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-.-^^' ifEB SISBB ALFRED P. WORKING PAPER SLOAN SCHOOL OF MANAGEMENT A Note on the Valuation of Stochastic Cash Flows Rajnish Mehra Sloan School of Management Massachusetts Institute of Technology and University of California Santa Barbara WP 7/1913-87 Revised July 1986 MASSACHUSETTS INSTITUTE OF TECHNOLOGY 50 MEMORIAL DRIVE CAMBRIDGE, MASSACHUSETTS 02139 A Note on the Valuation of Stochastic Cash Flows Rajnish Mehra Sloan School of Management Massachusetts Institute of Technology and University of California Santa Barbara WP-:-1913-87 Revised July 1986 I would like to thank John Donaldson for his helpful comments. Financial support from the National Science Foundation and the Faculty Research Fund of the Graduate School of Business, Columbia University is gratefully acknowledged. FEB 8 1988 Abstract This paper considers the implications of the equilibrium asset pricing model developed by Lucas (1978) for the valuation of risky cash flows. It is shown that even in the context of the simple economy studied, discounting stochastic cash flows becomes computationally complex and of limited practical use. . I. Introduction The paper by Lucas : Economy" initiated a "Asset Prices in an Exchange (1978) paradigmatic change in the Theory of Equilibriam Pricing of Risky Assets. It was followed by several papers including those by Brock (1979 and 1982), Prescott and Mehra and Mehra (1984) who generalize Lucas' model to a (1980) and Donaldson production setting. Research in the area of multiperiod valuation of risk assets has predominantly focussed on the conditions necessary to generalize the single period partial equilibrium model of Sharpe (1964), Lintner (1965) and Mossen (1966) to above, a multiperiod context. addition to the papers cited In related papers include the work of Bhattacharya (1981), Breeden (1979), Cox. (1985"), Ingersol and Ross (1985), Fama (1977), Mehra and Prescott Merton (1973), Rubinstein (1976) and Stapelton and Subrahmanyam (1978). This paper considers the logical implications of the equilibrium asset pricing model of Lucas flows. (1978) for the validation of risky cash Since this paper is extensively quoted m the Finance and Economics Literature, it seems appropriate that its implications be svstematically analysed to further its understanding. The paper is pedagogical in nature since much of the analysis such as tne existence, uniqueness and optimality of equilibrium has been developed earlier. The paper is divided into four sections. scribes the economy under consideration. developed in section comments 3 while in section Section 2 briefly de- The valuation formulae are 4 we provide some concluding The Economy II. We consider : pure exchange economy of the a As shown in Prescott and Mehra considered in Lucas (1978). example 2, such an economy can be cast as librium. a ty7)e (1980), recursive competitive equi- We have deliberately abstracted from the more general struc- tures considered by Brock fl979 and 1982) and Prescott and Mehra (1980) which allow for capital accumulation since they are not necessary to establish the results in this paper. The reader is referred to A brief sketch of the economy follows. Lucas (1978) for details.^ The economy has single representative ''stand in" household. a This agent orders its preferences over random consumption paths by X I 2 ^ ulc E; (1) )] t=0 where IS c is a stochastic process representing per capita consumption, the subjective discount factor, operator and u: R - R is Production be the output vector at P < 1, E|-} is the expectation the current period utility function. consumption good is produced by in period t. < is time firnis. n entirely "exogenous." t. X = Let X assumed to follow is The be the output of Let x. p a {x. .x, firm . i ..x Markov process defined by its transition function FIX'IXJ = priX^^^ 1 X'lX^ = Xj. Each period ownership in these productive units is determined in petitive stock market. equity share. a com- Each unit has outstanding one perfectly divisible A share entitles its owner as of the beginning of t to all of the unit's output period in t. Shares are traded ex-dividend at competitively determined price vector = p t = Let Z share holdings. consumed (c = ...z ,z^ (z, In this Ix denote economy, and all ) . ) m a a .)• (Pw,Po^---P t nt 1 1 i_ consumer's beginning of period equilibrium, all output will be shares held, that is = Z If we (1,1. ..1). 1 let the price of the consumption good be the numeraire, of the firms can be thought of as their "cash flows". then the output We will use the latter interpretation in the analysis that follows. III. (2) Valuation: The price dynamics of firm u'(c^)p^(X^) -- P J- u'U,,i)[x^_^^^ i are characterized by . P:'Vl^J m This IS the celebrated stochastic Euler equation sufficient to value risky cash flows from firm i. ''^ 'Vl'^t^ U97Sj and Lucas is The same equation is obtained in models incorporating capital accumulation developed by Brock (1979 and 1982), Donaldson and Mehra and others. (19S4j, Prescott and Mehra (1980) These models incorporate other equations that must be consistent with (2j; nevertheless, equation (2 > must necessarily be satisfied in each case. In the analysis Since there is follow we will value the cash flow froni firm i. little room for ambiguity, we will suppress the subscript so that at any time (3) to t equation (2) may be written as u'(c^) p(X^) - p J u'(c^^^)lx^^^ The object is to solve for p(^X future cash flows {x,X-, ...) ) which + P'\.i^l i_s of this firm . ^^ '\+l'-\) the present value of all i To this end leL us recursively define the sequence of conditional distribution functions {F (X^^ t+n ^2^\+2'^t^ |X according^ to )) t F(X^^JZJ dF (ZIX^) = / F3(X^^3|X^) = / F2(X^^3|ZJ dF (Z|X^) with F„(X^,JX^) = / Vl'^^.s'^^ '' ^2l-\'^ and hence ''Jh^n^h^'- {'\-l^\.n^^' Let n^ . t,j be '' the price of a riskless bond at time ^'^V t paving one unit ^ ^ - *^ of the consumption good at time t+j (4) t ,J t . Then "^ u (c^J and bv definition (5) ^'J where r^ . t,j is the return on To explicitly solve a (1 i for p(X + r, iJ period pure discount bond. ) in equation (3) we use the technique of recursive substitution. (6) Defi ne g(X^, -- f. J- u'lc^^jlx^^^ dF (X^.^IX^) and f(Xj_) (7) = p(X^)u'(c^) then equation (3J can be written as f(XJ (8) Lucas (.1978) = g(X^) + 3 / f(X^^^) dF (X^^jlX^) shows that equation (8) has a unique solution f(*) This implies that f(X^) (using equation ^) (9) To find fl*), observe that the operator T: C - C (where C is the space of bounded continuous functions] defined by (Tf)(x^) = g(x^j (10) is contraction. a + p ; Hence for any ^^\^i' ^^ f- e C. '^t+i'-'^t^ lim T'f- = f. n-3; (f^ = T"f.^ = T(T"'^f^l we can generate a n = 1,2,3. sequence of functions f^CX^J = g(X^) Letting *" jf f,, be the null (i .f..,...; using (10) function f^lX^) = g(X^) + p ; fj(X^^j) dF (X^^j|Xj_) and =8(X^) P/8(X^,^) Mu'(c^^^)x^^^ = ^ r Proceeding recursively since f(X ) t dF (X^^^IX^) = lim J u'(c^,,)x^^, dF, f n (x^.oi^t) can be expressed as ^ (X,) we have f n t f(X ,) ) -1 J = and using equation (9) we get (12) dF (X^^^IX^) f,(X^) = p J u'(c^^^)x^^^ dF (X^^jlX^) or (11) - P^E^lu'lc )x -^ l ] This can be written after some simplification (see the appendix) as * J where 6 t ,j = - [1 t" ^-J ^ (I l E - [x )> r Cov^(-"'(c^,J,x^^^ 6^^^ ^ - and 6 = e -EJu'(c^^^)J E^ Ix^,^ Equation (13) exhausts the implications for valuing stochastic cash m flows Lucasian economies -- economies characterized by "representative agents" who behave optimally in light of their objectives. All prices and price distributions are endogenous and are determined through market Expectations are formed rationally; i.e., the prices and price clearing. distributions on which the economic agents base their decisions coincide with those implied by their behavior. The economy is thus "informa- tionally efficient." Superficially equation (13) resembles the familiar discounting formulations of "text book finance" with adjustment. complex. being the certainty equivalent implications and applications are, however, far more Its The adiustment term 6,, J in the t+j from period to period. It 6 It is numerator will normallv vary' precisely this variability in difficult to generalize the Sharpe (1964), Lintner (1965 (1966) asset pricing model to generalization m a multiperiod context. effect constraint 6 to be constant, achieved by imposing restrictive conditions. Co\- (-u'(c •), t-^j x .)' t+j utility of consumption is = 0, ' is G j that makes and Mossin The attempts at which can only be In the case where I.e., where the covariance of marginal <= ' not correlated with the return from a project — J -- perhaps the project is an insignificant component of consumption basket -- then 6 = . 1 t+j the individuals were risk neutral. V a diversified This would also be true if i. Equation (13) could then be written as P(X (14) ) = z ^^ ^ { y (1+r j=l '- . 1 which is consistent with our intuition for risk neutral valuation. .Although equation (13) makes explicit the informational requirements for valuing stochastic cash flows in this simple economy, we can gain additional intuitive insights by rewriting it as (15) P(X^) = X 1 j=l E (X - ) \ Gov {^^^ (-u'(c ^ J {r. . '-' .} > X ^ ) jJ (1+r ^ where we have set t=0 ), J : > J the sequence of returns on pure discount is J bonds of various maturities and A. = J = —ry—ry f Eq[u'(c^)] is a constant for each i. If we take a Taylor's expansion of the utility function u(") around E(c.) - c. and retain only the second order terms, (15) can be rewritten as oc (16) P(X.) = 1 E (X {^—i ) - Cov (c ^ J 5-J X ) L] The approximation is exact if u(*) is quadratic or if the analysis is conducted in continuous time (Ross (1977)). 0. is the reciprocal of the coefficient of relative risk aversion and is Equation (16) is constant for each a j. "similar" to the various multiperiod valuation relations in the existing finance literature. this note to critique these studies, the purpose of is not It rather it illustrates a logical and cogent methodology for the valuation of stochastic cash flows in general equilibrium setting. a Clearly the information required to con- duct such an exercise in its general form is considerable. By imposing additional structure (restrictions) these requirements can be reduced. For example, has been postulated that the coefficient of relative it risk aversion is a constant; if this and (16) is further simplified. is then the case, ($). = $ for all j (1970) have argued that .Arrow and Kurz for evaluating public projects risk netural valuation is appropriate. In this case, equation (14) derived earlier would be the appropriate expression for valuation, significantly reducing the informational requirements. Clearly, other restrictions may be placed on (16) de- pending on the application to make it more "operational". 4 . Concluding Comments This note has examined the implications of Lucas' model for valuing stochastic cash flows. The assumption of consumer homogeneity is admittedly an unrealistic one. Mehra (1980) point out, asset pricing However, as Prescott and complete market setting the equilibrium in a process for economic aggregates and prices for heterogeneous consumer a economy will be observa tiona ly equivalent to that for some homogeneous 1 consumer economy. This provides our rationale for introducing a repre- sentative "stand-in" indivi.lual. It should be apparent that cash flows from a finitely lived firm (project) can be valued within the framework developed above by simply defining them to zero after the project terminates. 10 Finally, as observed by Constantinides (1980), it is evident that even in the context of the simple economy studied here, discounting stochastic cash flows is, in general, computationally complex and of limited practical use. 11 Footnotes For a succinct review and critique of the earlier work see 1. Constantinides (1980). For a comprehensive survey see LeRoy (1982). Conditions necessary for the existence and uniqueness of rium are also discussed in that paper. 2. 3. Lucas (1978), pp. a equilib- 1434-1435. See Arrow (1971). He further argues that the coefficient of relative risk aversion should be approximately one. 4. 5. For a formal proof see Constantinides (1982). 12 Appendix Rewriting equation (12) we have P(X^) » = / ( BM Z ._^ ^ E t u'(c^^.)\ I u \ rM-T U^) / )E / (x ) t' t+j' 1 u'(c^^,) + Substituting from equation (4) we get P(\) = ^^^ "t.j ^t^Vj)(^ where ' J ^.j c-t^-'^'^t.j^'Vj )) - E^iu'.c^^^jJ E^ [x^^^J or using equation (5) p(X ) 1 J = - \h^\^^^^^ r = \^^ Cov^(-u'(c^^^).x^^^^)) J l ^'''t,P or "" j where e,.. = [1 - =l 6^^^ I (1 + r, jJ ' CovJ-u'(c^^J,x^^^) This IS precisely equation (13) in the text. 13 References Arrow, K.J. "Essays in The Theory of Risk Bearing " North Holland (1971) ^_^ and M. Kurz "Public Investment, The Rate of Return and Optimal Fiscal Policy The John Hopkins University Press, Baltimore (1970). 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Ross: Interest Rates," Econometrica Cox, J., J. , , 55 (1982) 253-267, "A Theory of the Term Structure of (forthcoming, 1985). Donaldson, J.B. and R. Mehra: "Comparative Dynamics of an Equilibrium Intertemporal Asset Pricing Model," The Review of Economic Studies 51 (1984), 491-508. Fama , "Risk Adjusted Discount Rates and Capital Budgeting Under Uncertainty," Journal of Financial Economics 5 (19(/), 3---+. E. F.: . LeRoy, S.F.: "Expectations' Model of Asset Prices: Journal of Finance 37 (1982), 185-217. A Survey of Theory," , Lintner, J.: "The Valuation of Risky Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets," Review of Economics and Statistics 47 (1965), 13-37. , Lucas, R.E., J.: ".Asset Prices in an Exchange Economy," Econometrica 46 (1978), 1426-1445. Mehra, R. and E.C. Prescott: "The Equity Premium: of Monetary Economics 15 (1985), 145-161. , , A Puzzle," Journal 14 "An Intertemporal Asset Pricing Model," Econometrica Merton, R.C.: (1973), 867-887. "Equilibrium in Mossin, J.: (1966), 768-783. a Capital Asset Market," Econometrica , 41 , 34 Prescott, E.G. and R. Mehra: The "Recursive Competitive Equilibrium: Case of Homogeneous Households," Econometrica 48 (1980), 1365-1379. Ross, S.A.: "Risk Return and Arbitrage" in I. Friend and J. Bicksler (eds.) Risk and Return in Finance Ballinger, Cambridge (1977). , Rubinstein, M.: "The Valuation of Uncertain Income Streams and the Pricing of Options," Bell Journal of Economics 7 (1976), 407-425. , Sharpe, W.F.: "Capital Asset Prices: .A Theory of Market Equilibrium under Conditions of Risk," Journal of Finance 19 (1964), 425-442. , Stapelton, R.C. and M.G. Subrahmanyam: "A Multiperiod Equilibrium Asset Pricing Model," Econometrica 46 (1978), 1077-1096. , ^53/^ 067 >-. 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