FINM 6900 Finance Theory

Transcription

1 FINM 6900 Finance Theory Universiy of Queensland Lecure Noe 4 The Lucas Model 1. Inroducion In his lecure we consider a simple endowmen economy in which an unspecified number of raional invesors rade asses in securiies markes. All invesors derive uiliy from a single consumpion good, and each invesor solves a dynamic porfolio problem o arrive a an opimal holding of securiies. Thus, he invesor s problem is o imize lifeime uiliy of consumpion subjec o a series of one-period budge consrains. There are no ransacions coss, shor sales consrains, or oher barriers o rade. Each invesor is free o adjus he composiion of his or her porfolio a any poin in ime. 2. The Invesor s Problem We se he problem up as follows. subjec o where C X j + C = W + E B i U(C +i ) i=0 (1) X j 1( + ) (2) 1. E denoes he expecaion aken wih respec o he informaion available a ime. 2. X j is he amoun of securiy j purchased a ime. 3. is he price of securiy j a ime. 1

2 4. W is he represenaive invesor s wealh a ime. 5. C is he represenaive invesor s consumpion a ime. 6. is he dividend paid by securiy j a ime. 7. B is he ime discoun facor. Noe ha invesors are allowed o rade a each poin in ime. The budge consrain says ha payoff on he porfolio, X j 1( + ), plus he endowed wealh, W, is allocaed beween curren consumpion, C, and invesmen in he securiies marke, X j. The budge consrain in (2) can be solved for C, C = W + X j 1( + ) X j, (3) and he represenaive individual s problem in equaion (1) can be wrien as C L = E U + BU +. Subsiuing for C using equaion (3) yields X L = E U W + X 1( j + ) + BU W , wih X j N as he new se of choice variables. X j ( ) X j X+1 j +1 Differeniaing wih respec o X j gives he following firs order condiion. E U = E BU ( +1 + δ+1) j j. 2

3 Solving for or dividing by we find ha BU = E ( U ) BU 1 = E (1 + R j U +1) This resul is known as he Lucas model. j, (4) j. (4 ) 3. The Presen Value of Dividends Version Equaion (4) implies ha BU = E U +1 + BU U +1. (5) and also ha BU +1 = E +1 U ( +2 + δ+2) j. (6) We noe ha BU BU E E +1 U ( +2 + δ+2) j = E U ( +2 + δ+2) j. by he law of ieraed expecaions. Thus, subsiuing (6) ino (5) gives BU = E U +1 + BU ( BU ) U U ( +2 + δ+2) j, or BU = E U d j +1 + B2 U U If we coninue o subsiue for P +i T i=2, we obain = E T B i U (C +i ) U Le T and noe ha if 0 < B < 1 hen B T U (C +T ) lim T U The end resul is = E +2 + B2 U U BT U (C +T ) U +T. +T B i U (C +i ) U +i 3 0. j. (7)

4 4. Relaion o Sae Preference Theory In our discussion of sae preference heory, we derived he pricing relaion T S 0 = Φ s d j s =1 s=1 where d j s denoes he payoff on asse j a ime in sae s. Because he invesor could only rade a ime = 0 in his model, i is naural inerpre he payoff d j s as he dividend on asse j a ime in sae s. Thus, a ime, he sae preference model implies ha T S = Φ +i,s δ+i,s, j s=1 or, using our expression for he sae prices, = T S s=1 j, π +i,s B i U (C +i,s ) U +i,s If we assume ha he invesor s probabiliy assessmens are condiional on he informaion available a ime we have T B i U (C +i ) = E U +i j. (8) Le T go o infiniy and we ge he presen value version of he Lucas model. The saring ime was arbirary, so equaion (8) also implies ha T B i U (C +1+i ) +1 = E +1 U +1+i j. (9) Togeher, equaions (8) and (9) imply ha BU = E ( U ). (10) Proof Noe ha equaion (8) implies BU = E U +1 + BU U Apply he law of ieraed expecaions o ge BU = E U T +1 + BU T E U +1 Now subsiue for +1 using equaion (9) BU = E U 4 j. B i U (C +1+i ) U +1+i. B i U (C +1+i ) U +1+i. ( +1 + δ+1) j. (11)

5 5. The Disribuion of Reurns Consider an economy wih 1 securiy and a represenaive invesor. From equaion (7) we have P = E B i U (C +i ) δ +i U Assume U = ln and noe ha opimal consumpion will equal he dividend paid by he securiy o ge Thus we have and which implies B i δ P = E δ +i δ +i P = E B i δ P = E δ δ+1 P +1 = E +1 P = δ P +1 = δ +1 and ln(p +1 ) ln(p ) = ln(δ +1 ) ln(δ ). Suppose ha he dividend process akes he form ln(δ +1 ) = µ + ln(δ ) + ɛ +1. We hen have ha ln(p +1 ) ln(p ) = µ + ɛ +1, which is a random walk in reurns. Alernaively, suppose he dividend process is ln(δ +1 ) = µ + ρ ln(δ ) + ɛ +1. The resul is ha ln(p +1 ) ln(p ) = µ + (ρ 1) ln(δ ) + ɛ +1, which implies ha reurns follow a firs-order auoregressive process. 5