《投资学原理 Investments》课程教学资源（阅读资料）Lifetime Portfolio Selection By Dynamic Stochastic Programming

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LIFETIME PORTFOLIO SELECTION BY DYNAMIC STOCHASTIC PROGRAMMING Paul a. samuelson Introduction Third, being still in the prime of life, the portfolio selection businessman can“ recoup” any present losses nether they are of the Markowitz- in the future. The widow or retired man ney Tobin mean-variance or of more general type, ing life's end has no such "second or nt chance late and solve a many-period generalization, Fourth(and apparently related to the last corresponding to lifetime planning of consump- for so many periods. "the law of averages will tion and investment decisions. For simplicity of exposition I shall confine my explicit dis- even out for him, " and he can afford to act cussion to special and easy cases that suffice to almost as if he were not subject to diminishing illustrate the general principles involved marginal utility. As an example of topics that can be investi- What are we to make of these arguments? lted within the framework of the present It will be realized that the first could be purely model, consider the question of a "business- a one-period argument. Arrow, Pratt, an man risk"kind of investment. In the literature others 2 have shown that any investor who of finance, one often reads; Security A should faces a range of wealth in which the elasticity be avoided by widows as too risky, but is highly of his marginal utility schedule is great suitable as a businessman's risk "What is in- have high risk tolerance; and most writers First, the "businessman"is more afluent highest for rich-but not ultra-rich volved in this distinction? Many things. seem to believe that the elasticity is at than the widow; and being further removed people. Since the present model has no new from the threat of falling below some sub- insight to offer in connection with statical risk sistence level, he has a high propensity to tolerance, I shall ignore the first point here embrace variance for the sake of better yield. and confine almost all my attention to utility Second, he can look forward to a high salary functions with the same relative risk aversion in the future; and with so high a present dis- at all levels of wealth. Is it then still true that counted value of wealth, it is only prudent for lifetime considerations justify the concept of him to put more into common stocks compared a businessman's risk in his prime of life? to his present tangible wealth, borrowing if Point two above does justify leveraged in necessary for the purpose, the same thing by selecting volatile stocks that earnings. But it does not really involve any increase in relative risk-taking once we have related what is at risk to the proper larger base a Aid from the National Science Foundation is gratefully (Admittedly, if market imperfections make owledged. Robert C. Merton ch stimulus; and in a companion paper in this issue loans difficult or costly, recourse to volatile the review he is tackling the much harder problem of "leveraged"securities may be a rational pro- ptimal control in the presence of continuous-time sto- see for example harry markowitz [5] james tobin The fourth point can easily involve the in- for a pioneering treatment of the multi-period portfolio large numbers. "I have commented elsewhere 3 Robert C. Merton [13]. See however, James Tobin [15] problem; and Jan Mossin [7] whie erlaps with the on the mistaken notion that multiplying the 时包 e the basic dynamic same kind of ds to cancellation rather a See K. Arrow [1]; Pratt [9]: P. A. Samuelson and theorem that portfolio proportions will be invariant only R C. Merton [13] if the marginal utility function is iso-elastic P A Samuelson [11] This content downloaded from 202. 115.118.13 on Wed, 1I Sep 2013 02: 34: 15 AM All use subject to JSTOR Terms and Conditions

LIFETIME PORTFOLIO SELECTION BY DYNAMIC STOCHASTIC PROGRAMMING Paul A. Samuelson * Introduction M OST analyses of portfolio selection, whether they are of the MarkowitzTobin mean-variance or of more general type, maximize over one period.' I shall here formulate and solve a many-period generalization, corresponding to lifetime planning of consumption and investment decisions. For simplicity of exposition I shall confine my explicit discussion to special and easy cases that suffice to illustrate the general principles involved. As an example of topics that can be investigated within the framework of the present model, consider the question of a "businessman risk" kind of investment. In the literature of finance, one often reads; "Security A should be avoided by widows as too risky, but is highly suitable as a businessman's risk." What is involved in this distinction? Many things. First, the "businessman" is more affluent than the widow; and being further removed from the threat of falling below some sub-- sistence level, he has a high propensity to embrace variance for the sake of better yield. Second, he can look forward to a high salary in the future; and with so high a present discounted value of wealth, it is only prudent for him to put more into common stocks compared to his present tangible wealth, borrowing if necessary for the purpose, or accomplishing the same thing by selecting volatile stocks that widows shun. Third, being still in the prime of life, the businessman can "recoup" any present losses in the future. The widow or retired man nearing life's end has no such "second or nth chance." Fourth (and apparently related to the last point), since the businessman will be investing for so many periods, "the law of averages will even out for him," and he can afford to act almost as if he were not subject to diminishing marginal utility. What are we to make of these arguments? It will be realized that the first could be purely a one-period argument. Arrow, Pratt, and others2 have shown that any investor who faces a range of wealth in which the elasticity of his marginal utility schedule is great will have high risk tolerance; and most writers seem to believe that the elasticity is at its highest for rich - but not ultra-rich! - people. Since the present model has no new insight to offer in connection with statical risk tolerance, I shall ignore the first point here and confine almost all my attention to utility functions with the same relative risk aversion at all levels of wealth. Is it then still true that lifetime considerations justify the concept of a businessman's risk in his prime of life? Point two above does justify leveraged investment financed by borrowing against future earnings. But it does not really involve any increase in relative risk-taking once we have related what is at risk to the proper larger base. (Admittedly, if market imperfections make loans difficult or costly, recourse to volatile, "leveraged" securities may be a rational procedure.) The fourth point can easily involve the innumerable fallacies connected with the "law of large numbers." I have commented elsewhere 3 on the mistaken notion that multiplying the same kind of risk leads to cancellation rather * Aid from the National Science Foundation is gratefully acknowledged. Robert C. Merton has provided me with much stimulus; and in a companion paper in this issue of the REVIEW he is tackling the much harder problem of optimal control in the presence of continuous-time stochastic variation. I owe thanks also to Stanley Fischer. 'See for example Harry Markowitz [5]; James Tobin [14], Paul A. Samuelson [10]; Paul A. Samuelson and Robert C. Merton [13]. See, however, James Tobin [15], for a pioneering treatment of the multi-period portfolio problem; and Jan Mossin [7] which overlaps with the present analysis in showing how to solve the basic dynamic stochastic program recursively by working backward from the end in the Bellman fashion, and which proves the theorem that portfolio proportions will be invariant only if the marginal utility function is iso-elastic. 2 See K. Arrow [1]; J. Pratt [9]; P. A. Samuelson and R. C. Merton [13]. 3P. A. Samuelson [11]. [ 239 ] This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:34:15 AM All use subject to JSTOR Terms and Conditions

240 THE REVIEW OF ECONOMICS AND STATISTICS than augmentation of risk. I. e, insuring for prescribed (Wo, Wr+1. Differentiating many ships adds to risk(but only as v n); partially with respect to each W in turn, we hence, only by insuring more ships and by derive recursion conditions for a regular inte- also subdividing those risks among more people rior maximum is risk on each brought down (in ratio 1/V/n). (1+pvLwi-11+r writing this pa (7) that investing for each period is akin to agree. If U is concave, solving. these second-order ing to take a 1/ nth interest in insuring n inde- difference equations with boundary conditions pendent ships The present lifetime model reveals that in- lifetime consumption-investment program vesting for many periods does not itself in- Since there has thus far been one asset, and troduce extra tolerance for riskiness at early, that a safe one, the time has come to introduce or any, stages of life a stochastically-risky alternative asset and to face up to a portfolio problem. Let us postulate he existence alongside of the safe asset that Basic Assun akes d in it at time t return The familiar Ramsey model may be used as at the end of the period $1(1+r), a risk m oint of departure Let an individual maxi- that makes $1 invested in, at time t,return to you after one period $lZ+, where Z, is a random variable subject to the probability distribution e-pf vic(t)]du Pob{2t当2}=P(x).z≡0 subject to initial wealth Wo that can always be Hence, Zi+1-1 is the percentage"yield"of invested for an exogeneously-given certain each outcome. The most general probability rate of yield r; or subject to the constraint distribution is admissible: i.e., a probability rw(t)-w(t) (2)density over continuous s's, or finite positive If there is no bequest at death, terminal wealth shall usually assume independence between yields at different times so that P(20, 2l,, This leads to the standard calculus-of-varia- 2,..., 3r)=P(3+)P(31).P(ar) tions problem For simplicity, the reader might care to de J=Max/e-"Urw-响d (3)with th W(t)} Prob(Z=A)= 1/2 This can be easily related to a discrete- Prob(2=A-, A>1 time formulation In order that risk averters with concave utility Max (1+p)=U[C] (4) should not shun this risk asset when maximiz subject to ng the expected value of their portfolio, i must be large enough so that the expected value of (5)the risk asset exceeds that of the safe asset i.e Max (1+p) (6) λ>1+r+√2r+r2 'See P. A. Samuelson [12], p. 273 for Thus, for x= 1.4. the risk asset has a mean yield of 0.057, which is greater than a safe I assume that consumption, Ct, ta asset's certain yield of the beginning rather than at the end of the change alters slightly the appearance of the equilibriu At each instant of time. what will be the conditions, but not their substanc optimal fraction, wt, that you should put in This content downloaded from 202. 115.118.13 on Wed, 1I Sep 2013 02: 34: 15 AM All use subject to JSTOR Terms and Conditions

240 THE REVIEW OF ECONOMICS AND STATISTICS than augmentation of risk. I.e., insuring many ships adds to risk (but only as \In); hence, only by insuring more ships and by also subdividing those risks among more people is risk on each brought down (in ratio 1/V/n). However, before writing this paper, I had thought that points three and four could be reformulated so as to give a valid demonstration of businessman's risk, my thought being that investing for each period is akin to agreeing to take a 1/nth interest in insuring n independent ships. The present lifetime model reveals that investing for many periods does not itself introduce extra tolerance for riskiness at early, or any, stages of life. Basic Assumptions The familiar Ramsey model may be used as a point of departure. Let an individual maximize T e-Pt U[C(t)]dt (1) subject to initial wealth WO that can always be invested for an exogeneously-given certain rate of yield r; or subject to the constraint C(t) = rW(t) - W(t) (2) If there is no bequest at death, terminal wealth is zero. This leads to the standard calculus-of-variations problem T J = Max e-Pt U[rW - W]dt (3) {W(t)} ? This can be easily related I to a discretetime formulation T Max 1t0 (1+p)-t U[Ct] (4) subject to C= W Wt+1 (5) 1+r or, MaxEt= (1+p)t U [Wt- W+ ] (6) {w~~~O+ 1+r for prescribed (W0, WT+1). Differentiating partially with respect to each Wt in turn, we derive recursion conditions for a regular interior maximum (I +P) U wt1 +] 1+r1r =U' Wt - [w + (7) If U is concave, solving these second-order difference equations with boundary conditions (Won WT+1) will suffice to give us an optimal lifetime consumption-investment program. Since there has thus far been one asset, and that a safe one, the time has come to introduce a stochastically-risky alternative asset and to face up to a portfolio problem. Let us postulate the existence, alongside of the safe asset that makes $1 invested in it at time t return to you at the end of the period $1(1 + r), a risk asset that makes $1 invested in, at time t, return to you after one period $1Zt, where Zt is a random variable subject to the probability distribution Prob {Zt 1 (9) In order that risk averters with concave utility should not shun this risk asset when maximizing the expected value of their portfolio, X must be large enough so that the expected value of the risk asset exceeds that of the safe asset, i.e., - + A-1 > 1 + r, or 2 2 X > 1 + r + V2r + r2. Thus, for X = 1.4, the risk asset has a mean yield of 0.057, which is greater than a safe asset's certain yield of r = .04. At each instant of time, what will be the optimal fraction, Wt, that you should put in 'See P. A. Samuelson [12], p. 273 for an exposition of discrete-time analogues to calculus-of-variations models. Note: here I assume that consumption, Ct, takes place at the beginning rather than at the end of the period. This change alters slightly the appearance of the equilibrium conditions, but not their substance. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:34:15 AM All use subject to JSTOR Terms and Conditions

LIFETIME PORTFOLIO SELECTION 241 the risky asset, with 1-w going into the safe consider Phelps wage income, and even in asset? Once these optimal portfolio fractions the stochastic form that he cites Martin Beck are known, the constraint of (5) must be mann as having analyzed. More recently, writte Levhari and Srinivasan [4] have also treated the Phelps problem for T=oo by means of W [(1-w)(1+)+tz] the bellman functional equations of dynamic (10) programming, and have indicated a proof that concavity of U is sufficient for a maximum Now we use(10) instead of (4), and recogniz- Then, there is Professor Mirrlees' important ing the stochastic nature of our problem, work on the ramsey problem with Harrod specify that we maximize the expected value neutral technological change as a random vari of total utility over time. This gives us the able. Our problems become equivalent if I stochastic generalizations of (4) and (5)or replace W,-Wi+[(1+)(1-w,)+w,Zi]-1 (6) in(10)by Af (W/At)-nWt-(W1+1-Wi) Ct, W E 2(1+p)-U[CtI (11)let technical change be governed by the prob- ability distribution subject to Prob (At s At-12=P(n); C=L reinterpret my wt to be Mirrlees' per capita capital, K,/Lt, where Lt is growing at the nat- Wo given, Wr+1 prescribed ural rate of growth n; and posit that Atf(W/ If there is no bequeathing of wealth at death, A)is a homogeneous first degree, concave,neo- presumably Wr+1=0. Alternatively, we could classical production function in terms of cap- replace a prescribed Wr+ by a final bequest it should be remarked that I am confirming function added to(11), of the form B(Wr+1) and with W, a free decision variable to be myself here to regular interior maxima, and chosen so as to maximize(11)+BW+1). not going into the Kuhn-Tucker inequalities For the most part, I shall consider Cr=Wr that easily handle boundary maxima In(11), e stands for the"expected value Solution of the problem of, " so that, for example, The meaning of our basic problem EZ= adP(st) Jr(Wo)= Max e 2(1+p)-"U[C:] (11) (Ct, wt = In our simple case of (9), subject to Ct=Wt-Wi+1[(1-wt)(1 +, i- is not easy to grasp I act now at t =0 to select Co and wo, knowing Wo but not equation(11)is our basic stochastic program- yet knowing how Zo will turn out. I must act taneously for optimal saving-consumption and of Zos outcome will be known and that W1 will portfolio-selection decisions over time. then be known. Depending upon knowledge of Before proceeding to solve this problem, ref- W,, a new decision will be made for C,and erence may be made to similar problems that 01. Now I can only guess what that decision ve been dealt with explicitly in the will be economics literature. First there is the valu- As so often is the case in dynamic program- able naner by Phelps on the Ramsey problem ming, it helps to begin at the end of the plan in which capital's yield is a prescribed random ning period. This brings us to the well-known variable. This corresponds, in my notation, to the (wt) strategy being frozen at some frac- J.A.MirAs[8] tional level, there being no portfolio selection into a discrete version. Robert Merton's problem.(My analysis could be amplified to for d, throws light on Mirrlees'Brownian-mof This content downloaded from 202. 115.118.13 on Wed, 1I Sep 2013 02: 34: 15 AM All use subject to JSTOR Terms and Conditions

LIFETIME PORTFOLIO SELECTION 241 the risky asset, with 1 - wt going into the safe asset? Once these optimal portfolio fractions are known, the constraint of (5) must be written Ct =[Wt - Wt+1]. c[(1-wt) (1 +r) + wtZt] (10) Now we use (10) instead of (4), and recognizing the stochastic nature of our problem, specify that we maximize the expected value of total utility over time. This gives us the stochastic generalizations of (4) and (5) or (6) Max T {Ct, wt} E X (1 +p) -t U[Ct] (1 t=O subject to Ct =[ Wt(1 + r) (- wt) + wtZt] WO given, WT+1 prescribed. If there is no bequeathing of wealth at death, presumably WT+1 = 0. Alternatively, we could replace a prescribed WT+1 by a final bequest function added to (11), of the form B(WT+1), and with WT+1 a free decision variable to be chosen so as to maximize (11) + B(WT+D). For the most part, I shall consider CT = WT and WT+1 = 0? In (11), E stands for the "expected value of," so that, for example, E Zt = fztdP(zt) In our simple case of (9), EZt = 2 A + 1 A-1. 2 2 Equation ( 11 ) is our basic stochastic programming problem that needs to be solved simultaneously for optimal saving-consumption and portfolio-selection decisions over time. Before proceeding to solve this problem, reference may be made to similar problems that seem to have been dealt with explicitly in the economics literature. First, there is the valuable paper by Phelps on the Ramsey problem in which capital's yield is a prescribed random variable. This corresponds, in my notation, to the {wt} strategy being frozen at some fractional level, there being no portfolio selection problem. (My analysis could be amplified to consider Phelps' 5 wage income, and even in the stochastic form that he cites Martin Beckmann as having analyzed.) More recently, Levhari and Srinivasan [4] have also treated the Phelps problem for T = oo by means of the Bellman functional equations of dynamic programming, and have indicated a proof that concavity of U is sufficient for a maximum. Then, there is Professor Mirrlees' important work on the Ramsey problem with Harrodneutral technological change as a random variable.6 Our problems become equivalent if I replace W - Wt+1 [(1+r)(1-wt) + wtZtJ-1 in (10) byAtf(Wt/At) - nWt - (Wt+- Wt) let technical change be governed by the probability distribution Prob {At ? At-1Z} = P(Z); reinterpret my Wt to be Mirrlees' per capita capital, Kt/Lt, where Lt is growing at the natural rate of growth n; and posit that Atf(Wt/ At) is a homogeneous first degree, concave, neoclassical production function in terms of capital and efficiency-units of labor. It should be remarked that I am confirming myself here to regular interior maxima, and not going into the Kuhn-Tucker inequalities that easily handle boundary maxima. Solution of the Problem The meaning of our basic problem T JT(WO) = Max E X (1+P)-tU[Ct] (11) {ct,wt} t=O subject to Ct = Wt- Wt+1[(1-wt) (1+r) + w,Zt]-I is not easy to grasp. I act now at t = 0 to select C0 and w0, knowing W0 but not yet knowing how Z0 will turn out. I must act now, knowing that one period later, knowledge of Z0's outcome will be known and that W1 will then be known. Depending upon knowledge of W1, a new decision will be made for C1 and w1. Now I can only guess what that decision will be. As so often is the case in dynamic programming, it helps to begin at the end of the planning period. This brings us to the well-known 'E. S. Phelps [8]. 6 J. A. Mirrlees [6]. I have converted his treatment into a discrete-time version. Robert Merton's companion paper throws light on Mirrlees' Brownian-motion model for At. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:34:15 AM All use subject to JSTOR Terms and Conditions

242 THE REVIEW OF ECONOMICS AND STATISTICS one-period portfolio problem. In our terms, Substituting(C*r-1, 2*r-1)into the expres- this becomes sion to be maximized gives us J1(Wr-1)ex J1(Wr-1)= plicitly. From the equations in(12 ), we can by standard calculus methods, relate the de + E(1+p)-lUI(Wr-1-Cr-1) rivatives of u to those of J, namely, by the (1-m-1)(1+r) envelope relation J1(Wr-1)=U[Cr-1] Here the expected value operator E operates Now that we know J,[W,_11, it is easy to only on the random variable of the next period determine optimal behavior one period earlier nce current consumption CT-1 is known once namely by we have made our decision. Writing the second term as EF(Zr), this becomes J2(wr-2)= Max U[Cr-21 EF(∠Zr)=F(zr)dP(∠r{Zr-1,Zr-2,…,Zo) +E(1+p)-1J1[(Wr-2-Cr-2) (1-r-2)(1+r)+ (14) F(Zr)dP(Zr),by our independence Differentiating(14)just as we did(11) gives the following equations like those of(12) In the general case at a later stage of decision 0=U[CT-2] -(1+p)-1 E1[Wr-2] making, say t= T-1, knowledge will be avail (1-2-2)(1+n)+r-2zr-2}(15) ble of the outcomes of earlier random vari- 0= Ei[WT-1(WT-2-C-2)(Z7_2 ables, Zi-2,.; since these might be relevant the distribution of subsequent random vari J1[(Wr-2-Cr-2){(1-2r-2)(1+r) ables, conditional probabilities of the form t (Wr-2-Cr-2)(Zr-2-1-r) P(Zg-1ZT-2,.)are thus involved. How dP(z, ever, in cases like the present one, where in (15) dependence of distributions is posited, condi- These equations, which could by(13)be re tional probabilities can be dispensed within lated to U'[CT-1], can be solved simultaneous- favor of simple distributions ly to determine optimal(Ca Note that in (12)we have substituted fo Cr its value as given by the constraint in(11) Continuing recursively in this way for T-3 or(10) T-4. 2,1,0, we finally have our problem To determine this optimum (Cr-1, Wr_1), solved. The general recursive optimality equa we differentiate with respect to each separately, tions can be written as 0=U[co]-(1+p)-1E/r-1W] 0=U[Cr-1]-(1+p)-1EU"[Cr] (1-2)(1+r)+to2o} (1-r-1)(1+)+ n}(12)(0=Er-1[W](Wo-C0)(z0-1-r) 0=EU"[Cr](Wr-1-Cr-1)(Zr-1-1-r) 0=U[Cr-1]-(1+p)-1E/r-tW T (1-2-1)(1+r)+20+-121-1}(16) (1-r-1(1+r)-wm-12r-1} 0=Er'r-[Wt-1-Ct-1(Z (Wr-1-Cr-1)(Zr-1-1-r)dP(Zr-1) In(16), of course, the proper substitutions Solving these simultaneously, we get our must be made and the e operators must be optimal decisions(C*T-1, w0*t-1)as functions over the proper probability distributions. Solv- of initial wealth Wr-1 alone. Note that if ing(16") at any stage will give the optimal somehow C*?-1 were known,(12")would by decision rules for consumption-saving and for itself be the familiar one-period portfolio portfolio selection, in the form timality condition, and could trivially be re- C*,=S[WE; Zi-1,., Zo] written to handle any number of alternative =fr-tlWi if the Zs are independently assets This content downloaded from 202. 115.118.13 on Wed, 1I Sep 2013 02: 34: 15 AM All use subject to JSTOR Terms and Conditions

242 THE REVIEW OF ECONOMICS AND STATISTICS one-period portfolio problem. In our terms, this becomes JI (WT-1) = Max U[CT-1] {CT-1jWT-1} + E(l+p) 'U[ (WT-1 - CT-1) {(1-WT-1) (l+r) + WT-1ZT-1} 4].* (12) Here the expected value operator E operates only on the random variable of the next period since current consumption CT-1 is known once we have made our decision. Writing the second term as EF(ZT), this becomes EF(ZT) = J F(ZT)dP(ZTjZT-1,ZT-22 * , Zo) 0 /F (ZT) dP (ZT), by our independence postulate. In the general case, at a later stage of decision making, say t = T- 1, knowledge will be available of the outcomes of earlier random variables, Zt-2, ... ; since these might be relevant to the distribution of subsequent random variables, conditional probabilities of the form P(ZT-1IZT-2, .. .) are thus involved. However, in cases like the present one, where independence of distributions is posited, conditional probabilities can be dispensed within favor of simple distributions. Note that in (12) we have substituted for CT its value as given by the constraint in (11) or (10). To determine this optimum (CT-1, WT-1), we differentiate with respect to each separately, to get O = U' [CT-1] - (1+p)P EU' [CT] {(1-WT-1) (l+r) + WT-IZT-l} (12') O = EU' [CT] (WT-1 -CT_1) (ZT-1-1-r) - ,J'U' [ (WT-1 -CT-1) {(1-WT l(1+r) - WT-1ZT-1}] (WT-1-CT-1) (ZT-1 -r) dP (ZT-1) (12") Solving these simultaneously, we get our optimal decisions (C*T-1, W*T_1) as functions of initial wealth WT-1 alone. Note that if somehow C*T-1 were known, (12") would by itself be the familiar one-period portfolio optimality condition, and could trivially be rewritten to handle any number of alternative assets. Substituting (C*T-1, W*T_1) into the expression to be maximized gives us J1(WT-1) explicitly. From the equations in (12), we can, by standard calculus methods, relate the derivatives of U to those of J, namely, by the envelope relation JI'(WT-1) = U' [CT-1]. (13) Now that we know J1[WT_1], it is easy to determine optimal behavior one period earlier, namely by J2 (WT-2) = Max U[CT-2] {CT-21WT-2} + E(1 +p) -1JI [ (WT-2-CT-2) {( 1-WT-2) (1 +r) + WT-2ZT-2}]. (14) Differentiating (14) just as we did (11) gives the following equations like those of (12) O = U' [CT-2] - (1+p) - EJ1' [WT-2] { (1-WT-2) (I +r) + WT-2ZT-2} (15') 0 = EJ1' [WT-1] (WT-2 - CT-2) (ZT-2 - 1-r) = J fJ1' [ (WT-2 -CT-2) { ( 1-WT-2) (1 +r) + WT-2ZT-2}] (WT-2 -CT-2) (ZT-2 - 1-r) dP(ZT_2). (15") These equations, which could by (13) be related to U'[CT-1], can be solved simultaneously to determine optimal (C*T-2, W*T-2) and J2 (WT-2) . Continuing recursively in this way for T-3, T-4,...,2, 1, 0, we finally have our problem solved. The general recursive optimality equations can be written as { O = U'[Co] - (1 +p) -1 E'TT1 [Wo] { (1-wo) (l+r) + woZo} O = EJI'T-l[Wl] (WO -CO) (ZO -1-r) O = U'[CT-] - (1+P) EJ'T-t[Wt] { (I 1-wt-1) (I1 +r) + wt_IZt_I} ( 16') o = ETT-t [Wt-l -Ct-1) (Zt-1 r), (t =l,I...,IT-1I). (16tt) In (16'), of course, the proper substitutions must be made and the E operators must be over the proper probability distributions. Solving (16") at any stage will give the optimal decision rules for consumption-saving and for portfolio selection, in the form C*t = f [Wt; Zt-l, ... , Zo] = fT-t[Wt] if the Z's are independently distributed This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:34:15 AM All use subject to JSTOR Terms and Conditions

LIFETIME PORTFOLIO SELECTION 243 wet= glut;, Zo] Hence, equations(12)and(16)-(16")split gr-tlWt] if the Z's are independently into two independent parts and the Ramsey distributed Phelps saving problem becomes quite indepen Our problem is now solved for every case dent of the lifetime portfolio selection problem but the important case of infinite-time horizon. Now we have For well-behaved cases, one can simply let 0=(1/C)-(1+p)-1(W-C)-Ior T- o0 in the above formulas. Or, as often Cr-1=(1+p)(2+p)-iwr-1 (19″ happens, the infinite case may be the easiest of all to solve, since for itC*=f(W),v0*:=0=J(z-1-)[(1-)(1+) g(w,), independently of time and both these +wZ]-1 dP (Z)or unknown functions can be deduced as solutions wr-1= w* independently of Wr-1.(19 to the following functional equations These independence results, of the CT-1 and 0=U[f(W)]-(1+p)-1 zer_ decisions and of the dependence of wp- ∫r(m-r(m)(a+) on Wr,. hold for all u functions with iso- elastic marginal utility. I. e, ,(16) and(16) -g(W)(z-1-r)}[(1+r) g(w)(z-1-r)]dP(z) (17) become decomposable conditions for all U(C)=1/y Cy, (20) as well as for U(C)= log C, corresponding by 1+r-g(W)(Z-1-n) L'HOpital's rule to y =0 [Z-1-7]dpa) (17) To see this, write(12 )or(18) Equation(17"),y itself with g (w)pretended J(W(C ">+(1+e)-1(W-C)r Max (13)of Levhari and Srinivasan [4, p. f]. In deriving(17)-(17"), I have utilized the enve [(1-2)(1+)+2]dP(Z) lope relation of my(13), which is equivalent to (W一C) Levhari and Srinivasan's equation (12)[4 =Max-+(1+p)-1 Max/[(1-n)(1+) Bernoulli and Isoelastic Cases To apply our results, let us consider the in- 十2]vφ(Z).(21) teresting Bernoulli case where U log C. This Hence, (12")or(15")or(16)becomes does not have the bounded utility that Arrow [1] and many writers have convinced them selves is desirable for an axiom system. Since +t2]y-1(z-r-1)dP(Z)=0,(22″ I do not believe that Karl Menger paradoxes which defines optimal z*and gives terrors for the economist, I have no particular Max/[(1-w)(1+r)+wz]y dP(Z) interest in boundedness of utility and consider fae)o log C to be interesting and admissible. For this case, we have, from(12) [(1-t*)(1+n)+t*Z]?dP(Z) J1( w)= Max log C =[1+→*], for short Here, r* is the subjective or util-prob mean +E(1+P)-1log[(W-C) return of the portfolio, where diminishing mar (1-)(1+r)+wz ginal utility has been taken into account. To Max logC+(1+p)-og [W-C] get optimal consumption-saving, differentiate C ( 21)to get the new form of (12),(15),or Max/log [(1-z0)(1+r) (16) T See Samuelson and Merton for the util-prob concept +weZ]dP (4) (18)[13] This content downloaded from 202. 115.118.13 on Wed, 1I Sep 2013 02: 34: 15 AM All use subject to JSTOR Terms and Conditions

LIFETIME PORTFOLIO SELECTION 243 W*t = g[Wt; Zt_, ... *, Zo] = g-_t[Wt] if the Z's are independently distributed. Our problem is now solved for every case but the important case of infinite-time horizon. For well-behaved cases, one can simply let T -> oo in the above formulas. Or, as often happens, the infinite case may be the easiest of all to solve, since for it C*t = f(Wt), w*t = g(Wt), independently of time and both these unknown functions can be deduced as solutions to the following functional equations: 0 = U' [f(W)] -(1+p) fJ'[ (W - f(W)) {(1+r) -g(W) (Z- 1 -r)}] [ (1+r) -g(W)(Z - 1 - r)]dP(Z) (17') 00 0= fUu[{W - f(W)} {1 + r-g(W) (Z- 1 -r)}] [Z -1- r] d p (-i) ( 17"f) Equation (17'), by itself with g(W) pretended to be known, would be equivalent to equation (13) of Levhari and Srinivasan [4, p. f]. In deriving (17')-(17"), I have utilized the envelope relation of my (13), which is equivalent to Levhari and Srinivasan's equation (12) [4, p.5]. Bernoulli and Isoelastic Cases To apply our results, let us consider the interesting Bernoulli case where U = log C. This does not have the bounded utility that Arrow [1] and many writers have convinced themselves is desirable for an axiom system. Since I do not believe that Karl Menger paradoxes of the generalized St. Petersburg type hold any terrors for the economist, I have no particular interest in boundedness of utility and consider log C to be interesting and admissible. For this case, we have, from (12), J1(W) = Max logC {C,w} + E(l+p)-'log [(W - C) { (l-w) (l+r) + wZ}] = Max log C + (I +p) log [W-C] {C} + Max log [ (1-w) (1+r) {w} + wZ]dP(Z) (18) Hence, equations (12) and (16')-(16") split into two independent parts and the RamseyPhelps saving problem becomes quite independent of the lifetime portfolio selection problem. Now we have 0 = (1/C) - (1+p)-1 (W - C)-'or CT-1 = (l+p) (2+p) 'WT-1 (19) r00 o = (Z- 1 r)-[(l w) (l+r) + wZ] -1 dP(Z) or WT-1 = W* independently of WT-1. (19") These independence results, of the CT-1 and WT-1 decisions and of the dependence of WT_1 on WT_1, hold for all U functions with isoelastic marginal utility. I.e., (16') and (16") become decomposable conditions for all U(C) = 1/yCl, y < 1 (20) as well as for U(C) = log C, corresponding by L'Hopital's rule to y = 0. To see this, write (12) or (18) as J1 (W) = Max C + (1 +p) -1 (W-C)7 {C, w} Y Y rx f[(1 -w)(1+r) + wZ]Zi dP(Z) = Max-+ (+p)-1 (w-CYx {C} 'Y 'Y Max [(1-w) (1+r) w? + wZ] dp(Z). (21) Hence, (12") or (15") or (16") becomes rx [ (1 -w) (1+r) + wZ]Y-l (Z-r- 1) dP (Z) = O, (22") which defines optimal w* and gives rx Max [(1 -w)(1+r) +wZ]y dP(Z) {w} = f co (1 -w*) (1 +r) + w*Z]'ydP (Z) = [1 + r*]Y,for short. Here, r* is the subjective or util-prob mean return of the portfolio, where diminishing marginal utility has been taken into account.7 To get optimal consumption-saving, differentiate (21) to get the new form of (12'), (15'), or (16') See Samuelson and Merton for the util-prob concept [13]. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:34:15 AM All use subject to JSTOR Terms and Conditions

244 THE REVIEW OF ECONOMICS AND STATISTICS 0=Cr-1-(1+p)-(1+r*)?(W-C)Y-1. In the limiting case, as y-0 and we have (22) Bernoulli's logarithmic function, a,=(1+p) lving, we have the consumption decision rule independent of r*, and all saving propensities *r1=,a2-W=1 (23) depend on subjective time preference p only, ing independent of technological inves where opportunities (except to the degree that wt a1=[(1+r*)/(1+p)]1-1 (24)will itself definitely depend on those opportu Hence, by substitution, we find nities) J1(Wr-1)=b1Wr-1/ (25) We can interpret 1+r* as kind of a"risk corrected""mean yield; and behavior of a long- where lived man depends critically on whether b1=a1(1+a1)- +(1+p)-1(1+r*)?(1+a1)- (26)(1+r*)Y=(1+p), corresponding to ay 1. Thus,J1() is of the same elasticity form as U()was. Evaluating indeterminate forms for (i) For (1+r*)r=(1+p), one plans always to 0, we find J, to be of log form if U was consume at a uniform rate, dividing current w by remaining life 1/(1+i. If young enough, or ow, by mathematical induction, it is easy on the average; in the familiar "hump saving"fa to show that this isoelastic property must also one dissaves later as the end comes sufficiently close hold for J2 Wr-2), J3(Wr_3) Thenever it holds for Jn (Wr_n)it is deducible (i) For(1+r*)1>(1+p),a(1+), the perpetual lifetime problem 0=人[(1-m)(1+)+m2]1(z-1-)dP(Z)a=1-10, this case cannot arise. as i00. of all consumption-saving decisio *)r l, consumpti g to a constant w*, tl L Then optimal consumption decisions at each saving depends upon the size of r a- cra, or whether p (1+r*)/1- where one can deduce the recursion relations This ends the Theorem. Although many of C1 the results de on the no-bequest a1=[(1+p)/(1+r*)?]n-y paper shows(p. 247, this Review) we can easy sumption, Wr+1 =0, as Merton's compan (1+r*)?= (1-u*)(1+r) generalize to the cases where a bequest fun +u*Z17 dP(z) tion Br(Wr+) is added to 2To(1+p)-U(Ct acI- If Br is itself of isoelastic form Br= br(Wr+1)%/ <CI the algebra is little changed. Also, the same comparative statics put forward in Merton's a1(a1-1) ≠ continuous-time case will be applicable here g, the Bernoulli y =0 case is a watershed between cases where thrift is enhanced by risk iness rather than reduced etc This content downloaded from 202. 115.118.13 on Wed, 1I Sep 2013 02: 34: 15 AM All use subject to JSTOR Terms and Conditions

244 THE REVIEW OF ECONOMICS AND STATISTICS 0 = CY-1 _ (1+p)-l (1+r*)y (W-C)'-1. (22') Solving, we have the consumption decision rule C*T_1 = T1 1 (23) + aW where a, = [(1+r*)Y/(11+p)]/11tl. (24) Hence, by substitution, we find J1 (WT-1) = blW'YT-11/Y (25) where b, = aj'Y(1+aj)-Y + (1+p)-l (1+r*)y (1+al)'-y (26) Thus, J,(.) is of the same elasticity form as U(.) was. Evaluating indeterminate forms for y = 0, we find J, to be of log form if U was. Now, by mathematical induction, it is easy to show that this isoelastic property must also hold for J2(WT-2), I3(WT-3), ..., since, whenever it holds for JI(WT_") it is deducible that it holds for Jn+1(WT -1). Hence, at every stage, solving the general equations (16') and (16"), they decompose into two parts in the case of isoelastic utility. Hence, Theorem: For isoelastic marginal utility functions, U'(C) = C-1, 'y (l+p), corresponding to a, (1+p), a, (1+p), the perpetual lifetime problem, with T = oo, is divergent and ill-defined, i.e., JI(W) -- oo as i-- oo. For Y - 0 and p > 0, this case cannot arise. (iii) For (1+r*)Y 1, consumption at very early ages drops only to a limiting positive fraction (rather than zero), namely Lim c4 = 1- l/a1 1. - 00 Now whether there will be, on the average, initial hump saving depends upon the size of r* - c., or whether r*-- I- > O This ends the Theorem. Although many of the results depend upon the no-bequest assumption, WT+1 = 0, as Merton's companion paper shows (p. 247, this Review) we can easily generalize to the cases where a bequest function BT(WT+l) is added to X O (1+p) tU(Ct). If BT is itself of isoelastic form, BT- bT(WT+1Y"/y, the algebra is little changed. Also, the same comparative statics put forward in Merton's continuous-time case will be applicable here, e.g., the Bernoulli y = 0 case is a watershed between cases where thrift is enhanced by riskiness rather than reduced; etc. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:34:15 AM All use subject to JSTOR Terms and Conditions

LIFETIME PORTFOLIO SELECTION 245 Since proof of the theorem is straightfor- risk-tolerance as toward the end of life! The ward, i skip all details except to indicate how "chance to recoup"and tendency for the law the recursion relations for C and b i are derived, of large numbers to operate in the case of re- namely from the identities peated investments is not relevant. (Note b+1W°/y=J4+1(W) if the elasticity of marginal utility, -U(w) Max ( CW/y WU (W), rises empirically with wealth, and if he capital market is imperfect as far as lending bi(1+r*)(1-+p)-(W-C)/y) and borrowing against future earnings is con c+1+b;(1+r*) cerned. the seems to me to be likely (1+p)-1(1-c+1)"Wy that a doctor of age 35-50 might rationally and the optimality condition have his highest consumption then, and certain 0=Cr-1-bi(+r)7(1+p)-(W-C)7-1 ly show greatest risk tolerance then -in other =(c+1W)?-1-b(1+r*)(1+p) words be open to a‘ businessman' s risk?”But (1-G1+1)7-1Wy-1, not in the frictionless isoelastic model!) which defines ci+1 in terms of As usual, one expects w* and risk tolerance What if we relax the assumption of isoelastic to be higher with algebraically large y. One marginal utility functions? Then wet- be- expects Ct to be higher late in life when r ane comes a function of Wr_i-1(and, of course, r* is high relative to p. As in a one-period of r, p, and a functional of the probability dis- model, one expects any increase in"riskiness tribution P). Now the Phelps-Ramsey optimal of 2t, for the same mean, to decrease w*.One stochastic saving decisions do interact with the expects a similar increase in riskiness to lower optimal portfolio decisions, and these have to or raise consumption depending upon whether be arrived at by simultaneous solution of the marginal utility is greater or less than unity in nondecomposable equations(16)and(16) What if we have more than one alternative Our analysis enables us to dispel a fallacy asset to safe cash? Then merely interpret z, that has been borrowed into portfolio theory as a(column) vector of returns(22, 2u,...) from information theory of the Shannon type n the respective risky assets; also interpret Associated with independent discoveries by w, as a(row vector(w-t, w t interpret J. B. Williams [16], John Kelly [2], andH. A P(Z as vector notation for Latane [3 is the notion that if one is invest- Prob{22≤22,23≤23,…,} Ing for many p e pr (∠ )=P(Z) maximize the geometric mean of return rather as multiple integrals SG(4, ,.)dP(z2 za incorrect (except in the e i believe this to be interpret all integrals of the form G(z)dP(z) than the arithmetic mean Then(16")becomes a vector-set of case where it happens to be correct for reasons equations, one for each component of the vector .See Merton's cited companion paper in this issue, for the unknown 20, vector C*r and we*r functions as the parameters (p, y, r, r*, and If there are many consumption items, we results hold in the discrete-and-continuous-time model can handle the general problem by giving a See Latane [3, p. 151] for similar vector interpretation to C ewhat mystifying his footnote there Thus, the most general portfolio lifetime says, " As pointed out to me by Professor L.J. Say problem is handled by our equations or obviou eometric mean] the rule for maximum expected utility in extensions thereof nnection with bernoulli's fu rule is proximately valid for all utility functions. "[Latane, p. 151 n 13] The geometric mean criterion is definitely too con- We have now come full circle. Our model responding to positive y in my equation(20), and it is denies the validity of the concept of business- definitely too daring to maximize expected utility, when man's risk; for isoelastic marginal utilities, in 1969 position differs from the view attributed to him in your prime of life you have the same relative 1959 This content downloaded from 202. 115.118.13 on Wed, 1I Sep 2013 02: 34: 15 AM All use subject to JSTOR Terms and Conditions

LIFETIME PORTFOLIO SELECTION 245 Since proof of the theorem is straightforward, I skip all details except to indicate how the recursion relations for ci and bi are derived, namely from the identities b+l1W7/y = J+i(W) = Max {CY/y C + bj(1+r*)Y(1+p)-l(W-C)Y/y} {c'i+l + bi(l+r*)Y ( 1+p) -1 ( 1-Ci+1)_1} WY/y and the optimality condition 0 = CT-1_ b,(1+r*)Y(l+p)-l(W-C)7- = (c,+jW)8-1- bj(1+r*)7(1+p)1 ( 1-Cj+l ) T- W7-1j which defines cj+j in terms of bi. What if we relax the assumption of isoelastic marginal utility functions? Then WT_j becomes a function of WT>j1l (and, of course, of r, p, and a functional of the probability distribution P). Now the Phelps-Ramsey optimal stochastic saving decisions do interact with the optimal portfolio decisions, and these have to be arrived at by simultaneous solution of the nondecomposable equations (16') and (16"). What if we have more than one alternative asset to safe cash? Then merely interpret Zt as a (column) vector of returns (Z2,,Z3,, ..) on the respective risky assets; also interpret wt as a (row) vector (w2t,w3t, ...), interpret P(Z) as vector notation for Prob {Z2t _ Z2, Z3t ? Z3,... } = P(Z2 Z3,...) =P(Z), interpret all integrals of the form fG (Z) dP (Z) as multiple integrals fG (Z2,Z3, . ..) dP(Z2,Z3, ...). Then (16") becomes a vector-set of equations, one for each component of the vector Zt, and these can be solved simultaneously for the unknown wt vector. If there are many consumption items, we can handle the general problem by giving a similar vector interpretation to Ct. Thus, the most general portfolio lifetime problem is handled by our equations or obvious extensions thereof. Conclusion We have now come full circle. Our model denies the validity of the concept of businessman's risk; for isoelastic marginal utilities, in your prime of life you have the same relative risk-tolerance as toward the end of life! The ''chance to recoup" and tendency for the law of large numbers to operate in the case of repeated investments is not relevant. (Note: if the elasticity of marginal utility, - U' (W) / WU"(W), rises empirically with wealth, and if the capital market is imperfect as far as lending and borrowing against future earnings is concerned, then it seems to me to be likely that a doctor of age 35-50 might rationally have his highest consumption then, and certainly show greatest risk tolerance then - in other words be open to a "businessman's risk." But not in the frictionless isoelastic model!) As usual, one expects w* and risk tolerance to be higher with algebraically large y. One expects C, to be higher late in life when r and r* is high relative to p. As in a one-period model, one expects any increase in "riskiness" of Zt, for the same mean, to decrease w*. One expects a similar increase in riskiness to lower or raise consumption depending upon whether marginal utility is greater or less than unity in its elasticity.8 Our analysis enables us to dispel a fallacy that has been borrowed into portfolio theory from information theory of the Shannon type. Associated with independent discoveries by J. B. Williams [ 16], John Kelly [2 ], and H. A. Latane [3] is the notion that if one is investing for many periods, the proper behavior is to maximize the geometric mean of return rather than the arithmetic mean. I believe this to be incorrect (except in the Bernoulli logarithmic case where it happens I to be correct for reasons 'See Merton's cited companion paper in this issue, for explicit discussion of the comparative statical shifts of (16)'s C*t and W*t functions as the parameters (p, y, r, r*, and P(Z) or P(Z1,...) or B(WT) functions change. The same results hold in the discrete-and-continuous-time models. 'See Latane [3, p. 151] for explicit recognition of this point. I find somewhat mystifying his footnote there which says, "As pointed out to me by Professor L. J. Savage (in correspondence), not only is the maximization of G [the geometric mean] the rule for maximum expected utility in connection with Bernoulli's function but (in so far as certain approximations are permissible) this same rule is approximately valid for all utility functions." [Latane, p. 151, n.13.] The geometric mean criterion is definitely too conservative to maximize an isoelastic utility function corresponding to positive y in my equation (20), and it is definitely too daring to maximize expected utility when ,y < 0. Professor Savage has informed me recently that his 1969 position differs from the view attributed to him in 1959. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:34:15 AM All use subject to JSTOR Terms and Conditions

THE REVIEW OF ECONOMICS AND STATISTICS uite distinct from the williams-Kelly-Latane REFERENCES reasoning) [1] Arrow, K. J," Aspects of the Theory of Risk se writers must have in mind reasoning Bearing"(Helsinki, Finland: Yrjo Jahnssonin that goes something like the following: If one maximizes for a distant goal, investing and [2] Kelly, J, "A New \sgk reinvesting(all one's proceeds) many times on Rate,” Bell System the way, then the probability becomes great 1956),917-926 hat with a portfolio that maximizes the geo- [3] Latane, H.A., "Criteria for Choice Among Risky Ventures, Journal of Political Economy 67(Apr. metric mean at each stage will end up 959),144-155 with a larger terminal wealth than with any [4] Levhari, D, and T N Srinivasan, "Optimal Sav other decision strategy ings Under Uncertainty, Institute for Mathe This is indeed a valid consequence of the matical Studies in the Social Sciences, Technical eport No. 8, Stanford University, Dec. 1967. central limit theorem as applied to the addi- [5] Markowitz, H, Portfolio Selection: Eficient tive logarithms of portfolio outcomes. (I.e Diversification of Investment (New York: John maximizing the geometric mean is the same Wiley Sons, 1959 thing as maximizing the arithmetic mean of [6] Mirrlees, J. A,"Optimum Accumulation Under the logarithm of outcome at each stage if Uncertainty, "Dec. 1965, unpublished ach stage we get a mean log of m**> m*, L7JMossin, J,Optimal multiperiod portfolio po then after a large number of stages we will icies, "Journal of Business 41, 2(Apr. 1968), have m**T>>m*, and the properly no 215-229 malized probabilities will cluster around ital: A Sequential Utility Analysis, " Economet- er rica30,4(1962),729-743 There is nothing wrong with the logical [9] Pratt, J, "Risk Aversion in the Small and in the deduction from premise to theorem. But the arge, " Econometrica 32 (Jan. 1964) [ Samuelson, 10,p 3]. It is a mistake to think [7, zon Pap 3 P.A. "General Proof that Diversifica- plicit premise is faulty to begin with, as I have shown elsewhere in another connection ournal of Financial and quantitative alysis II (Mar. 1967),1-13 Risk and Uncertainty: A Fallacy of hat, just because a we**k decision ends up with Large Numbers, "Scientia, 6th Series, 57th yes almost-certain probability to be better than a (April-May, 1963) decision, this implies that 2u** must yield a [12J A Turnpike Refutation of the better expected value of utility. Our analysis Rule in a Welfare Maximizing M for marginal utility with elasticity differing Essay XIV Essays on the Theory of Economic Growth, Karl Shell (ed )(Caml from that of Bernoulli provides an effective Mass.: MIT Press, 1967) counter example, if indeed a counter example [13]-, and R C. Merton, "A Complete Model needed to refute a gratuitous assertion of Warrant Pricing that Maximizes Utility, In dustrial Management Review (in press) principle of selecting between two actions in [14] Tobin, I,p Liquidity Preference as Behavior terms of which has the greater probability of ⅹXV,67,Feb.1958,65-86 producing a higher result does not even possess [15] "The Theory of Portfolio Selection the property of being transitive. 0 By that The Theory of Interest Rates, F. H. Hahn and principle, we could have u***k better than we**, F.P. R. Brechling (eds )(London: Macmillan, and w ** better than 0* and also have 0* 1965) better than we*** [16] Williams, J. B,"Speculation and the Carryo terly Journal of Economics 50(May 19 See Samuelson [11] 455 This content downloaded from 202. 115.118.13 on Wed, 1I Sep 2013 02: 34: 15 AM All use subject to JSTOR Terms and Conditions

246 THE REVIEW OF ECONOMICS AND STATISTICS quite distinct from the Williams-Kelly-Latane reasoning). These writers must have in mind reasoning that goes something like the following: If one maximizes for a distant goal, investing and reinvesting (all one's proceeds) many times on the way, then the probability becomes great that with a portfolio that maximizes the geometric mean at each stage you will end up with a larger terminal wealth than with any other decision strategy. This is indeed a valid consequence of the central limit theorem as applied to the additive logarithms of portfolio outcomes. (I.e., maximizing the geometric mean is the same thing as maximizing the arithmetic mean of the logarithm of outcome at each stage; if at each stage, we get a mean log of m** > m*, then after a large number of stages we will have m**T > > m*T, and the properly normalized probabilities will cluster around a higher value.) There is nothing wrong with the logical deduction from premise to theorem. But the implicit premise is faulty to begin with, as I have shown elsewhere in another connection [Samuelson, 10, p. 3]. It is a mistake to think that, just because a w** decision ends up with almost-certain probability to be better than a w* decision, this implies that w** must yield a better expected value of utility. Our analysis for marginal utility with elasticity differing from that of Bernoulli provides an effective counter example, if indeed a counter example is needed to refute a gratuitous assertion. Moreover, as I showed elsewhere, the ordering principle of selecting between two actions in terms of which has the greater probability of producing a higher result does not even possess the property of being transitive.'0 By that principle, we could have w*** better than w**, and w** better than w*, and also have w* better than w***. REFERENCES [1] Arrow, K. J., "Aspects of the Theory of RiskBearing" (Helsinki, Finland: Yrj6 Jahnssonin Saati6, 1965). [2] Kelly, J., "A New Interpretation of Information Rate," Bell System Technical Journal (Aug. 1956), 917-926. [3] Latane, H. A., "Criteria for Choice Among Risky Ventures," Journal of Political Economy 67 (Apr. 1959), 144-155. [4] Levhari, D. and T. N. Srinivasan, "Optimal Savings Under Uncertainty," Institute for Mathematical Studies in the Social Sciences, Technical Report No. 8, Stanford University, Dec. 1967. [5] Markowitz, H., Portfolio Selection: Efficient Diversification of Investment (New York: John Wiley & Sons, 1959). [6] Mirrlees, J. A., "Optimum Accumulation Under Uncertainty," Dec. 1965, unpublished. [7] Mossin, J., "Optimal Multiperiod Portfolio Policies," Journal of Business 41, 2 (Apr. 1968), 215-229. [8] Phelps, E. S., "The Accumulation of Risky Capital: A Sequential Utility Analysis," Econometrica 30, 4 (1962), 729-743. [9] Pratt, J., "Risk Aversion in the Small and in the Large," Econometrica 32 (Jan. 1964). [10] Samuelson, P. A., "General Proof that Diversification Pays," Journal of Financial and Quantitative Analysis II (Mar. 1967), 1-13. [11] , "Risk and Uncertainty: A Fallacy of Large Numbers," Scientia, 6th Series, 57th year (April-May, 1963). [12] , "A Turnpike Refutation of the Golden Rule in a Welfare Maximizing Many-Year Plan," Essay XIV Essays on the Theory of Optimal Economic Growth, Karl Shell (ed.) (Cambridge, Mass.: MIT Press, 1967). [13] , and R. C. Merton, "A Complete Model of Warrant Pricing that Maximizes Utility," Industrial Management Review (in press). [14] Tobin, J., "Liquidity Preference as Behavior Towards Risk," Review of Economic Studies, XXV, 67, Feb. 1958, 65-86. [15] , "The Theory of Portfolio Selection," The Theory of Interest Rates, F. H. Hahn and F. P. R. Brechling (eds.) (London: Macmillan, 1965). [16] Williams, J. B., "Speculation and the Carryover," Quarterly Journal of Economics 50 (May 1936), "0See Samuelson [11]. 436-455. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:34:15 AM All use subject to JSTOR Terms and Conditions

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