Admissible uncertainty in the intertemporal asset pricing model

Journal of Fmanclal

Economxs

ADMISSIBLE

8 (1980) 71-86

0 North-Holland

Pubhshmg

Company

UNCERTAINTY IN THE INTERTEMPORAL ASSET PRICING MODEL* George Vnwerslty

Recc~ved October

M CONSTANTINIDES of Chuzago, Chtcago, IL 60637, USA 1978, final version

recc~ved January

1980

We embed the Sharp+Lmtner, two-parameter asset prlcmg theory m an mtertemporal general eqmhbrlum model The Investment opportunity set changes stochastlcally over time, rn general the short-term and long-term Interest rates and the dlstrlbutlon of the rate of return of the market portfoho are non-stationary This non-statlonarlty, which IS admissible m the SharpeLmtner model, has two Imphcatlons First, it may bias econometric methods which fad to exphcltly take mto account the non-statlonarlty Second, the sequential apphcatlon of the Sharpe-Lmtner model m the dlscountmg of stochastic cash flows becomes computationally complex and of little practical use

1. Introduction We develop a general eqmhbrmm theory of capital asset prlcmg which IS simple yet accommodates many stylized empn-lcal facts The advantages of this theory are threefold First, being a general eqmhbrmm theory, it IS defined m terms of fundamentals, which are production technologies and consumers’ resources, beliefs, and preferences Thus, for example, shifts m interest rates, m production levels, and m risk aversion are endogenously determined at the equlhbrmm Second, the theory 1s simple Although the theory allows for an investment opportunity set which changes stochastlcally over time, risk premla on asset returns are explamed solely m terms of the covarlance of asset returns with the market portfolio This feature of the theory contrasts with the theories of Merton (1973) and Long (1974) wherein risk premla on asset returns depend on the covarlance of asset returns not only with the market portfolio, but also with a number of hedging portfolios The ldentlficatlon of the hedging

*Earher versions of this paper were presented in seminars at Carnegie-Mellon Umverslty, the University of Chicago, Columbia Umverslty, Bell Laboratories, and the Western Finance Assoclatlon meetings I thank the partlclpants and, In particular, Sudlpto Bhattacharya for their helpful comments I also thank the edltor of this Journal and the referee, John B Long, Jr, for detaded and helpful comments I remam responsible for errors

12

G M

Constantznzdes,

The tntertemporal

asset pruxng

model

portfolios and the estlmatlon of asset covarlances with these portfolios are nontrivial empirical issues Third, the theory accommodates the followmg stylized emplrlcal facts (a) The probability dlstrlbutlon of asset returns and of the market portfolio is non-stationary Indeed, rates of return of assets and of the market portfolio may be serially correlated Also, the beta coefficients of assets and portfolios relative to the marked portfoho are non-stationary (b) The term structure of interest rates evolves stochastlcally over time In particular, specific hypotheses on the term structure, e g , various forms of the expectations hypothesis, may or may not be true (c) The empirical evidence with regard to the validity of the two-parameter asset pricing equation is inconclusive The failure of some tests to confirm the two-parameter asset pricing equation may be attributed to an incorrect proxy for the (as yet unobservable) market portfolio which was used m those tests ’ Given the empirical difficulty at this stage to reject the two-parameter asset pricing equatlpn m favor of a multi-parameter version, and to identify the various hedging portfolios, there 1s an advantage m theoretically developing a two-parameter asset pricing model which 1s m agreement with the empirical evidence cited m (a) and (b) (d) Consumers are heterogeneous and hold portfolios which are not identical in composition The theory has an important lmphcatlon which relates to the empirical testing of the SLM asset pricing model and to the apphcatlon of the model in empirical studies of market efficiency Even if the SLM model 1s the true description of the economy, as opposed to the multi-factor extensions of the model, and even if there are no substantive difficulties m the ldentlficatlon of the true market portfolio, yet the distribution of the market portfolio return, the return on the nskless asset, the security betas, and the market price of risk may well be non-stationary and may bias econometric methods which fad to exphcltly take mto account these non-statlonantles ’ The theory also illustrates the hmltatlons of the SLM asset pricing model m the dlscountmg of stochastic cash flows of multi-period proJects Nonstatlonarlty m the distribution of the market portfolio return, the return on the rlskless asset, the security betas, and the market price of risk IS admissible m the context of the SLM model Therefore, the sequential ‘The theory and emplrical findmgs on the asset prlcmg model are revlewed m Jensen (1972) An extensive crltlque of the empIrIcal work appears m Roll (19773, where the sensltlvlty of the empIrIcal results on the chosen proxy for the market portfoho IS stressed ‘Slmllar comments apply to multi-factor extensions of the SLM model and to Breeden’s (1979) consumption CAPM

G M Constantamdes,

The rntertemporal

asset przcmg model

13

apphcatlon of the SLM model m the dlscountmg of stochastic cash flows of multi-period prolects becomes computatlonally complex and of little practical use, unless one can produce convmcmg evidence to the effect that these nonstatlonarities are unimportant in practice 3 A preliminary and Informal dlscusslon of our main ideas serves as an mtroductlon to the formal development which begins m section 2 The single-penod capital asset pricing theory (SLM) of Sharpe (1964), Lmtner (1965), Mossm (1966), and Black (1972) rests on the assumption that each consumer’s utility 1s solely a function of consumption at the beginning of the period and of wealth at the end of the period, but IS otherwise Independent of the state of the economy at the end of the period Fama (1970) embedded the SLM theory m an mtertemporal economy where each consumer sequentially makes consumption and investment declslons over his hfetlme Fama noted that the SLM theory requires each consumer’s derived utlhty in every period to be a function of his wealth at that period (and possibly of his past consumption) but otherwise to be independent of the state of the economy State dependent derived utlhty has three possible sources (a) state dependent tastes, (b) stochastic relative prices of consumption goods, and (c) investment opportumtles which depend on preceding events Our model ehmmates the first source of state dependence by assuming that tastes are state independent It ehmmates the second source of state dependence by focusing on a angle-good economy wherein relative asset prices of consumption goods do not exist Fama (1970) and Merton (1973) eliminated the third source of state dependence by assuming that investment opportumtles do not depend on preceding events The followmg example illustrates that there exist economies where the SLM asset prlcmg equation may hold true even if investment opportumtles depend on preceding events Consider a single-good economy with homogeneous consumers, where each consumer’s utility function 1s state independent Assume that there 1s only one investment opportunity m positive net supply and that the dlstnbutlon of the rate of return on this investment opportunity 1s independent of preceding events Trade amongst the homogeneous consumers 1s redundant and therefore each consumer consumes a fraction of his wealth m every period and invests the remaining fraction of his wealth m the single mvestment opportunity which exists m positive net supply It can be shown that each consumer’s derived utility function m every period 1s state Independent and therefore the SLM asset %mdar comments apply to multi-factor extensions of the SLM model and to Breeden’s (1979) consumption CAPM Banz and Mdler (1978) and Breeden and Lltzenberger (1978) proposed an alternatlve computatlonal procedure which circumvents some of these ddllcult~es The prices of state contmgent claims are first estimated through the optlon prlcmg model and are used m dlscountmg future cash flows Bhattacharya (1979) proposed yet another procedure, the four asset hedge, which IS an extension of the Black-Scholes optlon hedge

14

G M Constantmades,

The intertemporal

asset pruzrng model

pricing equation holds true under appropriate additional assumptions Yet the eqmhbrmm rates for short-term and long-term borrowing and lending amongst consumers may well depend on the level of aggregate wealth and therefore may depend on preceding events Whereas the assumption of homogeneous consumers IS restrtctlve, we obtain the same kmd of slmphlicatlon even d consumers are heterogeneous, as long as prices are determined as If consumers were homogeneous Formally, an economy of heterogeneous consumers IS said to have the aggregation property if equdlbrmm prices are determined as if consumers were homogeneous m terms of resources, beliefs, and preferences Since we are primarily interested m asset pricing, if an economy possesses the aggregation property, we may derive the asset prices m a slmphfied economy where all consumers are homogeneous We shall refer to the latter consumers as aggregated consumers Contracts amongst aggregated consumers (I e , financial contracts) are m zero net supply As long as the dlstrlbutlon of the rates of return on assets m posltlve net supply IS independent of preceding events, the derived utility of aggregated consumers 1s state independent Under appropriate additional assumptions the SLM pricing equation still holds true These arguments are formalized m section 3 and stated as Proposltlons 1, 2 and 3 An example m section 4 illustrates these ideas and discusses their lmphcatlons on the theory, testing and apphcatlons of the asset pricing model There have been earlier attempts to enrich the economic environment m which the SLM asset pricing equation holds true Fama and MacBeth (1974) allowed both the investment opportunity set and consumers’ utility functions to be state dependent but forbade the consumers from engaging in contracts which hedged against changes m the investment opportunity set and m consumers’ tastes The latter assumption essentially requires that markets be incomplete Stapleton and Subrahmanyam (1978) assumed that consumers have additive exponential utility of consumption and that the interest rate IS non-stochastic and IS determined exogenously to the model A more promismg approach was taken by exploiting the myopic properties of logarithmic utility, which were noted by Mossm (1968) and Hakansson (1971) Under the assumption that consumers have logarithmic utility, Rubmstem (1976b) derived the SLM equation in an economy where the Investment opportumtles depend on preceding events 4 Rubmstem (1976a) “Rubmstem (1976b) formally assumed that the aggregated consumers alone have logarlthmlc utlhty and that the actual consumers have generahzed logarlthmlc utlhty By generahzed logarlthmlc utlhty we mean that the rth consumer’s utlhty of consumption c; at date t IS In(c; -$), where 13:1s a parameter specdic to the rth consumer It can beehown that the aggregated consumers’ utlhty of consumption c, at date t IS In(c,-Z!= 1c/I), where I IS the number of consumers m the economy If the aggregated consumers have logarlthmlc utlhty, I e , Z:=, ;:/I =O, yet at least some consumers have non-logarlthmlc utlhty, there must exist a consumer for whom $
GM

Constontmldes,

The mtertemporal

asset prtctng model

75

also derived the SLM equation m an economy where the consumers have power utthty and the rate of growth of aggregate consumption follows a random walk More recently, Breeden and Lltzenberger (1978) and Breeden (1979) showed that risk premla on asset returns can be explamed m terms of the covanance of asset returns with aggregate consumption alone, if consumers have addltlve power utility functions or asset prrces are governed by a dlffuslon process ’ In contrast to the above-mentloned theories, our theory does not rely on market mcompleteness, it does not rely on a nonstochastic Interest rate which IS exogenously determmed; it does not rely on logarlthmlc or power utility, which has the implication that consumers hold portfohos with ldentlcal composition, and it does not crltlcally rely on the dlffuslon process Sectlon 4 illustrates a model which 1s free from all of the abovementloned assumptions The theory developed m this paper 1s related to the recent work by Brock (1978), Cox, Ingersoll and Ross (1978), and Prescott and Mehra (1978), wherem the financial theory of capital asset prlcmg 1s embedded m a general equlhbrmm framework The goal of our paper, however, differs slgndicantly from the goals of the abovementloned three papers We seek sets of mmlmal condltlons on technologies and on consumers’ resources, beliefs and preferences, which retam the slmphclty of the two-parameter asset pricing equation. 2. Consumption-investment

behavior and equilibrium in tbe securities market

We employ the followmg notatron t = time subscript, of the economy at t, it summarizes all relevant available mformatlon , F,(+, 14, _ 1) = dlstrrbutlon function of 4, condlttonal upon 4, _ 1, IV, = number of investment opportumtles available at date t - 1, If a rlskless asset exists, it is included m the set N,, =one plus real rate of return on &(A)= #:(A), R:(A), ,Rfv(c#+)) investment opportunities over (t - 1, t), I = unit vector, 4, =state

For this consumer the margmal

utlhty of consumption

IS limte at zero consumption

level, I e ,

(d/dc:)ln (c, -C:)[,,=O = - l/C: < m Thus the non-negatlvtty of consumption constramt, c: 20 may well be bmdmg Rubmstem’s theory does not exphcltly consrder this constramt and I$ therefore consistent with an economy where c^;LO for all I Then If the aggregated consumers have logarlthmlc utility, all consumers must habe logarlthmlc utlhty, Ie , ?; =O, for all I See also related comments m Grauer (1976) and Rubmstem (1977, f 30) 5Bhattacharya (1979) also derived the consumption CAPM by consldermg the appropriate hmlt as the time interval tends to zero m Rubmstem’s (1976a) valuation equation

16

G M Constantmldes,

The mtertemporal

asset prrcrng model

T = consumer’s lifetime, , c,} =consumptlon vector until date t, Ct={~o,cl, U(C, I&) = utility of lifetime consumption, wt+&w:, ,wtNt+l} = consumer’s wealth invested m the N,, 1 investment opportumtles at date t and after consumption at date t, w=consumer’s wealth at date t and prior to consumption at date t

There IS no presumption that consumers are identical and, therefore, IT; C,, wt, WVU(C, IhI are m general different across consumers We make the followmg assumptions, (A l)-(A 3), which are mamtamed throughout the paper, unless otherwise stated (A 1) Perfect markets Investment opportunities are traded in perfect markets In particular, there are no transactlons costs or taxes, assets are mfimtely divisible, consumers are price takers, the market IS m equrhbrmm, If a rlskless asset exists, consumers can borrow and lend at the same interest rate, and short sales of all assets, with full use of proceeds, are allowed expectations (A 2) Homogeneous are expectations Consumers’ homogeneous Alternatively, consumers have heterogeneous expectations, but the aggregation problem 1s solved so that equlhbrlum prices are determined as If consumers were homogeneous (A 3) State consumption U(C, I &I=

Each consumer’s utility of lifetime independent utdzty does not exphcltly depend upon the state of the economy, I e, V(G)

A family of probablhty dlstnbutlons, defined by Ross (1978), and labeled ‘separating dlstnbutlons’, plays an important role m the theory of asset pncmg, as will be shprtly explained 6 Commonly employed dlstrlbutlons which are separatmg dlstrlbutlons include multlvarlate normal dlstrlbutlons of returns, stable Paretlan dlstrlbutlons of returns, market factor models, and diffusion processes, where each declslon period 1s mlimteslmal These dlstrlbutlons are labeled separatmg because m a one-period economy, where consumers have state Independent utlhtles, these dlstrlbutlons imply two-fund portfolio separation Our purpose m this sectlon IS to relax the followmg assumption which IS typically made m the derlvatlon of the SLM asset prlcmg equation N, IS at most a function of time, and rnvestment opportunrty set the probablhty dlstrlbutlon of R, IS independent of 4, _ 1 for all t

Constant

6The set of separating dlstributlons

IS defined In ROSS(1978% thm 2)

G M Constantmrdes, The rntertemporal asset prrcrng model

77

TypIcally, the theory assumes (A I)-(A 3), a constant mvestment opportunity set and either (a) the dlstrlbutlon of R,&) condltlonal upon 4,_1 1s a separating dlstnbutlon, or (b) each consumer’s utlhty function IS addltlve quadratic Under these assumptions one derives the SLM asset pricing equation7 E,_,[RI-R~]=C~~;~(~~flq)El_z[R:-R~], t1

(1)

f

where R,M zs the return on the market portfoho and Rf IS the return on the zero-beta portfolio, z e , cov, _ 1(Ry, RF) = 0 We shall prove that the SLM asset prlcmg equation can be derived even If we partly relax the assumption that the investment opportunity set 1s constant Consider a consumer who- consumes a fraction of his wealth m every period and invests the remammg fraction of his wealth m the N, investment opportunities Followmg Fama (1970) we formulate the consumer’s sequential optlmlzatlon problem as

(2)

for t= 1,2, , T Q,_1 decisions such that

zs the

OsC,_l

The boundary

condltlon

JT(C,-

196

consumption

(non-negative consumption), (budget constraint),

c,_1+w,_1I5PV:-, OS w,_ zR,(4,),

set of feasible

V4,

(no personal

and

investment

(3)

bankruptcy)

1s I &I=

U(G-

1, WT I&)

The objective (2) SubJect to constraints (3) and consumer’s optimization problem The following straightforward mductlon Lemma Consider a consumer’s optzmzzatzon Assume that the consumer optzmally znvests a subset of the first n, investment opportunztzes opportunztzes whzch are available at date optzmally invests zero wealth zn the remaining ‘See Shave (1964), Lmtner (1965), Mown (1973), and Ross (1976, 1978a)

(4) (4) completely defines the lemma can be proved by

problem defined by eqs (2)-(4) non-zero amount of wealth zn the out of the total N, investment t- 1, where t = 1,2, ,T, and N,-n, investment opportunztzes

(1966), Fama (1970, 1971). Black (1972), Merton

78

G M

Constantmrdes,

The mtertemporal

asset pndng

model

Assume also that the probabllrty dlstrlbutton of the rate of return on each one of the first n, rnvestment opportunltres 1s mdependent of IP,_ 1 for all t Fmally, assume (A 3), I e., U(CTI &)= V(C,) Then the consumer’s derived utrlrty function is state independent, 1e, J,_,(C,_2, K-, ~q3,_1)=J,-l(C,_,, w-1), for all t

In the next section the lemma 1s incorporated theory of asset prlcmg

m a general equlhbrmm

3. Teclmological uncertainty and admissible uncertainty in the asset p&ii model We first consider a production sector with constant returns to scale technologes At date t - 1 there exist n, competltlve firms. In general, the number of Investment opportumtles exceeds the number of firms, 1e., n, 6 N, These include bilateral contracts amongst consumers or groups of consumers, such as borrowmg-lending contracts of various matuntles, option contracts and insurance policies. Furthermore, a levered firm offers to consumers more than one type of investment opportumtles such as equity and bonds with different indenture provlslons. Without restrlctmg the investment opportunity set we assume that firms are unlevered, since equity and bonds of a levered firm can be manufactured through the writing of options on the equity of an unlevered firm which has the same technology The jth firm 1s endowed with a linear production function f (K:_,) =K{_ lR;, K:_ 1 1s the input at date t - 1 and K:_ lR: 1s the output at date t Since the firm 1s competitive, the market value of the firm at date t - 1 and after new capital 1s raised at t - 1 equals K;_ 1, and the value of the firm at date t and before new capital 1s raised at t equals K;__,R; By assumption, firms are unlevered and therefore, the one plus rate of return on the equity of the /th firm equals R: We denote the vector of returns of the nt competltlve lirms by ff, = {R:, . , RF} We note the dtstmctlon between ff, and R,, where R, IS the vector of returns of the n, firms and of the remammg N,-n, investment opportunities Production inputs are nonnegative, 1e , K;_ 12 0 for all J, t At some t, J, this constraint may well be binding, In which case the /th technology 1s inactive at date t - 1 and 1s independent of The probability dlstrlbutlon of i?, IS known preceding events, 1e , IS Independent of &J,_, By contrast, the dlstrlbutlon of returfis of the N,-nt Investment opportumtles may well depend on precedmg events For example, the equlhbrmm value of the short-term Interest rate may depend on precedmg events We shall collectively refer to the assumptions which we have made so far

G M Constantmuies,

on the productlon proposltion

The mtertemporal

asset prrcrng model

79

sector as assumption (A 4) We now prove the followmg

Proposrtlon 1 ’ Assume perfect markets (A I), homogeneous expectations (A2), state independent utthty (A3), and productton technologies ana’ firms conforming to assumptron (A4) Also assume that the aggregation problem 1s solved, re, equllrbrrum prices are determined as rf consumers were identical m terms of resources, beliefs and preferences’ And the drstrlbutlon of l?, 1s a separatmg dlstrrbutzon Then the asset pricing equation (1) holds for the rates of return on the equity of the n,firms Eq (1) also holds for the rates of return of the remauung N, -n, investment opportunrtres if 8, 1s generated by a drflusron process Proof

A relative asset prlcmg relatlonshlp ~111 be determmed m the aggregated economy, wherein consumers have homogeneous resources,

‘Two varlatlons of the assumptions followmg proposltlon

of ProposItIon

1 are stated

These varlatlons

lead to the

Proposltlon 1’ Assume perfect markets (A l), homogeneous expectations (A 2). state Independent urlltty (A 3). and productton technologres and firms confornung to assumptton (A 4) Also assume that one technology IS rrskless and consumers are sugictently risk averse that the rIskless technology IS always active Fmally, makr one of the followmg two assumptions (I) Dectsrons are made contmuously tn tune and the vector i?, 1s generated by a dlffuslon process (II) Consumers have addrtrve quadratIc utdaty functzons Then the asset prtcmg equation (1) holds for the rates of return on all mvestment opportunrtles ProposItIon l’, unhke Proposition 1 m the main text, lmphes interest rates evolves determunsbcally over time The proof of Proposltlon 1’ IS slmdar to the proof of ProposItIon The proof 1s avadable from the author upon request

that

the term

structure

1 and IS not presented

of here

‘Rubmstem (1974) provides suficlent condltlons for aggregation m a two-date economy where the rth consumer maximizes -U:(W)/U:(W)=A,+B,W, v,(c,)+~,W(c~) and - Vi ( W)/V;‘(W) = A, + 8, W Rubmstem’s condltlons are listed below (I) All consumers have (II) All consumers have (III) All consumers have (IV) All consumers have (v) A complete market (VI) A complete market and B=l

the same resources, beliefs and tastes p, II, V the same beliefs, rates of patience p, and taste parameters B #O the same behefs and taste parameters B=O the same resources, behefs, and taste parameters A =O, B = 1 exists and all consumers have the same taste parameter B =0 exists and all consumers have the same resources and tastes p, A=0

Furthermore, these condltlons easily generahze to an economy with more than two dates Brennan and Kraus (1978) demonstrate the necessity of these condltlons for aggregation Some qualifying remarks are called for Rubmstem’s proof assumes that the condltlons on non-negative consumption, c,ZO and c, 20 are not binding In order to ensure this, we shall further assume that 8>0 and ASO These conditions imply that margmal utlhty of consumption becomes mtimte at some non-negative level of consumption, I e , U’(W)= (A+BW)-s-’ becomes mlimte at W=-A/B20 Condltlons B>O and AjO ehmmate Rubmstem’s cases (III) and (v) Furthermore m order to ensure that there exist feasible consumption paths for the lth consumer we shall assume that his endowment at date zero exceeds -iI-’ ZG, (RI)-‘A,, where R’ IS the one plus rate of return on the rlskless technology

80

GM

Constantmtdes,

The mtertemporal

asset prwng

model

beltefs and preferences The assumed aggregation property Implies that the same relative asstt prlcmg relatlonshlp holds m the original economy also, because the aggregation property implies that the original economy and the aggregated economy are supported by the same prices Consider the aggregated Bilateral contracts amongst economy homogeneous consumers are redundant Given an equlhbrlum where the homogeneous consumers engage m bilateral contracts, there exists another equdrbrmm supported by the same prices, wherein consumers have the same consumption schedules, yet they do not engage m bilateral contracts This implies that equlhbrmm prices are determined as d consumers invest only m the n, firms at date t - 1 But & 1s independent of b,_ 1 The lemma implies that the derived utility of each aggregated consumer 1s state independent This observation, along with the other assumptions made m the proposltlon, imply that the asset pricing eq (1) holds for the rates of return on the equity of the n, firms If i, 1s generated by a dlffuslon process, then the rates of return on the remaining N, -n, investment opportumtles satisfy eq (1) also To see this, note that Merton’s (1973) two-parameter asset pricing equation applies to the rates of return of contracts m zero net supply as well as to the rates of return of firms The proof 1s complete Proposltlon 1 1s illustrated m the next section through an example In the remaining part of this section three generahzatlons of Proposltlon 1 are considered In the first generahzatlon we relax the requirement that production technologies have constant returns to scale Specifically we assume (A 5) Production technologies and firms

The output of the jth firm at date

t ISf,(K:_ 1, E;, t) where K;_ 1 1s the input and E; 1s a vector of random shocks which 1s independent of 4,_ 1 Each firm IS competltlve and profit-

maximizing In the particular case where production occurs contmuously m time and the production uncertainty IS generated by a dlffuslon process, the output of the /th firm at date t + dt 1s

where K; 1s the input at date t, g’(K:, t) IS a function of the input and time, pJ(& t) IS a function of the state and time, a’($, t) 1s an M-dlmenslonal vector whose elements are functions of the state and time, and dw(t) 1s the increment of the Wiener process w(t) m RM

G M Constantmrdes, The mtertemporal asset pruxng model

81

We now state Proposition 2 lo Assume perfect markets (A 1), homogeneous expectations (A2), state mdependent utlllty (A3), production technologies and firms conformrng to assumption (A5), the aggregation problem IS solved Also production occurs contmuously m time and the productton uncertamty 1s generated by a d@iusron process Then the asset pricing eq (1) holds for the rates of return on all investment opportunities

Proof outline Unlike the proof of Proposltlon 1, we may not assert that the rate of return on the securltles on the jth firm equals the rate of return f,(K{_ 1, E;, t)/K;_ 1 - 1 on the investment of the $h firm Since, however, the increment dw(t) of the Wiener process 1s independent of preceding events, the state of the aggregated economy IS summarized by the level of aggregate wealth The composite consumer’s derived utlhty 1s a function of aggregate wealth alone Arguments similar to those used m the proof of Proposltlon 1 complete the proof of Proposltlon 2 The line of reasoning which we employed m the proof of Proposltlon 2 also serves to slmphfy the asset prlcmg equation derived by Brock (1978, eq (2 31)) Brock assumed that consumers are homogeneous and that production technologies and firms conform to assumption (A 5) We conclude that the state of the economy at every pomt m time 1s summarized by Just one state vanable, namely aggregate wealth Thus, Brock’s K-factor asset pricing equation may be collapsed to a smgle-factor asset prlcmg equation In Brock’s model there exist, not K ‘Ross risk prices’, but Just one such price which 1s identified as the price of market risk In the second generahzatlon of Proposltlon 1 we allow the tastes of consumers and the production technologies to be state dependent and obtain the (m+2)-factor model of Merton (1973) and Long (1974) (A similar generalization apphes to Proposition 2)

“Two vartatlons of the assumptions followmg proposltlon

of ProposItIon

2 are noted

These varlatlons

lead to the

Proposition 2’ Assume per&t markets (A l), homogeneous expectatrons (A 2), state rndependent utrht)- (A 3), productlon technologies and firms conformmg to assumptron (A 5) Also assume that one technology IS rrskless and has constant returns to scale, and consumers are suflclently risk averse that thts technology IS always active Fmallj, make one of the followmg two assumptrons, (I) or (II) (I) Productron occurs contrnuously m trme and the productton uncertarnt] IS generated b) a dlffisron process (II) Consumers have addtttve quadratic utlirty functions Then the asset pruxng equation (1) holds for the rates of return on all mvestment opportumtres The proof is slmdar to earher proofs and IS left to the reader

82

G M

Constantmldes,

The mtertemporal asset pr~mg

model

Proposztzon 3 Assume perfect markets (A 1). homogeneous expectatzons (A 2). state dependent utzlzty U(C, I&)= U(C, 1xT), where x, zs an m-oector whzch zs a subset of the rnformatzon set $, such that F(x,l ~,_I)=F(x,~x,_,), production technologzes and firms descrzbed by (A4), where the technology 1x,_ 1), and the aggregatzon problem depends on x,_ 1, I e, F(i, 1#t_ ,)=F(k, IS solved Also, make one of the followzng assumptzons (1) iIt is multzoarzate normal (zz) Productron occurs contznuously zn time and the productzon IS generated by a dzfliszon process (zzz) Consumers have addztzve quadratic utzlzty jiinctzons

uncertainty

Then, zn case (1) the (m+2)-factor asset przczng equatzon holds for the rates of return on the equzty of the n, firms In cases (zz) and (zzz) the (m+2)-factor asset przczng equation holds for the rates of return on all investment opportunities The proof IS similar to the proof of Proposltlon 1 and IS omltted The proposltlon is easily generahzed to the case where consumers aggregate to L homogeneous classes Then an asset pricing equation with m + L + 1 factors holds

4. An illustration of admissible uncertainty and some concluding remarks We consider an economy where consumers have homogeneous the zth (I = 1,2, ,I) consumer has additive HARA utlhty

behefs and

(7) where p, y, 2 are parameters such that p ~0, y < 1 and c 20 for all z, t In order to ensure that there exist feasible consumption paths for the rth consumer, we assume that his endowment Wb at date zero satisfies Wi, > (1 -y)CTZO (RI)-‘c^:, where R’ >O 1s the one plus rate of return on a rlskless technology which may or may not be active Under these condltlons equlhbrmm prices are determmed as if consumers were identical and the composite consumer had resources N$ = C:=, Wb/I, utlhty (7) with parameter c?,= Zf= 1
G M Constantwades,

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83

The aggregation property ensures that the set of pnces, which supports the aggregated economy, also supports the original economy Therefore, the interest rate and the rate of return on the market portfolio m the aggregated economy equal the interest rate and the rate of return on the market portfolio m the original economy Without loss of generality then we shall study the behavior of these rates m the aggregated economy In the aggregated economy, the composrte consumer invests m technologies with one plus rates of return 8,, subject to the constraint that production inputs are nonnegative In general, technologies may be active or inactive Without loss of generality, we assume that the composite consumer does not invest m any assets m zero net supply, 1 e, does not engage m bilateral contracts, because such contracts are redundant m an economy with homogeneous consumers The efflclent frontier 1s Illustrated m fig 1, under the assumptton that i?, 1s multlvarlate normal and the rlskless technology IS inactive E(R,,)

Utihty functm of cCNnposlteconsumer Effwent framer of composite consumer

Fig 1 Illustration of the eficlent frontier and the derived utdlty function of the composite consumer at date t - 1 The tangency pomt M designates the market portfoho held at date t - 1 By constructlon, the eqmhbrmm short-term Interest rate r, IS such that the composite consumer does not borrow or lend R,! IS the rate of return on a rtskless technology which m general may or may not be active In this particular dlustratlon, the rlskless technology IS mactlve R, IS the rate of return on the portfoho of assets held by the composite consumer

Fig 1 also illustrates a utility indifference curve of the composite consumer for pairs of expected portfoho return, E(R,), and standard devlatlon of portfolio return, std(R,), which IS tangent to the efficient frontier The composite consumer chooses portfoho M which IS the pomt of tangency of his utility mdlfference curve and the efficient frontier Portfoho M 1s the value-welghted portfoho of the securltles of the n, firms This portfoho IS ldentlfied as the market p6rtfoho Note that the remammg N, -n, investment opportumtleq (bilateral contracts, optlons, Insurance contracts) are m zero net supply dnd therefore are not Included In the market portfoho The tangent at M to the utrhty mdlffelence curve and the effictent frontier meets

84

G M

Constantmldes,

The mtertemporal

asset prwng

model

the vertical axis at intercept r, > R,?, r, IS the shadow mterest rate If we allow the posslblhty of borrowmg/lendmg amongst the homogeneous consumers, r, IS the equlhbrmm interest rate at which the borrowmg/lendmg market clears In general, the slope of the composite consumer’s utlhty mdlfference curves is a function of the composite consumer’s wealth, @_ 1 =Ecf, 1 IV;_ JZ Only m the special case of power utility, I e , t, = Cf= 1 e/Z =O, are the composite consumer’s indifference curves independent of his wealth In general, however, & 20 and the point of tangency M depends on R_ l The composltlon of the market portfolio, the dlstrlbutlon of the rate of return on the market portfolio and the interest rate are all functions of l8$_l At dates prior to t - 1, the composite consumer’s wealth R_ 1 1s a stochastic variable because it depends on the reahzatlons of the production processes Therefore, the dlstrlbutlon of the market portfolio and the interest rate at date t- 1, are state dependent, m particular, they are functions of the state variable R_ 1 =Xf=, W;_,/Z Despite this state dependence, we have shown that the SLM asset pricing equation 1s satisfied The economy described m this section illustrates the first lmphcatlon of our theory, that the central role which the market portfolio occupies m the modern theory of portfolio selection and relative asset pricing may not be fully warranted In this economy, the covarlance matrix of the rates of return of the production processes 1s stationary Therefore, the covarlance matrix of the rates of return of the subset of firms, which are active m production, 1s also stationary Yet the excess return on the stock of any particular firm will be non-stationary because the interest rate will, m general, be non-stationary Furthermore, the covarlance of the return and the beta coefficient of the stock of any particular firm with respect to the market portfolio will be nonstationary because the dlstrtbutlon of the market portfolio return IS nonstationary I1 In contrast to the above, the arbitrage theory of asset pricing developed m Ross (1976) and tentatively tested m Roll and Ross (1979) IS free from the above crltlclsm Ross’s theory postulates a k-factor, linear stationary market model m which the market portfolio plays no special role Referring again to the economy described m this section, our theory implies that the market portfolio return may be an inappropriate factor in the market model, yet the stationary rates of return of the production processes may be adequately described by a k-factor, stationary market model, as suggested by Ross The economy described m this section also illustrates the second imphcation of our theory The market price of risk and the covarlance of the cash flows of a project with the market return will, m general, depend on “Llkewlse m Breeden’s (1979) consumption CAPM the expected return on a portfolio which IS perfectly correlated with changes m aggregate consumption, and the beta u&bent of the stock of any particular firm with respect to changes m aggregate consumptron ~111, m general, be non-stationary

G M Constantmldes, The mtertemporal asset pruzrng model

85

preceding events Therefore, the sequential apphcatlon of the SLM model m the dlscountmg of stochastic cash flows becomes computatlonally co,mplex and of little practical use

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G M Constantnudes. The rntertemporal asset prwng model

Roll. R, 1977, A cntlque of the asset prlcmg theory’s tests, Part I, Journal of Fmanaal E!conomw 4,129-176 Ross, S A, 1976, The arbitrage theory of capital asset pncmg, Journal of Economic Theory 13, December, 341-360 Ross, SA. 19784 Mutual fund separation m tinanclal theory - The separatmg dIstnbutlons, Journal of Economic Theory 17,25&286 Ross, SA, 1978b, The current status of the capital asset prlcmg model (CAPM), Journal of Fmance 33, June, 885901 Rubmstem, M , 1974, An aggregatron theorem for securltws markets, Journal of Fmanclal Economws 1,225-244 Rubmstem, M , 1976a, The valuation of uncertam Income streams and the prlcmg of optlons, Bell Journal of Economtcs 7, Autumn, 407-425 Rubmstem, M, 1976b. The strong case for the generahxed logarlthmrc utdlty model as the prenuer model of financml markets, Journal of Fmance 31, May, 551-571 An expanded version of the paper appears m 1977, Fmanctal decwon makmg under unccrtamty (Academic Press, New York) Sharpe, W F, 1964, Capital asset prices A theory of market eqmhbrwm under condltlons of risk, Journal of Fmance 19, Sept., 425-442 Stapleton, R C and M G Subrahmanyam, 1978, A multlpermd equlhbrmm asset pncmg model, Econometnca 46, Sept. 1077-1096