquidiy level premium 0.83% 0.71% 0.72% Toal liquidiy premium 0.80% 0.74% 0.78% 20

22 In he non-robus version of my model, he oal liquidiy premium generaed is 0.80% for a proporional liquidiy premium wih expeced value of 1% and sandard deviaion of 0.50%. The majoriy of his premium is due o he level of he ransacion cos: he liquidiy level premium is 0.83%. The liquidiy-risk premium is negaive, i is -0.03%. A firs his migh seem counerinuiive, bu i is acually very logical. A rigorous demonsraion of why he liquidiy premium is negaive if he correlaion beween he sock reurn and he ransacion cos is zero can be found in Driessen and Xing (2015). The inuiion is ha he sochasic naure of he ransacion cos gives addiional opporuniy o he invesor o choose when o rebalance her porfolio. If marke condiions are bad (i.e., he ransacion cos is high), she can pospone rebalancing, while if marke condiions are good (i.e., he ransacion cos is low), she can rade more. Driessen and Xing (2015) call his he Choice Effec. Inroducing a moderae level of robusness decreases he oal liquidiy premium by 0.06%. Though he liquidiy-level premium decreased, a liquidiy-uncerainy premium showed up, which is 0.05%. The decrease in he liquidiy level premium is due o he definiion of he hree componens of he oal liquidiy premium: insead of aking µ r = 8% as he baseline non-robus value and increasing i when inroducing robusness o calculae he liquidiy-uncerainy premium, I ake µ r = 8% as he robus baseline value. I do his because he µ r value is observed/esimaed independenly of he model specificaion, and if he represenaive invesor is assumed o be robus, hen his represenaive invesor generaes µ r = 8% expeced reurn, ha we can observe on he marke, and no a reurn which is higher han his by he liquidiy uncerainy premium. Thus, he main message of he hird column of Table 1 is: even a moderae level of uncerainy aversion can generae a liquidiy uncerainy premium of 0.05%, which is 25% of he oal liquidiy premium generaed in Consaninides (1986). Increasing he uncerainy aversion o a higher level, will produce a liquidiy-uncerainy premium of 0.10%. 21

23 Figure 5. Decomposiion of he liquidiy premium. The hree componens of he liquidiy premium: he liquidiy-level premium, he (negaive) liquidiy-risk premium, and he liquidiy-uncerainy premium. Figure 6. Decomposiion of he expeced sock reurn. Decomposiion of he expeced sock reurn ino he risk-free reurn, he (negaive) liquidiyrisk premium, he liquidiy-uncerainy premium, he liquidiy-level premium, and oher (non-liquidiy-relaed) premiums. 22

24 4 Conclusion I have shown ha uncerainy-aversion can explain a reasonable porion of he liquidiypremium puzzle. Wih a moderae level of uncerainy aversion my model can generae an addiional 0.05% liquidiy premium, which is even higher if I allow for a higher level of uncerainy aversion. I have also shown ha uncerainy-aversion has wo differen channels hrough which i affecs he opimal invesmen behavior, and he oal effec depends on he curren ransacion cos and he inheried invesmen raio. Besides hese wo economic conribuions, I provided wo echnical conribuions o efficienly solve robus dynamic asse allocaion problems involving ransacion cos. When I inroduced uncerainy aversion ino my model, I relied on he magniude of he leas-favorable ransacion cos values o assess he level of uncerainy aversion. This is an inuiive and appealing approach, however, more rigorous definiions of he level of uncerainy aversion, e.g., by using simulaion-based deecion error probabiliies can be a fruiful line of furher research. Moreover, developing a coninuous-ime version of he robus model and using he relaive enropy as a penaly erm migh provide opporuniies for inuiive inerpreaion of he uncerainy-aversion parameer in he conex of ambiguiy abou he fuure ransacion coss, and i can also give addiional insigh ino he working mechanism of he wo channels hrough which robusness affecs he invesor s decision. This is also a promising area of fuure research. 23

25 Appendix A Numerical procedure Solving he robus dynamic opimizaion Problem 1 numerically is compuaionally challenging for several reasons. Firs, due o he minimax seup of he problem, he soluion will be a saddle poin, hus I canno use he sandard numerical opimizaion echniques which can be used o efficienly solve maximizaion and minimizaion problems. Second, he value funcion is of class C 0, i.e., is parial derivaive wih respec o he invesmen raio is non-coninuous due o he presence of he no-rade region. Third, my model conains wo sources of uncerainy (boh he sock reurn and he ransacion cos are sochasic), hus calculaing he expeced value for he value funcion (and is parial derivaives) involves numerically approximaing a double inegral, where he funcion o be inegraed is no necessarily coninuous. To solve he problem wihin a reasonable ime, I ake several measures. Firs, I show in Theorem 2 ha he opimal consumpion raio, he opimal invesmen raio and he leas-favorable disorion a ime do no depend on he wealh level a ime. Since he wealh level is one of he sae variables, his heorem subsanially reduces he required compuaional ime o solve he opimizaion problem. Second, I develop a novel echnique o solve discree-ime robus dynamic asse allocaion problems wih ransacion cos. This involves rewriing Problem 1 in a non-recursive formulaion. I provide his reformulaion in Theorem 1. Then I wrie down he firs-order condiions in closed form, making use of he Benvenise-Scheinkman Condiion (Envelope Theorem). This resuls in a non-linear equaion sysem of hree equaions, wih hree variables. The equaions hemselves are he firs parial derivaives of he value funcion, hus some of hem are non-coninuous. To evaluae he double expecaions (which are equivalen o numerically evaluaing double inegrals), I explicily deermine he non-coninuiy lines, hen I apply he Gaussian quadraure rule o calculae he numerical inegral of he funcion par-wise. Moreover, I also parallelize he soluion algorihm o disribue he compuaional workload among several working unis. The seps of he soluion procedure are as follows. 24

26 1. Firs, I creae grids for possible ˆπ T 1 and Φ T 1 values. The grids for ˆπ T 1 lie in 0, 0.5, 1}, while he grids for Φ T 1 lie in 0, 0.01, 0.02, 0.06, 0.1}. I obain he opimal invesmen raio π T 1, he opimal consumpion raio c T 1, and he leas-favorable ransacion cos parameer u T 1 for each ˆπ T 1, Φ T 1 } pairs by numerically solving he firs order condiions wih respec o c T 1, π T 1, and u T 1. These firs-order condiions are (42), (43), and (44). 2. Now I go one period backwards. Since he soluion procedure will be he same for all 1, 2,..., T 2}, insead of T 2 I use he more general ime period in his sep, keeping in mind ha immediaely afer he above sep I have = T 2. Jus as in he previous sep, I creae grids for possible ˆπ and Φ values. The grids for ˆπ lie in 0, 0.5, 1}, while he grids for Φ lie in 0, 0.01, 0.02, 0.06, 0.1}. The firs order condiion on he Bellman equaion (12) wih respec o c is ] c = exp (δ) 1 + u2 (1 γ) (1 + I Φ ˆπ ) γ (1 + I Φ π ) 1 γ E u (R f + π (R +1 R f )) 1 γ v +1 (ˆπ +1, Φ +1 ) ]] 1 γ + (1 + I Φ ˆπ ) 1 } 1, (13) he firs order condiion on he Bellman equaion (12) wih respec o π is E u (1 γ) v+1 (ˆπ +1, Φ +1 ) (R f + π (R +1 R f )) γ ] I Φ R +1 R f (R f + π (R +1 R f )) 1 + I Φ π +R f R +1 (R f + π (R +1 R f )) 1 γ v +1 (ˆπ +1, Φ +1 ) ˆπ +1 } = 0, (14) and he firs order condiion on he Bellman equaion (12) wih respec o u is E u (R f + π (R +1 R f )) 1 γ v +1 (ˆπ +1, Φ +1 ) ( ) ]} 1 + u2 (1 γ) φ+1 µ φ u + u (1 γ) = 0. (15) θ σ 2 φ 25

27 Condiion (14) conains v +1 (ˆπ +1, Φ +1 ) / ˆπ +1, which no only makes he evaluaion of he expeced value in (14) compuaionally inensive, bu i also decreases he numerical accuracy of he obained resuls. To circumven his, I make use of he Benvenise-Scheinkman Condiion (Envelope Theorem). Le us inroduce he funcion ( ) ( ) 1 + u2 (1 γ) 1 + I Φ ˆπ c 1 γ 1 + I Φ π ] (R f + π (R +1 R f )) 1 γ v +1 (ˆπ +1, Φ +1 ), (16) v (c, π, u, ˆπ, Φ ) = E u he parial derivaive of which wih respec o ˆπ is v (c, π, u, ˆπ, Φ ) = ˆπ I Φ E u 1 + I Φ π ( ) 1 + u2 (1 γ) (1 γ) ( 1 + I Φ ˆπ c (R f + π (R +1 R f )) 1 γ v +1 (ˆπ +1, Φ +1 ) ) γ 1 + I Φ π ]. (17) Subsiuing (16) ino (12), he Bellman equaion becomes v (ˆπ, Φ ) = inf u } sup exp ( δ) c 1 γ + v (c, π, u, ˆπ, Φ ). (18) c,π } Then he following heorem (which I prove in Appendix B) holds. Theorem 3 (Benvenise-Scheinkman Condiion (Envelope Theorem)). If c = c, π = π, and u = u, hen v (ˆπ, Φ ) ˆπ = v (c, π, u, ˆπ, Φ ) ˆπ. (19) Using Theorem 3 and equaion (17), I rewrie he firs order condiion wih respec 26

28 o π, i.e., (14), as 0 =E u (Rf + π (R +1 R f )) γ v +1 (ˆπ +1, Φ +1 ) ] I Φ R +1 R f (R f + π (R +1 R f )) 1 + I Φ π ( ) u + R f R +1 (R f + π (R +1 R f )) (1 1 γ 2 ) + +1 (1 γ) ( 1 + I+1 Φ +1ˆπ +1 c γ +1) E u ( 1 + I+1 Φ +1 π +1) 1 γ I +1 Φ +1 (Rf + π +1 (R +2 R f ) ) 1 γ v+2 (ˆπ +2, Φ +2 )]}. (20) Appendix B Proofs Proof of Theorem 1. I will prove he heorem using backward inducion. Throughou he proof, I assume ha he choice variables, i.e., C, π, and u are always opimally chosen, hus I omi he inf and sup operaors. Moreover, since i will no cause any confusion, insead of V (W, ˆπ, Φ ) I simply wrie V. A ime T he value funcion is V T = exp ( δt ) C1 γ T 1 γ + u2 T (1 γ) V T +1 = exp ( δt ) C1 γ T 1 γ, (21) since V T +1 = 0. Going one sep backwards, he value funcion a ime T 1 is by definiion V T 1 = exp δ (T 1)} C1 γ T 1 1 γ + u2 T 1 (1 γ) = E T 1 exp δ (T 1)} C1 γ T 1 1 γ T = E T 1 exp ( δ) C1 γ 1 1 γ =T 1 C1 γ T + exp ( δt ) 1 γ s=t 1 V T + exp ( δt ) C1 γ T 1 γ 1 + u2 s (1 γ) } ( )} 1 + u2 T 1 (1 γ) ] }. (22) 27

29 Going one more sep backwards, I can wrie down he value funcion V T 2 in he same way: V T 2 =E T 2 =E T 2 T exp ( δ (T 2)) C1 γ T 2 1 γ + exp ( δt ) C1 γ T =T 2 ( ) C1 γ T 1 + exp ( δ (T 1)) 1 + u2 T 2 (1 γ) 1 γ ( ) ( )} 1 + u2 T 1 (1 γ) 1 + u2 T 2 (1 γ) 1 γ 1 ] } 1 + u2 s (1 γ). (23) exp ( δ) C1 γ 1 γ s=t 2 Progressing backwards in he same way, I can wrie he value funcion a ime as T V = E s= and he value funcion a ime 0 as exp ( δs) C1 γ s 1 γ s 1 m= T V 0 = E 0 exp ( δ) C1 γ 1 1 γ =0 This is he same as in Theorem 1, and i complees he proof. s=0 ] } 1 + u2 m (1 γ), (24) ] } 1 + u2 s (1 γ). (25) Proof of Theorem 2. I will prove he heorem using mahemaical inducion. Following Theorem 1, he value funcion a ime is V (W, ˆπ, Φ ) = inf u s sup E C s,π s} T s= exp ( δs) C1 γ s 1 γ From (26) follows ha he Bellman equaion is 11 V (W, ˆπ, Φ ) = inf u sup C,π } s 1 m= exp ( δ) C1 γ 1 γ + ] } 1 + u2 m (1 γ). (26) ] 1 + u2 (1 γ) E u V +1 (W +1, ˆπ +1, Φ +1 )]}, (27) where E u means ha he expecaion is aken assuming µ U φ. 11 The Bellman equaion can be also direcly derived from he definiion of he value funcion by rewriing (24). 28

30 The firs order condiions on he righ-hand side of he Bellman equaion in (27) wih respec o C, π, and u are 0 = exp ( δ) C γ ] u2 (1 γ) V+1 (W E u +1, ˆπ +1, Φ +1 ) C V+1 (W 0 = E u +1, ˆπ +1, Φ +1 ) π ], (28) ], (29) and u (1 γ) E u θ = 1 + u2 (1 γ) V +1 (W +1, ˆπ +1, Φ +1 )] ] E u φ +1 µ φ u σφ 2 V +1 (W +1, ˆπ +1, Φ +1 ) }, (30) respecively, where φ +1 = log (Φ +1 ). Now le us assume (I will prove his in Lemma 1) ha he value funcion V +1 (W +1, ˆπ +1, Φ +1 ) can be expressed in he form V +1 (W +1, ˆπ +1, Φ +1 ) = W 1 γ +1 1 γ v +1 (ˆπ +1, Φ +1 ). (31) Then he firs-order condiion wih respec o C, i.e., (28) becomes ] c = exp (δ) 1 + u2 (1 γ) (1 + I Φ ˆπ ) γ (1 + I Φ π ) 1 γ ]] E u (R f + π (R +1 R f )) 1 γ 1 γ v +1 (ˆπ +1, Φ +1 ) + (1 + I Φ ˆπ ) 1 } 1, (32) 29