A Brief Introduction to the Consumption Based Asset Pricing Model (CCAPM)

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1 A Brief Inroducion o he Consumpion Based Asse Pricing Model (CCAPM We have seen ha CAPM idenifies he risk of any securiy as he covariance beween he securiy's rae of reurn and he rae of reurn on he marke porfolio According o he CAPM, he uncerainy associaed wih he reurn on he marke porfolio is he sole source of risk in he economy bu CAPM has no heoreical srucure ha allows us o readily idenify wha i is ha causes he marke porfolio o be risky Macroeconomics does have such a heoreical srucure I ells us, for example, how he profis of firms are relaed o such hings as overall economic aciviy (GDP and he governmen's conduc of moneary and fiscal policies Macroeconomics provides us wih models ha enable us o no only idenify various sources of aggregae uncerainy bu o also undersand he mechanisms by which hese affec securiy reurns and prices The asse pricing model ha is embedded in sochasic models of macroeconomics is called he Consumpion Based Asse Pricing Model (CCAPM The name derives from he fac ha he equaions ha describe he behaviour of asse prices and reurns in he CCAPM devolve from he consumpion/saving and asse choice decisions of households In he CCAPM he economy is assumed o be populaed by a large number of households ha are idenical in all respecs, including preferences and endowmens This assumpion permis decision making o be analyzed by examining he behaviour of a single, represenaive household No maer wha he macroeconomic seing, one consequence of he CCAPM assumpion ha all households are idenical is ha households will never exchange asses wih one anoher For insance i will never be he case ha one household will borrow from anoher Why? All households are idenical; if one wishes o borrow, all will wish o borrow and here will be no household ha wishes o lend If here are any asses ha exis in posiive ne supply, hese mus come form ouside he household secor (eg from governmens, businesses, or he res of world Example, Perfec Cerainy The simples example of CCAPM is a wo period endowmen economy wih perfec cerainy Le us suppose ha he represenaive household has lifeime uiliy given by U = u c + βu( ( c Le us furher suppose ha he represenaive agen has endowmen Y unis of consumpion in he curren ime period and Y unis of consumpion in he fuure ime period Consider a discoun bond ha pays uni of consumpion in ime period and sells for he price p in ime period Le q denoe he quaniy of discoun bonds purchased by he represenaive household during ime period (q may be posiive or negaive The represenaive will choose he value of q so as o maximize lifeime uiliy subjec o he budge consrains: c = Y pq and c = Y + q Formally, he opimizaion problem is

2 Max u( Y pq + β u ( Y + q q The firs order condiion (FOC is ( Y pq p + β ( Y + q ( = We should regard he value of q in FOC ( as he quaniy of discoun bonds demanded by he represenaive household Now le us suppose ha he discoun bond under consideraion is issued by he governmen in his economy Suppose ha for each household in his economy he governmen issues a quaniy q s of he discoun bond Then for he bond marke o be in equilibrium, quaniies demanded and supplied mus be equal; hence he equilibrium price of he bond p mus be such ha he quaniy q demanded by he represenaive household is exacly equal o he quaniy q s supplied by he governmen The equilibrium price is he value of p ha solves FOC ( wih q = q s In oher words, FOC ( is an expression ha describes he equilibrium price of he discoun bond A numerical example will help o clarify his Suppose he uiliy funcion in our example is he logarihmic form u(c = ln(c and he ime discoun facor is β = 95 Also suppose he endowmens are Y = and Y = 7; and he quaniy of bonds supplied is q s = 2 per household Then FOC ( becomes p (' + 95( = ( 2 p The equilibrium value for he bond price is p = 876 And, since he price of a discoun bond is always equal o, where r is he rae of ineres, he equilibrium ineres ( + r rae in he economy is r = 473 (or 473% As a second numerical example, suppose ha preferences and endowmens are as given above bu here exiss no governmen so ha he equilibrium supply of bonds is q s = In his case FOC ( becomes ('' p + 95( = 7 The equilibrium bond price is p = 357 and he equilibrium ineres rae is r = 2632 (or negaive 2632% The inuiion for a negaive ineres rae here is ha wih endowmens of Y = and Y = 7 he represenaive household would have naural incenive o smooh consumpion by buying bonds (lending in ime period if he rae of 2

3 ineres were posiive Bu wih bonds in zero ne supply lending is impossible and he ineres rae mus be negaive o make he represenaive household choose ne demand for lending ha is exacly equal o zero End Example The general case is a muli-period model in which i assumed ha he represenaive household is infiniely-lived; ie he household lives forever To he uniniiaed i will seem odd ha a household is assumed o live forever, bu i is no so odd if we view a household as a consising of relaed individuals of differing ages As he older members of a household die off, hey are replaced by younger members of he nex generaion In his represenaion he household is, in effec, a never-ending "dynasy" The represenaive household is assumed o have expeced lifeime uiliy funcion given by i (2 [ U ] = u( c + E [ β u( c ] E + i i= Here E denoes he expeced value operaor aken during ime period and, herefore, condiional upon he informaion ha is available o he represenaive household during ha ime period u(c is he uiliy derived from consumpion underaken during ime period and E u(c +i is he expeced uiliy of consumpion in ime period +i β denoes a ime discoun facor I is assumed ha <β< so ha he weigh in U of consumpion during any fuure ime period +i is smaller, he larger is i Tha is, consumpion in he near fuure is valued more highly han consumpion in he more disan fuure Empirically, i is found ha he value of β is in fac only slighly less han, which implies ha consumpion in consecuive ime periods are very close subsiues and ha households desire o smooh consumpion so ha period-o-period variaions in he value of c are small In each ime period he represenaive household chooses curren period consumpion c and formulaes a plan for fuure consumpion c +, c +2,, subjec o a budge consrain The naure of he budge consrain can vary, depending upon he complexiy of he economic environmen in which he represenaive household is assumed o dwell However, for deermining equilibrium prices of asses he form of he budge consrain really does no maer We simply ask he quesion: If he represenaive household were able o purchase some securiy for a price of p x, in ime period and redeem (eg sell his asse for he quaniy of consumpion x + in ime period +, wha would be he opimal quaniy of his securiy ha he represenaive household would choose o purchase? The answer is given by he following Firs Order Condiion: (3 c p + β E [ x u ( c ] = ( x, + + Observe ha he firs erm in FOC (3 is he uiliy los in ime period from he purchase of a marginal uni of he securiy and he second erm is he expeced uiliy o be gained 3

4 in ime period + from he redempion of his marginal uni FOC (3 applies o any securiy x+ By definiion = ( + rx, +, where r x, + is he one-period rae of reurn o be received px, on he securiy in ime period + Using his definiion, FOC (3 may be rewrien as ( c + (4 = β E ( + rx, + ( c Equaion (4 is he sandard expression of CCAPM I ells us ha in economy-wide equilibrium he expeced value of (+ he one-period-ahead rae of reurn on any asse imes he marginal rae of subsiuion of consumpion beween ime periods and + mus equal This equaion mus be saisfied for all asses For a risk-free asse, he corresponding FOC is (5 = β ( + r f E u'( c+ u'( c We can subrac Equaion (5 from Equaion (4 and express he CCAPM relaionship in erms of excess reurns -- eliminaing in he process he parameer β: (6 E [( r ( c+ ] ( c x, + rf = To gain some inuiion ino his resul, le us make use of he saisical propery ha says ha he expeced value of he produc of wo random variables is he sum of heir covariance and he produc of heir respecive expeced values and rewrie Equaion (6 as ( c + ( c+ (7 rx, rf = Cov[ rx, +, ] [ ] ( c E ( c + Equaion (7 ells us ha he risk of any asse is proporional o he negaive of he covariance of is rae of reurn wih he marginal rae of subsiuion This makes some sense in so far as a main moivaion for acquiring asses is o faciliae he smoohing of consumpion (and uiliy across ime periods Suppose he asse x appearing in Equaion (7 has a rae of reurn ha is high whenever fuure consumpion c + is also high Then he marginal uiliy u ( c+ will be low when r x,+ is high, and he covariance in Equaion (7 will be negaive in sign In his case asse x does no help o smooh consumpion over ime and will command a posiive risk premium; ie r > r x, + f 4

5 Now suppose, insead, ha r x,+ is high whenever fuure consumpion c + is low In his case he asse does assis in smoohing consumpion over ime and he covariance in Equaion (7 will be posiive in sign, which implies r < r x, + f In CAPM a securiy's sysemaic risk depends on he covariance beween is reurn and he reurn on he marke porfolio In CCAPM a securiy's sysemaic risk depends on he covariance of is reurn wih fuure consumpion While Equaions (4 (7 deal only wih reurns one period ino he fuure, CCAPM can accommodae asses wih payoffs in any fuure ime period In his sense CCAPM is ruly a muli-period asse pricing heory Consider a securiy ha will pay y +j unis of consumpion in fuure ime period +j and nohing prior or afer ha ime period Le p y, denoe he price of his securiy in ime period Then he FOC ha describes he equilibrium value of his price is j (8 ( c p, + β E [ y ( c ] = y + j + j We will no furher pursue CCAPM in his general seing Insead, we will apply i in a very simple example Example 2 Le us suppose ha he curren ime period is period and ha he represenaive household has a curren endowmen of Y = 6 For each fuure ime period he endowmen is assumed o be random wih wo possible oucomes: wih probabiliy 6, Y = ; wih probabiliy 4, Y = 5 for all > The realizaions of Y are assumed o be independen from one ime period o he nex The uiliy funcion is u(c = ln(c and he ime discoun facor is β = 98 Le us suppose ha all asses are in zero ne supply so ha c = Y and c = Y for all values of We can use Equaion (4 o deermine he value of he one-period risk-free rae of ineres during ime period ha will cause he represenaive household o demand zero unis of he risk-free asse We know u ( c = = 667 and can easily 6 compue E[u'(c ] = 6( + 4( = 4 Then from Equaion (4, r f = Now le us deermine he period price of a discoun bond ha will pay uni of consumpion in ime period = 2 no maer which sae occurs Le P (2 denoe his price, which from Equaion (8 mus saisfy he following: 5

6 P (2 2 u'( c = β ( E u'( c 2 (2 E[u'(c 2 ] = 4 and u'(c = 667; consequenly P = 866 Noe ha he 2-year rae of ineres is he value r (2 (2 ha solves P = So here he 2-year rae is (2 2 ( + r (2 r = 35 Le q H denoe he price of he pure securiy ha pay will pay uni of consumpion in L ime period if he high consumpion sae occurs and le and le q o denoe he price of he pure securiy ha pay will pay uni of consumpion in ime period if he low consumpion sae occurs In deermining he values of hese wo prices we mus ake some care in evaluaing E[u'(c ] The pure securiy ha pays only in he high consumpion sae has E [ ( c ] = 6( = 6 The pure securiy ha pays only in he low consumpion sae has E [ ( c ] = 4( = 8 Then using Equaion (3, we 5 can infer ha q H = 3528 and q L = 473 We can confirm ha hese values are correc because we know ha (q H + q L is always equal o, and from our earlier + r f compuaions we found = = 823, which does indeed equal (q H + q L + r f 248 Observe ha here are wo possible saes in ime period and we have recovered he prices of he wo pure securiies; consequenly he securiy marke is complee here (Securiies markes are always complee under CCAPMNow le us deermine he price of a European pu opion wrien on c wih an exercise price of 85 (The mauriy dae is in ime period = [Noe: In he classroom we have no ye discussed pu and call opions Tha is sill o come Suffice i o say he pu opion under consideraion here is a securiy ha will pay if Y = and will pay 35 if Y = 5 Wih his informaion regarding he payoffs, he suden should have no difficuly deermining he equilibrium price of his securiy] There are wo differen ways of deermining he price of his pu opion Firs, we know ha he payoff of he opion will be non-zero only if c = Y = 5, in which case he payoff will be 35 (unis of period consumpion This is equivalen o owning 35 unis of he pure securiy wih price q L = 474 consequenly he ime period price of he pu mus be p pu, = 35(473 = 6464 Alernaively, we can use Equaion (3 o compue he equilibrium pu price direcly In his case Equaion (3 becomes 6

7 (3' ( p pu, + 98(35(4( =, 6 5 which yields p pu, = 6464 I leave for he suden o show ha he price of a European call opion wrien on c 2 wih an exercise price of 75 is p call, = 8644 (The mauriy is in period = 2 [In his case he call opion is a securiy ha will pay 25 a dae 2 if Y 2 = = and will pay a dae 2 if Y 2 = 5] Now le us inerpre he fuure endowmens o be received by he represenaive household as being he resul of boh labour effor and invesmen on he par of he household We will assume ha he represenaive household is o receive real wages of 5 unis of consumpion wih cerainy in every fuure ime period, saring wih ime period = The household is also assumed o own one share of sock in a corporaion and will receive real dividends in each fuure ime period in amoun 5 wih probabiliy 6 or in amoun wih probabiliy 4, also saring in ime period = In oher words, in he "high" sae (which occurs wih probabiliy 6 he endowmen will be wages = 5 plus dividends = 5; in he "low" sae (which occurs wih probabiliy 4 he endowmen will be wages = 5 plus dividends = (Noe ha we have no changed he oal endowmens here; we have jus idenified where he endowmens come from Le S be he ime period = price of he share of sock owned by he represenaive household and le Div be he value of he dividend o be received in ime period > I can be deduced from Equaion (8 ha he "presen value" of his risky dividend is given u ( c by he expression β E[ Div ], which promps he following generalizaion of ha u( c equaion: (9 S = E = ( c β Div ( c Equaion (9 says ha he share price is he sum of he presen discouned values of all fuure dividends ( c 2 Here E[ Div ] = (6(5 ( + (4( ( = 8 for all > ; hence u( c = 8 S 98 = 8(98 = 882 = 98 Tha is, he equilibrium price of one share of sock is 882 unis of consumpion in ime period 7

8 End Example 2 8

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