当前位置：和泉文库 > 经济 > 《投资学原理 Investments》课程教学资源（阅读资料）Optimal Multiperiod Portfolio Policies

《投资学原理 Investments》课程教学资源（阅读资料）Optimal Multiperiod Portfolio Policies

文件格式：PDF，文件大小：402.56KB，售价：4.74元

文档详细内容（约16页）

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

点击进入文档下载页（PDF格式）

共16页，可试读7页，点击继续阅读 ↓↓

您可能感兴趣的文档

点击购买下载（PDF）

下载及服务说明

购买前请先查看本文档预览页，确认内容后再进行支付；
如遇文件无法下载、无法访问或其它任何问题，可发送电子邮件反馈，核实后将进行文件补发或退款等其它相关操作；
邮箱：

文档浏览记录