1 Introduction The two most important static models of security markets the Capital Asset Pricing Model (CAPM), and the Arbitrage Pricing Theory (APT)

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1 Factor Pricing in Multidate Security Markets 1 Jan Werner Department of Economics, University of Minnesota December I have greatly benefited from numerous conversation with Steve LeRoy on the subject of this paper. The financial assistance of Deutsche Forschungsgemeinschaft, SBF 303, is gratefully acknowledged.

2 1 Introduction The two most important static models of security markets the Capital Asset Pricing Model (CAPM), and the Arbitrage Pricing Theory (APT) have a common feature that expected returns of a possibly large number of securities satisfy a linear relationship with measures of sensitivity of returns to few common factors. In the CAPM, there is a single factor the market return and the pricing relation is exact. In the APT, there are multiple factors and the pricing relation is approximate. The nature of static (two-date) security markets models is such that trade occurs only once, followed by terminal realization of agents' wealth and simultaneous maturity of all securities. Of course, when static models are used in empirical tests, time-series data on one-period security returns is used and various stationarity properties are assumed. In this paper we develop a general theory of factor pricing in intertemporal (multidate) security markets. The distinguishing feature of our approach is that measures of sensitivity of dividends to factors over life-time of securities are used, rather than sensitivity of one-period returns. We provide conditions under which security prices at each date depend in a linear fashion on sensitivity of dividends to factors. Our factor pricing relation says that the price of a security equals the sum of discounted expected future dividends plus a term that is proportional to sensitivity of dividends to factors. In other words, the net present value of a security depends linearly of sensitivity of dividends to factors. Formally, wehave p jt = fi=t+1 E t (x jfi ) R tfi + b jnt fl nt ; (1) where p jt denotes the (event-dependent) price of security j at date t, x jfi - the dividend at date fi, T - the maturity date, R tfi - the (gross) return on maturity-fi bond at t, b jnt - the measure of sensitivity of dividends of security j to the nth factor at t, and fl nt the coefficient of proportionality. We show that coefficient fl nt is a measure of date-t risk premium of the nth factor. In general, it depends on 1

3 date t and an event at t. The same is true for factor loading b jnt. It is important to emphasize that factor pricing (1) relates security prices at date t to expected future dividends and not to future prices which, in any equilibrium model, would be derived endogenously. For that reason, the pricing relation is stated in terms of prices and not one-period expected returns as it is common in dynamic asset pricing literature. For a security that pays dividends only at maturity, relation (1) can be restated using expected return-to-maturity (see section 3). Our approach to intertemporal factor pricing relies on methods developed for static factor pricing in LeRoy and Werner [7]. We introduce a multidate pricing kernel and show that a sufficient condition for exact multidate factor pricing is that the pricing kernel lies in the span of factors and risk-free payoffs of all maturities (factor span). Since, as we show, the pricing kernel lies in the multidate meanvariance frontier, it follows that it lies in the factor span if and only if the factor the mean-variance frontier is contained in the factor span. The multidate meanvariance frontier consists of payoffs with minimal discounted sum of variances for given date-0 price and given expected payoffs at each date. It is a multidate analogue of the mean-variance frontier plane in two-date security markets. The primary example of exact multidate factor pricing is the multidate CAPM derived in a companion paper by LeRoy and Werner [8]. In the multidate CAPM, the market payoff (defined as the multidate payoff of a market portfolio) lies on the mean-variance frontier. Exact factor pricing holds with the market payoff as a single factor. Factor loading b jt in the CAPM is a cumulative measure of (date-t conditional) covariance of dividends with the market payoff over the remaining life-time of the security, and hence is a counterpart of static beta. Coefficient fl t is the time-dependent market risk premium. Another example of exact factor pricing is a representative-agent equilibrium model of Connor and Korajczyk [3] which has been put forward in an attempt to extend the static Equilibrium APT of Connor [2] to intertemporal setting. 2 In 2 Another intertemporal extension of the APT has been proposed in Ohlson and Garman [9]. 2

4 that model dividends exhibit a factor structure with residual risks (i.e., residuals of a projection of dividends on factors) that are conditionally mean-independent of factors. There is a large number of securities which enable agents to diversify away residual risk. Further, the market portfolio is well-diversified, that is, it has no residual risk. We show, under the assumption of a representative agent, that exact factor pricing holds in a security markets equilibrium in the Connor and Korajczyk [3] model. A third example is the multidate security markets model with options on aggregate consumption of Breeden and Litzenberger [1]. Kim [6] proved that if options on aggregate consumption are taken as factors in two-date security markets, then exact factor pricing holds in equilibrium with risk-averse agents. Factor pricing with options as factors permits determination of security prices from the prices of options (see also LeRoy and Werner [7]). Here we extend this result to multidate security markets. As in Breeden and Litzenberger [1], we consider European call options on the aggregate endowment with all maturities. We show that in a security markets equilibrium, exact factor pricing holds with options as factors. The paper is organized as follows: In section 2 we present a multidate model of security markets and introduce the pricing kernel. In section 3 we define factor pricing and provide sufficient conditions for exact factor pricing at date 0. These results are extended to dates other than the initial date 0 in section 4. Section 5 summerizes some results from the multidate mean-variance analysis. The three models that lead to exact factor pricing - the CAPM, the Connor and Korajczyk [3] model, and the Breeden and Litzenberger [1] options model - are discussed in sections 6 and 7. 2 Multidate Pricing Kernel We consider multidate security markets extending over T + 1 dates t =0;1;:::;T. Uncertainty is described by a finite set of states of nature S and a sequence of For a continuous-time extension of the APT, see Reisman [10]. 3

5 partitions of S. Date-t partition F t represents the information about the state at date t. Date-0 partition is the trivial partition F 0 = fsg; date-t partition is total partition F T = ffsg : s 2 Sg. Partition F t+1 is finer than partition F t. An element of partition F t is called a date-t event. As usual, this information filtration can be interpreted as an event tree. We use Z to denote the space all multidate contingent claims, i.e., the space of all T -tuples z = fz t g T, where z t is a real-valued F t -measurable function on S. There are given a probability measure P on the set of states S and a sequence of deterministic discount factors fρ t g T. We assume that P (fsg) > 0 for every s 2 S, and ρ t > 0 for every t 1. There are J securities traded at all dates except the terminal date T. The dividend on security j is a multidate contingent claim x j = fx jt g T 2 Z. There are no dividends at date 0. The price of security j at date t is denoted by p jt and the holding of security j by h jt. Both are F t -measurable random variables. In order to simplify notation we have date T prices p jt which are set equal to zero. We will frequently assume that risk-free zero-coupon bonds with maturities t = 1;:::;T are traded. The dividend of maturity-t bond is denoted by e t deterministic unit payoff at date t. The return on maturity-t bond is denoted by R t (assumed strictly positive) so that 1=R t is date-0 price of the bond. Of course, the sequence fr t g T describes term structure of risk-free returns. The payoff of a portfolio strategy h = fh t g T t=0 at date t is the cum-dividend payoff of the portfolio chosen at date t 1 minus the price of the portfolio chosen at t. The set of payoffs available via trades on security markets is the asset span and is defined by M = fz 2Z:z t =(p t +x t )h t 1 p t h t ; 8;:::;T; for some hg: (2) For each payoff z 2Mwedefine date-0 price of z, denoted by q(z), by setting q(z) =p 0 h 0 ; (3) where h is a portfolio strategy that generates z. We assume that the law of one price holds, that is, that all portfolio strategies that generate a payoff have the 4

6 same date-0 price. So defined q is a linear functional on the asset span M, and is called the payoff pricing functional. The asset span M is a Hilbert space under the discounted-expectations inner product X <z;y> T ρ t E(z t y t ) (4) for z; y 2M. The Riesz Representation Theorem implies that q can be represented by a kernel» 2Mso that for every z 2 M, or more explicitly q(z) =<z;»> (5) q(z) = for every z 2 M. Kernel» is the multidate pricing kernel. Eq. (6) implies that For maturity-t bond, (7) implies p j0 = R t = 3 Date-0 Factor Pricing ρ t E(» t z t ) (6) ρ t E(» t x jt ): (7) 1 ρ t E(» t ) : (8) Consider N multidate contingent claims f n 2Z,;:::;N, called factors. The factors may or may not lie in the asset span. The linear span of the factors and the risk-free payoffs is the factor span and is denoted by F spanfe 1 ;:::;e T ;f 1 ;:::;f N g. It is assumed that the N factors and the risk-free claims are linearly independent. Let ^f n be the nth factor rescaled to have zero expectation at every date, that is ^f nt = f nt E(f nt ). Clearly, rescaled factors and risk-free payoffs form a basis of the factor span. Projecting dividend stream x j on F (using the discountedexpectations inner product) we obtain the following decomposition: x j = E(x jt )e t + 5 b jn ^fn + ffi j (9)

7 for some ffi j such that E(ffi jt ) = 0 for every t and P t ρ t E( ^f nt ffi jt ) = 0 for every n. Decomposition (9) can also be written as for t =1;:::;T. x jt = E(x jt )e t + b jn ^fnt + ffi jt (10) Coefficient b jn in (9) is the factor loading of security j. If factors f n are uncorrelated with each other at every date, then one can show (by taking the inner product with ^f n on both sides of (9)) that P T b jn = ρ tcov(x jt ;f nt ) P T ρ : (11) tvar(f nt ) Thus, b jn measures sensitivity of dividend stream x j to factor f n. Exact factor pricing holds at date 0 if security prices satisfy p j0 = E(x jt )ff t + b jn fl n ; (12) for some scalars fl 1 ;:::;fl N and ff 1 ;:::;ff T, for every j. Eq. (12) is a linear relation between date-0 security prices and factor loadings. We have 3.1 Theorem If the pricing kernel» lies in the factor span F, then date-0 exact factor pricing (12) holds with ff t = ρ t E(» t ) and fl n = Further, if risk-free bonds are traded, then ρ t E(» t ^fnt ): (13) ff t = 1 R t : (14) Proof: Taking the inner product with» of both sides of (9), we obtain p j0 = ρ t E(x jt )E(» t )+ b jn [ 6 ρ t E(» t ^fnt )] + ρ t E(» t ffi jt ): (15)

8 Since» lies in the factor span F, it is orthogonal to ffi j and therefore P T ρ te(» t ffi jt )= 0. Hence exact factor pricing (12) holds with coefficients given by (13). If maturityt bond is traded, then (8) implies 14). 2 Theorem 3.1 is an extension of a similar result for two-date security markets in LeRoy and Werner [7]. As an inspection of the proof reveals, for the result to hold true it is sufficient that each residual ffi j according to the discounted-expectations inner product. is orthogonal to the pricing kernel This is a significantly weaker condition that the assumed» 2F. If the residuals lie in the asset span this happens if all factors lie in the asset span then orthogonality to the pricing kernel means that the residuals have zero price. We emphasize that Theorem 3.1 does not require that factors or residuals lie in the asset span. With coefficients ff t given by (14) date-0 exact factor pricing becomes p j0 = E(x jt ) R t + b jn fl n : (16) On the right-hand side of (16) we have the sum of discounted expected dividends of security j and a term proportional to factor loadings. Thus factor pricing relation (16) says that the net present value depends linearly on factor loadings. More can be said about coefficient fl n given by (13). If risk-free bonds are traded, then fl n = ρ t E(» t f nt ) T X ρ t E(» t )E(f nt )= If factor f n lies in the asset span, then we have fl n = q(f n ) T X ρ t E(» t f nt ) T X E(f nt ) R t : (17) E(f nt ) R t ; (18) so that fl n is the (negative of the) net present value of factor n, or the risk premium on factor n. If security pays dividends only at a single date, then factor pricing (16) can expressed in terms of return-to-maturity. Suppose that security j has zero dividends 7

9 at all dates except for the terminal date T. Then p j0 = E(x jt) R T + b jn fl n : (19) Dividing both sides by p j0 and multiplying by R T, and using the return-to-maturity R jt = x jt p j0 we have E(R jt ) R T = R T fi jn fl n ; (20) where fi jn = b jn p j0. In (20) the expected return-to-maturity in excess of the risk-free return is proportional to factor loadings. 4 Factor Pricing Dynamics Our focus has been thus far on date-0 security prices. Security prices at other dates obey similar relations. It is important to present these relations explicitly so as to see the stochastic nature of factor pricing. For the use later we note that date-t price of security j satisfies p jt = 1 ρ t» t fi=t+1 ρ fi E t (» fi x jfi ) (21) In equation (21) both sides are F t -measurable random variables with E t denoting conditional expectation with respect to F t. Eq. (21) obtains by applying (6) to the payoff of portfolio strategy of buying security j in an event at date t and holding it in every successor event thereafter. For each date t and each date-t event, let M t be the set of payoffs of portfolio strategies in all successor events at dates t +1through T and let x t j = fx t fig T fi =t+1 denote the dividend stream of security j in all successor events from t + 1 through T. 3 In similar fashion we have f t n,» t, and e t fi for fi > t. Finally, we have date-t conditional factor span F t as the linear span of risk-free payoffs e t t+1 ;:::;et T conditional factors f t 1 ;:::;ft N. 3 Our notation does not explicitly reflect the dependence on a date-t event. Such variables as M t, x t j and others are to be considered as random variables or their realizations, as the context requires. 8 and

10 Let ^f t nfi = f t nfi E t (f nfi )bethenth factor rescaled to have zero date-t conditional expectation for every fi > t. The following decomposition results from projecting x t j on the factor span F t : 4 with ffi t j x t j = fi =t+1 E t (x jfi )e t fi + b jnt ^ft n + ffi t j ; (22) such that E t (ffi t jfi) = 0 for every fi > t and P T fi =t+1 ρ fi E t ( ^f t nfi ffit jfi) = 0 for every n. Eq. (22) can be written as for each fi > t. Coefficient b jnt x jfi = E t (x jfi )e fi + and hence is the date-t conditional factor loading. b jnt ^ft nfi + ffi t jfi ; (23) depends not only on t but also on date-t event, If factors are conditionally uncorrelated with each other at every date (i.e., cov t (f nfi ;f n0 fi) = 0 for n 6= n 0 and fi >t), then P T fi=t+1 b jnt = ρ fi cov t (x jfi ;f nfi ) P T fi ρ : (24) =t+1 fi var t (f nfi ) Thus, b jnt results from updating date-0 factor loading b jn according to information available at date t. Exact factor pricing holds at date t if p jt = for some scalars ff tfi and fl nt. fi=t+1 E t (x jfi )ff tfi + b jnt fl nt (25) If the pricing kernel» lies in the factor span F, then the conditional kernel» t lies in the factor span F t for every t. Using (21) we obtain exact factor pricing (25) with ff tfi = ρ fi E t (» fi ) ρ t» t and fl nt = Further, if risk-free bonds are traded, then fi =t+1 ρ fi E t (» fi ^f t nfi) ρ t» t : (26) ff tfi = 1 R tfi ; (27) 4 Here we use the conditional version of the inner product (4); that is, P T fi =t+1 ρ fi E t (z fi y fi ) for z; y 2M t. 9

11 where R tfi denotes the return on maturity-fi bond at date t, for fi >t. Thus if» lies in the factor span F, then exact factor pricing holds not only at date 0 (Theorem 3.1), but also at every date t. With coefficients ff tfi given by (27) date-t exact factor pricing becomes p jt = fi=t+1 E t (x jfi ) R tfi + b jnt fl nt : (28) The price of security j equals the sum of discounted conditional expected dividends of security j plus a term proportional to conditional factor loadings. As in (18), coefficient fl nt can be interpreted as date-t risk premium on factor n. 5 Multidate Mean-Variance Frontier We assume in this section that risk-free bonds with all maturities are traded. A multidate payoff z 2Mis a mean-variance frontier payoff if there does not exist any other payoff z 0 2 M with the same date-0 price, the same expectation at each date, and lower discounted sum of variances, that is, z 0 E(z 0 t)=e(z t ) for each t 1, and P T ρ tvar(z 0 t) < P T ρ tvar(z t ). with q(z 0 ) = q(z), Let E be a(t+ 1)-dimensional hyperplane in Z spanned by the pricing kernel and the risk-free payoffs e 1 ;:::;e T, that is, 5.1 Theorem: E = spanf»; e 1 ;:::e T g: (29) E is the set of mean-variance frontier payoffs. Proof: Taking the orthogonal projection (with respect to the discounted-expectations inner product) of an arbitrary payoff z 2Monto E results in z = z E + ffl; (30) with z E 2E and ffl 2E?. The latter implies that E(ffl t )=0for every t, that ffl has zero price at date 0, and that P t ρ t cov(ffl t ;z E t)= P tρ t E(ffl t z E t)=0. Consequently, 10

12 Pt ρ t var(z t ) = P t ρ t var(z E t )+ P tρ t var(ffl t ) and thus P t ρ t var(z E t )» P t ρ t var(z t ), with strict inequality if ffl 6= 0. payoff lies in E. This implies that every mean-variance frontier The converse inclusion follows from the fact that any twopayoffs in E with the same date-0 price and the same expectations at each date must be equal. 2 The hyperplane E is the analogue of the mean-variance frontier plane in the two-date case. Theorem 5.1 implies that the pricing kernel lies in the factor span if and only if the mean-variance frontier hyperplane is contained in the factor span. 6 Multidate CAPM Let μ h t denote the outstanding portfolio of securities at date t. Portfolio μ h t may vary with date-t events. We refer to portfolio strategy μ h = f μ h t g T t=0 as the market portfolio strategy. Let m =(m 1 ;:::;m T )2Mbe the payoff of μ h, that is, m t =(p t +x t ) μ h t 1 p t μ ht : (31) We call m the market payoff. In the multidate CAPM with risk-free bonds (see LeRoy and Werner [8]), the market payoff lies on the mean-variance frontier. It follows from Theorem 3.1 that exact factor pricing holds with the market payoff as the single factor and we have p j0 = E(x jt ) R t + b j0 fl 0 ; (32) with b j05 given by P T b j0 = ρ tcov(x jt ;m t ) P T ρ : (33) tvar(m t ) In (32) the net present value of security j is proportional to the market-payoff loading which in turn is equal to the discounted sum of covariances of dividends 5 Note that we have relabeled coefficients of date-0 factor pricing by adding date-0 subscript. This is more consistent with the notation of date-t factor pricing. 11

13 with the market payoff divided by the discounted sum of variances of the market payoff. Coefficient fl 0 in (32) can be written as fl 0 = q(m) T X E(m t ) R t ; (34) where q(m) is date-0 price of the market payoff. Thus, coefficient fl 0 is (negative of) the net present value of the market payoff and is the market risk premium. Date-t security prices in multidate CAPM satisfy p jt = fi=t+1 E t (x jfi ) R tfi + b jt fl t ; (35) with b jt given by P T fi=t+1 b jt = ρ ficov t (x jfi ;m fi ) P T ρ ; (36) fi=t+1 fivar t (m fi ) and with time-dependent market risk premium fl t. 6 Multidate CAPM obtains in equilibrium in security markets (see for instance LeRoy and Werner [7], Chapter 21) if agents have multidate mean-variance preferences, that is, preferences over multidate consumption plans (c 0 ;:::;c T ) that can be represented by a utility function of the form v 0 (c 0 )+v(e(c 1 );:::E(c T ); ρ t var(c t )); (37) where function v is strictly decreasing in the discounted sum of variances. Under (37), agent's preferences over future consumption plans depend only on expected values of consumption and the discounted sum of variances. An example of multidate mean-variance preferences is discounted expected utility with quadratic period utility function: v 0 (c 0 )+ ρ t E[ (ff c t ) 2 ] (38) 6 There is a strong empirical evidence of time-dependent risk premia, e.g., Fama and French [4], [5]. 12

14 7 Mean-Independent Factor Structure Dividends have mean-independent factor structure if the residuals ffi t j in decomposition x jfi = E t (x jfi )e fi + b jnt ^ft nfi + ffi t jfi ; for fi >t; (39) see (23), satisfy the following mean independence condition: E t (ffi t jfijf 1fi ;:::;f Nfi )=0; (40) for every security j and all dates t and fi such that t<fi. 7 We consider an equilibrium model of multidate security markets with I agents. Agents' preferences over multidate consumption plans are represented by discounted expected utility functions of the form t=0 ρ t E[v i (c t )] (41) with differentiable, strictly increasing von Neumann-Morgenstern utility functions v i : R +! R for i = 1;:::;I. Utility functions are assumed strictly concave so that agents are strictly risk averse. Agents have initial portfolios of securities, P but no other endowments. Let h μi 0 be agent i's initial portfolio, and let h μ = μ i h i 0 be the market portfolio. The market payoff is (see (31)) m t = x th μ and does not depend on security prices. Further, we assume that risk-free bonds with all maturities are traded. We show that exact factor pricing holds in an equilibrium with security dividends exhibiting the factor structure (40) under two different sets of additional assumptions. The first set of assumptions corresponds to a finite-time, finitenumber-of-securities version of the model of Connor and Korajczyk [3]. These assumptions are (A1) There is a representative agent (i.e., I = 1). (A2) E t [(ffi t μ fi h) 2 ]=0for every t and fi with t<fi. 7 We recall that ^f t nfi = f nfi E t (f nfi ). 13

15 (A3) For each n, there exists a (time- and event-independent) portfolio h n such that f nt = x t h n for all t. Assumption A2 says that the market portfolio has no residual risk. Assumption A3 requires that each factor f n can be replicated by a buy-and-hold portfolio strategy. It assures that f n lies in the asset span regardless of security prices. In a security market equilibrium with a representative agent, the agent's consumption plan is equal to the market payoff and equilibrium prices of securities must satisfy the standard first-order conditions of the agent's optimal choice. If each factor lies in the asset span (A3) and all risk-free payoffs lie in the asset span, then it can be seen from (9) that each residual ffi j lies in the asset span. Therefore we can meaningfully speak of date-0 equilibrium price q(ffi j ) of residual ffi j. We show that q(ffi j ) = 0 which in light of our discussion in section 3 is sufficient for date-0 exact factor pricing. for c t That q(ffi j ) = 0 follows from the following first-order condition: q(ffi j )= 1 v 0 (c 0 ) ρ t E[v 0 (c t )ffi jt ]; (42) = m t. This condition reflects optimality of the consumption plan c when date-0 investment in a portfolio replicating ffi j is considered. By assumption A2 the market payoff, and hence the consumption plan c, lie in the span of factors. The assumption of factor structure (40) implies that E[v 0 (c t )ffi jt ] = 0 for every t 1 and hence that q(ffi j ) = 0. Date-t exact factor pricing can be demonstrated in an analogous way. Thus exact factor pricing holds under assumptions A1, A2, A3, and the factor structure of dividends. The second set of assumptions under which exact factor pricing holds in an equilibrium corresponds to the model of Breeden and Litzenberger [1] of security markets with options on the market payoff. The assumption is (B1) European call options on the market payoff with all maturities t = 1;:::;T and arbitrary exercise prices are traded. Assumption B1 may seem to imply an infinite number of traded options. This is, however, not the case since market payoff takes only finitely many values and 14

16 so there are only finitely many nonredundant options. We take payoffs of options as factors. One can show using the same arguments as in two-date security markets with options, see LeRoy and Werner [7], that dividends of securities have a mean-independent factor structure. We show that exact factor pricing holds under B1 using some of the arguments developed for the set of assumptions A1 - A3. Since the market payoff lies in the linear span of options, assumption A2 is satisfied. Assumption A3 follows directly from B1. However, A1 does not apply since we have not assumed a representative agent. Nevertheless, the argument of (42) for proving zero price of residual risks can be applied provided that we show that equilibrium consumption plan of some agent lies in the span of factors (options). Since the aggregate consumption equals the market payoff, it follows from Theorem 1 in Breeden and Litzenberger [1] that every Pareto optimal consumption allocation lies in the span of options. Since options are traded, Pareto optimal allocations are attainable through security markets. Therefore, equilibrium allocation in security markets must be Pareto optimal, and hence lie in the span of options/factors. The argument of (42) implies that date-0 prices of residual risks are zero. Consequently, date-0 factor pricing holds in an equilibrium in security markets satisfying assumption B1. 15

17 References [1] Douglas T. Breeden and Robert H. Litzenberger. Prices of State-Contingent Claims Implicit in Option Prices. Journal of Business, 51(4): , [2] Gregory Connor. A Unified Beta Pricing Theory. Journal of Economic Theory, 34:13 31, [3] Gregory Connor and Robert A. Korajczyk. An Intertemporal Equilibrium Beta Pricing Model. Review of Financial Studies, , 1989 [4] Eugene F. Fama and Kenneth R. French. Common Risk Factors in the Returns on Stocks and Bonds. Journal of Financial Economics, 33:3 56, [5] Eugene F. Fama and Kenneth R. French. Multifactor Explanations of Asset Pricing Anomalies Journal of Finance, 51:55 84, [6] Chongmin Kim. Stochastic Dominance, Pareto Optimality and Equilibrium Asset Pricing. Review of Economic Studies, 65: , [7] Stephen F. LeRoy and Jan Werner. Principles of Financial Economics. Cambridge University Press, [8] Stephen F. LeRoy and Jan Werner. Multidate Beta Pricing and Capital Budgeting. mimeographed, 2001 [9] James Ohlson and Mark Garman. Dynamic Equilibrium for the Ross Arbitrage Model. Journal of Finance, 35: , [10] Haim Reisman. Intertemporal Arbitrage Pricing Theory. Review of Financial Studies, 1992 [11] Stephen A. Ross. The Arbitrage Theory of Capital Asset Pricing. Journal of Economic Theory, 13: ,