A simple Consumption-based Capital Asset Pricing Model

Transcription

1 simle Consumtion-based Caital sset Priing Model Integrated wit MCandless and Wallae Kjetil Storesletten Setember 3, Introdution Purose of leture: understand te onsumtion-based aital asset riing model CCPM. Questions:. ow does risk affet te rie of assets and te equity remium? 2. wi assets ay a ig exeted return? 6. Environment ssume te oulation is onstant, N t =N for all t, and tat endowments are idential aross individuals and time: ere are two assets in tis eonomy, ω t =[ω,ω 2 ]. rivate borrowing and lending, aying a risk-free rate of return r t. 2. units of land, yielding an unertain ro dt er unit of land. ssume tat te yieldineaeriodt is stoasti: were σ [0,d. us, dt = d + ε t +σ wit robability ε t 2 σ wit robability 2 Eε t = 0 varε t = σ 2

2 ssume referenes are time-searable and idential aross individuals and time: u t t t, t t +=u t t + u t t +. e funtion u. is assumed to be differentiable and onave tese roerties of u guarantee tat u t is onvex and differentiable, see leture. 6.2 Solving te individual s roblem Wen te return on land is stoasti and referenes are time-additive, agents maximize exeted utility. at is, agents solve max E n t u t t + u {a t,l t t + o t} subjet to budget onstraints: t t ω l t ta t t t + ω 2 + rtl t+t ++d + εt +a t Note tat te exetation oerator E t is te exetation onditional on information at time t, i.e. te exetation of te stoasti variable εt +. e solution to te individual s roblem an be found by substituting te budget onstraints into te utility funtion and differentiating te exeted utility wit reset to individual demand for lending l t and land a t For lending l t we ave tat for land a t: d 0 = dl t E n t u t t + u t t + o u t t t t t t l t + u t t + t t + t t + l t u t t + r t u t t + t t t t +. d 0 = da t E n t u t t + u t t + o u t t t t t t a t + u t t + t t + t t + a t t u t t + [t ++d + εt +] u t t + t t t t

3 Note tat in order for te Þrst order equations to be suffiient onditions for otimum, tere must be no borrowing or sortsale onstraints for te agents. Sine t t isknownineriodt, any funtion of t t an be taken outside te exetations sign, and equations and 2 an be rewritten to get wat is alled te fundamental asset riing equations: r t t u t+ t t t+ u t t t t u0 [t ++d + εt +] t t + u 0 t t u t t+ t t+ u t t t t u0 t t + [t ++d + εt +] u 0 t t Note also tat sine equations 3 and 4 must old for any agent, all agents agree on te ries in equilibrium. 6.3 Solving for te ometitive equilibrium e ometitive equilibrium requires i individual otimization, and ii market learing see DeÞnition in leture. ree markets are required to lear for ea eriod t:. te market for rivate lending, wi requires 2. te market for land, wi requires 3. tegoodsmarket,wirequires Nt = t t+ Nt = Nt = Nt = t t += l t =0. a t =. Nt = ω t t+ Nt = ω t t ++dt. Sine all individuals {, 2,...,Nt} in any generation t are idential same endowments and same referenes, teir otimal demand for lending, l t, and land, a t, 3 4 3

4 must be te same for all. Hene, in any ometitive equilibrium it must be tat asset demands for all are given by l t = 0 5 a t = Nt = N. 6 Claim: market learing for land and rivate lending guarantees goods market learing. Proof: In tis ase, total onsumtion for te young and te old in eriod t are given by Nt = Nt = t t = t t = Nt = Summing bot equations, we get Nt = Ã Nt = ω t Nt Ã! ω 2 +t+d + εt = Nω 2 +t+d + εt = Nω 2 + t + dt t t+ Nt = i.e. market learing in te goods market. QED = Nω t! Nt t t =Nω + Nω 2 + dt, Wat remains now is to Þnd ries t and rt for ea t su tat asset demands are given by 5 and 6. Sine te environment is stationary, we guess on a stationary equilibrium, i.e. t = and rt =r for all t. Moreover, exloiting te market learing onditions for l t and a t i.e. equations 5-6, we an rewrite te asset riing equations 3-4 as r u 0 ω 2 +[ + d + εt +]/N = E t u 0 ω /N = E t u 0 ω 2 +[ + d + εt +]/N u 0 ω /N [ + d + εt +] Sine equations 7 and 8 inororates individual otimization for all agents, and sine tey also imly market learing in all markets, tese equations are now our equilibrium onditions. 6.4 Imosing furter restritions In order to get sarer results, we need to make some assumtions about referenes, ro, and endowments

5 6.4. isk neutral agents isk neutral referenes means tat te utility funtion is linear in onsumtion, so tat marginal utility is a onstant: u = α u 0 = u = α e equilibrium onditions equations 7 and 8 ten beome ¾ ½ α = E t = r α ½ ¾ α = E t α [ + d + ε t +] = [ + d]+e t {ε t +} = [ + d], wi imlies te same relation between rie of land and te interest rate on rivate lending, namely r = + d =. One useful alternative way of exressing te differenes between te risky asset and te safe asset is to onsider te differenes in terms of exeted return. Let ˆr denote te exeted risk remium, i.e. te exeted return on land minus te return on lending: ˆr + d + εt + = + d r r + E t εt + Bottom line: unertainty don t matter beause agents are indifferent to risk risk neutral isk averse agents more interesting ase is wen agents are risk averse i.e. dislike risk. In order to solve tis ase in a simle way, we make two restritions. Preferenes are assumed to be logaritmi, i.e.: u = log u 0 =. is is also labeled te exeted exess return on land relative to te safe asset. 5 =0

6 Given our notation, tis means tat u t is given by t t, t t + =log t t+log t t + u t 2. Endowments are assumed to be ω t =[ω, 0] for all agents. Given restritions and 2, te seond equilibrium ondition 8 simliþes to ω /N = E t [ + d + εt +] [ + d + εt +]/N wi yields = wi yields r = µ N ω N +, ω.eþrst equilibrium ondition 7 ten simliþes to ω /N = E t [ + d + εt +]/N = E t = 2 = + ω + ω +d + εt +/N + ω ω + +d + σ /N ω + + d/n ω d 2 N + ω r = + + ω = + d + ω + ω +d σ /N 2, 9 N σ 2 + N d ω N ω + + /N dσ2 σ 2 + d. Note tat in order to arrive at equation 9, we used te deþnition of te stoasti variable ε t +from setion 6. above. Note te following fats:. ere an only be one stationary equilibrium. 2. If land is risk-free, i.e. σ =0,ten ereturnonlandequalstereturnonlending,r = +d,asinterisk neutral ase. e rie of land equals te disounted value of te future endowments, = d. r 6

7 3. e only asets of te risky asset tat matter for ries are te exeted ro, d, and te variane of te ro, σ e exeted equity remium is now ˆr + d + εt + r + d + εt + + d + σ 2 εt + = + d + E t = + + ω N ω /N + dσ2 σ 2 + d Bottom line, te equity remium is dereasing in d and inreasing in σ Wat assets get ig exeted return? Suose we introdue a new asset in te eonomy, a tree yielding a ro ft er tree. ssume tat te yield in ea eriod t is stoasti: us, ft = d + δ t +s wit robability δ t 2 s wit robability 2 E f t = d = E d t Eδ t = 0 varδ t = s 2 Moreover, assume tat te yield on te tree is orrelated wit te yield on te land, orr δ t,ε t =M. o failitate notation, deþne m t+ u t t+ t t+ u t t t t u 0 t t + = = u 0 t t MS t,t+ e asset riing equations ten beome r t {m t+ } 0 t {[t ++d + εt +]m t+ } 7

8 Using te asset riing equation 4, te rie of te tree, t, an be omuted as n t t ++d + δt + i o m t+ n t+ m t+ o were t+ t ++d + δt + t istereturnontetreeineriodt +. Using te deþnition we ave ov x, y =E x y E x E y, n t+ m t+ o ewriting and using equation 0, we get Substituting in m t+,weave = ov t t+,m t+ + Et n t+ o Et {m t+ } n o E t t+ = E t {m t+ } ov t t+,m t+ E t {m t+ } = r t ov t t+,m t+ E t {m t+ } t+,u 0 t t + n o ov E t t t+ = r t E t {u 0 t t +} = r t std t+ orr t+,u 0 t t + std u0 t t + E t {u 0 t t +} Note tat te term u 0 t t anels beause it an be taken outside of te onditional exetations terms and so it aears in bot te denominator and te enumerator, i.e. if k is a onstant and x and y are random variables, ten ov,y x k E x k = Wat do we lean from equation? ov x, y k E x = k ov x, y. E x. e risk remium on an asset is linear in std t+. 8

9 2. n asset wit ig variane var t+ but zero orrelation wit u 0 t t +gets no remium over te risk free rate. e only reason an asset gets a ositive or negative risk remium i.e. an exeted return larger or smaller tan r t istat it rovides ositive or negative insurane against onsumtion ßutuations. 3. n asset wi rovides a low return wen u 0 t t + is ig as a negative orrelation, orr t+,u 0 t t + < 0, and terefore gets a ig exeted return. If te utility funtion is onave agents are risk averse, and referenes are onvex, as deþnedinleture,tenu 0 is ig wen is low. us, tis asset ays a low return wen onsumtion is low. No investors would want to old tis asset if te exeted return was equal to te risk-free rate! us, it must ave a ositive risk remium in order to indue eole to old it. 4. Conversely, an asset wit ositive orrelation wit u 0 t t +as negative orrelation wit t t +and terefore els to smoot onsumtion. Peole would terefore like to old it, even if te exeted return is lower tan te riskfree rate. Insurane is an examle of an asset wit tese arateristis; i.e., it ays off only wen onsumtion is low, and it as a negative exeted return. 5. e risk remium is inreasing in std u 0 t t +. us, if eiter t t +is very variable or te marginal utility u 0 t t + is very stee i.e. te utility funtion u is very onave, ten te risk remium beomes big. 9