W CHICAGO JOURNALS Optimal Multiperiod Portfolio Policies Author(s): Jan Mossin Source: The Journal of Business, Vol. 41, No. 2(Apr, 1968), pp. 215-229 Published by: The University of Chicago Press StableurL:http://www.jstororg/stable/2351447 Accessed:11/09/20130233 Y of the JSTOR archive indicates your acceptance of the Terms Conditions of Use, available http://www.jstor.org/page/info/about/policies/terms.jsp is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@ jstor. org The University of Chicago Press is collaborating with JSTOR to digitize, preserve and extend access to The Journal of Business. 的d http://www.jstororg This content downloaded from 202. 115.118.13 on Wed, I I Sep 2013 02: 33: 00 AM All use subject to STOR Terms and Conditions
Optimal Multiperiod Portfolio Policies Author(s): Jan Mossin Source: The Journal of Business, Vol. 41, No. 2 (Apr., 1968), pp. 215-229 Published by: The University of Chicago Press Stable URL: http://www.jstor.org/stable/2351447 . Accessed: 11/09/2013 02:33 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. . The University of Chicago Press is collaborating with JSTOR to digitize, preserve and extend access to The Journal of Business. http://www.jstor.org This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:33:00 AM All use subject to JSTOR Terms and Conditions
OPTIMAL MULTIPERIOD PORTFOLIO POLICIES+ JAN MOSSINT mulation of the decision problem(even in the A. BACKGROUND e)in terms of por Most of the work in portfolio theory to folio rate of return tends to obscure an date! has taken what may be called a important aspect of the problem, namely, the role of the absolute size of the port- mean variability approach-that is, the folio. In a multiperiod theory the devel investor is thought of as choosing among opment through time of total wealth be- alternative portfolios on the basis of the mean and variance of the portfolios'rate comes crucial and must be taken into nt. a formula lecting this of return. A recent contribution by Ar- can easily become misleading row prepares the ground for a consider- In order to bring out and resolve the ably more general approach. 2 problems connected with a rate-of-return Although there would seem to be an formulation, it is therefore necessary to obvious need for extending the one-peri- start with an analysis of the one-period od analysis to problems of portfolio man agement over several periods, Tobin problem. Thus prepared, the extension to appears to be one of the first to make an plished, essentially by means of a dy will be demonstrated in this article, the validity of portions of this analysis ap- B. RISK-AVERSION FUNCTIONS pears to be doubtful. The explanation is The Pratt-Arrow measures An earlier as CORE Discus. in the analysis. They are abse the au- aversion Research and Econometrics, Unive Louvain, Belgium. Ra(r)=-v(yy, nomics and Business Administration, Bergen, Nor- relative risk aversion, way uidity Preference as behavior R+(Y)= U(Y) New York: Wiley, 1959);J. Tobin, "The Theory of where U is a utility function representing Portfolio selection in F H. Hahn and e. P. r. preferences over probability distribu don: Macmillan,1965),. Mossin,"Equilibrium in tions for wealth Y. Discussions of the a Capital Asset Market, "Econometrica(1966), pp. significance of these functions are found 768-83. Arrow and pratt. 4 a K. J. Arrow, Aspects of the Theory of risk Bearing (Yrjo Jahnsson Lectures [Helsinki: The rjo Jansson Foundation, 1965) 4 Arrow, op. cil. i J. Pratt, "Risk Aversion in the Small and in the Large, " Econometrica(1964), pp 3 Tobin, " Theory of Portfolio Selection. " 122-36. his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 33: 00 AM All use subject to JSTOR Terms and Conditions
OPTIMAL MULTIPERIOD PORTFOLIO POLICIES* JAN MOSSINt I. INTRODUCTION A. BACKGROUND Most of the work in portfolio theory to date' has taken what may be called a mean variability approach-that is, the investor is thought of as choosing among alternative portfolios on the basis of the mean and variance of the portfolios' rate of return. A recent contribution by Arrow prepares the ground for a considerably more general approach.2 Although there would seem to be an obvious need for extending the one-period analysis to problems of portfolio management over several periods, Tobin appears to be one of the first to make an attempt in this direction.3 However, as will be demonstrated in this article, the validity of portions of this analysis appears to be doubtful. The explanation is partly to be found in the fact that a formutation of the decision problem (even in the one-period case) in terms of portfolio rate of return tends to obscure an important aspect of the problem, namely, the role of the absolute size of the portfolio. In a multiperiod theory the development through time of total wealth becomes crucial and must be taken into account. A formulation neglecting this can easily become misleading. In order to bring out and resolve the problems connected with a rate-of-return formulation, it is therefore necessary to start with an analysis of the one-period problem. Thus prepared, the extension to multiperiod problems can be accomplished, essentially by means of a dynamic programing approach. B. RISK-AVERSION FUNCTIONS The Pratt-Arrow measures of risk aversion are employed at various points in the analysis. They are absolute risk aversion, Ra( Y) U" ( Y) relative risk aversion, Rr( Y) = U( Y) Y U'(Y)I where U is a utility function representing preferences over probability distributions for wealth Y. Discussions of the significance of these functions are found in Arrow and Pratt.4 * An earlier version appeared as CORE Discussion Paper No. 6702. It was written during the author's stay as visitor to the Center for Operations Research and Econometrics, University of Louvain, Louvain, Belgium. t Assistant professor, Norwegian School of Economics and Business Administration, Bergen, Norway. IJ. Tobin, "Liquidity Preference as Behavior towards Risk," Review of Economic Studies (1957- 58), pp. 65-86; H. Markowitz, Portfolio Selection (New York: Wiley, 1959); J. Tobin, "The Theory of Portfolio Selection," in F. H. Hahn and F. P. R. Brechling (eds.), The Theory of Interest Rates (London: Macmillan, 1965), J. Mossin, "Equilibrium in a Capital Asset Market," Econometrica (1966), pp. 768-83. 2K. J. Arrow, Aspects of the Theory of RiskBearing (Yrj6 Jahnsson Lectures [Helsinki: The Yrj6 Jahnsson Foundation, 1965]). 3 Tobin, "Theory of Portfolio Selection." 4 Arrow, op. cit.; J. Pratt, "Risk Aversion in the Small and in the Large," Econometrica (1964), pp. 122-36. 215 This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:33:00 AM All use subject to JSTOR Terms and Conditions
216 THE JOURNAL OF BUSINESS II. SINGLE-PERIOD MODELS points a =0 or a= A; the condition for By a single-period model is meant a the former is that dalu(r)l da is nega theory of the following structure The tive at a=0, which is seen to imply and investor makes his portfolio decision at require E(X).s 0. Thus, the investor the beginning of a period and then waits will hold positive amounts of the risky until the end of the period when the rate asset if and only if its expected rate of of return on his portfolio materializes. return is positive He cannot make any intermediate If the maximum occurs at an interior changes in the composition of his port- value of a, we have at this point folio. The investor makes his decision EU(Y)Ⅺ=0 with the objective of maximizing ex pected utility of wealth at the end of the To see how such an optimal value-of a period (final wealth) depends upon the level of initial wealth we differentiate(2)with respect to A and A. THE SIMPLEST CASE obtain In the simplest possible case there are only two assets, one of which yields a da--eluinxai (3) random rate of return(an interest rate) It is possible to prove that the sign of of X per dollar invested, while the other this derivative is positive, zero, or nega- asset( call it"cash")gives a certain rate tive, according as absolute risk aversion analyzed in some detail in Arrow is decreasing, constant, or increasing. I. one which he invests an amount a in the might consider, preferences over prob risky asset, his final wealth is the random terms of means and variances only. If variable to ar- Y=A+aX (1) bitrary probability distributions,the With a preference ordering U(Y) over utility function must clearly be of the levels of final wealth, the optimal valt f a is the one which maximizes ElU(nI U(n=Y-arz subject to the condition0≤a≤A Then the optimal a is the one which General analysis. The first two de- maximizes rivatives of EU(F]are EIU()]= EA+aX-a(A+ aX)? dElu(r)l-Elu(Y)XI (A-aA)2+(1-2aA)Ea and a(v+e2)a2 dElU(r)I=eu()x. where e and v denote expectation and variance of X, respectively. An interior Assuming general risk aversion(U aximum is then given by 0), the second derivative is negative, so (1-2aA)E that a unique maximum point is guaran (5) teed. This might occur at one of the end Thus, the optimal a depends on the level Arrow, op cit of initial wealth. The same is also true of his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 33: 00 AM All use subject to JSTOR Terms and Conditions
216 THE JOURNAL OF BUSINESS II. SINGLE-PERIOD MODELS By a single-period model is meant a theory of the following structure: The investor makes his portfolio decision at the beginning of a period and then waits until the end of the period when the rate of return on his portfolio materializes. He cannot make any intermediate changes in the composition of his portfolio. The investor makes his decision with the objective of maximizing expected utility of wealth at the end of the period (final wealth). A. THE SIMPLEST CASE In the simplest possible case there are only two assets, one of which yields a random rate of return (an interest rate) of X per dollar invested, while the other asset (call it "cash") gives a certain rate of return of zero. This model has been analyzed in some detail in Arrow.5 If the investor's initial wealth is A, of which he invests an amount a in the risky asset, his final wealth is the random variable Y = A + aX. (1) With a preference ordering U(Y) over levels of final wealth, the optimal value of a is the one which maximizes E[U(Y)], subject to the condition 0 < a < A. General analysis.-The first two derivatives of E[U(Y)] are dE[ U ( -) =E[ U'( Y)X] da and d2E[U(Y)] =E[U"(fY)X . da2 Assuming general risk aversion (U" < 0), the second derivative is negative, so that a unique maximum point is guaranteed. This might occur at one of the end points a 0 or a = A; the condition for the former is that dE[U(Y)]/da is negative at a = 0, which is seen to imply and require E(X) < 0. Thus, the investor will hold positive amounts of the risky asset if and only if its expected rate of return is, positive. If the maximum occurs at an interior value of a, we have at this point E[U'(Y)X] = 0. (2) To see how such an optimal value-of a depends upon the level of initial wealth, we differentiate (2) with respect to A and obtain da E [U"( Y)X] dA E[ U"( Y)X2] (3) It is possible to prove that the sign of this derivative is positive, zero, or negative, according as absolute risk aversion is decreasing, constant, or increasing. Quadratic utility.-In particular, one might consider preferences over probability distributions of Y being defined in terms of means and variances only. If such a preference ordering applies to arbitrary probability distributions, the utility function must clearly be of the form U(Y)= Y-aY2. (4) Then the optimal a is the one which maximizes E[U(Y)] = E[A + aX - a(A + aX)2] = (A - aA)2 + (1 - 2aA)Ea - a(V +EI)a2 , where E and V denote expectation and variance of X, respectively. An interior maximum is then given by (1 -2aA)E a 2a(V+E) (5) Thus, the optimal a depends on the level I Arrow, op. cit. of initial wealth. The same is also true of This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:33:00 AM All use subject to JSTOR Terms and Conditions
OPTIMAL MULTIPERIOD PORTFOLIO POLICIES the proportion a/A of initial wealth held mization problem, the level of initial in the risky asset. It is seen that da/da wealth has somehow slipped out the back 0; this is the disconcerting property men- door. Also, the resulting maximum level tioned above of any utility function ex- of expected utility would seem to be inde hibiting increasing absolute risk aver- pendent of initial wealth So it appears to be a conflict between With the optimal value of a given by the two formulations. A little reflection (5), maximum expected utility will be shows, however, that when initial wealth E2 is taken as a given, constant datum(say maxElL (Y)]=4a(v+E) 100), any level of final wealth can ob- (6)viously be equivalently described eitl in absolute terms(say, 120)or as a rate of return( 2). But Tobin's formulation.,s formu- wealth level of 120, it is immaterial to the lation is somewhat different 6 he also investor whether this is a result of assumes quadratic utility, but the argu- initial wealth of 80 with yield. 5 or an ment of the utility function is taken as initial wealth of 100 with yield. 2(or any one plus the portfolio rate of return. Sec- other combination of A and R such that ond, he takes as decision variable the AR= 120). The explanation of the ap- proportion of initial wealth invested in parent confict is now very simple: When the risky asset. If this fraction is called using a quadratic utility function in R, k, he thus wishes to maximize expected the coefficient p is not independent of A utility of the variateR= 1+kX. In the if the function shall lead to consistent symbols used above, decisions at different levels of wealth This is seen by obs Y-A+ax=1+x=1+kX. so that ervin thatR= Y/A, R Then with a quadratic utility function V(R)=R-BR', (7) which is equivalent, as a utility function, k is determined such that EV(R)] is a to Y-(B/A)Y. What this then, is that a utility function of the form R-βR2 cannot be used with the sameβ ELV(R)]= E[1+kX-B(1+kx] at different levels of initial wealth.The (1一)+(1-2)Ek appropriate value of B must be set such β(V+B)2 that B/A= athat is, B must be changed in proportion to A. But when An interior maximum is given by the this precaution is taken, Tobin's formu decision h(1-28)E lation will obviously lead to the correct (8)decision; with B= aA substituted in equation( 8), we get The important point to be made here is that the way( 8)is written, it seem k (1-2aA)E the optimal k is independent of 2aA(v+E2 weealth. In the formulation of the tha nat is (1-2aA)E 6 Tobin,“ Theory of Portfolio Selection.” his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 33: 00 AM All use subject to JSTOR Terms and Conditions
OPTIMAL MULTIPERIOD PORTFOLIO POLICIES 217 the proportion a/A of initial wealth held in the risky asset. It is seen that da/dA < 0; this is the disconcerting property mentioned above of any utility function exhibiting increasing absolute risk aversion. With the optimal value of a given by (5), maximum expected utility will be maxE[U(Y)] =4a(V+E2) (6) + V (A- A ) V+E 2 Tobin's formulation.-Tobin's formulation is somewhat different.6 He also assumes quadratic utility, but the argument of the utility function is taken as one plus the portfolio rate of return. Second, he takes as decision variable the proportion of initial wealth invested in the risky asset. If this fraction is called k, he thus wishes to maximize expected utility of the variate R = 1 + kX. In the symbols used above, Y A+aX=1+ a X=+kX. A A A Then with a quadratic utility function V(R)=R- 3R2, (7) k is determined such that E[V(R)] is a maximum: E[V(R)] = E[1 + kX - (1 + kX)2] = (1 - A) + (1 - 23)Ek - 3(V + E2)k2 . An interior maximum is given by the decision (1- 2f3)E The important point to be made here is that the way (8) is written, it seems as if the optimal k is independent of initial wealth. In the formulation of the maximization problem, the level of initial wealth has somehow slipped out the back door. Also, the resulting maximum level of expected utility would seem to be independent of initial wealth. So it appears to be a conflict between the two formulations. A little reflection shows, however, that when initial wealth is taken as a given, constant datum (say, 100), any level of final wealth can obviously be equivalently described either in absolute terms (say, 120) or as a rate of return (.2). But in considering a final wealth level of 120, it is immaterial to the investor whether this is a result of an initial wealth of 80 with yield .5 or an initial wealth of 100 with yield .2 (or any other combination of A and R such that AR = 120). The explanation of the apparent conflict is now very simple: When using a quadratic utility function in R, the coefficient f is not independent of A if the function shall lead to consistent decisions at different levels of wealth. This is seen by observing that R = Y/A, so that V(R) =V = _: I which is equivalent, as a utility function, to Y - (3/A) Y2. What this implies, then, is that a utility function of the form R - OR2 cannot be used with the same 13 at different levels of initial wealth. The appropriate value of : must be set such that 13/A = a-that is, : must be changed in proportion to A. But when this precaution is taken, Tobin's formulation will obviously lead to the correct decision; with A = aA substituted in equation (8), we get a (1-2aA)E A 2aA (V+E2)' that is, ( 1-2aA)E a 2a(V+E2) 6 Tobin, "Theory of Portfolio Selection." This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:33:00 AM All use subject to JSTOR Terms and Conditions
218 THE JOURNAL OF BUSINESS which is the same solution as(5). For ex- sent the same preference ordering, there ample, if the utility function for final exist constants b and c such that v wealth is Y-(1/400)Y2, it may be per- bU + c. Therefore, if a utility function fectly acceptable to maximize the expec- U determines an ordering of probability tation of R-iR, but only if initial distributions for rate of return and this s It should be kept in mind that when probability distributions for final wealth, ordering is identical with the ordering of we are here speaking of different levels of then U(R)and U(Y)=U(AR)must wealth, this is to be interpreted strictly represent the same ordering. This must in terms of comparative statics; we are mean that U (R) and U(AR only assering that if the investor had transformations of each othe are linear had an initial wealth different from A then his optimal k would have been dif- U(AR)=bU(R)+c ferent from( 1- 2aA)E/2aA(V+E?). Here b and c are independent of R,but When we consider different levels of they may depend upon A health at different points in time (in a se- Differentiation of (9 )with respect to R quence of portfolio decisions), other fac- gives tors may also affect the decisions, as we U(AR)A= bU(R).( shall see later. And it will also become clear that attempting to use a utility Then differentiating (10)with respect to function of the form of equation(7)in A, we have such a setting may easily cause difficul- U"(AR)AR+ U(AR)=bU(R),(11) Uility functions implying constant as- where b denotes derivative with respect set proportions.--If attention is not re- to A. From(10) the right-hand side is stricted to quadratic utility functions, (6A/bU(AR), so that(11)can be writ- however, it may be possible to get invest- ten ment in the risky asset strictly propor- U/(F)Y+U(Y b′A U'(Y) tional to initial wealth Requiring that a/A= k is seen to be or U(Y)Y the same as requiring that choices among U(F)1~b′A (12) portfolios be based upon consideration of the probability distribution for the t, of variations in Y and A, both sides are Since this must hold for independent folio's rate of return independently initial wealth: the choice of a probability constant. This means that relative risk distribution for R=1+kX consists in aversion must be constant, equal to, say, a choice of a value of k, this choice being y. It is easily verified that the only solu- made independently of A. Therefore, the tions to this condition are linear trans- problem of finding the class of utility formations of the function functions with the property that a/A U(Y)= In Y if y=1(13a) k is equivalent to the problem of deter- and mining the class of utility functions with rty that choices among distri- butions for rate of return on the portfolio Thus, utility functions belonging to this are independent of initial wealt class are the only ones permitted if con If two utility functions U and V repre- stant asset proportions are to be optimal his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 33: 00 AM All use subject to JSTOR Terms and Conditions
218 THE JOURNAL OF BUSINESS which is the same solution as (5). For example, if the utility function for final wealth is Y - (1/400) Y2, it may be perfectly acceptable to maximize the expectation of R - 'R2, but only if initial wealth happens to be 100. It should be kept in mind that when we are here speaking of different levels of wealth, this is to be interpreted strictly in terms of comparative statics; we are only asserting that if the investor had had an initial wealth different from A, then his optimal k would have been different from (1 -2 aA)E/2 aA (V + E2). When we consider different levels of wealth at different points in time (in a sequence of portfolio decisions), other factors may also affect the decisions, as we shall see later. And it will also become clear that attempting to use a utility function of the form of equation (7) in such a setting may easily cause difficulties. Utility functions implying constant asset proportions.-If attention is not restricted to quadratic utility functions, however, it may be possible to get investment in the risky asset strictly proportional to initial wealth. Requiring that a/A = k is seen to be the same as requiring that choices among portfolios be based upon consideration of the probability distribution for the portfolio's rate of return independently of initial wealth: the choice of a probability distribution for R = 1 + kX consists in a choice of a value of k, this choice being made independently of A. Therefore, the problem of finding the class of utility functions with the property that a/A = k is equivalent to the problem of determining the class of utility functions with the property that choices among distributions for rate of return on the portfolio are independent of initial wealth. If two utility functions U and V represent the same preference ordering, there exist constants b and c such that V = bU + c. Therefore, if a utility function U determines an ordering of probability distributions for rate of return and this ordering is identical with the ordering of probability distributions for final wealth, then U(R) and U(Y) = U(AR) must represent the same ordering. This must mean that U(R) and U(AR) are linear transformations of each other: U(AR) = bU(R) + c. (9) Here b and c are independent of R, but they may depend upon A. Differentiation of (9) with respect to R gives U'(AR)A = bU'(R). (10) Then differentiating (10) with respect to A, we have U"(AR)AR + U'(AR) = b'U'(R), (1 1) where b' denotes derivative with respect to A. From (10) the right-hand side is (b'A/b)U'(AR), so that (11) can be written bA U"(Y)Y+U'(Y)= b U'(Y) or U"(Y)Y bI A U'( = 1---. (12) Since this must hold for independent variations in Y and A, both sides are constant. This means that relative risk aversion must be constant, equal to, say, Ay. It is easily verified that the only solutions to this condition are linear transformations of the functions U(Y) = In Y if '=1 (13a) and U (Y) =y1-- if Py 0 . (1 3b) Thus, utility functions belonging to this class are the only ones permitted if constant asset proportions are to be optimal. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:33:00 AM All use subject to JSTOR Terms and Conditions
OPTIMAL MULTIPERIOD PORTFOLIO POLICIES 219 To see that these functions indeed sat- Maximum expected utility is then isfy our requirem when relative risk aversion is constant nax E(n Y)=In A hat is, whe 十E[n(1+1.(14) U(Y)Y Similarly, with U=Y-y, k will be de U'(Y) termined by E(1+kX)X]=0 (YX=-YU(nX and so correspondingly max e(Y-m)=A-vEll+ kx)l-.(15) EU"(Y)XⅥ=-YEU(Y)X] B. MORE GENERAL CASES At an interior maximum point we have almost all the analysis above is easily EU()X]=0 generalized to the case where the yield on he certain asset is non-zero or to the case EU"()X1]=0, where the yields on both dom. Since the analyses are in both cases completely parallel, we shall only give AElU"()X]+aE[U"(n)X2]=0; the results for the more general of two (both yields random). Results for the former case are then obtained simply by replacing the random yield X2 by a non random variable r to represent the inter But from( 3)the left-hand side is da/dA; est on the certain asset hence da/dA a/A, implying a= kA Generalization to an arbitrary number The conclusion is therefore that th of assets would be trivial and add little here m ay exist preferences which can be repre- theoretical interest sented by a utility function in rate of If the random rates of return on the return only, but then it must be of the two assets are X1 and X2, and a is the form In R or Rl-y(n Y and Yl-r are amount in vested in the first asset, ther final wealth and R-1). Other forms, like the quad- Y=(1+X2)A+a(x1-X2) ratic(7)with constant B, are ruled out. By so to say substituting(1+ X,)A for U=In y, the maximum condition b A and X1-X2 for X throughout, most of the conclusions from the discussion of comes the simplest case are readily obtained X Thus, in the general case, an interior maximum point would be one where so that k is determined by the condition EU(Y)(X1-X2)=0,(16) X and the corresponding expression for 十kX da/ dA would be da_B[U"(Y)(x1=x2)(1+x2)] d a EIU"(Y)(X1-X2)2] his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 33: 00 AM All use subject to JSTOR Terms and Conditions
OPTIMAL MULTIPERIOD PORTFOLIO POLICIES 219 To see that these functions indeed satisfy our requirement, we observe that when relative risk aversion is constant, that is, when U"f( Y) Y U'( Y) then U"(Y)YX = -yU'(Y)X, and so E[U"(Y)XY] = --yE[U'(Y)X]. At an interior maximum point we have E[U'(Y)X] = 0, and so E[U"(Y)XY] = 0, or AE[U"(Y)X] + aE[U"(Y)X21=0; thus E[U"(Y)XJ a E[ U"( Y)X2] A' But from (3) the left-hand side is da/dA; hence da/dA = a/A, implying a = kA. The conclusion is therefore that there may exist preferences which can be represented by a utility function in rate of return only, but then it must be of the form In R or R1- (In Y and Y1Fo are equivalent, as utility functions, to In R and R1zz). Other forms, like the quadratic (7) with constant A, are ruled out. We note for later reference that when U = In Y, the maximum condition becomes so that k is determined by the condition E \ Maximum expected utility is then max E (In Y) = InA (14) + E [ln (1 + kX) (4 Similarly, with U = YF1', k will be determined by E[(1 + kX)-YX] = 0, and so correspondingly maxE(Y'-Y) = A-'E[1 + kX)'-z] . (15) B. MORE GENERAL CASES Almost all the analysis above is easily generalized to the case where the yield on the certain asset is non-zero or to the case where the yields on both assets are random. Since the analyses are in both cases completely parallel, we shall only give the results for the more general of the two (both yields random). Results for the former case are then obtained simply by replacing the random yield X2 by a nonrandom variable r to represent the interest on the certain asset. Generalization to an arbitrary number of assets would be trivial and add little of theoretical interest. If the random rates of return on the two assets are X1 and X2, and a is the amount invested in the first asset, then final wealth is Y= (1 + X2)A + a(Xi - X2) . By so to say substituting (1 + X2)A for A and X1 - X2 for X throughout, most of the conclusions from the discussion of the simplest case are readily obtained. Thus, in the general case, an interior maximum point would be one where E[U'(Y)(X1- X2)] = 0, (16) and the corresponding expression for da/dA would be da E[ U"(Y)(X1-X2)(1 +X2)] d A Et " ( ( X1-X2 ) 2 ](17) This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:33:00 AM All use subject to JSTOR Terms and Conditions
THE JOURNAL OF BUSINESS It is clear that in general nothing can be said about the sign of -random, the depender the slope of the absolt function is exactly as before, however. In the case where utility is quadratic in wealth(and assuming Xi and X to be independently distributed), the optimal a will be given b E1-E2-2aA[(1+E)(E1-E2)一V2 and correspondingly max ElU(r)1=1(1+B)+2(1+B) (19 4-4是2小++画 With the Tobin formulation, however, the optimal h would be expressed as E1-E2-2B[(1+E2)(E1-E2)-V2] (20 2B[V1+V2+(E1-E2 and, again, if decisions are to be consist- IIL, MULTIPERIOD MODELS ent at different levels of initial wealth, B A. GENERAL METHOD OF SOLUTIOI must be proportional to A The derivation of the utility functions, By a multiperiod model is meant a (13a)and (13b), is clearly independent of theory of the following structure: The the specific setting of the decision prob- investor has determined a certain future lem. The sufficiency part of the proof is point in time(his horizon)at which he also completely analogous. plans to With the utility function U=In y, k then available. He will still make his in- rould now be determined by the condi- vestment decisions with the objective of maximizing expected utility of wealth at X1-X 1+X2+k(X1-X2) that the time between the present and giving his horizon can be subdivided into n peri- ods(not necessarily of the same length E(In[1+ X2+ k(Xi-Xel.(21) at the end of each of which return on the max E(In Y)=In A izes and he can make a new decision Similarly, with U =Yl-y,k is deter- the composition of the portfolio to be mined by held during the next period E(1+X2+k(Xr-X]-(X -X2))=0, This formulation of the problem de- media max E(r-m portfolio decisions are clearly interre (22) lated, and no defense for leaving con- AE{1+X2+k(X1-X2)-? sumption decisions out of the picture can his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 33: 00 AM All use subject to JSTOR Terms and Conditions
220 THE JOURNAL OF BUSINESS It is clear that in general nothing can be said about the sign of this derivative; for X2 non-random, the dependence on the slope of the absolute risk-aversion function is exactly as before, however. In the case where utility is quadratic in wealth (and assuming X1 and X2 to be independently distributed), the optimal a will be given by a E1-E2-2aA [ (1 +E2) (E1-E2) -V21 18) 2ca[VI+V2+(El-E2)2( and correspondingly maxE[ U ( Y) V1( 1 +E2) +V2 (1 +E1) VI1+V2 + (El -E2 )2 (19) XF V1 (l1+E2)2?+V2 (l?E1)2 +V1V2A21+ (E1-E2)2 [ V1V( 1 +E2) +V2 (1 +E1) 4a[Vi+V2 + (El-Ei)2] With the Tobin formulation, however, the optimal k would be expressed as E1-E2-2f [ (1 +E2)(E1-E2) -V2( 2fl[V1+V2+(E1-E2)2( and, again, if decisions are to be consistent at different levels of initial wealth, A must be proportional to A. The derivation of the utility functions, (13a) and (13b), is clearly independent of the specific setting of the decision problem. The sufficiency part of the proof is also completely analogous. With the utility function U = In Y, k would now be determined by the condition + 1 -X2 ) giving max E(ln Y) =nA (21) + E{ln [1 + X2+ k(Xi-X2)]} ( Similarly, with U = Y1Fo, k is determined by E{[1 + X2+ k(X -X2)]-'(X1 -X2)}= 0, giving max E(Y'Y) (22) = Al-'E{[l + X2 + k(X1 - X2)'y1I . III. MULTIPERIOD MODELS A. GENERAL METHOD OF SOLUTION By a multiperiod model is meant a theory of the following structure: The investor has determined a certain future point in time (his horizon) at which he plans to consume whatever wealth he has then available. He will still make his investment decisions with the objective of maximizing expected utility of wealth at that time. However, it is now assumed that the time between the present and his horizon can be subdivided into n periods (not necessarily of the same length), at the end of each of which return on the portfolio held during the period materializes and he can make a new decision on the composition of the portfolio to be held during the next period. This formulation of the problem deliberately ignores possibilities for intermediate consumption. Consumption and portfolio decisions are clearly interrelated, and no defense for leaving consumption decisions out of the picture can This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:33:00 AM All use subject to JSTOR Terms and Conditions
OPTIMAL MULTIPERIOD PORTFOLIO POLICIES 221 be offered except for the simple(but ad- Rather, any sequence of portfolio deci vantageous)strategy of taking one thing sions must be contingent upon the out at a time. Such a partial analysis has comes of previous periods and at the been justified by picturing the investor same time take into account information as providing for a series of future on future probability distributions, It is sumption dates by dividing total wealth only when the last period is reached, and into separate portfolios for each con- the final decision is to be taken, that the sumption date, with each such portfolio simple models of the preceding section to be managed independently. This pro- are applicable cedure must be rejected as clearly sub- At the beginning of the last period (n) optimal and hardly represents a satisfac- the investor's problem is simply to make tory solution. Neither the decision on a decision(call it dn), dividing his wealth how much to consume in any given pe- as of that time, An-1, among the different riod nor the management of any given assets such that Em[U(A,)] is maximized subportfolio could generally be independ -(the notation En indicates expectation ent of actual performance of other port- with respect to probability distributions folios. The first attempts to consider of yields during the nth period). But once squarely the interrelations between con- he has thus chosen his optimal decision sumption and portfolio decisions appear (depending, in general, upon An-1),the be represented by the still unpublished maximum o of expected utility of final papers by Dreze and Modigliani and by wealth is determined solely in terms of In our version of the theory, an in vestor, starting out with a given initial max En[U(An)=φn(An) wealth Ao, will make a first-period deci- The function n-1 is referred to as the sion on the allocation of this wealth to indirect""or "derived"utility functio different assets, then wait until the end of the period when a wealth level A1 ma- and is the appropriate representation of terializes. He then makes a second-period preferences over probability distributions decision on the allocation of Al, and so for An-1. Therefore, the optimal decision d,-1 to choose at the beginning of period 1 is the one which maximizes It is clear that for such a multiperiod planning problem it is rarely optimal, if En-1lom-1(An-1)]=En-lmax En[U(An)Il at all possible, to specify a sequence of single-period decisions once and for all Is way Nor could it generally be optimal to the next-to-last decision as a simple one- simply make a first-period decision that period problem, granted that the objec- would maximize expected utility of tive is appropriately defined in terms of wealth at the end of that period while the "derived" utility function. But to do disregarding the investment opportuni- so obviously requires the investor to ties in the second and later periods. specify the optimal last-period decision for every possible outcome of yield dur- eriod n is by means of such J. Dreze and F. Modigliani, "Consumption a backward-recursive procedure that it is Decisions under uncertai possible to determine an optimal first and Portfolio Choice"(manuscript in preparation ). period decision his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 33: 00 AM All use subject to JSTOR Terms and Conditions
OPTIMAL MULTIPERIOD PORTFOLIO POLICIES 221 be offered except for the simple (but advantageous) strategy of taking one thing at a time. Such a partial analysis has been justified by picturing the investor as providing for a series of future consumption dates by dividing total wealth into separate portfolios for each consumption date, with each such portfolio to be managed independently.7 This procedure must be rejected as clearly suboptimal and hardly represents a satisfactory solution. Neither the decision on how much to consume in any given period nor the management of any given subportfolio could generally be independent of actual performance of other portfolios. The first attempts to consider squarely the interrelations between consumption and portfolio decisions appear to be represented by the still unpublished papers by Dreze and Modigliani and by Sandmo.8 In our version of the theory, an investor, starting out with a given initial wealth Ao, will make a first-period decision on the allocation of this wealth to different assets, then wait until the end of the period when a wealth level A 1 materializes. He then makes a second-period decision on the allocation of A1, and so on. It is clear that for such a multiperiod planning problem it is rarely optimal, if at all possible, to specify a sequence of single-period decisions once and for all. Nor could it generally be optimal to simply make a first-period decision that would maximize expected utility of wealth at the end of that period while disregarding the investment opportunities in the second and later periods. Rather, any sequence of portfolio decisions must be contingent upon the outcomes of previous periods and at the same time take into account information on future probability distributions. It is only when the last period is reached, and the final decision is to be taken, that the simple models of the preceding section are applicable. At the beginning of the last period (n) the investor's problem is simply to make a decision (call it d.), dividing his wealth as of that time, An-1 among the different assets such that E4[U(An)] is maximized (the notation En indicates expectation with respect to probability distributions of yields during the nth period). But once he has thus chosen his optimal decision (depending, in general, upon An-1), the maximum of expected utility of final wealth is determined solely in terms of An-1, that is, max En[U(An)] =na (An-l) . dn The function 4n-1 is referred to as the "indirect" or "derived" utility function and is the appropriate representation of preferences over probability distributions for An-1. Therefore, the optimal decision d._1 to choose at the beginning of period n - 1 is the one which maximizes En-J[?n-i(An-] = En-,{max En[U(An)] } dn In this way it is possible to consider the next-to-last decision as a simple oneperiod problem, granted that the objective is appropriately defined in terms of the "derived" utility function. But to do so obviously requires the investor to specify the optimal last-period decision for every possible outcome of yield during period n - 1. It is by means of such a backward-recursive procedure that it is possible to determine an optimal firstperiod decision. 7lbid. 8 J. Dreze and F. Modigliani, "Consumption Decisions under Uncertainty" (manuscript in preparation); A. Sandmo, "Capital Risk, Consumption, and Portfolio Choice" (manuscript in preparation). This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:33:00 AM All use subject to JSTOR Terms and Conditions
THE JOURNAL OF BUSINESS Both for a theoretical development The optimal decision for the second nd for purposes of practical computa- period is obtained from (18)as tion, the solution method is very much complicated if statistical dependence 5.7A1-400 4.84 (23 among yields in different periods (i.e, serial correlation) is to be allowed for. This expression defines the best possible We shall therefore assume throughout decision, for any value of wealth at the that such dependence is absent, although beginning of the second period. When this decision rule is adopted, the ex- TABLE 1 pected value of final wealth will be, ac- rding to(19) AssEt 1 中(41)=aaxB2[U(4x2) Yield Yield 2.94 A 2.7675 +const E1=1 Vi=.01 Ea=5 v2=2.25 It is the expectation of this function hich is to be maximized by the first Initial wealth: Ao=200 period decision, which we achieve by Utility function: U=A2-(1/1,000)41 TABLE 2 this certainly means some loss of ger erality. The basic nature of the approach is the same however, and the conclusions we are to derive are certainly unaffected by this simplification Dependence among yields within any period would be rela ively easy to handle, but for a theoreti- cal development it does not seem worth maximizing the expectation of the func the extra trouble. Also, we shall ignore tion in parentheses transaction costs E1(A1 2.7675 B. A TWO-PERIOD EXAMPLE WITH QUADRATIC UTILITY Using formula ( 18)again (with a To illustrate the procedure, we shall 2.7675/2490), we then get the optimal an develop in some detail a numerical ex- as ample with two assets with random 57A0-360=1612.(24) yields. To simplify the notation as much 4.84 as possible, it is assumed that the yields Thus, the optimal decision to be effected X1and X2 are independent and that their immediately is to invest about 80.6 per distributions are the same in both peri cent in asset 1 and the remainder in asset ods. We take a1 and ay to be investment 2 in the first asset in periods 1 and 2, re- The possible outcomes of the spectively. The data of the example are period portfolio are then as show given in Table 1 Table 2(each with probability t) his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 33: 00 AM All use subject to JSTOR Terms and Conditions
222 THE JOURNAL OF BUSINESS Both for a theoretical development and for purposes of practical computation, the solution method is very much complicated if statistical dependence among yields in different periods (i.e., serial correlation) is to be allowed for. We shall therefore assume throughout that such dependence is absent, although this certainly means some loss of generality. The basic nature of the approach is the same, however, and the conclusions we are to derive are certainly unaffected by this simplification. Dependence among yields within any period would be relatively easy to handle, but for a theoretical development it does not seem worth the extra trouble. Also, we shall ignore transaction costs. B. A TWO-PERIOD EXAMPLE WITH QUADRATIC UTILITY To illustrate the procedure, we shall develop in some detail a numerical example with two assets with random yields. To simplify the notation as much as possible, it is assumed that the yields X1 and X2 are independent and that their distributions are the same in both periods. We take a, and a2 to be investment in the first asset in periods 1 and 2, respectively. The data of the example are given in Table 1. TABLE 1 ASSET 1 ASSET 2 Yield P Yield P 0.0 .5 -1.0 .5 0.2 .5 2.0 .5 E1= . 1 V1=.01 E2=. 5 V2=2.25 Initial wealth: A o=200 Utility function: U=A2-(1/1,OOO)A2 The optimal decision for the second period is obtained from (18) as a2, -=.A 400, (23) 4.84 This expression defines the best possible decision/for any value of wealth at the beginning of the second period. When this decision rule is adopted, the expected value of final wealth will be, according to (19): k1(Ai) =maxE2[U(A2) I a2 2.94 2.Al 27675 7 A2 + const. 2.42 2490 / It is the expectation of this function which is to be maximized by the firstperiod decision, which we achieve by TABLE 2 Xi X2 Ai a 2 a2 in % Xi X: Ai ~~~~~~~ of Al 0.0 -1 161.2 107.2 66.5 0.2 -1 193.4 145.1 75.0 0.0 2 277.6 244.3 80.0 0.2 2 309.8 282.2 91.1 maximizing the expectation of the function in parentheses: 2.76 752 maxE1 Al- - A1l a, 2490 / Using formula (18) again (with a= 2.7675/2490), we then get the optimal a, as 5.7Ao-360 (2) a14.84 Thus, the optimal decision to be effected immediately is to invest about 80.6 per cent in asset 1 and the remainder in asset 2. The possible outcomes of the firstperiod portfolio are then as shown in Table 2 (each with probability '). This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:33:00 AM All use subject to JSTOR Terms and Conditions
OPTIMAL MULTIPERIOD PORTFOLIO POLICIES 223 The example illustrates several impor- be utility functions, however, for which tant points. First, although it is possible, such a procedure is optimal and indeed necessary, to specify in ad We shall say that if the investors se- vance an optimal decision rule for the quence of decisions is obtained as a series second-period investment, it is not pos- of single-period decisions(starting with sible to determine the specific decision to the first period), where each period is be taken. This will depend upon the out- treated as if it were the last one, then he ome of the"experiment"" performed in behaves myopically. With myopia, the investor bases each period's decision on Second, the decision rules are generally that period s initial wealth and probabil- different in different periods: the rela- ity distribution of yields only, with the tionship in equation(23)between the objective of maximizing expected utility optimal a2 and Al is different from the of final wealth in that period while dis- relationship in equation(24)between the regarding the future completely. It is ob- optimal a1 and Ao. The reason for this is vious that if it were optimal to make de that in period 1 the investor must take cisions in this manner, the problem of into account the probability distribu- portfolio management would be grerical tions for the second period, because the simplified. But, also on the theoret "derived"utility function depends on it. level, it is interesting to isolate those util More explicitly, it is seen that in period 1 ity functions for which such behavior is he still maximizes a quadratic utility optimal function, but the coefficient a is now re- The set of utility functions for which myopia is optimal will generally change V1(1+E2)2+V2(1+E1)2+vv2 according to assumptions about the na V1(1+E2)+V2(1+E1) ture of the asset yields. Here we shall analyze the case with one riskless asset (f[19 ) Thus, even if the investor hap- with yield r. For arbitrary r, it will be pens to end up at the end of the first pe- shown that the only utility functions al- riod with the same wealth as he had at lowing myopic decision making are the the beginning(this is not actually pos- logarithmic and power functions which ible in the example), he will make a dif- we have encountered earlier. For r=0 ferent decision. In the example, with the set includes other functions; it is ger 41=200, he would have invested a erally characterized by the condition maller amount in asset 1(a2= 152.8) U'(r) thus making up for the loss of time left F)=+λr before the horizon by playing more It now turns out that this larger set be possible to identify such a"time ef- still requires only a very modest amount fect, "and we shall return to a discussion of foresight even when r=0.All that of that problem below(Sec. III D) needs to be known about subsequent pe- riods is the value( or values)of r, while C. UTILITY FUNCTIONS ALLOWING information about the yield distribution for the risky asset is unnecessary. In such As noted, it is generally non-optimal cases, the investor can make his immedi to make decisions for one period at a ate decision as if the entire resulting ime without looking ahead. There may wealth would have to be invested at the his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 33: 00 AM All use subject to JSTOR Terms and Conditions
OPTIMAL MULTIPERIOD PORTFOLIO POLICIES 223 The example illustrates several important points. First, although it is possible, and indeed necessary, to specify in advance an optimal decision rule for the second-period investment, it is not possible to determine the specific decision to be taken. This will depend upon the outcome of the "experiment" performed in the first period. Second, the decision rules are generally different in different periods: the relationship in equation (23) between the optimal a2 and A1 is different from the relationship in equation (24) between the optimal a1 and A o. The reason for this is that in period 1 the investor must take into account the probability distributions for the second period, because the "derived" utility function depends on it. More explicitly, it is seen that in period 1 he still maximizes a quadratic utility function, but the coefficient a is now replaced by V1( 1 +E2 ) 2 +V2 ( 1 +E1) 2 + V1V2 V10(+E2) +V2(1+E1) a (cf. [19]). Thus, even if the investor happens to end up at the end of the first period with the same wealth as he had at the beginning (this is not actually possible in the example), he will make a different decision. In the example, with Al 200, he would have invested a smaller amount in asset 1 (a2 = 152.8), thus making up for the loss of time left before the horizon by playing more boldly. Under certain conditions, it may be possible to identify such a "time effect," and we shall return to a discussion of that problem below (Sec. III D). C. UTILITY FUNCTIONS ALLOWING MYOPIC DECISIONS As noted, it is generally non-optimal to make decisions for one period at a time without looking ahead. There may be utility functions, however, for which such a procedure is optimal. We shall say that if the investor's sequence of decisions is obtained as a series of single-period decisions (starting with the first period), where each period is treated as if it were the last one, then he behaves myopically. With myopia, the investor bases each period's decision on that period's initial wealth and probability distribution of yields only, with the objective of maximizing expected utility of final wealth in that period while disregarding the future completely. It is obvious that if it were optimal to make decisions in this manner, the problem of portfolio management would be greatly simplified. But, also on the theoretical level, it is interesting to isolate those utility functions for which such behavior is optimal. The set of utility functions for which myopia is optimal will generally change according to assumptions about the nature of the asset yields. Here we shall analyze the case with one riskless asset with yield r. For arbitrary r, it will be shown that the only utility functions allowing myopic decision making are the logarithmic and power functions which we have encountered earlier. For r = 0, the set includes other functions; it is generally characterized by the condition __ U'( Y )_ +Xy U'if(Y) It now turns out that this larger set still requires only a very modest amount of foresight even when r 5 0. All that needs to be known about subsequent periods is the value (or values) of r, while information about the yield distribution for the risky asset is unnecessary. In such cases, the investor can make his immediate decision as if the entire resulting wealth would have to be invested at the This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:33:00 AM All use subject to JSTOR Terms and Conditions
