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Lifetime Portfolio Selection By Dynamic Stochastic ProgrammingAuthor(s): Paul A. SamuelsonSource: The Review of Economics and Statistics, Vol. 51, No. 3 (Aug., 1969), pp. 239-246Published by: The MIT PressStable URL: http://www.jstor.org/stable/1926559

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LIFETIME PORTFOLIO SELECTIONBY DYNAMIC STOCHASTIC PROGRAMMING

Paul A. Samuelson*Introduction

M OST analyses of portfolio selection,whether they are of the Markowitz-Tobin mean-varianceor of more general type,maximize over one period.' I shall here formu-late and solve a many-period generalization,corresponding o lifetime planningof consump-tion and investment decisions. For simplicityof exposition I shall confine my explicit dis-cussion to special and easy cases that sufficetoillustrate the general principles involved.As an example of topics that can be investi-gated within the framework of the presentmodel, consider the question of a "business-man risk" kind of investment. In the literatureof finance,one often reads; "SecurityA shouldbe avoided by widowsas too risky, but is highlysuitable as a businessman'srisk." What is in-volved in this distinction? Many things.First, the "businessman" is more affluentthan the widow; and being further removedfrom 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 salaryin the future; and with so high a present dis-counted value of wealth, it is only prudent forhim to put more into commonstocks comparedto his present tangible wealth, borrowing ifnecessary for the purpose, or accomplishingthe same thing by selecting volatile stocks thatwidows shun.

Third, being still in the prime of life, thebusinessman can "recoup" any present lossesin the future. The widow or retired man near-ing life's end has no such "second or nthchance."Fourth (and apparently related to the lastpoint), since the businessmanwill be investingfor so many periods, "the law of averages willeven out for him," and he can afford to actalmost as if he were not subject to diminishingmarginalutility.What are we to make of these arguments?It will be realizedthat the first could be purelya one-period argument. Arrow, Pratt, andothers2 have shown that any investor whofaces a range of wealth in which the elasticityof his marginal utility schedule is great willhave high risk tolerance; and most writersseem to believe that the elasticity is at itshighest for rich - but not ultra-rich! -people. Since the present model has no newinsight to offer in connectionwith statical risktolerance, I shall ignore the first point hereand confine almost all my attention to utilityfunctions with the same relative risk aversionat all levels of wealth. Is it then still true thatlifetime considerations justify the concept ofa businessman'srisk in his prime of life?Point two above does justify leveraged in-vestment financedby borrowing against futureearnings. But it does not really involve anyincrease in relative risk-taking once we haverelatedwhat is at risk to the properlargerbase.(Admittedly, if market imperfections makeloans difficult or costly, recourse to volatile,"leveraged" securities may be a rational pro-cedure.)The fourth point can easily involve the in-numerable fallacies connectedwith the "law oflarge numbers." I have commented elsewhere3on the mistaken notion that multiplying thesame kind of risk leads to cancellation rather

* Aid from the National Science Foundation is gratefullyacknowledged. Robert C. Merton has provided me withmuch stimulus; and in a companion paper in this issue ofthe REVIEW he is tackling the much harder problem ofoptimal control in the presence of continuous-time sto-chastic variation. I owe thanks also to Stanley Fischer.'See for example Harry Markowitz [5]; James Tobin[14], Paul A. Samuelson [10]; Paul A. Samuelson andRobert C. Merton [13]. See, however, James Tobin [15],for a pioneering treatment of the multi-period portfolioproblem; and Jan Mossin [7] which overlaps with thepresent analysis in showing how to solve the basic dynamicstochastic program recursively by working backward fromthe end in the Bellman fashion, and which proves thetheorem that portfolio proportions will be invariant onlyif the marginal utility function is iso-elastic.2 See K. Arrow [1]; J. Pratt [9]; P. A. Samuelson andR. C. Merton [13].3P. A. Samuelson [11].

[ 239 ]

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240 THE REVIEW OF ECONOMICS AND STATISTICSthan augmentation of risk. I.e., insuringmany ships adds to risk (but only as \In);hence, only by insuring more ships and byalso subdividingthose risks amongmore peopleis risk on each brought down (in ratio 1/V/n).However, before writing this paper, I hadthought that points three and four could bereformulatedso as to give a valid demonstra-tion of businessman's risk, my thought beingthat investing for each period is akin to agree-ing to take a 1/nth interest in insuring n inde-pendent ships.The present lifetime model reveals that in-vesting for many periods does not itself in-troduce extra tolerance for riskiness at early,or any, stages of life.

BasicAssumptionsThe familiar Ramsey model may be used asa point of departure. Let an individual maxi-mize

Te-Pt U[C(t)]dt (1)

subject to initial wealth WO hat can always beinvested for an exogeneously-given certainrate of yield r; or subject to the constraintC(t) = rW(t) - W(t) (2)If there is no bequest at death, terminalwealthis zero.This leads to the standard calculus-of-varia-tions problem

TJ = Max e-Pt U[rW - W]dt (3){W(t)} ?This can be easily relatedI to a discrete-time formulation

TMax 1t0 (1+p)-t U[Ct] (4)subject to

C= W Wt+1 (5)1+ror,MaxEt= (1+p)t U [Wt- W+ (6){w~~~O+ 1+r

for prescribed (W0, WT+1). Differentiatingpartially with respect to each Wt in turn, wederive recursion conditions for a regular inte-rior maximum(I +P) U wt1 +]1+r1r=U' Wt -w + (7)

If U is concave, solving these second-orderdifference equations with boundary conditions(Won WT+1) will suffice to give us an optimallifetime consumption-investmentprogram.Since there has thus far been one asset, andthat a safe one, the time has come to introducea stochastically-risky alternative asset and toface up to a portfolioproblem. Let us postulatethe existence, alongside of the safe asset thatmakes $1 invested in it at time t return to youat the end of the period $1(1 + r), a risk assetthat makes $1 invested in, at time t, return toyou after one period$1Zt,whereZt is a randomvariable subject to the probability distributionProb{Zt < z} = P(z). z- (8)Hence, Zt+1 - 1 is the percentage "yield" ofeach outcome. The most general probabilitydistribution is admissible: i.e., a probabilitydensity over continuous z's, or finite positiveprobabilities at discrete values of z. Also Ishall usually assume independence betweenyields at different times so that P(zo, z1, ... ,Z...* , ZT) = P(zt)P(Z1) ... P (zT)For simplicity, the readermight care to dealwith the easy case

Prob {Z = X} = 1/2=Prob{Z=X-}, x> 1(9)In order that risk averters with concave utilityshould not shun this risk asset when maximiz-

ing the expectedvalue of theirportfolio, Xmustbe large enough so that the expected value ofthe risk asset exceeds that of the safe asset, i.e.,- + A-1 > 1 + r, or2 2X > 1 + r + V2r + r2.

Thus, for X = 1.4, the risk asset has a meanyield of 0.057, which is greater than a safeasset's certain yield of r = .04.At each instant of time, what will be theoptimal fraction, Wt, that you should put in

'See P. A. Samuelson [12], p. 273 for an exposition ofdiscrete-time analogues to calculus-of-variations models.Note: here I assume that consumption, Ct, takes place atthe beginning rather than at the end of the period. Thischange alters slightly the appearance of the equilibriumconditions, but not their substance.

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LIFETIME PORTFOLIO SELECTION 241the risky asset, with 1 - wt going into the safeasset? Once these optimal portfolio fractionsare known, the constraint of (5) must bewritten

Ct =[Wt - Wt+1].c[(1-wt) (1 +r) + wtZt] (10)Now we use (10) instead of (4), and recogniz-ing the stochastic nature of our problem,specify that we maximize the expected valueof total utility over time. This gives us thestochastic generalizations of (4) and (5) or(6)

Max T{Ct, wt} E X (1 +p) -t U[Ct] (1t=O

subjecttoCt =[ Wt(1 + r) (- wt) + wtZt]

WOgiven,WT+1 prescribed.If there is no bequeathing of wealth at death,presumablyWT+1 = 0. Alternatively,we couldreplace a prescribed WT+1 by a final bequestfunction added to (11), of the form B(WT+1),and with WT+1 a free decision variable to bechosen so as to maximize (11) + B(WT+D).For the most part, I shall consider CT = WTand WT+1 = 0?In (11), E stands for the "expected valueof," so that, for example,

E Zt = fztdP(zt)In our simple case of (9),

EZt = 2 A+ 1 A-1.2 2Equation ( 11) is our basic stochastic program-ming problem that needs to be solved simul-taneously for optimal saving-consumptionandportfolio-selectiondecisions over time.Before proceedingto solve this problem,ref-erence may be made to similar problems thatseem to have been dealt with explicitly in theeconomics literature. First, there is the valu-able paper by Phelps on the Ramsey problemin which capital's yield is a prescribedrandomvariable. This corresponds, n my notation, tothe {wt} strategy being frozen at some frac-tional level, there being no portfolio selectionproblem. (My analysis could be amplified to

consider Phelps' 5 wage income, and even inthe stochastic form that he cites Martin Beck-mann as having analyzed.) More recently,Levhari and Srinivasan [4] have also treatedthe Phelps problem for T = oo by means ofthe Bellman functional equations of dynamicprogramming,and have indicated a proof thatconcavity of U is sufficient for a maximum.Then, there is Professor Mirrlees' importantwork on the Ramsey problem with Harrod-neutral technologicalchange as a randomvari-able.6 Our problems become equivalent if Ireplace W - Wt+1 (1+r)(1-wt) + wtZtJ-1in (10) byAtf(Wt/At) - nWt - (Wt+- Wt)let technical change be governed by the prob-ability distributionProb {At ? At-1Z} = P(Z);reinterpret my Wt to be Mirrlees' per capitacapital, Kt/Lt, where Lt is growing at the nat-ural rate of growth n; and posit that Atf(Wt/At) is a homogeneous irst degree,concave,neo-classical production function in terms of cap-ital and efficiency-unitsof labor.It should be remarked that I am confirmingmyself here to regular interior maxima, andnot going into the Kuhn-Tucker inequalitiesthat easily handle boundarymaxima.

Solutionof the ProblemThe meaning of our basic problemTJT(WO) = Max E X (1+P)-tU[Ct] (11){ct,wt} t=Osubject to Ct = Wt- Wt+1[(1-wt) (1+r)+ w,Zt]-I is not easy to grasp. I act now att = 0 to select C0 and w0, knowing W0 but notyet knowing how Z0 will turn out. I must actnow, knowing that one period later, knowledgeof Z0'soutcomewill be known and that W1willthen be known. Depending upon knowledgeofW1, a new decision will be made for C1 andw1. Now I can only guess what that decisionwill be.As so often is the case in dynamic program-ming, it helps to begin at the end of the plan-ning period. This brings us to the well-known'E. S. Phelps [8].6 J. A. Mirrlees [6]. I have converted his treatmentinto a discrete-time version. Robert Merton's companionpaper throws light on Mirrlees' Brownian-motion modelfor At.

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242 THE REVIEW OF ECONOMICS AND STATISTICSone-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 operatesonly on the randomvariable of the next periodsince current consumptionCT-1 is known oncewe have made our decision. Writing the secondterm as EF(ZT), this becomes

EF(ZT) = 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 decisionmaking, say t = T- 1, knowledgewill be avail-able of the outcomes of earlier random vari-ables, Zt-2, ... ; since these might be relevantto the distributionof subsequentrandomvari-ables, conditional probabilities of the formP(ZT-1IZT-2, .. .) are thus involved. How-ever, in cases like the present one, where in-dependence of distributions is posited, condi-tional probabilities can be dispensed withinfavor of simple distributions.Note that in (12) we have substituted forCT its value as given by the constraint in (11)or (10).To determine this optimum (CT-1, WT-1),we differentiatewith respect to each separately,to getO= 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 ouroptimal decisions (C*T-1, W*T_1) as functionsof initial wealth WT-1 alone. Note that ifsomehow C*T-1 were known, (12") would byitself be the familiar one-period portfolio op-timality condition, and could trivially be re-written to handle any number of alternativeassets.

Substituting (C*T-1, W*T_1) into the expres-sion to be maximized gives us J1(WT-1) ex-plicitly. From the equations in (12), we can,by standard calculus methods, relate the de-rivatives of U to those of J, namely, by theenvelope relation

JI'(WT-1) = U' [CT-1]. (13)Now that we know J1[WT_1], it is easy todetermine optimal behavior one period earlier,namely byJ2 (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 followingequationslike 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)= 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 re-lated to U'[CT-1], can be solved simultaneous-ly to determine optimal (C*T-2, W*T-2) and

J2 (WT-2) .Continuingrecursivelyin this way for T-3,T-4,...,2, 1, 0, we finally have our problemsolved. The general recursiveoptimality equa-tions can be written asO = 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 -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 substitutionsmust be made and the E operators must beover the properprobabilitydistributions. Solv-ing (16") at any stage will give the optimaldecision rules for consumption-savingand forportfolio selection, in the form

C*t = f [Wt; Zt-l, ... , Zo]= fT-t[Wt] if the Z's are independentlydistributed

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LIFETIME PORTFOLIO SELECTION 243W*t = g[Wt; Zt_, ...*, Zo]= g-_t[Wt] if the Z's are independentlydistributed.Our problem is now solved for every casebut the importantcase of infinite-timehorizon.For well-behaved cases, one can simply letT -> oo in the above formulas. Or, as oftenhappens, the infinite case may be the easiest ofall to solve, since for it C*t = f(Wt), w*t =g(Wt), independently of time and both theseunknown functionscan be deducedas solutionsto the following functionalequations: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')000= 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) pretendedto be known, would be equivalent to equation(13) of Levhari and Srinivasan [4, p. f]. Inderiving (17')-(17"), I have utilized the enve-lope relationof my (13), which is equivalenttoLevhari and Srinivasan's equation (12) [4,p.5]. Bernoulli and IsoelasticCases

To apply our results, let us consider the in-teresting Bernoullicase where U = log C. Thisdoes not have the bounded utility that Arrow[1] and many writers have convinced them-selves is desirable for an axiom system. SinceI do not believe that Karl Menger paradoxesof the generalizedSt. Petersburgtype hold anyterrors for the economist, I have no particularinterest in boundednessof utility and considerlog C to be interestingand admissible. For thiscase, 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") splitinto two independent parts and the Ramsey-Phelps saving problem becomes quite indepen-dent of the lifetime portfolio selectionproblem.Now we have0 = (1/C) - (1+p)-1 (W - C)-'orCT-1 = (l+p) (2+p) 'WT-1 (19)r00o = (Z- 1 r)-[(l w) (l+r)

+ wZ] -1 dP(Z) orWT-1 = W* independentlyf WT-1. (19")

These independence results, of the CT-1 andWT-1 decisionsand of the dependence f WT_1on WT_1, hold for all U functions with iso-elastic marginal utility. I.e., (16') and (16")become decomposableconditions for all

U(C) = 1/yCl, y < 1 (20)as well as for U(C) = log C, correspondingbyL'Hopital's rule to y = 0.To see this, write (12) or (18) asJ1(W) = Max C + (1+p) -1 (W-C)7{C, w} Y Yrxf[(1 -w)(1+r) + wZ]ZidP(Z)

= Max-+ (+p)-1 (w-CYx{C} 'Y 'YMax [(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 optimalw* and givesrxMax [(1 -w)(1+r) +wZ]y dP(Z){w}co(1 -w*) (1+r) + w*Z]'ydP Z)

= [1 + r*]Y,for short.Here, r* is the subjective or util-prob meanreturnof the portfolio, where diminishingmar-ginal utility has been taken into account.7 Toget optimal consumption-saving, differentiate(21) to get the new form of (12'), (15'), or(16')See Samuelsonand Merton for the util-prob concept[13].

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244 THE REVIEW OF ECONOMICS AND STATISTICS0 = CY-1 _ (1+p)-l (1+r*)y (W-C)'-1. (22')

Solving,we have the consumptiondecision ruleC*T_1 = T1 1 (23)

+ aWwherea, = [(1+r*)Y/(11+p)]/11tl. (24)Hence, by substitution, we findJ1(WT-1) = blW'YT-11/Y (25)whereb, = aj'Y(1+aj)-Y+ (1+p)-l (1+r*)y (1+al)'-y (26)Thus, J,(.) is of the same elasticity form asU(.) was. Evaluating indeterminate forms fory = 0, we find J, to be of log form if U was.Now, by mathematical induction, it is easyto show that this isoelastic property must alsohold for J2(WT-2), I3(WT-3), ..., since,whenever it holds for JI(WT_") it is deduciblethat it holds for Jn+1(WT -1). Hence, atevery stage, solving the generalequations (16')and (16"), they decompose into two parts inthe case of isoelastic utility. Hence,Theorem:For isoelastic marginalutility functions, U'(C)

= C-1, 'y < 1, the optimal portfolio decisionis independent of wealth at each stage and in-dependent of all consumption-savingdecisions,leading to a constant w*, the solution tor0o= f[(1-w)(l+r)+wZ]T-h(Z-1-r)dP(Z).Then optimal consumption decisions at eachstage are, for a no-bequest model, of the form

C*T-i = CiWT-iwhere one can deduce the recursion relations

a,Cl = _.SA.la1 = +W1a, [ (1+p)1( 1+r*)'Y] 1/1_7rX(1+r*)y = J'[(1 w*)(1+r)

+ w* Z]y dP (Z)alc_-c1s=- 1+a1jc_j

1+al+a 21+ ... +ailail (a -1)=ai+1-1+i

In the limiting case, as y -O 0 and we haveBernoulli's logarithmic function, a, = (1+p),independent of r*, and all saving propensitiesdepend on subjective time preference p only,being independent of technological investmentopportunities (except to the degree that Wtwill itself definitely depend on those opportu-nities).We can interpret 1+r* as kind of a "'risk-corrected"mean yield; and behaviorof a long-lived man depends critically on whether(1+r*)y > (l+p), corresponding o a, < 1.

(i) For (1+r*)'y = (1+p), one plans always toconsume at a uniform rate, dividing current WT-i evenlyby remaininglife, 1/(1+i). If young enough, one saveson the average; in the familiar "hump saving" fashion,one dissaves later as the end comes sufficiently closeinto sight.(ii) For (1+r*)7 > (1+p), a, < 1, and investmentopportunities are, so to speak, so tempting comparedto psychological time preference that one consumesnothing at the beginning of a long-long life, i.e., rigor-ouslyLim c4 = 0, a. < 1

i -- ooand again hump saving must take place. For (1+r*)y> (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+p), a,. > 1, consumptionat very early ages drops only to a limiting positivefraction (rather than zero), namelyLim c4 = 1- l/a1< 1, a> 1.

- 00Now whether there will be, on the average, initial humpsaving depends upon the size of r* - c., or whetherr*-- I- > OThis ends the Theorem. Although many ofthe results depend upon the no-bequest as-sumption, WT+1 = 0, as Merton's companionpapershows (p. 247, this Review) we can easilygeneralize to the cases where a bequest func-tion 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 samecomparative statics put forward in Merton'scontinuous-time case will be applicable here,e.g., the Bernoulli y = 0 case is a watershedbetween cases where thrift is enhancedby risk-iness rather than reduced; etc.

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LIFETIME PORTFOLIO SELECTION 245Since proof of the theorem is straightfor-ward, I skip all details except to indicate howthe recursionrelations for ci and bi are derived,namely from the identitiesb+l1W7/y = J+i(W)= Max {CY/yC

+ 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/yand 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- W7-1jwhich definescj+j in terms of bi.What if we relax the assumptionof isoelastic

marginal utility functions? Then WT_j be-comes a function of WT>j1l (and, of course,of r, p, and a functional of the probability dis-tributionP). Now the Phelps-Ramsey optimalstochastic saving decisions do interact with theoptimal portfolio decisions, and these have tobe arrived at by simultaneous solution of thenondecomposableequations (16') and (16").What if we have more than one alternativeasset to safe cash? Then merely interpret Ztas a (column) vector of returns (Z2,,Z3,, ..)on the respective risky assets; also interpretwt as a (row) vector (w2t,w3t, ...), interpretP(Z) as vector notation for

Prob {Z2t _ Z2, Z3t ? Z3,... }= P(Z2 Z3,...) =P(Z),interpretall integralsof the formfG (Z) dP(Z)as multiple integrals fG (Z2,Z3, . ..) dP(Z2,Z3,...). Then (16") becomes a vector-set ofequations,one for each componentof the vectorZt, and these can be solved simultaneously forthe unknown wt vector.If there are many consumption items, wecan handle the general problem by giving asimilar vector interpretationto Ct.Thus, the most general portfolio lifetimeproblemis handledby our equationsor obviousextensions thereof.

ConclusionWe have now come full circle. Our modeldenies the validity of the concept of business-man's risk; for isoelastic marginal utilities, inyour prime of life you have the same relative

risk-tolerance as toward the end of life! The''chance to recoup" and tendency for the lawof large numbers to operate in the case of re-peated investments is not relevant. (Note:if the elasticity of marginal utility, - U'(W) /WU"(W), rises empiricallywith wealth, and ifthe capital marketis imperfect as far as lendingand borrowing against future earnings is con-cerned, then it seems to me to be likelythat a doctor of age 35-50 might rationallyhave his highest consumptionthen, and certain-ly show greatest risk tolerance then - in otherwords be open to a "businessman'srisk." Butnot in the frictionless isoelastic model!)As usual, one expects w* and risk toleranceto be higher with algebraically large y. Oneexpects C, to be higher late in life when r andr* is high relative to p. As in a one-periodmodel, one expects any increase in "riskiness"of Zt, for the same mean, to decrease w*. Oneexpects a similar increase in riskiness to loweror raise consumption depending upon whethermarginalutility is greater or less than unity inits elasticity.8Our analysis enables us to dispel a fallacythat has been borrowed into portfolio theoryfrom information theory of the Shannon type.Associated with independent discoveries byJ. B. Williams [ 16], John Kelly [2 ], and H. A.Latane [3] is the notion that if one is invest-ing for many periods, the proper behavior is tomaximize the geometric mean of return ratherthan the arithmetic mean. I believe this to beincorrect (except in the Bernoulli logarithmiccase where it happensI to be correct for reasons

'See Merton's cited companion paper in this issue, forexplicit discussion of the comparative statical shifts of (16)'sC*t and W*t functions as the parameters (p, y, r, r*, andP(Z) or P(Z1,...) or B(WT) functions change. The sameresults hold in the discrete-and-continuous-time models.'See Latane [3, p. 151] for explicit recognition of thispoint. I find somewhat mystifying his footnote there whichsays, "As pointed out to me by Professor L. J. Savage (incorrespondence), not only is the maximization of G [thegeometric mean] the rule for maximum expected utility inconnection with Bernoulli's function but (in so far as cer-tain approximations are permissible) this same rule is ap-proximately valid for all utility functions." [Latane, p. 151,n.13.] The geometric mean criterion is definitely too con-servative to maximize an isoelastic utility function cor-responding to positive y in my equation (20), and it isdefinitely too daring to maximize expected utility when,y < 0. Professor Savage has informed me recently that his1969 position differs from the view attributed to him in1959.

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246 THE REVIEW OF ECONOMICS AND STATISTICSquite distinct from the Williams-Kelly-Latanereasoning).These writers must have in mind reasoningthat goes something like the following: If onemaximizes for a distant goal, investing andreinvesting (all one's proceeds) many times onthe way, then the probability becomes greatthat with a portfolio that maximizes the geo-metric mean at each stage you will end upwith a larger terminal wealth than with anyother decision strategy.This is indeed a valid consequence of thecentral limit theorem as applied to the addi-tive logarithms of portfolio outcomes. (I.e.,maximizing the geometric mean is the samething as maximizing the arithmetic mean ofthe logarithm of outcome at each stage; if ateach stage, we get a mean log of m** > m*,then after a large number of stages we willhave m**T > > m*T, and the properly nor-malized probabilities will cluster around ahigher value.)There is nothing wrong with the logicaldeduction from premise to theorem. But theimplicit premise is faulty to begin with, as Ihave shown elsewhere in another connection[Samuelson, 10, p. 3]. It is a mistake to thinkthat, just because a w** decision ends up withalmost-certain probability to be better than aw* decision, this implies that w** must yield abetter expected value of utility. Our analysisfor marginal utility with elasticity differingfrom that of Bernoulli provides an effectivecounter example, if indeed a counter exampleis needed to refute a gratuitous assertion.Moreover, as I showed elsewhere, the orderingprinciple of selecting between two actions interms of which has the greater probability ofproducinga higher result does not even possessthe property of being transitive.'0 By thatprinciple,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 Risk-Bearing" (Helsinki, Finland: Yrj6 JahnssoninSaati6, 1965).[2] Kelly, J., "A New Interpretation of InformationRate," Bell System Technical Journal (Aug.1956), 917-926.[3] Latane, H. A., "Criteria for Choice Among RiskyVentures," Journal of Political Economy 67 (Apr.1959), 144-155.[4] Levhari, D. and T. N. Srinivasan, "Optimal Sav-ings Under Uncertainty," Institute for Mathe-matical Studies in the Social Sciences, TechnicalReport No. 8, Stanford University, Dec. 1967.[5] Markowitz, H., Portfolio Selection: EfficientDiversification of Investment (New York: JohnWiley & Sons, 1959).[6] Mirrlees, J. A., "Optimum Accumulation Under

Uncertainty," Dec. 1965, unpublished.[7] Mossin, J., "Optimal Multiperiod Portfolio Pol-icies," Journal of Business 41, 2 (Apr. 1968),215-229.[8] Phelps, E. S., "The Accumulation of Risky Cap-ital: A Sequential Utility Analysis," Economet-rica 30, 4 (1962), 729-743.[9] Pratt, J., "Risk Aversion in the Small and in theLarge," Econometrica 32 (Jan. 1964).[10] Samuelson, P. A., "General Proof that Diversifica-tion Pays," Journal of Financial and QuantitativeAnalysis II (Mar. 1967), 1-13.[11] , "Risk and Uncertainty: A Fallacy ofLarge Numbers," Scientia, 6th Series, 57th year(April-May, 1963).[12] , "A Turnpike Refutation of the GoldenRule in a Welfare Maximizing Many-Year Plan,"Essay XIV Essays on the Theory of OptimalEconomic Growth, Karl Shell (ed.) (Cambridge,Mass.: MIT Press, 1967).[13] , and R. C. Merton, "A Complete Modelof Warrant Pricing that Maximizes Utility," In-dustrial Management Review (in press).[14] Tobin, J., "Liquidity Preference as BehaviorTowards 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 andF. P. R. Brechling (eds.) (London: Macmillan,1965).[16] Williams, J. B., "Speculationand the Carryover,"QuarterlyJournal of Economics 50 (May 1936),436-455.0See Samuelson [11].