Problem Set 6 Finance 1,

Transcription

1 Carnege Mellon Unversty Graduate School of Industral Admnstraton Chrs Telmer Wnter 2006 Problem Set 6 Fnance, (representatve agent constructon) Consder the followng two-perod, two-agent economy. Two perods: today and tomorrow Two equally lkely states-of-the-world are possble tomorrow: state and state 2. Today wll be referred to as state 0. Two agents, labeled agent and agent 2, derve utlty from the consumpton of a sngle good. Denote c R 3 ++ as agent s consumpton and d R 3 ++ as agent 2 s consumpton. Smlarly, c and d, = 0,, 2, denote the state-specfc consumpton of agent and 2, respectvely. Preferences are as follows. U (c) = log(c 0 ) + β 2 (log(c ) + log(c 2 )) U 2 (d) = log(d 0 ) + β 2 2 (log(d ) + log(d 2 )) Agent s and 2 have rates of tme preference such that β = 0.8 and β 2 = 0.9 The agents have endowments as follows. c = [ c 0 c c 2 ] = [ 2 ] d = [ d 0 d d2 ] = [ 2], where c denotes agent s endowment and d denotes agent 2 s endowment. Assume that there are two securtes marketed n ths envronment. Securty pays off one unt of the good f state one occurs and zero f state two occurs. Securty 2 pays off one unt of the good f state 2 occurs and zero f state occurs. Construct a representatve agent for ths economy and use your constructon to characterze the compettve equlbrum. That s, f we denote the aggregate endowment as e R 3 ++, you are to construct a functon, U λ (e),

2 λ R ++, such that no-trade s an equlbrum for ths aggregate economy, gven the prces whch solve the decentralzed problem. Your answer should use the representatve agent constructon to characterze the equlbrum allocaton of consumpton amongst the two agents. You should also clearly demonstrate the valdty of the representatve agent constructon for ths economy (.e., show that no-trade s an equlbrum at the market clearng prces). 2. (representatve agent dervaton of CAPM) Consder the statc exchange economy we studed n class. There are S states of the world. D s an N S matrx of payoffs whch corresponds to the N marketed assets whch have prces q R N. A portfolo s some θ R N. Agents have endowments, e R S ++ and e m e. Recall that f m agents each have expected utlty, U (c) = E(u (c)), then we can construct an autarkc representatve agent equlbrum where the representatve agent s preferences are U λ (c) = E(u λ (c)), wth u λ (y) = max x R m m λ u (x ) subject to m x y. Suppose that we have the followng addtonal structure: There are postve constants, c and {b } m = such that, for any state s, c s < c for each agent and u (x) = x b x 2 for all x c. The followng are all well defned: the return R θ D θ/q θ. R M, the return on a portfolo, θ M, wth payoff equal to e. R 0, the return on a portfolo, θ 0, wth Cov(D θ 0, e) = 0. β θ Cov(R θ, R M )/Var(R M ) (a) Show that u λ (e) = k Ke for some postve constants k and K. From ths derve the CAPM, q = AE(D) BCov(D, e), (b) where A and B are postve constants and Cov(D, e) s the vector of covarances, Cov(D, e). Show that ths mples the zero-beta securty market lne, E(R θ R 0 ) = β θ E(R M R 0 ). Ths queston s based on Duffe,.2, page 5. 2

3 3. (computaton of general equlbrum n lnear quadratc economy ) Ths queston uses the same data as an earler problem. There are 0 states of the world whch are equally lkely. The followng table gves frm-specfc cash flows n each state. Path Frm Frm Frm The followng nformaton s also relevant for ths problem: There are two classes of nvestors n the economy wth preferences: Type I : U I ( W ) = W W 2 Type II : U II ( W ) = W W 2 Investors are endowed wth both securtes as well as cash. There are four types of endowments: Shares Investor Type Cash Frm Frm 2 Frm 3 Type Type Type Type Investor s wth endowment types,2 and 3 have type I preferences. The dstrbuton of endowment types s such that for each type, 2 and 3 nvestor there are three type 4 nvestors. The rsk-free nterest rate pad on cash balances s 2%. Your job s to characterze the general equlbrum for ths economy. That s, you are to solve for the equlbrum frm values, market prce of rsk, optmal demands (verfyng that demand = supply) and frm specfc β s. Verfy that a securty market lne holds for ths economy. The attached appendx should be helpful. 3

4 Carnege Mellon Unversty Graduate School of Industral Admnstraton Chrs Telmer Wnter 2006 APPENDIX Problem Set 5 CAPM General Equlbrum These notes are ntended to help-out wth the the queston on computaton of general equlbrum n lnear quadratc economy. We ll solve a smple general equlbrum model, drven by a two-fund result. The two fund result wll n turn be drven by quadratc utlty. To foreshadow thngs slghtly recall that, wth a rskless asset, the two-fund theorem says that all nvestors hold the same portfolo of rsky assets. For ths exercse t wll be useful to note that ths s equvalent to the statement that an nvestor holds equal proportons of each frm n the economy (.e., the fracton of frm held by nvestor k equals that of frm j held by ths nvestor). You should verfy ths for yourself. Once agan t wll prove convenent to express thngs slghtly dfferently. A portfolo wll be expressed as the fracton of frm j held by some nvestor. Notaton: W k termnal wealth of nvestor k. W k 0 ntal wealth of nvestor k. z k j the proporton of the value of frm j held by nvestor k. V j value of frm j. x j (s) state contngent cash flow from frm j. R one plus the rsk-free nterest rate. p(s) probablty of state s. µ j E(x j (s)). σ j Cov(x, x j ). u k (W k ) = W k c k (W k ) 2. So each agent wll maxmze E(u(W k )). These notes are based on notes wrtten by John O Bren and Sanjay Srvastava.

5 We can now wrte wealth dynamcs as follows: W k (s) = R(W k 0 z k V ) + x (s)z k () The term n parentheses s smply the monetary value of wealth whch s not nvested n the rsky assets. The second term s smply the state contngent cash flow from the total nvestment n the rsky technologes. We can rewrte ths as, W k (s) = RW k 0 + z k (x (s) RV ) (2) Subject to the wealth constrant, nvestor k wll solve, The frst order condtons are, max {z k j }N j= p(s)u k (W k (s)) (3) s E( 2c k W k )(x j RV j ) = 0 j (4) Next, plug () nto the FOC and use the fact that E(x x j ) = µ µ j + σ j. Ths gves an expresson for demand functons, z k [σ j + (µ RV )(µ j RV j )] = (µ j RV j )( 2c k RW k 0 ). (5) The key thng here s that the only nvestor specfc terms asde from the choce varables, z k, enter multplcatvely (the last term on the rght). Therefore, f we solve, z [σ j + (µ RV )(µ j RV j )] = (µ j RV j ), (6) for z, =, 2,..., N, we wll have that, z k = z ( 2c k RW k 0 ). (7) It s ths lnear homogenety whch wll drve the separaton result. Next, snce k zk =, we can use (7) to get that, = z j k ( 2c k RW k 0 ), (8) z k j = (2ck ) RW k 0 k [(2ck ) RW k 0 ]. (9) 2

6 Equaton (9) s the separaton result. It says that z k s ndependent of. Let z k denote the proporton of every frm held by nvestor k. Now, use the demand equatons, (5), to get z k Market clearng mples that, [σ j + (µ RV )(µ j RV j )] = (µ j RV j )( 2c k RW k 0 ). (0) Therefore, snce k W k = V, σ j + (µ j RV j ) (µ RV ) = (µ j RV j )( k A lttle algebra gves, whch mples that, σ j + (µ j RV j )( V j = R [µ j z k = () k µ k 2c k R k W k 0 ). (2) 2c k ) = 0, (3) σ j k (2ck ) µ ]. (4) Equaton (4) s a valuaton model for the frm. Note the nterpretaton as the expected cash flow less a rsk adjustment term. Fnally, note that t may prove more useful n certan crcumstances (HINT), to solve for the z s drectly..e., z [σ j + (µ RV )(µ j RV j )] = (µ j RV j ) (5) can be wrtten (omttng the astersks) as where, E (µ RV ) z [z, z 2,..., z N ] [Ω] j σ j Defnng φ (Ω + EE ), the soluton s, z Ω + E z E = E, (6) z = φ E (7) 3